Endogenous structural transformation in economic development
aa r X i v : . [ ec on . T H ] N ov Endogenous Structural Transformation in EconomicDevelopment
Jistin Yifu Lin a and Haipeng Xing b ∗ Abstract :The structures, including technologies, industries, infrastructure and institutions, in acountry have three attributes, namely, structurality, durationality, and transformality. Thepaper proposes a novel method to model the structural transformation in a market economy.With the common knowledge assumption, the paper proposes a generic model combiningoptimal control and optimal switching to study the static equilibrium and dynamic equilib-rium of resource allocations under a given industrial structure and the equilibrium when thecondition for transforming industrial structures arises by a social planner to maximize therepresentative household’s utility in a market economy. The paper establishes the mathemat-ical underpinning of the static equilibrium, dynamic equilibrium and structural equilibrium.The generic model and its equilibria are then extended to economies with complicated eco-nomic structures consisting of hierarchical production, composite consumption, technologyadoption and innovation, infrastructure, and economic and political institutions. The paperconcludes with a brief discussion of applications of the proposed methodology to economicdevelopment problems in other scenarios.
Keywords:
Endogenous structural transformation, structural equilibrium, Hamilton-Jacobi-Bellman equations with variational inequalities, viscosity solutions, structurality, durational-ity, transformality ∗ Corresponding author a Institute of New Structural Economics, Peking University, Beijing, China. b Department of Applied Mathematics and Statistics, State University of New York, StonyBrook, NY 11794, USA. Email: [email protected]
We are grateful to Jones Chad, Gene Grossman, Joseph Kaboski, Edmund Phelps,Thomas Sargent, Joel Sobel, and Yi Wen for their helpful comment on an earlier version ofthe paper. 1 ontents
References 62 Introduction
The task of this paper is to develop a method for modeling endogenous structural trans-formation and efficient resource allocation in a market economy from the early, catching-upstage to the advanced, sustained stage. Following North (1981), we define “structure” as thecharacteristics of an economy, which are the basic determinants of economic activities withthree attributes, namely, structurality, durationality, and transformality, on which we willelaborate in the next section. “Structural transformation” refers to changes in the basicdeterminants of the composition and organization of economic activities during the processof economic development. “Endogenous structural transformation” means that, for the pur-pose of economic development, the social planner in the economy makes use of knowledge oneconomic structures and chooses optimal economic structures to transform during economicdevelopment. Finally, “from the early, catching-up stage to the advanced, sustained stage”means the process of how a less developed economy moves from inside the world productionpossibilities frontier toward its frontier.To illustrate the idea and present our framework, we begin with the study of structuraltransformation and economic growth in the neoclassical sense. Structural transformation,or structural change, has been referred to as the process of a country’s productive resourcesrelocating from low-productivity to high-productivity economic activities during the pastdecades. Lewis (1954) argues the importance of structural transformation from a dual econ-omy to an industrialized market economy in the first stages of economic development, andKuznets (1966, 1973) explains the economic growth as a sustained increase in per capitaincome accompanied by “sweeping structural changes.” Kuznets (1966, 1973) further doc- The literature has not yet provided a widely accepted definition of the economic structure of an economy.In this paper, economic structure refers to a collection of basic determinants of the composition and orga-nization of production, consumption, distribution, exchange, and other economic activities in an economy.This mainly includes production structures, technological structures, market structures, consumer prefer-ence or consumption structures, population structures, financial system structures, trade structures, andeven institutional and cultural structures in an economy.In addition to a country’s economic characteristics, economists consider noneconomic characteristics as de-terminants of economic activities. For instance, Kuznets (1966, p.437) assumes that “the economic and manynoneconomic characteristics of social structure are interrelated as both causes and effects,” and illustrates“some of the noneconomic characteristics associated with differences in economic development and struc-ture between underdeveloped and developed countries bear upon: (1) demographic patterns, (2) politicalstructure, and (3) cultural aspects.”In his book explaining the stability or change of economic structures in economic history, North (1981, p.1)explains “structure” as follows. “By ‘structure’ I mean those characteristics of a society which we believe tobe the basic determinants of performance. Here I include the political and economic institutions, technology,demography, and ideology of a society.” Kuznets (1966, p.1) explains the economic growth as follows: “We identify the economic growth ofnations as a sustained increase in per capita or per worker product, most often accompanied by an increasein population and usually by sweeping structural changes. In modern times these were changes in theindustrial structure within which product was turned out and resources employed–away from agriculturetoward nonagricultural activities, the process of industrialization; in the distribution of population betweenthe countryside and the cities, the process of urbanization; in the relative economic position of groups within and lists it as one of the six major stylized factos of long-term growth. To explainthese sectoral reallocation processes, multisector models are developed to explore the linkagebetween sectoral structural transformation and balanced or non-balanced growth and explainsectoral structural transformation as the result of changes in income and/or relative pricesof goods. These multisector models generate results that are consistent with the “styl-ized facts” of sectoral structural transformation and provide insights on several interestingeconomic issues, such as economic development, regional income convergence, and aggre-gate productivity trends. However, they do not feature Kuznets’ (1961) sweeping structuralchange, due to the following two reasons.First, Kuznets’ sweeping structural change involves changes in basic determinants ofthe composition and organization of economic activities, such as agrarian production andother institutions related to modern industrialized production and related institutions. Bycontrast, sectoral structural transformation in the multisector models deals only with varia-tion in resource allocation among three given sectors under a specific economic structure. Zooming in on structural transformation from the sector level to the industry level, changesin resource allocation among the three given sectors—agriculture, industry, and services—result from the transformation of industrial structures, and such transformation consists ofthe birth and decay of various industries during economic development, which the multi-sector growth models cannot describe. Second, Kuznets’ sweeping structural change isreferred to as structural transformation involving changes in the composition of industries; the nation distinguished by employment status, attachment to various industries, level of per capita income,and the like; in the distribution of product by use–among household consumption, capital formation, and thegovernment consumption, and within each of these major categories by further subdivisions; in the allocationof product by its origin within the nation’s boundaries and elsewhere; and so on.” Kuznets (1973, p. 248) concludes that ”major aspects of structural change include the shift away fromagriculture to nonagricultural pursuits and, recently, away from industry to services.” Stylized facts on sectoral structural transformation have been documented in the literature, which istoo large to be entirely listed here. Sectoral structural transformation is usually demonstrated via sectoralshares of employment, value added, and final consumption expenditure; see a recent summary on this inHerrendorf, Rogerson, and Valentinyi (2014, section 2). See Acemoglu and Guerrieri (2008), Herrendorf, Rogerson, and Valentinyi (2014), and the referencestherein. Acemoglu (2009, pp. 693-696) refers to “structural change” as changes in the composition of productionand employment and uses the term “structural transformation” to describe changes in the organization andefficiency of production accompanying the process of development. Acemoglu explains that, “one might ex-pect Kuznets’ structural change to be accompanied by a process that involves the organization of productionbecoming more efficient and the economy moving from the interior of the aggregate production possibilitiesset toward its frontier.” Therefore, “we would like to develop models that can account for both the structuralchanges and transformations at the early stages of development and the behavior approximated by balancedgrowth at the later stages.” Ju, Lin and Wang (2015) provide an empirical study on this and further explain the phenomenon by aninfinite-industry growth model that assumes geometrically distributed production functions and exogenousgoods prices. Realizing the limits of the multisector growth models, some theoretical economistshighlight the importance of characterizing “sweeping structural changes” and advocate fora unified theoretical framework to characterize the process of structural transformation andresource allocation during economic development and growth. However, such a frameworkhas never been developed.To bridge this gap, this paper starts with a summary of three attributes of economicstructures and proceeds step by step to develop a theoretical framework that characterizesan economy’s development and growth via endogenous structural transformation (EST) andeffective resource allocation. Secton 2 argues that all structures possess three importantattributes: structurality, durationality, and transformality. In addition to illustrating anddiscussing these three attributes, the section discusses the implications of the attributes onmethods of modeling structural transformation.Section 3 presents a baseline EST model, by assuming that knowledge on the world’sindustrial structures can be freely obtained and a unique final good can be produced bydifferent aggregate production functions. The economic structure of the economy is featuredwith a specific aggregate production function, and the social planner in the economy mustoptimally and dynamically choose an economic structure, resource allocation within the cho-sen structure, and a consumption level over time to maximize the representative household’stotal utility subject to the endowment constraint. To solve the social planner’s maximization problem, section 4 studies a general class ofcombined infinite-horizon optimal control and optimal switching problems, which involvescontinuous and discrete dynamics and continuous and discrete controls, and establishes theviscosity solution theory for the general maximization problem. We derive the dynamicprogramming equation that is satisfied by the value function of the combined control problem,which is in the form of a system of
Hamilton-Jacobi-Bellman equations and quasi-variationalinequalities (HJBQVI). Then we discuss the continuity of the value function and show that Early efforts characterizing economic development in underdeveloped countries include Hirschman’s(1958) emphasis on unbalanced growth, Nurkse’s balanced growth, and Rosenstein-Rodan’s big push model.Dual economic models were developed to describe the process of structural changes that occurred in theearly stages of economic development; see Lewis (1954), Ranis and Fei (1961), and others. Acemoglu (2009, pp. 693-696) argues as follows: “A useful theoretical perspective might therefore beto consider the early stages of economic development taking place in the midst of—or even via—structuralchanges and transformations. We may then expect these changes to ultimately bring the economy to theneighborhood of balanced growth, where our focus has so far been. If this perspective is indeed useful, thenwe would like to develop models that can account for both the structural changes and transformations atthe early stages of development and the behavior approximated by balanced growth at the later stages.” Note that the maximization probem here is distinct from the representative household’s maximizationproblem in the neoclassical economic model, which only deals with resource allocation (or more specifically,optimal consumption) and can be solved mathematically by the theory of infinite-horizon optimal control.
3t is the unique viscosity solution to the HJB-QVI system. Since stationary forms ofcombined control problems are often encountered in macroeconomics, we further consider astationary form of the combined control problem and establish the solution theory of optimaldecisions and value functions.With the developed solution theory for the combined control problem, section 5 solvesthe social planner’s maximization problem in the generic EST model and establishes theassociated competitive equilibrium theory. We show that, in addition to the static and dy-namic equilibria, which constitute the competitive equilibrium in the neoclassical growthmodel, the competitive equilibrium in the EST model contains a third type of equilibrium,which we refer to as the structural equilibrium . While the former two equilibria determine ef-ficient resource allocation and imply optimal paths of economic activities in a given econoimcstructure, the structural equilibrium characterizes optimal economic structures at each timeperiod and hence their path. In the special case that the economy has only a single eco-nomic structure to choose, the structural equlibrium degenerates so that the EST modelreduces to the neoclassical growth model. From this perspective, the EST model extendsthe neoclassical economy from the case of a single structure to the case of multiple struc-tures with possible transformation among them. Section 5 further discusses the impacts ofthe factor endowments on the optimal structures and transformation regions of the factorendowments. To illustrate this idea, the section provides examples of the EST and presentstheir implications for economic development.The generic EST model in section 3 and its associated competitive equilibrium in sec-tion 5 deal with transformation of simple structures (or aggregate production functions ofthe final goods), but the idea of the EST model and its mathematical characterization insection 4 are general enough to be applied to economies that have complex structures atdifferent development stages. Section 6 illustrates this by describing EST in different typesof structures and discussing their patterns of stagewise development and growth. The firsttype contains a class of hierarchical production structures with which intermediate and fi-nal goods are produced with exogenous technological progress. The second type involves acombined production and consumption structure with which consumption and investmentgoods are produced under exogenous technological progress and a composite of consump-tion goods is consumed. The third type deals with structures of endogenous technologicalprogress achieved by adoption and/or research and development (R&D) and their trans-formation. The fourth type deals with transformation of economic institutions involvinginfrastructures and economic policies. The fifth type handles structural transformation of The necessity of developing mathematical control theory here can be seen from the mathematical foun-dation of neoclassical economic theory. Provided the dynamic programming equations or the HJB equationsderived from the infinite-horizon optimal control problem, mathematicians have proved that the value func-tion associated with the optimal control problem is the solution to the HJB equation. However, since thesolution of the HJB equation may not be continuous in many cases, solutions to the HJB equations in aweak sense or viscosity solutions are developed under mild conditions on state equations and loss/rewardfunctions. Therefore, based on this theory, economists can directly develop macroeconomic models to studyvarious types of economic phenomena, under the assumption that conditions for the existence and uniquenessof viscosity solutions to the HJB equation are always satisfied.
In the literature, an economy’s structures usually consist of economic and noneconomiccharacteristics of activities. The former include a set of specifications on the compositionand organization of production and other activities, and the latter consist of a collection ofrules on economic and political institutions and even the norms, values, and ideologies ofthe society. Since these characteristics determine the ways in which economic activities areorganized and managed, they are different from numerical economic variables that measurethe levels of input and output of economic activities. In particular, three specific attributesare found in an economy’s structures—structurality, durationality, and transformality.
The first attribute is structurality , which refers to the organic relationships of all the economicand noneconomic structures, each with specific characteristics, in the overall structure of aneconomy or society. An economy’s overall structure is a collection of all the structures, whichare organized and interconnected according to certain rules. Different schools of thought haveproposed different theories to explain the fundamental determinant of an economy’s overallstructure and the change, say, from an agrarian economy to a modern industrialized economy.For example, according to Weber (1904, 1905), it is value; according to Marx and Engels(1848), it is productive force; and according to North (1981), it is the land-to-labor ratio thatdetermines the overall structure of a society. In neoclassical economics, the competitiveequilibrium suggests that, among all the endogenous variables, capital intensity (or generally,the factor endowments) determines efficient resource allocation, which includes production,consumption, and endogenous technological changes. In institutional economics, the role ofinstitutions in shaping economic behavior is highlighted, and institutional structure seemsto be more fundamental than the factor endowment. In new structural economics the en-dowment structure determines the comparative advantages, that is, the industrial structureand appropriate hard infrastructure and soft institutions of an economy in the process ofdevelopment. In the literature, the word “structure” is sometimes used to refer to the overallstructure or alternatively the totality of all (sub)structures in a society; at other times, it In Marx’s historical materialism, the superstructure, or institutions, is determined by and impacts theeconomic base, which consists of the productive forces and the production relations determined by theproductive forces. The productive forces are determined by the prevailing technology and industries in aneconomy.
5s used to refer to a specific (sub)structure in the overall structure. In the latter case, anadjective is often added, for example, the endowment structure, the industrial structure, thepreference structure, the financial structure, the legal structure, the institutional structure,the political structure, and so forth.The second attribute is durationality , which means that the overall structure and eachof its substructures in an economy will not change instantaneously and will have differentlevels of stability. Compared with numerical economic variables that measure the input andoutput levels of economic activities, economic structures describe how economic activitiesare organized and managed and/or how economic variables are generated and, hence, do notchange as fast as numerical economic variables. For instance, an economy’s industrial struc-ture does not change instantaneously with variation in firms’ output and technology levels.Similarly, a firm’s structure, including its equipment and organization, does not change in-stantaneously with its output level. Furthermore, different (sub)structures show differentextents of durationality. One example of this is that quality improvement in Schumpeteriangrowth occurs during the lifetime of each variety of machine; hence, the characteristics (orstructure) of improvement in the quality of each type of machine are less durable than thestructure of production that determines whether a particular type of machine should beproduced. Another example is that structures of political institutions in an economy aremore durable than other kinds of economic (sub)structures, such as technological progressstructures, industrial structures, and even structures of economic institution. In addition,the overall structure of an agrarian society and its related substructures, including economic,social, and political structures, may have existed for millennia in a society, and change ofthe overall structure to an industrialized society has taken place only in modern times.The third attribute is transformality , which means that a substructure or even the over-all structure is not constant forever; it can transform into another structure under certainconditions during the process of economic development and growth. Transformation of eco-nomic (sub)structures can be easily found in countries’ development and growth processesand has been widely discussed by economists over the past centuries. Since the literaturedocumenting structural transformation is too large to be entirely reviewed here, we empha-size the difference between the transformality of economic structures and the variability ofnumerical economic variables. In contrast to numerical economic variables, which changealmost instantaneously, economic structures change on a much longer time scales due totheir durationality. Furthermore, economic (sub)structures have different degrees of trans-formality in economic development. For instance, it is typically much easier for a country totransform the structure of industry from labor intensive to capital intensive and technolog-ical progress from technological borrowing to indigenous innovation than to transform theinstitutional structures of the economy. 6 .2 Implication for the methodology of economic modeling
The three attributes distinguish economic structures from economic variables of resourceallocation and also from themselves, and such distinction indicates that economic structuresshould be characterized by a method different from that of modeling economic variables. Inthe following, we briefly discuss several issues related to modeling economic structures.
Decoupling economic structures and economic variables
Since a country’s economic activities are measured by their input and output levels anddetermined by their composition and organization, their characterization can be decoupledinto two types of variables. The first type is a set of functional variables that describeeconomic (sub)structures or determinants of economic activities, and the second type is aset of numerical variables that represent the inputs and outputs of economic activities. Asan example of this, consider an economy that has a unique final good produced by aggre-gate production functions. Capital stock, labor, and output are the numerical variablesthat serve as inputs and outputs of the production functions, and the aggregate productionfunctions are functional variables that determine the production process and hence repre-sent the production structure of the economy. If exogenous or endogenous technologicalprogress is considered in the economy, the structure of technological progress in productionis represented by functionals that determine the technological progress.The decoupling idea separates economic structures from measurements in economic ac-tivities and allows us to deal with functional and numerical variables differently in economicmodeling. Specifically, given a functional variable (or an economic sub-structure) that de-termines specific economic activities, the dynamics of numerical economic variables can beanalyzed by existing economic methods. When an economic (sub)structure transforms fromone to another, it may be considered as a change of the functional variables. Economic(sub)structures with different extents of durationality may be represented as functional vari-ables on different time scales.
Structures of economic (sub)structures
By representing each economic substructure as a functional variable, a country’s economicstructure becomes a collection of functional variables that determine various economic activi-ties and are organized by certain economic rules. For instance, consider an economy that hasintermediate and final goods. The production structure of the economy is represented as afunctional that aggregates the production functions of the intermediate and final goods, andthe structure of consumers’ preference is described by a functional of intermediate (and/orfinal) goods. These two substructures are aggregated as part of the economic structure ofthe economy under the resource constaint.However, the economic rule of organizing economic (sub)structures may not be unique,and sometimes the cause and effect of economic (sub)structures is controversial among7conomists. As exploration of such rules goes beyond the scope of this paper, we postulatethat the cause and effect of economic (sub)structures are known by economists; hence, thestructure of economic (sub)structures can be well specified by economic researchers. Oncethe cause and effect of economic (sub)structures are clear, the pivot or central structure ofall the economic (sub)structures may be identified. knowledge on economic structures
The transformation of economic structures indicates that more than one economic structureappears in a country’s development and growth process. To describe the transformation pro-cess, we briefly discuss how knowledge on economic structures is generated for a country. Inprinciple, knowledge on economic structures is created or summarized by countries’ political,economic, and intellectual elites. Take the structure of technological progress as an examplewhich refers to the organization of economic activities to improve an economy’s technologylevel. For countries inside the world production possibilities frontier, their knowledge onstructures of technological progress consists of not only ways to carry out R&D by them-selves, but also the paths of other countries’ technological progress for adoption. However,this is not the case for countries on the world production possibilities frontier, since theymay not need to know paths of technological progress in countries inside the world produc-tion possibilities frontier. Thus, their knowledge on structures of technological progress onlyconsists of R&D and/or learning by doing during their development process.Similar argument can be applied to knowledge on other kinds of structures. But forstructures involving noneconomic characteristics, the knowledge on them may be obtainedin different ways. Take the knowledge on structures of legal institutions as an example. Sinceeach country’s development, culture, and social environment is unique, his/her knowledgeon structures of legal institutions is also unique and may not be useful for other countries.
Role of the social planner
Once the knowledge on economic structures is obtained, the next concern is about who willmake use of the knowledge and make decisions on the transformation of economic structures.It may be assumed that such decisions are made by interactions and joint efforts of thecountries’ political, economic, and intellectual elites at different levels. For example, when anew structure of technological progress for a machine type needs to be adopted in countriesinside the global production possibilities frontier, an association of entrepreneurs may makethe decision; if such decision needs to be supported by workers with different skill sets trainedin the schools or through subsidies provided by the government, the government will decidewhether a new technology structure should be adopted and provide education and subsidiesaccordingly. Decison makers for other types of economic substructures usually depend on A potentially confusing issue here is the difference between the technological structure and the struc-ture of technological progress. As the former usually refers to the composition and level of technology inproduction, we refer to the latter as how the level of technology is determined in economic activities.
This section considers a closed economy in which the social planner makes decisions on re-source allocation for production activities within an industrial structure to meet the house-holds’ consumption demands and transformation of industrial structure within the givenoverall structure of the economy.
Households, firms, production, and technology
Let H be the set of households in the economy and suppose that the economy admits arepresentative household. That is, the demand side of the economy can be representedas if there were a single household making the aggregate consumption and saving de-cisions subject to an aggregate budget constraint. Denote this characteristic by H = { households in H are representative } . The behavior of households in the economy is char-acterized by the pair ( H , H ).Similarly, let F be the set of firms in the economy and suppose that all firms are repre-sentative. That is, all firms in the economy access the same aggregate production functionfor the final good. Denote this characteristic by F = { firms in F are representative } . Thebehavior of firms in the economy is characterized by the pair ( F , F ).Suppose that firms use the aggregate production function Y ( t ) = F ( K ( t ) , L ( t ) , A ( t ))to produce the final good, where K ( t ), L ( t ), and A ( t ) are, respectively, the capital stock,employment, and technology used in production at time t . Let Y = { Y ( t ) ∈ R + } representthe output of production and Y = { F } represent the composition and organization ofproduction. Then, ( Y , Y ) characterizes production and its organizational structure.The level of the technology A is exogenously determined by a function of technologicalprogress, A ( t ). A = { A ( t ) ∈ R + } denotes the level of technology at time t and A = { A : R + → R + be exogenously given. } represents how the technology is exogenously generated.Then, technology and its structure of production can be characterized by ( A , A ). A ( t ) maydepend on the production function. 9 nstitution, labor, resource constraint, and price To specify a way of allocating resources, we assume that all goods and factor markets arecompetitive and complete. Let M = { the demand and supply of labor, capital, and thegoods } and M = { households and firms are price-takers and pursue their own goals andprices clear markets and all markets are complete. } . Then the composition and institutionalstructure of the market can be expressed as ( M , M ).Given ( M , M ), we explore the firms’ demand for labor and capital in the aggregateproduction ( Y , Y i ) and their structures. Assume the population L grows exponentially atrate π ( π > L ( t ) := L π ( t ) = L (0) exp( πt ) , π > . (3.1)Hence, the population structure is expressed as ( L , L ) = ( { L ( t ) } , { L π } ), where L π representsthe functional (3.1). To describe the labor market, note that L ( t ) ∈ R + is the amount ofdemand for labor and L ( · ) is a functional characterizing such demand; hence, the labormarket and its structure are represented by ( L , L ) = ( { L ( t ) } , { L ( · ) } ). When the market iscompetitive, the labor market-clearing condition is L ( t ) = L ( t ), suggesting that the labormarket structure coincides with the population structure, that is, ( L , L ) ∼ = ( L , L ). The households own the capital stock of the economy and rent it to firms. Underthe capital market-clearing condition, the demand for capital by firms equals the supply ofcapital by households, which is denoted K ( t ). The aggregate resource constraint, which isequivalent to the budget constraint of the representative household, requires that˙ K ( t ) = Y ( t ) − δK ( t ) − C ( t ) , (3.2)where C ( t ) := c ( t ) L ( t ) is the total consumption, and investment consists of new capital˙ K ( t ), and replenishment of depreciated capital δK ( t ). The constraint (3.2) suggests that thecapital market and its structure can be characterized by ( K , K ), in which K = { K ( t ) ∈ R + } ,and K = { K ( · ) | K ( · ) satisfies (3.2) } .Under appropriate assumptions (or specifically, (A1) and (A2) in section 3.2), the factorprices can be obtained by solving the profit maximization problem of the representative firm.Denote by R ( t ) and w ( t ) the rental price of capital and wage at time t , respectively. Let P ( t ) := 1 be the normalized price of the final good. Factor and goods prices and theirstructures can be expressed as ( P , P ), in which P = { R ( t ) , w ( t ) , P ( t ) } and P = { factorprices are determined by firms to maximize their profit, and P ( t ) = 1 } . Furthermore, therental price and wage under ( P , P ) can be expressed as R ( t ) = ∂F∂K ( K, L, A ) , w ( t ) = ∂F∂L ( K, L, A ) . This further implies that, if there exist two growth rates π > , π > π = π , then ( { L ( t ) } , { L π } )and ( { L ( t ) } , { L π } ) are two different representation of the labor market in the economy. onsumption and utility function We now consider consumption in the economy and its characteristic. Suppose that house-holds only consume the unique final good. Then total consumption is given by C ( t ) = c ( t ) L ( t ) and the characteristic of the consumption in this economy is C = { households con-sume the final good } . Therefore, consumption and its characteristic are expressed as ( C , C ).Given the consumption per capita c ( t ), we assume that the representative household hasan instantaneous utility function u ( c ), which represents the preference of the household’sconsumption. Then, the preference of consumption and its structure can be expressed as( U , U ) := ( { u ( c ) } , { u ( · ) } ). Economic structure
The above argument describes the following agents’ behavior and economic activity compo-nents in the economy and their characteristics, ( H , H ), ( F , F ), ( Y , Y ), ( M , M ), ( L , L ),( K , K ), ( A , A ), ( P , P ), ( C , C ), and ( U , U ). To represent these in a more concise way, welet E := ( H , F , M , Y , L , K , A , P , C , U ) represent agents’ behavior and economic activitiesin the economy, and E := ( H , F , M , Y , L , K , A , P , C , U ) represent the economic andnoneconomic characteristics of E . Then the overall structure and activities of the economycan be characterized by ( E , E ). Suppose that the final good in the economy can be produced by I aggregate productionfunctions or industrial structures in the world: Y ( t ) = F i [ K ( t ) , L ( t ) , A ( t )] , i ∈ I := { , . . . , I } , (3.3)and knowledge on these industrial structures can be freely obtained by economies in theworld. The knowledge on the world’s industrial structures for the social planner is thengiven by I Y = { Y i | Y i = { F i } , i ∈ I } . When the social planner of the economy chooses F i to produce the final good, the industrial structure of the economy is described by the pair( Y , Y i ) = ( { Y ( t ) } , { F i } ), and correspondingly, the economy with industrial structure Y i inits overall structure is characterized by ( E , E i ). The knowledge on the industrial structuresfor the social planner can now be expressed as I E := { E i | i ∈ I } .Provided the economy and its economic structure ( E , E i ), equations (3.3) and (3.2) implythe following aggregate resource constraint in the economy˙ K ( t ) = F i [ K ( t ) , L ( t ) , A ( t )] − δK ( t ) − C ( t ) . (3.4)Suppose that the production function F i [ K, L, A ] exhibits constant returns to scale in K and L . The output per capita is given by y ( t ) ≡ Y ( t ) /L ( t ) ≡ f i ( t, k ( t )) , f i ( t, k ( t )) = F i [ k ( t ) , , A ( t )] , (3.5)11here k ( t ) ≡ K ( t ) /L ( t ). Then, the accumulated capital per capita is given by the equation˙ k ( t ) = f i ( t, k ( t )) − ( δ + π ) k ( t ) − c ( t ) . (3.6)We assume that f i ( i ∈ I ) satisfy the following conditions for later discussion:(A1) For each i ∈ I , the production function f i ( t, k ) is twice differentiable, strictlyincreasing, and concave in k . (A2) For each i , f i satisfies the Inada conditions: lim k → ∂f i ( t, k ) /∂k = + ∞ and lim k → + ∞ ∂f i ( t, k ) /∂k = 0 . Moreover, f i ( t,
0) = 0 for all t . Furthermore, we assume that the utility function u ( c ) satisfies the following condition:(A3) The utility function u : R + → R is strictly increasing, concave, and twice differ-entiable, with derivatives u ′ ( c ) > and u ′′ ( c ) < for all c in the interior of its domain. At each time t , the social planner determines an economic structure (or more precisely, anindustrial structure) for the economy and allocate resources under the given overall structure.Let θ ( t ) be the economic structure (or with a slight abuse of notation, industrial structure)chosen at time t by the social planner from the information set of economic structures I E = { E i | i ∈ I } , and denote by e ( t ) ∈ E households’ and firms’ behaviors and economicactivities at time t . Then, given the industrial structure prior to time t , θ ( t − ), the socialplanner decides whether the industrial structure θ ( t ) should be the same as the previous one.If yes, then θ ( t ) = θ ( t − ); otherwise, the social planner chooses an industrial structure θ ( t )( = θ ( t − )) from I E and transforms the economic structure from θ ( t − ) to θ ( t ). After choosingthe industrial structure, the social planner will allocate resouces and decide the level of e ( t ).Since all the economic structures in I E are different only in industrial structures I Y , e ( t )can be expressed as a vector of numerical economic variables ( k ( t ) , R ( t ) , A ( t ) , w ( t ) , c ( t )).The above argument indicates that the path of industrial structures { θ ( t ) } is piecewiseconstant and satisfies the durationality and transformality of structures discussed in section2. Then, to present the idea, we represent { θ ( t ) } in another way. Let τ n denote the time ofthe n th structural transformation and κ n ∈ I E denote the transformed structure at time τ n ,for n ∈ { , , . . . , } . Assume that this decision problem starts at time t , and let τ = t and κ = E i ∈ I E (or i ∈ I ). The series of τ n and κ n satisfy(A4) t = τ ≤ τ < τ < · · · < τ n < · · · , and lim n → + ∞ τ n = + ∞ . κ n ∈ I E for all n ≥ τ ≥ τ means that the social planner may decide to transform the industrialstructure at the starting time t . Let ξ = { ( τ n , κ n ) n ≥ } be the double series of transformationtime and transformed structures, and denote by A the set of all series ξ satisfying (A4). Thengiven an ξ ∈ A , the industrial structure of the economy at time t is expressed as θ ( t ) = X n ≥ κ n [ τ n ,τ n +1 ) ( t ) , for t ≥ τ = t , [ τ n ,τ n +1 ) ( t ) is an indicator function of t , taking value 1 if t ∈ [ τ n , τ n +1 ) and 0 otherwise.By definition, θ ( t ) are right continous and have left limits at each τ n , and are piecewiseconstant over time t . Suppose there is no market failure during the structural transformation, so that the socialplanner does not need to intervene the transformation. Then, given an ξ = { ( τ n , κ n ) n ≥ } ∈ A ,the capital stock K ( · ) is continuous at transformation times τ n , that is, K ( τ n − ) = K ( τ n ).Combining this with (3.4) yields the capital accumulation process (cid:26) ˙ K ( t ) = F κ n [ K ( t ) , L ( t ) , A ( t )] − δK ( t ) − C ( t ) , τ n ≤ t < τ n +1 ,K ( τ n − ) = K ( τ n ) , t = τ n , n = 1 , , . . . , (3.7)Accordingly, the capital accumulation process per capita is expressed as (cid:26) ˙ k ( t ) = f κ n ( t, k ( t )) − ( δ + π ) k ( t ) − c ( t ) , τ n ≤ t < τ n +1 ,k ( τ n − ) = k ( τ n ) , t = τ n +1 , n = 1 , , . . . (3.8)Thus, provided the initial overall structure θ ( t ) = E i , the initial level of capital intensity k ( t ) = k , a path of structural transformation ξ = { ( τ n , κ n ) n ≥ } , and a path of consumption { c ( t ) } t ≥ , the total utility for the representative household starting at time t is expressedas J i ( t , k ; { c ( t ) , ξ } ) = Z ∞ t e − ( ρ − π ) t u ( c ( t )) dt. (3.9)The social planner’s objective is to solve the maximization problem V i ( t , k ) = max { c ( t ) ,ξ } J i ( t , k ; { c ( t ) , ξ } )subject to (3.8) and k ( t ) = k ∈ R + , θ ( t ) = E i ∈ I E . (3.10)Since markets are complete and competitive, given an initial industrial structure θ ( t ) = i and an initial capital intensity k ( t ) = k , the competitive equilibrium is defined as the pathsof structures and the amount of consumption and savings { θ ( t ) , c ( t ) , ˙ k ( t ) } t ≥ t that maximizethe household’s total utility (3.9) subject to the constraint (3.8). To solve the social planner’s optimal structure transformation and optimal resource allocationproblems in section 3, this section studies a general class of optimal switching and optimalcontrol problems and establishes the mathematical underpinning of the solution theory forthe problem. 13 .1 Combined optimal switching and optimal control problem
Consider a class of optimal switching and optimal control problems on infinite horizon [ t , ∞ )specified as follows. Denote by I = { , . . . , I } the index set of regimes. A switching control isa double sequence ξ = { ( τ n , κ n ) n ≥ } , where { τ n } is an increasing sequence in [ t , ∞ ) ( t ≥ τ n → ∞ , representing the decision on “when to switch,” and κ n is a piecewise constantc´adl´ag (right continuous with left limits) function taking values in I and representing thenew value of the regime at time τ n until time τ n +1 or the decision on “where to switch.”Denote by A the set of all double sequences ξ . Given an initial regime value i ∈ I anda control ξ = { ( τ n , κ n ) n ≥ } ∈ A , the regime value at any time t ( > t ) is expressed as θ i ( t ) = P n ≥ κ n [ τ n ,τ n +1 ) ( t ), t ≥ τ = t , and θ i ( t − ) = κ = i. State equations
Let Q = [ t , ∞ ) × R p and U ⊂ R q . For each i ∈ I , function µ i ( t, k, c ) : Q × U → R satisfiesthe following condition:(B1) For each i ∈ I , µ i ( t, k, c ) is continuous and satisfies a uniform Lipschitz condition:for any k, h ∈ R p , c ∈ U , and t ≥ t , there exists a constant D i > such that | µ i ( t, k, c ) − µ i ( t, h, c ) | ≤ D i · || k − h || . (4.1)Let U [ t , ∞ ) denote the set of Lebesgue measurable functions c ( t ) : [ t , ∞ ) → U such that Z T | µ i ( s, , c ( s )) | ds < ∞ , for all T > , i ∈ I . (4.2)Then, given an initial state k ( t ) = x ∈ R p , an initial regime θ ( t − ) = i ∈ I , controls ξ ∈ A and c ( · ) ∈ U [ t , ∞ ), a controlled dynamic system is given by (cid:26) ˙ k ( t ) = µ κ n ( t, k ( t ) , c ( t )) , τ n ≤ t < τ n +1 ,k ( τ n − ) = k ( τ n ) , t = τ n , n = 1 , , . . . . (4.3) Theorem 4.1.
Assume that µ i ( t, k, c ) satisfy condition (B1) for each i ∈ I . Given an initialstate ( i, x ) ∈ I × R p and a joint control ( ξ, c ( · )) ∈ A × U [ t , ∞ ) , system (4.3) has solutions k i,x ( t, c ) (or k i,x ( t ) if no confusion may arise) of the form k i,x ( t, c ( t )) = x + X n : τ n ≤ t Z τ n τ n − µ κ n ( s, k i,x ( s ) , c ( s )) ds, almost surely for t > , (4.4) and such a solution is unique. Moreover, | k i,x ( t, c ) | ≤ (1 + | x | ) e M ( t − t ) for all c ∈ U and t ≥ t . (4.5) If k i,y ( t, c ) is the solution with the initial state ( i, y ) , then | k i,x ( t, c ) − k i,y ( t, c ) | ≤ e D i ( t − t ) | x − y | for all c ∈ U and t ≥ t . (4.6)14 eward function Let φ ( t, k, c ) : Q × U → R be a measurable function.(B2) Assume that φ ( t, k, c ) is continuous and satisfies the Lipschitz condition, that is forany t ≥ t , k ∈ R p , and c ∈ U , there exists a constant D φ > such that | φ ( t, k, c ) − φ ( t, h, c ) | ≤ D φ | k − h | . (4.7) Moreover, φ ( t, k, c ) satisfies the following growth condition: there exists a ρ > such that,for all t ≥ t , k ∈ R p , and c ∈ U , | φ ( t, k, c ) | ≤ e − ρ ( t − t ) (1 + | k | ) . (4.8)With a little abuse of notation, we use ρ here to represent the discount rate ρ − π insection 3. Define for i ∈ I , x ∈ R p , and t ≥ U i ( t, x ) := { c ( · , k ) ∈ U [ t , ∞ ) (cid:12)(cid:12) k i,x (0) = x, Z ∞ t | φ ( s, k ( s ) , c ( s, k )) | ds < ∞} . We assume that U i ( t, x ) is not empty for all ( i, x ). We now specify for the initial data( t, x ) a set U i ( t, x ) of admissible controls, which means that, if we replace an admissiblecontrol by another admissible control after a certain time, then the resulting control is stilladmissible. More precisely, let c ( s, k ) ∈ U i ( t, x ) and c ( · ) ∈ U i ( r, x ( r )) for some r ∈ [ t, T ].Define a new control by e c ( s ) = c ( s ) I { s ∈ [ t,r ] } + c ( s ) I { s ∈ ( r,T ] } . Let e k ( s ) be the solution to (4.3)corresponding to control e c ( · ) and initial condition e k ( s ) = x and θ ( s − ). Then we assume that e c s ( · ) := e c ( · ) | [ s,T ] ∈ U i ( s, e k ( s )), s ∈ [ t, T ], where e c s ( · ) denotes the restriction to [ s, ∞ ] of e c ( · ).Note that e c s ( · ) implies that an admissible control always stays admissible. Total utility
Given an initial state k ( t ) = x ∈ R p , θ ( t − ) = i ∈ I , and a control ( ξ, c ( t, k )) ∈ A × U i ( t, x ),a controlled dynamic system is obtained and the total utility function may be defined. Instandard optimal switching problems, it is assumed that an instantaneous switching cost η ij ( t ) is incurred when the system switches from regime i to regime j , and η ij ( t ) satisfy thefollowing condition(B3) η ij ( t ) + η j,l ( t ) > η i,l ( t ) for j = i, l , η ii ( t ) = 0 , and | η ij ( t ) | ≤ e − ρt C for some ρ > and C > . This condition can be interpreted as follows. First, the triangular inequality in (B3) meansthat it is more costly to switch in two steps via an intermediate regime j than to swtichdirectly in one step from i to l . Second, let i = l and notice that η ii ( t ) = 0. The inequality15mplies that η ij ( t )+ η ji ( t ) >
0, which prevents arbitrarily switching back and forth at the sametime. Mathematically, the instantaneous costs distinguish the I regimes from each other andavoid degenerated switchings in which the controlled system switches at the beginning time t or ∞ . From the economic perspective, these costs can be considered as transformationbarriers that are generated by market failures due to the need for coordination in investmentsin related industries (Murphy, Shleifer, and Vishny 1989) or improvements in hard and softinfrastructure in the overall structure of the economy.Then given an initial state, k ( t ) = x ∈ R p , θ ( t − ) = i ∈ I , and a control, ( ξ, c ( t, k )) ∈ A × U i ( t, x ), consider the total utility function defined as J i ( t, x ; { ξ, c ( t, k ) } ) = Z ∞ t φ ( s, k, c ) ds − X n ≥ η κ n − ,κ n ( τ n ) . (4.9)The objective is to maximize the total utility function over A × U i ( t, x ). Accordingly, wedefine the value function V i ( t, x ) = sup ( ξ,c ( s,k )) ∈ A × U i ( t,x ) J i ( t, x ; { ξ, c ( s, k ) } ) , i ∈ I , x ∈ R p . (4.10) We first state the linear growth and Lipschitz continuity of the value function.
Lemma 4.1.
There exists some positive constant ρ ′ > and C > such that for t ≥ t , x ∈ R p , | V i ( t, x ) | ≤ C e − ρ ′ ( t − t ) (1 + || x || ) . (4.11) Moreover, the value function V i ( t, k ) ( i ∈ I ) is Lipschitz continuous in k : For any x, y ∈ R + and any t ≥ t , | V i ( t, x ) − V i ( t, y ) | ≤ C e ρ ′ ( t − t ) || x − y || . In the sequel, we assume that ρ is large enough so that V i ( t, x ) is Lipschitz continuous.To derive a partial differential equation characterization of V i ( t, x ), we shall use the followingdynamic programming principle: Lemma 4.2.
For any ( i, x ) ∈ I × R p and any r ≥ t , V i ( t, x ) = sup ξ,c ( s,k ): s ≥ r n sup ξ,c ( s,k ): t ≤ s Let w ( t, k ) ∈ C ( Q ) and fix i ∈ I .(i) We say that w ( t, k ) is a viscosity sub-solution of (4.16) if for any ϕ ∈ C , ( Q ) and any( t , k ) such that ( t , k ) is a local maximum of w ( t, k ) − ϕ ( t, k ), we havemax n sup c ∈ U h L c ϕ ( t , k ) + φ ( t , k , c ) i , M i V i ( t , k ) − V i ( t , k ) o ≥ . (4.17)17ii) We say that w ( t, k ) is a viscosity super-solution of (4.16) if for any ϕ ∈ C , ( Q ) andany ( t , k ) such that ( t , k ) is a local minimum of w ( t, k ) − ϕ ( t, k ), we havemax n sup c ∈ U h L c ϕ ( t , k ) + φ ( t , k , c ) i , M i V i ( t , k ) − V i ( t , k ) o ≤ . (4.18)(iii) We say that w ( t, k ) is a viscosity solution of (4.16) if w ( t, k ) is both a viscosity sub-solution and a viscosity super-solution of (4.16). Theorem 4.2. For each i ∈ I , the value function V i ( t, k ) defined by (4.10) is a viscositysolution to (4.16) . We next show the comparison principle for the HJB-QVI system (4.16). Theorem 4.3. Let U i ( t, k ) (respectively, W i ( t, k ) ), i ∈ I , be viscosity subsolutions (respec-tively, viscosity supersolutions) to (4.16) and satisfy the linear growth condition (4.11) . Then U i ( t, k ) ≤ W i ( t, k ) on Q = [0 , ∞ ) × R + for all i ∈ I . Combining Lemma 4.1, Theorem 4.2, and Theorem 4.3, we obtain the following. Theorem 4.4. For each i ∈ I , if the value functions V i ( t, k ) and W i ( t, k ) are viscositysolutions to (4.16) , then V i = W i on Q . We now consider a special class of the above combined control problems. In particular,assume that, for all i ∈ I , ( t, k ) ∈ Q and c ∈ U , µ i ( t, k, c ) ≡ e µ i ( k, c ) , φ ( t, k, c ) = e − ρt e φ ( k, c ) , η ij ( t ) = e − ρt e η ij . Then, by modifying the proofs for the results in sections 4.2 and 4.3, we can obtain resultssimilar to those in sections 4.2 and 4.3. Since the ideas of the proofs are similar, we omitthe proofs of the results in this section. We proceed by stating the conditions for e µ i and e φ modified from conditions (B1)-(B4).(B1) ′ For each i ∈ I , e µ i ( k, c ) is continuous and satisfies a uniform Lipschitz condition:for any k, h ∈ R p , c ∈ U , there exists a constant D i > such that | e µ i ( k, c ) − e µ i ( h, c ) | ≤ D i || k − h || . (4.19)(B2) ′ Assume that e φ ( k, c ) is continuous and satisfies the Lipschitz condition, that is, forany t ≥ t , k ∈ R + , and c ∈ U , there exists a constant D e φ > such that | e φ ( k, c ) − e φ ( h, c ) | ≤ D e φ || k − h || (4.20)18 oreover, e φ ( k, c ) satisfies the following growth condition: there exists a C > such that,for all k ∈ R p , and c ∈ U , | e φ ( k, c ) | ≤ C (1 + || k || ) . (4.21)(B3) ′ e η ij + e η jl > e η il for j = i, l and e η ii = 0 . Let U denote the set of Lebesgue measurable functions c ( k ) : R + → U such that Z T | e µ i (0 , c ( s )) | ds < ∞ , for all T > , i ∈ I . (4.22)Then, given an initial state k (0) = x ∈ R + , an initial regime θ (0 − ) = i ∈ I , controls ξ ∈ A and c ( · ) ∈ U , we define a controlled dynamic system (cid:26) ˙ k ( t ) = e µ κ n ( k ( t ) , c ( t )) , τ n ≤ t < τ n +1 ,k ( τ n − ) = k ( τ n ) , t = τ n , n = 1 , , . . . . (4.23)Define for i ∈ I , and x ∈ R , e U i ( x ) := { c ( k ) ∈ U (cid:12)(cid:12) k i,x (0) = x, Z ∞ t e − ρs | e φ ( k ( s ) , c ( s, k )) | ds < ∞} . Then, given an initial state k (0) = x , θ (0 − ) = i , and a control ( ξ, c ( k )) ∈ A × e U i (0 , x ), thetotal utility function is defined as e J i ( x ; { ξ, c ( k ) } ) = Z ∞ e − ρs e φ ( k, c ) ds − X n ≥ e − ρτ n e η κ n − ,κ n . (4.24)The social planner must solve the maximization problem v i ( x ) = sup ( ξ,c ( k )) ∈ A × e U i ( x ) e J i ( x ; { ξ, c ( k ) } ) , i ∈ I , x ∈ R p , (4.25)subject to (4.23) and k (0) = x and θ (0 − ) = i . The original value function defined by (4.25)can be obtained via V i ( t, x ) = e − ρt v i ( x ) . We can use arguments that are analogous to those in sections 4.2 to obtain the Lipschitzcontinuity for v i ( x ) and the dynamic programming principle. Lemma 4.3. There exists some positive constant ρ ′ > and C > such that for x ∈ R + , | v i ( x ) | ≤ C (1 + x ) . (4.26) Moreover, the value functions v i ( k ) ( i ∈ I ) is Lipschitz continuous in k : for any x, y ∈ R p , | v i ( x ) − v i ( y ) | ≤ C || x − y || . emma 4.4. For any ( i, x ) ∈ I × R + and any r ≥ t , v i ( x ) = sup ξ,c ( k ): s ≥ r n sup ξ,c ( k ): t ≤ s Let w ( k ) ∈ C ( R ) and fix i ∈ I .(i) We say that w ( k ) is a viscosity sub-solution of (4.29) if for any ϕ ∈ C ( R p ) and any( k ) such that ( k ) is a local maximum of w ( k ) − ϕ ( k ), we havemax n sup c ∈ U he L c ϕ ( k ) + e φ ( k , c ) i , f M i v i ( k ) − v i ( k ) o ≥ . (4.30)(ii) We say that w ( t, k ) is a viscosity super-solution of (4.29) if for any ϕ ∈ C ( R p ) andany ( t , k ) such that ( t , k ) is a local minimum of w ( t, k ) − ϕ ( t, k ), we havemax n sup c ∈ U he L c ϕ ( k ) + e φ ( k , c ) i , f M i v i ( k ) − v i ( k ) o ≤ . (4.31)(iii) We say that w ( k ) is a viscosity solution of (4.29) if w ( k ) is a viscosity sub-solution anda viscosity super-solution of (4.29).Following the arguments in Section 4.3, we can show the existence and uniqueness ofthe viscosity solutions to (4.29). Theorem 4.5. For each i ∈ I , the value function v i ( k ) defined by (4.25) is a viscositysolution to the HJB-QVI system (4.29) . Theorem 4.6. Let v i ( k ) (respectively, w i ( k ) ), i ∈ I be viscosity sub-solutions (respectively,viscosity super-solutions) to the HJB-QVI system (4.29) and satisfy the linear growth condi-tion (4.26) . Then v i ( k ) ≤ w i ( k ) on R + for all i ∈ I . Theorem 4.7. For each i ∈ I , if the value functions v i ( k ) and w i ( k ) are viscosity solutionsto the HJB-QVI system (4.29) , then v i = w i on R + . .5 Vanishing switching costs We now consider the limiting behavior of the value function (4.10) of the combined controlproblem when the positive switching cost η ij ( t ) → + uniformly. Let ǫ be the supremumof η ij ( t ) over I and [0 , ∞ ), that is, ǫ = max i,j ∈ I ,t ∈ [0 , ∞ ) η ij ( t ). Assumption (B3) implies that ǫ < ∞ . For convenience, denote V ǫi ( t, k ) by the value function V i ( t, k ) when switching costs η ij ( t ) are upper bounded by ǫ .Let ǫ → + in (4.13), we obtain the limit of the value function e V i ( t, k ) = lim ǫ → V ǫi ( t, k ),which should satisfymax n sup c ∈ U h ∂ e V i ∂t ( t, k ) + µ i ( t, k, c ) ∂ e V i ∂k ( t, k ) + φ ( t, k, c ) i , max j = i e V j ( t, k ) − e V i ( t, k ) o = 0 , (4.32)for all i ∈ I . The above equation indicates that max j = i e V j ( t, k ) ≤ e V i ( t, k ) for all i ∈ I so that e V ( t, k ) = · · · = e V I ( t, k ) , t ∈ [0 , ∞ ) , k ∈ R + . Denote by V ( t, k ) the common value of e V i ( t, k ), i ∈ I . Then (4.32) implies that V ( t, k ) is asolution to ∂V∂t ( t, k ) + max i ∈ I n sup c ∈ U h µ i ( t, k, c ) ∂V∂k ( t, k ) + φ ( t, k, c ) io = 0 . (4.33)Then this is the HJB equation for the infinite horizon control problem V ( t, x ) = sup c ( s,k ) ∈ U ( t,x ) Z ∞ t φ ( s, k, c ) ds, k = x ∈ R p . (4.34)The above heuristic argument can be summarized by the following result. Theorem 4.8. When η ij ( t ) = 0 , the value function V ( t, k ) defined by (4.34) is a uniqueviscosity solution of (4.33) . The above argument can be easily extended to the stationary form the problem. Inparticular, we have µ i ( t, k, c ) ≡ e µ i ( k, c ) and φ ( t, k, c ) = e − ρt e φ ( k, c ). Let ǫ = max i,j ∈ I e η ij .Let ǫ → + , we obtain the limit of the value function e v i ( k ) = lim ǫ → + v i ( t, k ), which shouldsatisfy max n sup c ∈ U h − ρ e v i ( k ) + e µ i ( k, c ) e v ′ i ( k ) + e φ ( k, c ) i , max j = i e v j ( k ) − e v i ( k ) o = 0 . (4.35)Noting that max j = i e v j ( k ) ≤ e v i ( k ) for all i, j , we have e v ( k ) = · · · = e v I ( k ) for all k . Denoteby v ( k ) the common value of e v i ( k ), i ∈ I . Then (4.35) implies that v ( k ) is a solution to − ρv ( k ) + max i ∈ I n sup c ∈ U he µ i ( k, c ) v ′ ( k ) + e φ ( k, c ) io = 0 , (4.36)21hich is the HJB equation for the infinite horizon control problem v ( x ) = sup c ( k ) ∈ e U i ( x ) Z ∞ e − ρs e φ ( k, c ) ds, k = x ∈ R p . (4.37)In parallel to Theorem 4.8, we have the next result Theorem 4.9. When e η ij ≡ , the value function v ( k ) defined by (4.37) is a unique viscositysolution of (4.36) . The combined control problem and its solution theory in section 4 provide the mathematicalunderpinning for a class of structural transformation and resource allocation problems. Wenow use the theory developed in section 4 to analyze the generic EST model in section 3 andestablish its competitive equilibrium. The generic EST model in section 3 assumes that p = q = 1 and µ i ( t, k, c ) = f i ( t, k ) − ( δ + π ) k − c, φ ( t, k, c ) = e − ( ρ − π ) t u ( c ) , Note that the competitive equilibrium in the generic EST model consists of paths of con-sumption, capital stock, wage rates, rental rates of capital, and industrial structures, suchthat (i) the social planner maximizes the representative household’s utility given the initialeconomic structure and initial capital intensity, wage rates and rental rates, and (ii) thepaths of wage rates and rental rates are such that given the paths of capital stock and labor,all markets clear. Using Theorems 4.2, 4.3, and 4.8, we can show the following. Proposition 5.1. For each i ∈ I , V i ( t, k ) defined by (3.10) is a viscosity solution to max n sup c ∈ U h ∂V i ∂t ( t, k ) + (cid:2) f i ( t, k ) − ( δ + π ) k − c (cid:3) ∂V i ∂k ( t, k ) + e − ( ρ − π ) t u ( c ) i , max j = i (cid:2) V j ( t, k ) − η ij ( t ) (cid:3) − V i ( t, k ) o = 0 , (5.1) and such solutions are unique on [0 , ∞ ) × R + . The above HJB-QVI system contains two components and their economic interpreta-tion is clear. The first component determines optimal consumption and characterizes thetransitional dynamics of the capital intensity and optimal consumption, given that the cur-rent economic structure E i (or the current production structure Y i ) is optimal. The second22omponent compares the value functions associated with each of the economic structuresand chooses the optimal one. In particular, given economic structure E i prior to time t andanother economic structure E j ( j = i ), the social planner compares the total utility asso-ciated with i and the total utility after transforming instantaneously from E i to E j . If theformer is larger for any j ∈ I and j = i , the economy should stay with economic structure E i at time t and then use the HJB component to determine the resource allocation within thegiven industrial structure and optimal consumption. Otherwise, the social planner shouldtransform the industrial structure and overall economic structure from E i to E j at time t .Proposition 5.1 has also the following economic implication for some special cases of I E = { E , . . . , E I } . When I = 1 or I E = { E } , that is, there is only one economic structureavailable to the social planner, hence i ≡ c ∈ U h ∂V ∂t ( t, k ) + (cid:2) f ( t, k ) − ( δ + π ) k − c (cid:3) ∂V ∂k ( t, k ) + e − ( ρ − π ) t u ( c ) i = 0 . (5.2)Equation (5.2) is the same as the standard HJB equation in the neoclassical growth model.Hence, the competitive equilibrium associated with HJB-QVI system (5.1) with I = 1 isthe same as the one associated with the neoclassical growth model. When I = { , } or I E = { E , E } , HJB-QVI equations (5.1) hold for i, j = 1 , , i = j , and the social planner ofthe economy only needs to decide when to transform from the current industrial structureand, thus, the economic structure, to the other one. When I = { , . . . , I } and I ≥ 3, thesocial planner needs to choose not only when but also which industrial structure to transform.The latter choice involves comparing two industrial structures other than the current one.Another issue is that the two components in HJB-QVI system (5.1) can not attain 0at the same time. This can be seen from the proof of Theorem (4.2) and has an intuitiveeconomic interpretation. Intuitively, the social planner decides at each time t not to trans-form or to the transform her economic structure, which corresponds to the first or secondcomponent in system (5.1), respectively. Hence only one of the two components can attain0 at any time t > Proposition 5.1 implies that there exist three types of equilibria in the EST model—static,dynamic, and structural. We now discuss these equilibria and their implications. Static and dynamic equilibria and the Euler equation of consumption The first two types of equilibria are the static and dynamic equilibria when the economicstructure E i prior to time t is still optimal at time t . The static and dynamic equilibria are23haracterized by ( sup c ∈ U h ∂V i ∂t ( t, k ) + (cid:2) f i ( t, k ) − ( δ + π ) k − c (cid:3) ∂V i ∂k ( t, k ) + e − ( ρ − π ) t u ( c ) i = 0 , max j = i (cid:2) V j ( t, k ) − η ij ( t ) (cid:3) − V i ( t, k ) < . (5.3)The second component, sup j = i V j ( t, k ) − V i ( t, k ) < 0, in (5.3) indicates that there does notexist an economic structure that is better than the current economic structure E i . Conse-quently, economic structure E i (or production structure Y i ) prior to time t is still optimal attime t . Then, θ ( t − ) = θ ( t ) = E i and the equationsup c ∈ U h ∂V i ∂t ( t, k ) + (cid:2) f i ( t, k ) − ( δ + π ) k − c (cid:3) ∂V i ∂k ( t, k ) + e − ( ρ − π ) t u ( c ) i = 0 (5.4)characterizes the static and dynamic equilibria that are associated with the current economicstructure E i and, hence, optimal consumption can be determined. The first-order conditionof (5.4) implies that optimal consumption c ∗ i ( t, k ) solves the equation e − ( ρ − π ) t u ( c ∗ i ( t, k )) = ∂V i ∂k ( t, k ) . (5.5)Since c ∗ i ( t, k ) is the optimal consumption associated with economic structure E i , optimalconsumption c ∗ i ( t, k ) and c ∗ j ( t, k ) under two different economic structures E i and E j ( i = j )are usually different.To study the dynamics of optimal consumption, we first take the derivative of equation(5.5) with respect to k and compare the result with (5.5). Then we have˙ k · ∂ V i ∂k ( t, k ) = − ǫ u ( c ∗ i ) · c ∗ i ∂c ∗ i ∂t ( t, k ) · ∂V∂k ( t, k ) . (5.6)where ǫ u ( c ) = − u ′′ ( c ) · c/u ′ ( c ) is the elasticity of the marginal utility u ′ ( c ). Then taking thederivative of equation ∂V i ∂t ( t, k ) + (cid:2) f i ( t, k ) − ( δ + π ) k − c ∗ i ( t, k ) (cid:3) ∂V i ∂k ( t, k ) + e − ( ρ − π ) t u ( c ∗ i ( t, k )) = 0 (5.7)with respect to k and simplifying the result by (5.6), we obtain the Euler equation of con-sumption associated with economic structure E i :1 c ∗ i ∂c ∗ i ∂t ( t, k ) = 1 ǫ u ( c ∗ i ) h ∂f i ∂k ( t, k ) − δ − π − ∂ V i ∂t∂k . ∂V i ∂k i . (5.8)This can be viewed as a nonstationary version of the Euler equation in the neoclassicalgrowth models; see Acemoglu (2009, section 8.2.2). Since assumption (A3) suggests that theutility function is monotonically increasing with c , the procedure of maximizing the total24tility by choosing economic structures can be considered a procedure of comparing (5.8)for different economic structures. We will elaborate on this in section 5.4.Competitive factor markets imply that, when the economic structure at time t is E i , therental rate of capital R i ( t ) and the wage rate w i ( t ) are given by R i ( t, k ) = ∂F i ∂K [ K ( t ) , L ( t ) , A ( t )] = ∂F i ∂k [ k ( t ) , , A ( t )] = ∂f i ∂k ( t, k ) , (5.9)and w i ( t, k ) = ∂F i ∂L [ K ( t ) , L ( t ) , A ( t )] = f i ( t, k ( t )) − k ( t ) ∂f i ∂k ( t, k ) . (5.10)Given economic structure E i and optimal consumption (5.5), the dynamic equilibrium of thecapital-labor ratio and optimal consumption is characterized by the equation˙ k ( t ) = f i ( t, k ( t )) − ( δ + π ) k ( t ) − c ∗ i ( t, k ( t )) . (5.11)The static and dynamic equilibria in the EST model are not exactly the same as those inthe neoclassical growth models. In the neoclassical growth models, the economic structureis fixed with E i , and hence, static and dynamic equilibria are defined whether the currenteconomic structure E i is optimal or not. By contrast, in the EST model, the static anddynamic equilibria at time t depend on the optimal economic structure E i at that time.That is, if an economic structure E i is not optimal at time t and hence is not chosen bythe social planner of the economy, static and dynamic equilibria associated with economicstructure E i at time t do not exist. Structural equilibria and optimal industrial structures HJB-QVI system (5.1) also characterizes a third type of equilibrium, which we refer to as the structural equilibrium . Under such an equilibrium, ( sup c ∈ U h ∂V i ∂t ( t, k ) + (cid:2) f i ( t, k ) − ( δ + π ) k − c (cid:3) ∂V i ∂k ( t, k ) + e − ( ρ − π ) t u ( c ) i < , max j = i (cid:2) V j ( t, k ) − η ij ( t ) (cid:3) − V i ( t, k ) = 0 . (5.12)The HJB inequality in (5.12) implies that, if the current economic structure were E i , theassociated economy would not attain the static and dynamic equilibria no matter how con-sumption was chosen. Mathematically, the optimality principle fails when the economy isassociated with the industrial structure embodied in economic structure E i . In contrastto this, the equality in (5.12) shows that there exists another industrial structure Y j (orequivalently, an economic structure E j = E i ) such that its associated value function (i.e., themaximized total utility) is greater than that associated with economic structure E i . Denotingthe new optimal economic structure by E j ∗ ( t,k ) , j ∗ satisfies the following condition: j ∗ ( t, k ) = arg max j ∈ I ,j = i (cid:2) V j ( t, k ) − η ij ( t ) (cid:3) , V j ∗ ( t, k ) − η i,j ∗ ( t ) = V i ( t, k ) (5.13)25his suggests that the optimal economic structure at time t is expressed as θ ( t ) = (cid:26) E i if equation(5.3) holds , E j ∗ ( t,k ) if equation(5.12) holds , (5.14)in which j ∗ ( t, k ) is given by condition (5.13).The discussion above assumes positive switching costs that satisfy assumption (B3). Inthe degenerate case of vanishing switching costs η ij ( t ) ≡ 0, the discussion in Section 4.5implies that the value function V ( t, k ) = · · · = V I ( t, k ) = V ( t, k ) is a solution of ∂V∂t ( t, k ) + max i ∈ I n sup c ∈ U h(cid:0) f i ( t, k ) − ( δ + π ) k − c (cid:1) ∂V∂k ( t, k ) + e − ( ρ − π ) t u ( c ) io = 0 . (5.15)This suggests that the optimal economic structure at time t is given by θ ( t ) = E j ∗ ( t,k ) , inwhich j ∗ ( t, k ) satisfies the following two conditions: j ∗ ( t, k ) = arg max i ∈ I n ∂V∂t ( t, k ) + sup c ∈ U h(cid:0) f i ( t, k ) − ( δ + π ) k − c (cid:1) ∂V∂k ( t, k ) + e − ( ρ − π ) t u ( c ) io , (5.16)and ∂V∂t ( t, k ) + sup c ∈ U h(cid:0) f j ∗ ( t, k ) − ( δ + π ) k − c (cid:1) ∂V∂k ( t, k ) + e − ( ρ − π ) t u ( c ) io = 0 . (5.17)The discussion above implies the following property: Proposition 5.2. The optimal industrial structure in an overall structure at any given time t and with any given capital intensity k is determined by structural equilibrium (5.1) at ( t, k ) .Moreover, the optimal industrial structure is a function of t and k ( t ) , and hence endogenousto the capital intensity (or more generally, the factor endowments of the economy). Furthermore, since E j ∗ ( t,k ) is the optimal industrial structure for the given ( t, k ), the socialplanner should transform the economic structure of the economy from E i to E j ∗ ( t,k ) at t .Consequently, the optimal consumption, rental rate of capital, and wage rate will shift from c ∗ i ( t, k ), R i ( t, k ), and w i ( t, k ) to c ∗ j ∗ ( t, k ), R j ∗ ( t, k ), and w j ∗ ( t, k ), respectively. The structural equilibrium tells us how an optimal economic structure is determined for thegiven ( t, k ). We now consider the issue of finding the set of capital intensities to supportstructural transformation. Specifically, let E i ∈ I E be the economic structure prior to time t . We define the transformation region of capital intensities from E i as follows: S i ( t ) = n k ∈ (0 , ∞ ) : max j = i (cid:2) V j ( t, k ) − η ij ( t ) (cid:3) − V i ( t, k ) = 0 o . (5.18) When η ij ( t ) ≡ 0, we can make use of equation (5.15) and define transformation regions and no-transformation regions in a similar way. i ( t ) is a closed subset of (0 , ∞ ) and represents a set of capital intensities with which thesocial planner should transform the economic structure away from E i . Consider an economicstructure E j , which is different from E i , and define S i,j ( t ) = (cid:8) k ∈ S i ( t ) : V j ( t, k ) − η ij ( t ) = V i ( t, k ) (cid:9) . (5.19)Since assumption (A4) implies that η ij + η ji > 0, the static and dynamic equilibria associatedwith economic structures E i and E j imply that S i,j ( t ) ∪ S j,i ( t ) ( (0 , ∞ ). It is easy to see that S i ( t ) = [ j ∈ I ,j = i S i,j ( t ) . (5.20)We also define N i ( t ) as the complement set of S i ( t ) in (0 , ∞ ), which is the so-called contin-uation region or the no-transformation region associated with economic structure E i . N i ( t ) = n k ∈ (0 , ∞ ) : sup j = i V j ( t, k ) − η ij ( t ) < V i ( t, k ) o . (5.21) N i ( t ) is an open set and represents a collection of capital intensities with which economicstructure E i is optimal. In this open domain, the value function V i ( t, k ) is continuous differ-entiable and satisfies equation (5.4). Then, by definition, S i ( t ) ∪ N i ( t ) = h [ j ∈ I ,j = i S i,j ( t ) i ∪ N i ( t ) = (0 , ∞ ) . (5.22)The definition of transformation regions suggests a measure for comparing the compar-ative advantage of an economic structure over another. Assume that the current economicstructure is E i and consider economic structures E j and E l ( j = l, j = i ). Let H j,li ( t, k ) := [ V j ( t, k ) − η ij ( t )] − [ V l ( t, k ) − η il ( t )] , l = j, j = i. (5.23) H j,li ( t, k ) measures the comparative structural advantage of E j over E l with respect to thecurrent economic structure E i . Therefore, we say that economic structure E j dominateseconomic structure E l with respect to current economic strucutre E i if H j,li ( t, k ) > The discussion so far has focused on the general or nonstationary form of the maximizationproblem. We now consider a stationary form of the problem by assuming that f i ( t, k ) = f i ( k )for all i ∈ I and e η ij → + uniformly for all i, j ∈ I . In such case, Proposition 5.1 becomesthe following: 27 roposition 5.3. For each i ∈ I , v i ( k ) defined by (4.27) is a viscosity solution to max n sup c ∈ U h − ( ρ − π ) v i ( k ) + (cid:2) f i ( k ) − ( δ + π ) k − c (cid:3) v ′ i ( k ) + u ( c ) i , max j = i [ v j ( k ) − e η ij ] − v i ( k ) o = 0 , (5.24) and such a solution is unique on R + . In this economy, the competitive equilibrium consists of time paths of industrial struc-tures, consumption, capital stock, wage rates, and rental rates of capital, such that the socialplanner maximizes the representative household’s total utility, and the time paths of wagerates and rental rates of capital are taken so as to make all markets clear. As an analog ofsection 5.2, Proposition 5.3 implies that the competitive equilibrium of the economy can becharacterized by its three components, the static, dynamic and structural equilibria. Beforewe characterize the competitive equilibrium, we first note the following results obtained byarguments analogous to those in sections 5.1 and 5.2:(i) When I E = { E } , HJB-QVI equation (5.24) degenerates tosup c ∈ U h − ( ρ − π ) v i ( k ) + (cid:2) f i ( k ) − ( δ + π ) k − c (cid:3) v ′ i ( k ) + u ( c ) i = 0 , which is the same as the stationary form of the HJB equation for the neoclassicalgrowth model.(ii) The static and dynamic equilibria of the economy are characterized by the first compo-nent in equation (5.24), and the structural equilibrium is characterized by the secondcomponent when switching costs satisfy assumption (B3)’.(iii) When e η ij ≡ 0, the discussion in Section 4.5 implies that the value function v i ( k ) satisfiesmax i ∈ I n sup c ∈ U h − ( ρ − π ) v i ( k ) + (cid:2) f i ( k ) − ( δ + π ) k − c (cid:3) v ′ i ( k ) + u ( c ) io = 0 . (5.25)(iv) Since the value function v i ( k ) has a stationary form, the optimal industrial structureis completely determined by capital intensity k , or generally, by the economy’s factorendowments.(v) Since the value function has a stationary form, regions of transformation and no-transformation in the case of e η ij > S i,j = (cid:8) k ∈ (0 , ∞ ) : v j ( k ) − e η ij = v i ( k ) (cid:9) , S i = ∪ j = i S i,j and N i = (cid:8) k ∈ (0 , ∞ ) : sup j = i [ v j ( t ) − e η ij ] < v i ( k ) (cid:9) , respectively.For the vanishing case e η ij ≡ 0, regions of transformation and no-transformation canbe similarly defined by making use of (5.25).28 uler consumption equation Since the stationary form of the value function has the form V i ( t, k ) = e − ( ρ − π ) t v i ( k ), theEuler equation of consumption (5.8) can be simplified as˙ c ∗ i c ∗ i = 1 ǫ u ( c ∗ i ) (cid:2) f ′ i ( k ) − δ − ρ (cid:3) . (5.26)Choosing a constant relative risk aversion utility function, that is, u ( c ) = c − γ − γ for γ = 1 , γ ≥ u ( c ) = log c for γ = 1, the elasticity of the marginal utility of consumption is givenby the constant γ , that is, ǫ u ( c ∗ i ) ≡ γ . In such case, the increase in consumption dependson the increase in production f ′ i ( k ), and hence choosing an optimal economic structureis equivalent to choosing an optimal industrial structure, or more specifically, an optimalproduction function. To explain this idea, we study an economy of I industrial structures and its competitiveequilibrium and structural transformation regions. Assume that L ( t ) ≡ 1, so that π ≡ f i ( k ) = A i ( k − x i ) + for i ∈ I , where A i is the technology level, ( k − x i ) + = max( k − x i , x i is a threshold for the i th production function. Assume that 0 < A < A < · · · < A I and 0 = x < x < · · · < x I .Given industrial structures Y i = { f i } at time t , the state equation (or the capitalaccumulation process) (4.23) becomes (cid:26) ˙ k ( t ) = ( A i − δ )( k ( t ) − x i ) + − c ( t ) , τ n ≤ t < τ n +1 ,k ( τ n ) = k ( τ n − ) , n = 1 , , . . . (5.27)Assume that the representative household has a power utility function u ( c ) = c − γ / (1 − γ ), γ > 1. Then, given the initial condition k (0) = k and θ (0 − ) = i ( i ∈ I ), the representativehousehold’s total utility is given by (4.24). The social planner must solve the maximizationproblem (4.25) subject to the resource constraint (5.27). By Proposition 5.3, we have that,for i ∈ I , the value function defined by (4.25), v i ( k ), is a unique viscosity solution tomax n sup c n [( A i − δ )( k − x i ) + − c ] v ′ i ( k ) − ρv i ( k ) + c − γ − γ o , sup j = i v j ( k ) − v i ( k ) o = 0 . (5.28)Note that the argument in Section 4.5 suggests that v ( k ) = · · · = v I ( k ) for all k .We first consider the static and dynamic equilibria when the economy is associated withoverall structure E i or industrial structure f i ( k ). Denote the value function in this case by29 v i ( k ). The static equilibrium is implied by the first supremum in equation (5.28), whichsuggests the optimal consumption when the economy has industrial structure Y i = { f i } ,then we have c ∗ i = (cid:2) v ′ i ( k ) (cid:3) − /γ . Plugging this into the first supremum of equation (5.28), weobtain a partial differential equation for the dynamic equilibrium: − ρv i ( k ) + ( A i − δ )( k − x i ) + v ′ i ( k ) + γ − γ (cid:2) v ′ i ( k ) (cid:3) γ − γ = 0 . (5.29)Solving equation (5.29) yields the value function when the economy stays with industrialstructure { f i } : e v i ( k ) = Q i ( k − x i ) − γ + , Q i := γ γ − γ (cid:2) ρ + ( A i − δ )( γ − (cid:3) − γ . (5.30)Accordingly, the optimal consumption associated with industrial structure { f i } is c ∗ i = ρ + ( A i − δ )( γ − γ ( k − x i ) + . Plugging c ∗ into (5.29) yields the dynamics of the capital intensity˙ k ( t ) = A i − ρ − δγ ( k ( t ) − x i ) + , and hence ( k ( t ) − x i ) + = ( k ( t ) − x i ) + e ( A i − ρ − δ )( t − t ) /γ , t ≥ t . We next discuss the structural transformation regions of the capital intensities. For conve-nience, we denote for 1 ≤ i < j ≤ I , a ij := (cid:16) Q j Q i (cid:17) − γ = (cid:16) A j − A i )( γ − ρ + ( A i − δ )( γ − (cid:17) γγ − > ,k ij := x j + x j − x i a ij − > x j . (5.31)Obviously, a ij = a − ji and k ij = k ji . Economy with I = 2 industrial structures Consider an economy in which only one of I = 2 production functions can be chosen by thesocial planner. Since f ≺ f , we can show the following: Proposition 5.4. Assume that I = { , } . The structural equilibrium in equation (5.28) implies that S = [ k , ∞ ) . Moreoever, the value function v ( k ) is given by v ( k ) = (cid:26) e v ( k ) if < k < k e v ( k ) if k ≤ k (5.32) where k is defined by (5.31) . f ) and initial capital intensity k , if k 0. Then when k < k , the social planner shouldimmediately transform the industrial structure from { f } to { f } , if the transforma-tion does not require structural adjustment costs. In such case, the path of optimaleconomic structures is given by θ (0 − ) = E , θ ( t ) = E for t ∈ [0 , t ) and θ ( t ) = E for t ∈ [ t , ∞ ). 31 conomy with I = 3 industrial structures We now consider an economy with three industrial structures, Y i = { f i } , i ∈ I = { , , } .The competitive equilibrium of the economy in this case is a little more complicated thanthat with two industrial structures, as the social planner in this economy needs to choosenot only when to transform the industrial structure, but where to transform as well. Proposition 5.5. Suppose that I = { , , } and k < min { k , k } . Then S = [ k , min { k , k } )and S = [max { k , k } , ∞ ). v ( k ) = e v ( k ) if k ∈ [ x , k ) ∪ [min { k , k } , max { k , k } ) e v ( k ) if k ∈ S e v ( k ) if k ∈ S . (5.35)Proposition 5.5 further concludes that an economy’s optimal path of structural transfor-mation is determined by the economy’s capital intensity (or factor endowment) at each timepoint. In addition to similar interpretations as those implied by Proposition 5.4, Proposition5.5 further implies the following:(i) An interesting problem of structural transformation in the presence of more than twoindustrial structures is which structure is more appropriate for an economy. Proposition5.5 shows that, when the accumulated capital intensity is not high enough, say k ∈ S ,the economy should not choose industrial structure { f } which requires very highcapital intensity ( f requires k ≥ x ).(ii) If an economy starts with industrial structure { f } but a high level of capital intensity k ∈ [min { k , k } , max { k , k } ), although the high level of capital intensity k exceedsthe maximum of transformation region S , that is, max { k | k ∈ S } , it does not suggestthat the economy should immediately transform to industrial structure { f } . In suchcase, the optimal growth path of the economy is to keep growing for a while so thatthe capital intensity accumulates to a higher level, max { k , k } , and then transformfrom industrial structure { f } to industrial structure { f } directly.(iii) Only if the economy starts with industrial structure { f } but with a very high capitalintensity k ∈ S or k ≥ max { k , k } should the economy transform from industrialstructure 1 to 3 directly.The above result should be very carefully interpreted in economic applications. Forinstance, although (ii) seems to suggest that an economy can accumulate capital, increasecapital intensity, and then choose a time to upgrade and skip intermediate industrial struc-tures, it actually suggests not to do so due to the following reasons. Mathematically, whenwe consider the development path of an economy, the value function of the economy in thereal world does not have the stationary form; hence, at each time point, the optimal path32tarting at that time should be investigated from the original HJB-QVI equation. The abovecase may occur in resource abundant countries, such as some oil rich countries in the MiddleEast. In most other cases, a developing country usually starts with relatively low capitalintensity k ∈ k and accumulates capital stock continuously. This suggests that the de-veloping economy cannot get around region S in which the economy should upgrade theeconomic structure to E , except when the economy is small and there is a sudden and bigpositive inflow of capital, which is possible but rare. The generic EST model and its competitive equilibrium argument can be extended toeconomies with more complicated structures. The EST model provides us a general frame-work to “paste together” models of economic development and growth at different devel-opment stages via transformation of economic structures. Hence, various economic issuesand different economic ideas can be discussed on the same platform. In this section, weexplain this point by extending the EST to several complex economic substructures, includ-ing composite production structures, composite consumer preference, choice of exogenoustechnology, switching between technology adoption and R&D, and institutional structures. The economy in the generic EST model produces the unique final good. We now extend itto an economy in which intermediate goods and a unique final good are produced and thefinal good uses intermediate goods as inputs. When a production structure is fixed in thiseconomy, a variation of the non-balanced growth model (Acemoglu and Guerrieri, 2008) isobtained. Hence, the extended EST model here describes an economy of stagewise economicgrowth. Intermediate goods and technology The economy is similar to that in section 3, except intermediate goods are also produced.The final good is produced competitively by combining m intermediate goods with elasticityof substitution ǫ ∈ [0 , ∞ ), that is, Y ( t ) = F [ Y ( t ) , . . . , Y m ( t ); ω ] = (cid:16) m X j =1 w j Y j ( t ; ω ) ǫ − ǫ (cid:17) ǫǫ − , (6.1)where w > , . . . , w m > w + · · · + w m = 1, and ω represents the vector of parameters inthe production functions and will be specified later. The intermediate goods Y , . . . , Y m are33roduced competitively with production functions Y j ( t ; ω ) = F i ( K j , L j , A j ; ω ) = A j ( t ) L j ( t ) − α j K j ( t ) α j , (6.2)where A j ( t ) is the technology of the j th intermediate good at time t . Technological progressin all sectors is exogenous and takes the form˙ A j ( t ) = ϑ j A j ( t ) , ϑ j > , j = 1 , . . . , m. (6.3)For convenience, we assume that α ≤ · · · ≤ α m , that is, the sectors with larger α ’s are morecapital intensive. Capital and labor market clearing requires that at each time K ( t ) + · · · + K m ( t ) ≤ K ( t ) , L ( t ) + · · · + L m ( t ) ≤ L ( t ) , (6.4)where K denotes the aggregate capital stock and L is total population. L j ( t ) and K j ( t )( j = 1 , . . . , m ) are nonnegative. Labor L ( t ) is supplied inelastically and follows the process(3.1) or L ( t ) = L (0) exp( πt ).Let P j ( t ; ω ) be the price of Y j good j ( j = 1 , . . . , m ) at time t for the given industrialstructure ω . We normalize the price of the final good, P , to one at all points, so that1 ≡ P ( t ; ω ) = (cid:16) m X j =1 w ǫj P j ( t ; ω ) − ǫ (cid:17) / (1 − ǫ ) . (6.5)Denote the rental price of capital, wage rate, and interest rate by R ( t ; ω ), w ( t ; ω ), and r ( t ; ω ),respectively. Production structures We now represent the production structure of the economy, using the notation introducedin section 3. Since m intermediate goods and a final good are produced, production andits structure can be expressed as ( Y , Y ( ω )), where Y = { ( Y ( t ) , Y ( t ) , . . . , Y m ( t )) } and Y ( ω ) = { F, F , . . . , F m } , and ω = ( w , . . . , w m , α , . . . , α m ) is an element of the set Ω Y = { ω | w , . . . , w m ≥ , P mj =1 w j = 1 , ≤ α ≤ · · · < α m < } . Here, Y ( ω ) describes thecomposition and organization of industrial structures, producing the final good and inter-mediate goods. Note that w j = 0 implies that intermediate goods j are not produced in theeconomy. Hence, the change of w j from w j = 0 to w j > A , A ), where A = { ( A ( t ) , . . . , A m ( t ) } , A ( e ϑ ) = { ( A ,ϑ ( · ) , . . . , A m,ϑ m ( · )) | A j ( · ) are given by (6.3) } , and e ϑ = ( ϑ , . . . , ϑ m ) ∈ Ω A = { e ϑ | ϑ j > , j = 1 , . . . , m } . Since at most m intermediategoods are produced, labor allocation in production and its structure are given by ( L , L ),where L = { ( L ( t ) , L ( t ) , . . . , L m ( t ) } and L = { ( L ( · ) , L ( · ) , . . . , L m ( · )) | L ( · ) + · · · + L m ( · ) ≤ ( · ) , L ( · ) satisfies (3.1) } . Similarly, the allocation of capital stock and its structure are ex-pressed as ( K , K ), where K = { ( K ( t ) , K ( t ) , . . . , K m ( t )) | K ( t ) + · · · + K m ( t ) ≤ K ( t ) } and K = { ( K ( · ) , K ( · ) , . . . , K m ( t )) | K ( · ) + · · · + K m ( · ) ≤ K ( · ) , K ( · ) satisfies the budget con-straint (3.4) } . Goods and factor prices and their structure can be represented as ( P , P ),where P = { ( R ( t ) , w ( t ) , P ( t ) , . . . , P m ( t ) , P ( t )) } and P = { factor prices are determined byfirms to maximize their profit, and P ( t ) , . . . , P m ( t ) , P ( t ) satisfy (6.5) } .Then with ( H , H ), ( F , F ), ( M , M ), ( C , C ), and ( U , U ) defined as those in section 3,agents’ behavior and economic activities and the economic structure in the economy can stillbe expressed as ( E , E ). To highlight the structures of production and exogenous technologicalprogress, which are described by ω ∈ Ω Y and e ϑ ∈ Ω A , respectively, the economic structureof the economy may be expressed as E ( ω, e ϑ ). Social planner’s objective and competitive equilibrium Suppose the social planner’s information set of economic structures is I Y × A = { E i | E i = E ( ω i , e ϑ i ) , i ∈ I } . The social planner can choose an economic structure for the economy andallocate resources under the chosen economic structure. Assume that an economic structure E i ∈ I Y × A (or equivalently, i ∈ I ) is chosen for the economy. The aggregate resourceconstraint, which is equivalent to the budget constraint of the representative household, hasthe same form as (3.4). Define the shares of capital and labor allocated to industry i as ξ j ( t ) ≡ K j ( t ) K ( t ) , and λ j ( t ) ≡ L j ( t ) L ( t ) . Then output per capita at time t can be expressed as y ( t ; ω ) ≡ f ( t, k ( t ); ω i ) ≡ n m X j =1 w ij (cid:2) k ( t ) α ij A j ( t ) ξ j ( t ) α ij λ j ( t ) − α ij (cid:3) ǫ − ǫ o ǫǫ − . (6.6)At each time t , the social planner in the economy must optimally choose an eco-nomic structure θ ( t ) = E i ∈ I Y × A , consumption c ( t ), and allocations of capital and labor( ξ j ( t ) , λ j ( t )). The aggregate resource constraint per capita is (cid:26) ˙ k ( t ) = f κ n ( t, k ( t ); ω i ) − ( δ + π ) k ( t ) − c ( t ) , τ n ≤ t < τ n +1 ,k ( τ n − ) = k ( τ n ) , t = τ n +1 , n = 1 , , . . . (6.7)The representative household’s total utility, starting at time t , has the same form as (3.9)and is given as J i ( t , k ; { c ( t ) , ξ } ) = Z ∞ t e − ( ρ − π ) t u ( c ( t )) dt (6.8)The social planner’s objective is to solve the maximization problem V i ( t , k ) = max { c ( t ) ,ξ } J i ( t , k ; { c ( t ) , ξ } )subject to (6.7) and k ( t ) = k ∈ R + , θ ( t ) = E i ∈ I Y × A . (6.9)35iven the above specification, the competitive equilibrium of the economy consists ofpaths for factor and intermediate goods prices { r ( t ; θ ( t )) , w ( t ; θ ( t )) , p j ( t ; θ ( t )) } , employmentand capital allocation { ξ j ( t ; θ ( t )) , λ j ( t ; θ ( t )) } , consumption and savings decisions { c ( t ) , ˙ K ( t ) } ,and economic structures { θ ( t ) } such that the utility of the representative household is max-imized, firms maximize profits, and markets clear. To characterize the competitive equilib-rium mathematically, we first express the maximization problem in the form of the combinedoptimal control and optimal switching problem in section 4.1 in which µ i ( t, k, c ) = f ( t, k ( t ); ω i ) − ( δ + π ) k ( t ) − c ( t ) , τ n ≤ t < τ n +1 , and the control space U ≡ { ( c ( t ) , ξ ( t ) , . . . , ξ I ( t ) , λ ( t ) , . . . , λ I ( t )) | c ( t ) : [0 , ∞ ) → U, ξ i ( t ) : [0 , ∞ ) → [0 , ,λ i ( t ) : [0 , ∞ ) → [0 , 1] are Lebesgue measurable functions } . According to Theorems 4.2-4.4 and Proposition 5.1, we have the following: Proposition 6.1. For each i ∈ I , V i ( t, k ) defined by (3.10) is a viscosity solution to max n sup c ( t ) ,λ j ( t ) ,ξ j ( t ) h ∂V i ∂t ( t, k ) + (cid:2) f i ( t, k ) − ( δ + π ) k − c (cid:3) ∂V i ∂k ( t, k ) + e − ( ρ − π ) t u ( c ) i , max j = i V j ( t, k ) − V i ( t, k ) o = 0 , (6.10) and such solutions are unique. In particular, the static and dynamic equilibria of the economyare characterized by the supremum in (6.10) , and the structural equilibrium is characterizedby the maximum over j = i in (6.10) . Furthermore, the path of optimal production structuresin the overall economic structure can be determined as in Theorem 4.4. Our model extends the two-sector nonbalanced growth model in Acemoglu and Guerrieri(2008) to the case of an m -sector stagewise nonbalanced growth model with EST. The staticand dynamic equilibria in (6.10) can be analyzed similarly as in Acemoglu and Guerrieri(2008). Economic implication The extended EST model of stagewise growth characterizes an economy’s endogenous trans-formation among various production structures and has different variants if more detailedassumptions about the economy can be provided. We now briefly discuss several EST sce-narios of the model.First, consider a problem of economic development in which the economy is at the earlydevelopment stage and produces only a few intermediate goods. For example, assume theeconomy produces two intermediate goods, that is, ω = ( w , w , , . . . , < α < < 1, so the production structure is Y ( ω ). This is the case of non-balanced growth studiedby Acemoglu and Guerrieri (2008). When the number of intermediate goods in the world isgreater than two, that is, I > 2, development of the economy can be achieved by transformingthe production structures from two to three intermediate products. Suppose the economytransforms from Y ( ω ) to Y ( ω ′ ), where ω ′ = ( w ′ , w ′ , w ′ , , . . . , 0) and 0 < α < α < α < As an alternative to the EST with hierarchical production structures, we consider an ESTmodel in which production and consumption structures are involved in the transformation.The model here is extended from a benchmark approximately balanced growth model inHerrendorf, Rogerson, and Valentinyi (2014), which describes proportional structural trans-formation at the sector level and an economy with stagewise structural transformation withapproximately balanced development and growth.37 omposite consumption and production of investment and consumption goods Assume that consumption at time t , c ( t ), is a composite of m consumption goods, c ( t ; ω c ) := c ( c ( t ) , . . . , c m ( t )) = (cid:16) m X i =1 w i ( c i ( t ) + ¯ c i ) ǫ − ǫ (cid:17) ǫǫ − (6.11)where ω c = ( w , . . . , w m ) ∈ Ω C , Ω C = { ω c | w , . . . , w m ≥ , w + · · · + w m = 1 } . As discussedin Herrendorf, Rogerson, and Valentinyi (2014, section 3.2), (6.11) can capture two featureson the demand side: how household demand reacts to changes in income and relative prices.The term ¯ c i allows for the period utility function to be non-homothetic and, hence, changesin income may lead to changes in expenditure shares even if relative prices are constant.We assume the following Cobb-Douglas production function, for each of the m consump-tion goods and a unique investment good: c j ( t ) = k j ( t ) α ( A i ( t ) n j ( t )) − α , j = 1 , . . . , m, x ( t ) = k x ( t ) α ( A x ( t ) n x ( t )) − α , (6.12)in which k j ( t ) and n j ( t ) are the capital and labor allocated for good j = 1 , . . . , m and x .Technological progress in all sectors is exogenous and takes the form˙ A j ( t ) = ϑ j A j ( t ) , ϑ j > , j = 1 , . . . , m, x. (6.13)We assume that capital and labor are freely mobile between the m +1 goods, so that feasibilityrequires that in each period: k ( t ) = k x ( t ) + m X j =1 k j ( t ) , n x ( t ) + m X j =1 n j ( t ) . (6.14)Total output is given by y ( t ; ω ) = p x ( t ) x ( t ) + m X j =1 p j ( t ) c j ( t ) , (6.15)where p x ( t ) and p j ( t ) are prices of the investment good and the j th consumption good,respectively. Since all consumption goods are consumed at time t , capital accumulates inthe form ˙ k ( t ) = p x ( t ) x ( t ) − δk ( t ) , (6.16)where δ ∈ (0 , 1) denotes the depreciation rate.38 conomic structures To represent the economic structure of this economy, we modify the composition of ( E , E )as follows. Let C = { ( c ( t ) , c ( t ) , . . . , c m ( t )) } and C ( ω c ) = { c ( · ; ω c ) } , where ω c ∈ Ω C . Theconsumption preference and its structure are expressed as ( C , C ( ω c )). The production andits structure are then given by ( Y , Y ), where Y = { ( c ( t ) , . . . , c m ( t ) , x ( t )) } and Y ( α ) = { ( c ( · ) , . . . , c m ( · ) , x ( · )) | c ( t ) , . . . , c m ( t ) and x ( t ) are given by (6.12) } for α ∈ (0 , m consumption goods and one investment good are produced, the technol-ogy in production and its structure can be expressed as ( A , A ( e ϑ )), where A = { ( A ( t ), . . . , A m ( t ), A x ( t )) } , A ( e ϑ ) = { ( A ( · ) , . . . , A m ( · ) , A x ( · )) | A ( · ) , . . . A m ( · ) and A x ( · ) are given by(6.13) } , and e ϑ = ( ϑ , . . . , ϑ m , ϑ x ) ∈ Ω A = { e ϑ | ϑ , . . . , ϑ m , ϑ m > } . Labor allocation in pro-duction and its structure are expressed as ( L , L ), where L = { ( n ( t ) , . . . , n m ( t ) , n x ( t )) } and L = { ( n ( · ) , . . . , n m ( · ) , n x ( · )) | n ( · ) + · · · + n m ( · ) + n x ( · ) = 1 } . Capital allocation in produc-tion and its structure can be similarly expressed. Specifically, let K = { ( k ( t ) , k ( t ) , . . . , k m ( t ) , k x ( t )) } and K = { ( k ( · ) , . . . , k m ( · ) , k x ( · )) | k ( · )+ · · · + k m ( · )+ k x ( · ) = k ( t ), k ( t ) satisfies (6.16) } . Cap-ital allocation and its structure are then given by ( K , K ). Goods and factor prices and theirstructure can be represented as ( P , P ), where P = { ( R ( t ) , w ( t ) , p ( t ) , . . . , p m ( t ) , p x ( t )) } and P = { factor prices are determined by firms to maximize their profit, and p ( t ) , . . . , p m ( t ) , p ( t )satisfy (6.14) } .Then with ( H , H ), ( F , F ), ( M , M ), ( C , C ), and ( U , U ) defined as in section 3, agents’behavior and economic activities and the economic structure in the economy are expressedas ( E , E ). The structures of production, exogenous technological progress, and consumptioninvolve parameters α ∈ (0 , 1) =: Ω Y , e ϑ ∈ Ω A , and ω c ∈ Ω C , respectively, and the economicstructure of the economy may be expressed as E ( α, e ϑ, ω c ). Objective of the social planner and the competitive equilibrium Suppose the social planner’s information set of economic structures is I Y × A × C = { E i , i ∈ I | E i = E ( α i , e ϑ i , ω c,i ) , α i ∈ Ω Y , e ϑ ∈ Ω A , ω c ∈ Ω C } . At each time t , the social planner ofthe economy must choose an economic structure E i ∈ I Y × A × C and determine the allo-cation of total income between total consumption and savings and total consumption ex-penditure between m consumption goods, that is, e ( t ) := ( c ( t ) , . . . , c m ( t ), x ( t ), k ( t ), . . . , k m ( t ) , n ( t ) , . . . , n m ( t ) , k x ( t ) , n x ( t ) } . Denote ξ = { τ n , κ n } n ≥ the sequence of decisions on“when to transform” and “where to transform,” and let A be the set of all such sequences.The aggregate resource constraint is given by (cid:26) ˙ k ( t ) = p x ( t ) x ( t ) − δk ( t ) , τ n ≤ t < τ n +1 ,k ( τ n +1 ) = k ( τ − n +1 ) , t = τ n +1 , n = 0 , , , . . . . (6.17)The representative household’s total utility, starting at time t , is given by J i ( t, k ; { ξ, { e ( s ) } ) = Z ∞ t e − ρs u ( c ( s )) ds, (6.18)39n which k ( t ) = k and θ ( t − ) = E i . The social planner must solve the maximization problem V i ( t, k ) = max { ξ, { e ( s ) }} J i ( t, k ; { ξ, { e ( s ) }} )subject to (6.17) and k ( t ) = k, θ ( t − ) = E i . (6.19)Given the above argument, the competitive equilibrium of the economy consists of pathsof consumption and savings decisions { c ( t ) , ˙ k ( t ) } , consumption expenditure of m consump-tion goods and the investment good { c ( t ) , . . . , c m ( t ) , c x ( t ) } , and economic structures { θ ( t ) } ,such that the utility of the representative household (6.18) is maximized, firms maximizeprofits, and markets clear. To characterize the competitive equilibrium, the economy can beexpressed using the notation from section 4.1 as µ i ( t, k, c ) = p x ( t ) x ( t ) − δk ( t ) , i ∈ I and the control space U ≡ { ( c ( t ) , . . . , c m ( t ) , x ( t )) | x ( t ) : [0 , ∞ ) → U, c j ( t ) : [0 , ∞ ) → U are Lebesgue measurable functions. Then according to Theorems 4.2-4.4, we obtain thefollowing result: Proposition 6.2. For every i ∈ I , the value function V i ( t, k ) defined by (6.19) is a viscositysolution to max n sup c ,...,c m ,c x n ∂V i ∂t ( t, k ) + ( p x ( t ) x ( t ) − δk ) ∂V i ∂k ( t, k ) + e − ρt u ( c ( t )) o , sup j = i V j ( t, k ) − V i ( t, k ) o = 0 . (6.20) and such solutions are unique. Furthermore, the static and dynamic equilibria of the economyare characterized by the supremum in (6.20) , and the structural equilibrium is determined bythe maximum over j = i in (6.20) . A nice property in Herrendorf, Rogerson, and Valentinyi’s (2014) balanced growth modelis that it allows analytical expressions for many economic variables in the economy. Forexample, factor and goods prices can be computed analytically as R ( t ) = αk ( t ) α − A x ( t ) − α , w ( t ) = (1 − α ) K ( t ) α A x ( t ) − α , and p i ( t ) = ( A x ( t ) /A i ( t )) − α for i = 1 , . . . , m and x . Weskip the discussion on this and, instead, highlight that the extended EST model describes astagewise approximately balanced development and growth process. Economic history shows that technological progress in countries on the world’s productionpossibility frontier is usually achieved via R&D, whereas technological progress in countriesinside the frontier may be attained via technology adoption and/or R&D. As previous ESTmodels assume exogenous technological progress, we extend the EST to endogenize struc-tural transformation of different types of technological progress and discuss its economicimplications. 40 roduction with variety and quality ladders The literature usually considers two types of technological change, namely process and prod-uct innovation, or the introduction of a new product and innovations to reduce the costs ofproduction of existing products. Two commonly used canonical models for these changesare product-variety models (Romer, 1990) and Schumpeterian models (Aghion and Howitt,1992). For convenience, we extend production function (6.1) and assume that the uniquefinal good is produced competitively by combining the output of m ( t ) sectors with elasticitiesof substitution ǫ ∈ [0 , ∞ ), that is, Y ( t ; ω ) = F [ Y ( t ) , . . . , Y m ( t ) ( t )] = (cid:16) Z m ( t )0 w ( j, t ) q ( j, t ) Y ( j, t ) ǫ − ǫ dj (cid:17) ǫǫ − , (6.21)where the weights w ( j, t ) satisfy w ( j, t ) ≥ t and j and R m ( t )0 w ( j, t ) dj = 1, m ( t )denotes the number of varieties of inputs at time t , q ( j, t ) represents the “quality ladder”for machine type j , and ω is a vector of parameters related to the production functions andwill be specified later. The intermediate goods of machine type j , Y ( j, t ), are produced bythe production function Y ( j, t ) := F ( j, t ) = K ( j, t ) α ( j,t ) L ( j, t ) − α ( j,t ) . (6.22)The technology level of machine type j is represented by the quality ladder q ( j, t ); hence,there is no need to keep the term A ( j, t ) in (6.22). Let ξ ( j, t ; ω ) = K ( j, t ; ω ) /K ( t ) and λ ( j, t ; ω ) = L ( j, t ; ω ) /L ( t ). The output per capita at time t is f ( t, k ( t ); ω ) = n Z m ( t )0 w ( j, t ) q ( j, t ) (cid:2) k ( t ) α ( j,t ) ξ j ( t ) α ( j,t ) λ j ( t ) − α ( j,t ) (cid:3) ǫ − ǫ o ǫǫ − . (6.23)Capital and labor market clearing requires that Z m ( t )0 K ( j, t ) dj ≤ K ( t ) , Z m ( t )0 L ( j, t ) dj ≤ L ( t ) . (6.24)Then production and its structure of the economy can be expressed as ( Y , Y ( ω )), where Y = { Y ( t ) , { Y ( j, t ) } ≤ j ≤ m ( t ) } describes the output levels of the intermediate and final goods, Y ( ω ) = { F ( · ) , { F ( j, · ) } ≤ j ≤ m ( t ) } represents the composition and organization of all produc-tion, and ω = ( { w ( j, · ), α ( j, · ) } ≤ j ≤ m ( t ) ) is an element of the set Ω Y = { ω | w ( j, t ) ≥ R m ( t )0 w ( j, t ) dj = 1, 0 < α ( j, t ) < j } . Endogenous technological progress We first consider endogenous technological progress via technology adoption. Suppose e m ( t )and e q ( j, t ) are the number of machines and quality ladder for machine type j on the world’s41roduction possibility frontier at time t , respectively. By definition, e m ( t ) and e q ( j, t ) areexogenously determined and monotonically nondecreasing over t . The economy in our studyhas m ( t − ) machines and machine type j is on the quality ladder q ( j, t − ) prior to time t .Obviously, m ( t − ) ≤ e m ( t ) and q ( j, t − ) ≤ e q ( j, t ).At time t , the social planner of the economy decides whether the number of machinetypes and the quality of each machine type should be improved via technology adoption. If yes, the number of machine types m ( t ) and the quality ladder q ( j, t ) of machine type j attime t should be chosen from the sets ( m ( t − ) , e m ( t )] and ( q ( j, t − ) , e q ( j, t )], respectively. Thenthe level of technology is given by A = { ( m ( t ) , { q ( j, t ) } ≤ j ≤ m ( t ) ) , where m ( t ) ∈ ( m ( t − ) , e m ( t )] and q ( j, t ) ∈ ( q ( j, t − ) , e q ( j, t )] } , and the technological structure in production is expressed as A adopt ( ǫ ) = { ǫ = ( ǫ m , { ǫ j } ≤ j ≤ ǫ m ), ǫ m = m ( · ), ǫ j = q ( j, · ) } , which is an element of the information set I A , adopt = { A adopt ( ǫ ) | ǫ ∈ Ω A , adopt } and Ω A , adopt = { ( ǫ m , { ǫ j } ≤ j ≤ ǫ m ) | ǫ m ∈ ( m ( t − ) , e m ( t )] , ǫ j ∈ ( q ( t, j ) , e q ( t, j )] } . Amathematical concern here is whether such a decision process leads to continuous variationof m ( t ) and q ( j, t ), which indicates that the technological structure violates the durationalityattribute. This concern only arises in theoretical analysis, since the improvement from m ( t − )to m ( t ) and/or from q ( j, t − ) to q ( j, t ) in reality involves some cost, which prevents the socialplanner from choosing m ( t ) and q ( j, t ) continuously. In theoretical analysis, the diminishingmarginal returns of production functions ensure durationality and transformality. Hence,the functionals A adopt , chosen optimally by the social planner, are still piecewise constant.Then given production ( Y , Y ( ω )) and technology ( A , A adopt ), the resource constraint of theeconomy at time t is ˙ K ( t ) = Y ( t ; ω ) − δK ( t ) − C ( t ) . (6.25)Accordingly, the allocation of capital stock and its structure are expressed as ( K , K adopt ( ω, ǫ )),where K is defined similarly as in section 6.1 and K adopt ( ω, ǫ ) = { K ( j, · ) | R m ( t )0 K ( j, · ) dj ≤ K ( · ), K ( · ) satisfies (6.25) } .In addition to technology adoption, the social planner may also decide to improve thetechnology level via R&D. To explain how to describe the structure of technological progressvia R&D, we suppose that Z ( t ) is expenditure on R&D at time t , and the number of machinetypes m ( t ) and the quality of the j th machine type q ( j, t ) at time t satisfy the following˙ m ( t ) = ι m Z ( t ) , lim ∆ t → q ( j, t + ∆ t ) − q ( j, t )∆ t = ι j Z ( t ) for all j and t, (6.26)where ι m and ι j are nonnegative parameters and satisfy the constraint ι m + R m ( t )0 ι j dj = 1.Then the technology level of production is A = { ( m ( t ) , { q ( j, t ) } ≤ j ≤ m ( t ) ) | m ( t ) and q ( j, t ) sat-isfy (6.26) } . To represent the technological structure, let ι = ( ι m , { ι j } ) and Ω A , r&d = { ι | ι m ≥ Recall that the social planner is defined as the social elite of an economy, which can collect informationand make decision on production and other economic activities. , ι j ≥ j, ι m + R m ( t )0 ι j dj = 1 } . Then the structure of technology can be expressed as A r&d ( ι ), which is an element of the set of functionals I A , r&d = { ( m ( · ) , { q ( j, · ) } ≤ j ≤ m ( · ) ) | ι ∈ Ω A , r&d , m ( · ) and q ( j, · ) satisfy (6.26) } . Provided production ( Y , Y ( ω )) and technology( A , A r&d ( ι )), the resource constraint of the economy at time t is˙ K ( s ) = Y ( t ; ω ) − δK ( t ) − C ( t ) − Z ( t ) . (6.27)Consequently, the allocation of capital stock and its structure are expressed as ( K , K r&d ( ω, ι )),where K is defined similarly as in section 6.1 and K r&d ( ω, ι ) = { K ( j, · ) | R m ( t )0 K ( j, · ) dj ≤ K ( · ), K ( · ) satisfies (6.27) } .Therefore, the total information set of technology and its structure for the social planneris I A := I A , adopt ∪ I A , r&d , and the corresponding total information set of capital allocationand its structure is I K := I K , adopt ∪ I K , r&d . Once the social planner chooses structuresof technology and capital allocation, technology level A and capital alloation K can bedetermined by corresponding mechanisms.Other structures ( H , H ), ( F , F ), ( M , M ), ( L , L ), ( P , P ), ( C , C ), and ( U , U ) canbe defined similarly with the necessary modifications as in section 6.1. Then we may stilluse ( E , E ) to represent agents’ behavior and economic activities and their structures in theeconomy. To highlight production structure ω ∈ Ω Y and technological structure ǫ ∈ Ω A , adopt or ι ∈ Ω A , r&d , one may express the economic structure of the economy as E ( ω, ǫ ) or E ( ω, ι ). Social planner’s maximization problem Suppose the social planner’s information set of economic structures is I Y × K × A = { E i | E i = E ( ω i , e ϑ i ) , e ϑ i ∈ Ω A , adopt ∪ Ω A , r&d , i ∈ I } . The social planner can choose an economic structurefor the economy and allocate resources under the given economic structure. When E ( ω i , e ϑ i )( e ϑ i ∈ Ω A , adopt ) is chosen, equation (6.25) implies that the resource constraint per capita is˙ k ( t ) = f ( t, k ( t ); ω i ) − ( δ + π ) k ( t ) − c ( t ) , (6.28)and when E ( ω i , e ϑ i ) ( e ϑ i ∈ Ω A , r&d ) is chosen, equation (6.27) implies that the resource con-straint per capita is ˙ k ( t ) = f ( t, k ( t ); ω ) − ( δ + π ) k ( t ) − c ( t ) − z ( t ) . (6.29)Suppose the economy starts with economic structure E ( ω i , e ϑ i ) at time t , Then the socialplanner’s total utility starting at time t is given by J i ( t , k ; { c ( t ) , ξ } ) = Z ∞ t e − ( ρ − π ) t u ( c ( t )) dt. The social planner’s objective is to solve the maximization problem V i ( t , k ) = max { c ( t ) ,ξ } J i ( t , k ; { c ( t ) , ξ } ) subject to (6.28) or (6.29)and k ( t ) = k ∈ R + , θ ( t ) = E i ∈ I Y × K × A . (6.30)43iven the above specification, the competitive equilibrium of the economy consists ofpaths for factor and intermediate goods prices, employment and capital allocation { ξ j ( t ; θ ( t )) , λ j ( t ; θ ( t )) } ,consumption and savings decisions { c ( t ) , ˙ K ( t ) } , and structures of production, capital alloca-tion, and technological progress { θ ( t ) } such that the utility of the representative householdis maximized, firms maximize profits, and markets clear.To characterize the competitive equilibrium mathematically, we express the maximiza-tion problem in the form of the combined optimal control and optimal switching problemsin section 4.1 in which, for τ n ≤ t < τ n +1 and κ n = E ( ω i , e ϑ i ), µ i ( t, k, c ) = ( f ( t, k ( t ); ω i ) − ( δ + π ) k ( t ) − c ( t ) , e ϑ i ∈ Ω A , adopt f ( t, k ( t ); ω i ) − ( δ + π ) k ( t ) − c ( t ) − z ( t ) , e ϑ i ∈ Ω A , r&d and the control space U := { ( c ( t ) , { ξ ( j, t ) , λ ( j, t ) } ) | c ( t ) : [0 , ∞ ) → U, ξ ( j, t ) : [0 , ∞ ) → [0 , ,λ ( j, t ) : [0 , ∞ ) → [0 , 1] are Lebesgue measurable functions } . Then according to Theorems 4.2-4.4, we obtain the following result: Proposition 6.3. For each E i = E ( ω i , e ϑ i ) ∈ I Y × K × A , let V i ( t, k ) be defined by (6.9) . Thenfor e ϑ i ∈ Ω A , adopt , V i ( t, k ) is a unique viscosity solution to max n sup e ( t ) h ∂V i ∂t ( t, k ) + (cid:2) f i ( t, k ) − ( δ + π ) k − c (cid:3) ∂V i ∂k ( t, k ) + e − ( ρ − π ) t u ( c ) i , max j = i (cid:8) V j ( t, k ) (cid:12)(cid:12) e ϑ j ∈ Ω A , r&d ∪ Ω A , adopt (cid:9) − V i ( t, k ) o = 0 , (6.31) where e ( t ) = ( c ( t ) , { ξ ( j, t ) , λ ( j, t ) } ) . For e ϑ i ∈ Ω A , r&d , V i ( t, k ) is a unique viscosity solutionto max n sup e ( t ) h ∂V i ∂t ( t, k ) + (cid:2) f i ( t, k ) − ( δ + π ) k − c − z (cid:3) ∂V i ∂k ( t, k ) + e − ( ρ − π ) t u ( c ) i , max j = i (cid:8) V j ( t, k ) (cid:12)(cid:12) e ϑ j ∈ Ω A , r&d ∪ Ω A , adopt (cid:9) − V i ( t, k ) o = 0 , (6.32) where e ( t ) = ( c ( t ) , z ( t ) , { ξ ( j, t ) , λ ( j, t ) } ) . Furthermore, the static and dynamic equilibria ofthe economy are characterized by the supremum in the two equations, and the structuralequilibrium is characterized by the second-line maximum of the two equations. Economic implications Proposition 6.3 integrates several scenarios of endogenous technological progress into a singleframework and has interesting implication for economic development and growth. We nextbriefly discuss scenarios and implications of Proposition 6.3.44he first scenario is countries on the global production possibility frontier and their tech-nological progress. Since these countries or economies are on the world production possibilityfrontier, the social planners of these economies will rule out the possibility of technologicaladoption and the information set of technological structure reduces from I A , adopt ∪ I A , r&d to I A , r&d . Hence, the HJB-QVI system (6.32) for the value function of the representativehousehold in Proposition 6.3 becomes that, for e ϑ i ∈ Ω A , r&d ,max n sup e ( t ) h ∂V i ∂t ( t, k ) + (cid:2) f i ( t, k ) − ( δ + π ) k − c − z (cid:3) ∂V i ∂k ( t, k ) + e − ( ρ − π ) t u ( c ) i , max j = i (cid:8) V j ( t, k ) (cid:12)(cid:12) e ϑ j ∈ Ω A , r&d (cid:9) − V i ( t, k ) o = 0 . (6.33)This indicates that R&D is the only way to achieve technological progress for countries onthe global production possibility frontier, and the structure of technological progress viaR&D is characterized by the rate ι = ( ι m , { ι j } ) ∈ Ω A , r&d . If a rate of R&D is fixed with ι and no transformation of the R&D structure is needed, then (6.33) further reduces tosup e ( t ) h ∂V i ∂t ( t, k ) + (cid:2) f i ( t, k ) − ( δ + π ) k − c − z (cid:3) ∂V i ∂k ( t, k ) + e − ( ρ − π ) t u ( c ) i = 0 , (6.34)which is in the form of technological progress via R&D in the neoclassical economy.The second scenario is developing countries that are inside the global production pos-sibility frontier. Two situations might occur. One is that the developing country is wellconnected with the world’s economy, so that the information of the world’s technology canbe accessed by the economy freely or with low cost. In such case, the social planner of theeconomy would naturally consider improving the country’s technology via adoption insteadof R&D, as inside-the-frontier indigenous innovation is not efficient. Thus, the HJB-QVI sys-tem (6.32) for the value function of the representative household in Proposition 6.3 becomesthat, for e ϑ i ∈ Ω A , adopt ,max n sup e ( t ) h ∂V i ∂t ( t, k ) + (cid:2) f i ( t, k ) − ( δ + π ) k − c (cid:3) ∂V i ∂k ( t, k ) + e − ( ρ − π ) t u ( c ) i , max j = i (cid:8) V j ( t, k ) (cid:12)(cid:12) e ϑ j ∈ Ω A , adopt (cid:9) − V i ( t, k ) o = 0 . (6.35)Some discussion starts from the perspective of the social planners in developed countries andfocuses on technology diffusion. The social planners in these discussions play a passive roleand accept the technology diffused from developed countries without the process of choosing“appropriate” technology levels (or structures). Equation (6.35) avoids this issue and high-lights the active role of social planners in developing countries in choosing “appropriate”technological structures. Another situation in the second scenario is that the developingcountry is somehow isolated from the world economy, so that inside-the-frontier innovationis necessary. In such case, the value function of the representative household in the isolatedeconomy is characterized by the HJB-QVI system (6.34).45he third scenario is countries that are inside but near the global production possibilityfrontier. As the technology level in those countries is near the global production possibilityfrontier, due to the fear of competition, the countries on the global production possibilityfrontier may embargo their technology knowhow. Therefore, R&D must be carried outfor further economic development. Then the main issue for the social planners in thesecountries is when to switch from technology adoption to R&D. To fix the idea, suppose thesocial planner’s information set of technological structures is I A = { E ( ω , e ϑ ) , E ( ω , e ϑ ) } with e ϑ ∈ Ω A , adopt and e ϑ ∈ Ω A , r&d . Then Proposition 6.3 implies that the value functions V ( t, k ) and V ( t, k ) of the representative household with corresponding initial economicstructures E ( ω , e ϑ ) and E ( ω , e ϑ ), respectively, and times of switching from the mode oftechnology adoption to that of R&D are characterized by the following HJB-QVI system max n sup e ( t ) h ∂V ∂t ( t, k ) + (cid:2) f ( t, k ) − ( δ + π ) k − c (cid:3) ∂V ∂k ( t, k ) + e − ( ρ − π ) t u ( c ) i ,V ( t, k ) − V ( t, k ) o = 0 , e ( t ) = ( c ( t ) , { ξ ( j, t ) , λ ( j, t ) } );max n sup e ( t ) h ∂V ∂t ( t, k ) + (cid:2) f ( t, k ) − ( δ + π ) k − c − z (cid:3) ∂V i ∂k ( t, k ) + e − ( ρ − π ) t u ( c ) i ,V ( t, k ) − V ( t, k ) o = 0 , e ( t ) = ( c ( t ) , z ( t ) , { ξ ( j, t ) , λ ( j, t ) } ) . (6.36) The discussion so far has focused on structures of production and consumption and theirtransformation in the process of economic development. In general, in addition to the pro-cess of industrial upgrading and technological progress, which involves households’ and firms’decisions on the supply and demand of factors of production, an economy’s development alsoinvolves the production of public goods and infrastructure, which are supplied by govern-ments or require collective actions and cannot be internalized in the decisions of individualhouseholds or firms. We now consider production structures of public goods and infrastruc-ture.Infrastructure includes hard infrastructure and soft infrastructure. Hard infrastructureconsists of the physical infrastructure of highways, port facilities, airports, telecommuni-cation systems, electricity grids, and other public utilities. Soft infrastructure consists ofinstitutions, regulations, social capital, and other social and economic arrangements. Mosthard infrastructure and almost all soft infrastructure is exogenously provided to individualfirms in the form of public goods and cannot be internalized in their production decisions. Toillustrate the idea, we consider an approach to model infrastructure as public goods availableto firms and a government’s tax policy as a simplified economic institution. Economic institutions have different meanings in different contexts, and we refer to them as taxes, thesecurity of property rights, contracting institutions, and other economic arrangements. This is different fromthe political institutions discussed in the next subsection, which refer to the rules and regulations affectingpolitical decision making. roduction, infrastructure, and economic institution Suppose the economy has a unique final good, and the representative firm produces output Y t according to the following production function: Y ( t ) = F [ K ( t ) , L ( t ) , A ( t ) , G ( t )] , or y ( t ) := f ( t, k ( t )) = F i [ k ( t ) , , A ( t ) , G ( t )] , (6.37)where G ( t ) is the aggregate stock of public goods (or infrastructure) available to all firms attime t . The public good G ( t ) is a common external input to each firm’s production function.The government impose a tax rate τ ∈ [0 , 1) on output. Hence, equation (3.6) for capitalaccumulation per capita changes to˙ k ( t ) = (1 − τ ) f ( t, k ( t )) − ( δ + π ) k ( t ) − c ( t ) . (6.38)Accordingly, the accumulation of public goods is described by˙ G ( t ) = τ f ( t, k ( t )) , (6.39)and firms hire capital and labor to maximize (1 − τ ) f ( t, k ( t )) − w ( t ) − R ( t ) k ( t ). Economic structures In addition to the components discussed in the previous sections, the economic structureof the economy includes structures (functionals) of infrastructure investment and economicinstitutions. We characterize the level of infrastructure investment and its structure by( I , I ), where I = { G ( t ) } ∈ R and I := { G ( · ) satisfies (6.39) } . Infrastructure invest-ment is determined by the tax rate τ , and its level and structure can be described bythe pair ( T , T ), where the tax level T = { τ f ( t, k ( t )) } and the structure (or the func-tional) of the tax T := { τ f ( · , · ) } . Provided ( I , I ) and ( T , T ), the production structure( Y , Y ) must be modified accordingly. Thus, activities in the economy can be summa-rized as E := ( H , F , M , Y , L , K , A , P , C , U , I , T ) and its structure can be represented by E := ( H , F , M , Y , L , K , A , P , C , U , I , T ). Then, given a set of specific structures ofinfrastructure investment and economic institution { I i , T i } , ( i ∈ I ) at an initial time, thediscussion on the social planner’s maximization problem and its solution is analogous to thatin the previous sections and hence is skipped here. Economic implications In constrast to infrastructure that can be represented as public goods produced via explicitproduction functions, most soft infrastructure cannot be described in this way. However,when the mechanism of the impact of soft infrastructure on economic activities can be de-scribed via explicit functions, the EST approach can be extended to characterize the trans-formation of soft infrastructue and the corresponding competitive equilibrium and optimal47nfrastructure. This would help us understand the diversity of certain soft infrastructure indifferent countries. Take a country’s financial structure as an example. Modern financialinstitutions, such as the stock market, venture capital, corporate bonds, and large bankshave the functions of mobilizing large amounts of capital and/or diversifying risks. Theindustrial structure of developed countries consists of large-scale capital intensive industriesthat rely on risky R&D for achieving endogenous technological innovation. Hence, modernfinancial institutions are appropriate for serving the needs of the real sector in developedcountries. However, such financial institutions may not be appropriate for developing coun-tries, due to the differences in the structures of industries and technological progress betweendeveloping and developed countries. Since developing countries possess mostly small-scale,labor-intensive industries and rely on the adoption of mature technologies for technologicalprogress, small local or community-based financial institutions may be better suited (Lin,2011).For other types of soft infrastructure or institutions, such as economic policies, protectionof intellectual property, and other regulations and rules, the EST framework can embedthem and their transformation into an economy’s development process, which provides thefollowing implications for studies of economic policies and institutions.First, most studies of economic policies focus on the design, implementation, and eval-uation of specific policies and seldom deal with their dynamic changes. As the time scalesof economic policies (or institutions) are usually larger than those of economic variablesand even some economic substructures, transformations of economic institutions and theircompetitive equilibria can be characterized via the EST framework. In particular, as manymodels in the neoclassical sense concentrate on static and dynamic equilibria under a giveneconomic policy or institution, the EST framework characterizes the structural equilibriumof policies and institutions in the development process.Second, the transformation or birth and decay process of a specific economic institutionhas been largely discussed by economic historians, but it has not been studied via theoreticalmodels. When different development stages of a particular economic institution are modeledas different economic structures, the EST framework helps us understand the structures ofeconomic policies and institutions and their transformations, for example, the optimal entryand exit times of specific policies and institutions.Third, recent studies show that economic policies can be categorized as structural andnonstructural policies (Abdel-Kader, 2013). Since structural and nonstructural policies servedifferent purposes in an economy’s development process, it is difficult to study both types ofpolicies with the existing economic models. The EST framework can overcome this difficultyby modeling structural and non-structural policies as different institutional structures andcharacterizing transformations among structural and nonstructural policies.48 .5 Political regimes and institutions Another related and interesting issue is the impact of political institutions on economic de-velopment. As institutions have different meanings in different contexts, we define politicalinstitutions as a system of laws on the organizational form and methods of political decisionmaking, such as the organization of state power, structural form of the state, political partysystem, and so on. It is not difficult to see that the structures of an economy’s politicalinstitutions still have the attributes of durationality and transformality. Furthermore, po-litical institutions change on a time scale much longer than that of economic activities andstructures.To provide an illustration of the transformation of political institutions via the ESTframework, we extend the discussion as follows. Suppose that at initial time t , the initialinstitutional structure and initial economic structure of the country are N i and E i , respec-tively, and the initial capital intensity is k . The dynamic equation for capital intensity underthe given institutional and economic structures ( N i , E i ) may be given by˙ k ( t ) = f N i , E i ( t, k ) − ( δ + π ) k − c ( t ) . The total utility and social planner’s objective are still defined by equations (6.8) and (6.9).The competitive equilibrium of the economy can be defined similarly as in earlier sections.Then Theorems 4.2-4.4 imply the following: Proposition 6.4. For each institutional structure N i ∈ N and each economic structure E i ∈ E , the value function V N i , E i ( t, k ) is a unique viscosity solution of the HJBQVI system max n sup c ∈U h ∂V N i , E i ∂t ( t, k ) − (cid:2) f N i , E i ( t, k ) − ( δ + π ) k − c ( t ) (cid:3) ∂V N i , E i ∂k ( t, k ) + e − ρt u ( c ) i , max h max E j = E i { V N i , E j ( t, k ) } , max N j = N i { V N j , E j ( t, k ) } i − V N i , E i ( t, k ) o = 0 . (6.40)The static and dynamic equilibria of the economy are still characterized by the supremumin (6.40), and the structural equilibrium is accounted for by the second line of the equation.Then, using an argument analogous to that for Proposition 5.2, we obtain the following. Proposition 6.5. The optimal institutional and economic structures at any given time t and with any given capital intensity k are characterized by the structural equilibrium (6.40) at ( t, k ) . Moreover, the optimal institutional and economic structures are given functionsof t and k ( t ) and, hence, endogenous to the capital intensity (or more generally, the factorendowments of the economy) at time t . We next briefly discuss the implications of equation (6.40) for the transformation ofinstitutional structures in different cases. The first case is for developed countries and isrelatively simple. Since developed countries are mostly industrialized, and their income per49apita is usually higher than that in developing countries, the social planner (or the economicand political elites) in developed countries will not consider it necessary to transform theirpolitical institutions. Or equivalently, the social planner in developed countries may havecompared different types of institutional structures but concludes that the country’s currentinstitutional and economic structures are better than those in other countries. In such case,the structural equilibrium of equation (6.40) is not necessary and (6.40) degenerates to theequation of the supremum, which describes the static and dynamic equilibria of the economywith fixed institutional and economic structures.The second case is developing countries focusing on “economic development” or “eco-nomic transition” and trying to figure out the optimal economic and political institutionsfor the economy. The structural equilibirum in equation (6.40) ismax h max E j = E i { V N i , E j ( t, k ) } , max N j = N i { V N j , E j ( t, k ) } i − V N i , E i ( t, k ) = 0 . (6.41)This implies that, for the social planner of the economy, there are two types of structuraltransformation involving institutional and economic structures. The first type is that theeconomic structure E i transforms into another economic structure, while the political insti-tution N i remains the same, that is,max E j = E i { V N i , E j ( t, k ) } = V N i , E i ( t, k ) . (6.42)The second type is that political institution N i and economic structure E i transform intoother structures in N × E , which is characterized bymax N j = N i { V N j , E j ( t, k ) } = V N i , E i ( t, k ) (6.43)Equation (6.42) corresponds to a scenario in which a developing country develops the econ-omy by reforming the economic structures, but the political institution remain invariant. Bycontrast, equation (6.43) indicates that the developing country transforms both the politicalinstitutions and economic structures.The choice of equation (6.42) or (6.43) for the social planner (or the social elite) of theeconomy to solve may lead to different development paths. Take for example China’s and theformer Soviet Union’s economic transition processes. China’s economic reform process canbe described as the process of solving equation (6.42), whereas the economic reform processin the former Soviet Union and other Eastern European countries can be characterized as theprocess of solving equation (6.43). Although these two processes are completely different,they can be described by a unified EST framework. Structural transformation has been discussed intensively in the literature on economic growthand development over the past decades. Although sectoral structural transformation models50ave been developed to study changes in numerical economic variables across different sec-tors, a general theoretical framework is still missing to characterize a country’s full processof structural transformation.The EST framework proposed in this paper bridges this gap and makes the followingcontributions. First, three fundamental attributes of structures—struaturality, durationality,and transformality—are summarized from empirical observations and the literature of eco-nomic history. Second, with the necessary assumptions on the information set of economicstructures, a theoretical framework is proposed to model the dynamics of economic activitiesand their structures in different time scales and characterize the endogenous transformationprocess of structures. Third, to characterize competitive equilibrium in the EST framework,we study a class of combined infinite-horizon optimal control and optimal switching prob-lems and demonstrate that the value function of the combined control problem is a uniqueviscosity solution to a system of Hamilton-Jacobi-Bellman equations and quasi-variationalinequalities. Fourth, we use the developed mathematical method to solve the social planner’soptimization problem in the EST model and establish the associated competitive equilib-rium theory. We show that, in addition to the static and dynamic equilibria that constitutecompetitive equilibrium in neoclassical growth models, competitive equilibrium in the ESTframework suggests the existence of a third type of equlibrium, the structural equilibrium.To demonstrate the flexibility of the proposed EST framework, we have discussed extensionsof the EST framework that deal with hierarchical production structures, composite struc-tures of consumer preference, technological structures via adoption and R&D, changes ininfrastructure and economic institutions, and switching of political institutions.The EST framework provides a method to model the structural differences and endo-geneity of those differences for countries at different levels of development and sheds newlight on many interesting and oftentimes debated issues. We consider a few examples. Theimport-substitution strategy failed in most developing countries in the 1950s and 1960s,despite the coordination provided by their governments’ big pushes (Murphy, Shleifer, andVishny, 1989). The failure was due to the industrial structure targeted in the strategy beingtoo capital intensive while capital was scarce in the countries. The growth driven by capitalaccumulation without total factor productivity in Singapore and other East Asian economiesin their catching-up stage was sustainable instead of being doomed to fail, as predicted byKrugman (1994). This was because capital accumulation is required for upgrading industrialstructure and technology adoption in the process of catching up and returns to capital will notdiminish before they reach the global production possibility/technology frontier and switchto technological innovation through R&D and growth driven by total factor productivity.From the perspective of EST, the poverty trap or middle-income trap for many developingcountries is a result of their inability to implement dynamic structural transformation due tothe lack of sufficient capital accumulation to cross the capital thresholds required for struc-tural transformation, or the lack of government facilitation to overcome coordination failuresto make the required improvements in hard and soft infrastructure for the transformation.The EST framework in this paper can be extended to incorporate other types of macroe-51onomic models with specific structures, such as models of heterogeneous households, het-erogeneous firms, overlapping generations, trade structures in an open economy, stochasticgrowth and so forth. Mathematically, as long as the economic problem involves differenttypes of structures, an extended version of the EST model can be obtained. Economically,the extended EST model and associated competitive equilibrium can be characterized bythe mathematical method and theory developed here, and hence a structural equilibriumcan be obtained for the extended EST problem. In addition, since the proposed EST frame-work can integrate different types of economic structures and turn a macroeconomic modelwith a single type of economic structures into one with multiple economic structures, theEST framework can integrate models of economic development and growth at different de-velopment stages into a stagewise development and growth model. Hence, the EST modelprovides a unified framework to account for a country’s development and growth processwith structural transformation.The mechanism of structural transformation characterized in the EST framework as-sumes that the market is complete, information can be freely obtained, and the transfor-mation is frictionless, which are certainly not true in the real world. Instead, in the realworld, different kinds of market and information incompleteness and different structures re-quire different hard infrastructure and economic as well as political institutions to unleashits economic potentials in practice. The EST cannot occur spontaneously. Instead, policyintervention is usually needed to facilitate the transformation of economic structures (Lin,1989), which is a topic for further discussion in subsequent research. A Proofs of theorem and proposition A.1 Proof of Theorem 4.1 We first show the following two lemmas. Lemma A.1. Assume µ i ( t, k, c ) satisfy (B1) for all i ∈ I . Fix k (0) = x ∈ R . Given any t ′ ≥ 0, set M x = sup {| µ i ( t, k, c ) | : | k − x | ≤ , c ∈ U, i ∈ I } . Then for any ξ ∈ A and c ∈ U there exists a Lipschitz solution k of (4.4) defined on [ t ′ , t ′ + M − x ] that satisfies | k i,x ( t, c ) − x | ≤ M x ( t − t ′ ) for all c ∈ U and t ∈ [ t ′ , t ′ + M − x ] . (A.1) Proof of Lemma A.1. First we show that, for all t ≥ c ∈ U , there exists a constant M i,x > | µ i ( t, , c ) | ≤ M i,x . Otherwise, there exists a sequence { t j } j ≥ suchthat lim j →∞ | µ i ( t j , , c j ) | = ∞ . By the continuity of µ i , there exists an ǫ > | t − t ′ | < ǫ , | µ i ( t, , c ) − µ i ( t ′ , , c ′ ) | < 1. Then there exists a j such that for all j > j , | µ i ( t j , , c j ) | > /ǫ . The continuity of µ i implies that, for | t − t j | < ǫ , | µ i ( t, , c ) | > / (2 ǫ ), hence R t j + ǫt j − ǫ | µ i ( t, , c ) | dt > ǫ/ (2 ǫ ) = 1 / (2 ǫ ). Since ǫ is arbitrary, it contradicts thedefinition of U . Hence, there exists a constant M i,x > | µ i ( t, , c ) | ≤ M i,x . By the52ipschitz continuity of µ i , | µ i ( t, k, c ) | ≤ D i | k | + | µ i ( t j , , c j ) | ≤ D i | k | + M i,x for all t ≥ c ∈ U . Then M x ≤ max i { D i + M i,x } < ∞ .For any t ′ ≥ t , set t = t ′ + 1 /M x and G := { g : [ t ′ , t ] × U → { y : | y − x | ≤ } continuous, g ( t ′ , c ( t ′ )) = x } . For any g ∈ G we define H ( g )( t, c ) := x + X n : τ n ≤ t Z τ n τ n − µ κ n ( s, g k ( τ n − ) ( s ) , c ( s )) ds, for t ∈ [ t ′ , t ], because the function to be integrated is bounded by M x and it is measurable.Moreover, | H ( g )( t, c ) − x | ≤ M x ( t − t ′ ) ≤ t , thus H : G → G . Furthermore, H is continuous with respect to the uniform convergence. In fact, if g n , g ∈ G , then for all t ∈ [ t ′ , t ], | H ( g n )( t, c ) − H ( g )( t, c ) | ≤ Z t t ′ | µ i ( s, g n ( s ) , c ( s )) − µ i ( s, g ( s ) , c ( s )) | ds, and the right-hand side tends to 0 as n → ∞ by Lebesgue’s dominated convergence theoremif g n → g . Thus H ( g n ) → H ( g ) uniformly.Note that F ( G ) is equicontinuous since | H ( g )( t, c ) − H ( g )( τ, c ) | = | X τ ≤ τ n ≤ t Z tτ µ i ( s, k ( s ) , c ( s )) ds | ≤ M x | t − τ | for all g ∈ G , t, τ ∈ [ t ′ , t ]. Then H ( G ) has compact closure by the Ascoli-Arzel´a The-orem (Bardi and Capuzzo-Dolcetta, 1997, section III.5, Theorem 5.2). Since H satisfiesthe assumption of Schauder’s Theorem (Bardi and Capuzzo-Dolcetta, 1997, section III.5,Theorem 5.3), it has a fixed point k that is a solution of (4.4) in [ t ′ , t ]. The estimate | H ( g )( t, c ) − H ( g )( τ, c ) | ≤ M x | t − τ | gives the Lipschitz continuity of g with Lipschitz con-stant M x,t , so (A.1) holds as well. (cid:3) Lemma A.2. Assume µ i ( t, k, c ) satisfy (B1) for all i ∈ I .(1) For any t ′ ≥ k ∈ R and c ∈ U , there is a unique solution k i,x : [0 , ∞ ) → R of (4.4),and it satisfies | k i,x ( t, c ) | ≤ ( | x | + p M ( t − t ′ )) e M ( t − t ′ ) for all c ∈ U and t ≥ t ′ , (A.2)where M := max i { D i , M i,x } and M i,x is defined in the proof of Lemma A.1. Moreover,there exists an M > M such that | k i,x ( t, c ) | ≤ (1 + | x | ) e M ( t − t ′ ) for all c ∈ U, t ≥ t ′ . (A.3)532) If k i,y is the solution with the initial condition ( i, y ), then | k i,x ( t, c ) − k i,y ( t, c ) | ≤ e D i ( t − t ′ ) | x − y | for all c ∈ U and t ≥ t ′ . (A.4) Proof of Lemma A.2. (1) Note that (B1) implies that | µ i ( s, k, c ) − µ i ( s, , c ) | ≤ D i,̺ | k | and therefore | µ i ( t, k, c ) k | ≤ ( D i | k | + M i,x ) | k | ≤ M (1 + k ). For any control c ∈ U thesolution of (4.4) satisfies d ( | k ( t, c ) | ) /dt = 2 kµ i ( t, k, c ) ≤ M (1 + k ), thus integratingfrom t to t | k ( t, c ) | ≤ x + 2 M ( t − t ) + 2 M R tt | k ( s, c ) | ds . The Gronwall inequalitygives for t ≤ s , | k ( t, c ) | ≤ ( | x | + 2 M ( t − t ′ )) e M ( t − t ′ ) , which implies (A.2) for all s suchthat the solution exists. Estimate (A.3) can be obtained by taking M > M such that M ( t − t ′ ) < e ( M − M )( t − t ′ ) / .(2) To show the global existence, consider the supremum s of the times t such thatthere exists a solution of (4.4) defined on [ t ′ , t ]. Assume by contradiction that s < ∞ . If asolution exists in [ t ′ , s ] we have a contradiction since the solution can be continued on theright of s by Lemma A.1. So, we are left with the case that the maximal interval of existenceis [ t ′ , s ). We claim that lim t → s k ( t ) = k ∈ R . Since we have shown that | k i,x ( t, c ) | ≤ ( | x | + p M ( s − t ′ )) e M ( s − t ′ ) := M , for t ∈ [ t ′ , s ) . Set M = sup {| µ i ( t, k, c ) | : | k | ≤ M , c ∈ U, t ∈ [ t ′ , s ) , i ∈ I } . We have M < ∞ . For s < t < s , denote τ m − < s ≤ τ m < · · · < τ n < t ≤ τ n +1 and c, c ′ ∈ U , | k i,x ( t, c ) − k i,x ( s, c ′ ) | ≤ (cid:16) Z τ m s + Z τ m +1 τ m + · · · + Z τ n τ n − + Z tτ n (cid:17) | µ κ n ( τ, k, c ) − µ κ n ( τ, k, c ′ ) | dτ ≤ M | t − s | , which implies the existence of lim t → s k ( t ) = k . Then it is easy to see that the extension of k ( t ) obtained by setting k ( s ) = k is a solution of (4.4) in [ t ′ , t ], which completes the proof.Next, we show (A.4) which implies the uniqueness of the solution of (4.4) by taking x = y . Note that ddt | k i,x ( t, c ) − k i,y ( t, c ) | = 2 | k i,x ( t, c ) − k i,y ( t, c ) | · | µ i ( t, k i,x ( t, c ) , c ) − µ i ( t, k i,y ( t, c ) , c ) |≤ D i | k i,x ( t, c ) − k i,y ( t, c ) | . Since ψ ( t ) = | k i,x ( t, c ) − k i,y ( t, c ) | is absolutely continuous, by integrating both sides from t to t we get ψ ( t ) ≤ ψ ( t ) + 2 D i R tt ψ ( s ) ds , and thus by Gronwall’s inequality, ψ ( t ) ≤ ( x − y ) e D i ( t − t ) , which shows (A.4) for all t such that both solutions k i,x and k i,y exist. (cid:3) Proof of Theorem 4.1 . Lemma A.1 shows the local existence of solutions of (4.4) and LemmaA.2 proves the global existence of solutions of (4.4) for all times. (cid:3) .2 Proofs of Lemmas 4.1 and 4.2 Proof of Lemma 4.1 . By definition and (B2), for some constant C > ρ > M , J i ( t, x ) ≤ Z ∞ t e − ρs (1 + | k i,x ( s ) | ) ds ≤ C + (1 + | x | ) Z ∞ t e ( M − ρ ) s ds. Hence V i ( t, x ) ≤ C + (1 + | x | ) Z ∞ t e ( M − ρ ) s ds ≤ C e − ( ρ − M ) t (1 + x )This implies the finiteness of the value function if ρ > M , in which case the value functionssatisfy the linear growth condition.To show the Lipschitz continuity of the value function, we use (B2) and note that | V i ( t, x ) − V i ( t, y ) | ≤ sup ξ ∈ A ,c ∈ U i ( t ) Z ∞ t | φ ( t, k t,x ( s ) , c )) − φ ( t, k t,y ( s ) , c ) | ds ≤ C sup ξ ∈ A ,c ∈ U i ( t ) Z ∞ t e − ρ ( s − t ) | k i,x ( s, c ) − k i,y ( s, c ) | ds ≤ C Z ∞ t e − ( D i − ρ )( s − t ) | x − y | ds ≤ C | x − y | for ρ > max i ∈ I D i . (cid:3) Proof of Lemma 4.2 . By definition V i ( t, x ) ≥ J i ( t, x ; { ξ, c ( s, k ) } ) = Z rt φ ( s, k, c ) ds − X n : t ≤ τ n Proof of Theorem 4.2 . (i) We first show that V i ( t, k ) is a viscosity sub-solution. Choose ϕ ∈ C ([0 , ∞ ) × R + ) and ( t , k ) such that ( t , k ) is local maximum of V i ( t, k ) − ϕ ( t, k ). Weneed to show thatmax n sup c ∈ U h L c V i ( t , k ) + φ ( t , k , c ) i , M i V i ( t , k ) − V i ( t , k ) o ≥ . (A.5)By Lemma 4.2, if M i V i ( t , k ) ≥ V i ( t , k ), then the above inequality holds trivially. So wemay assume that M i V i ( t , k ) < V i ( t , k ).Choose ǫ > 0, let ( ξ, c ( · , · )) be an ǫ -optimal control, that is, V i ( t , k ) < J i ( t , k ; { ξ, c ( · , · ) } ) + ǫ. (A.6)We claim that the first transformation time τ > t . Otherwise, τ = t , J i ( t , k ; { ξ, c ( · , · ) } ) = J κ ( t , k ; {{ τ n , κ n } n ≥ , c ( · , · ) } ) ≤ V κ ( t , k ) ≤ max j = i { V j ( t , k ) } = M i V i ( t , k ) . (A.7)Inequalities (A.6) and (A.7) indicate that 0 < V i ( t , k ) − M i V i ( t , k ) < ǫ . This is acontradiction if ǫ < V i ( t , k ) − M i V i ( t , k ). This shows that we must have τ > t .Now choose a R < ∞ , ̺ > 0, and define τ := τ ∧ R ∧ inf { t > || k i,k ( t, c ) − k | ≥ ̺ } .Since ( t , k ) is a local maximum of V i ( t, k ) − ϕ ( t, k ), that is, V i ( τ, k τ ) − ϕ ( τ, k τ ) ≤ V i ( t , k ) − ϕ ( t , k ), then V i ( τ, k τ ) − V i ( t , k ) ≤ ϕ ( τ, k τ ) − ϕ ( t , k ) = Z τt L c ϕ ( t, k ) dt. (A.8)56y Lemma 4.2, for each ǫ > 0, there exists a control e c such that V i ( t , k ) ≤ Z τt φ ( s, k i,k ( s ) , e c ( s )) ds + V i ( τ, k τ ) + ǫ (A.9)Combining (A.8) and (A.9) yields − ǫ ≤ R τt (cid:2) L e c ϕ ( t, k ) + φ ( t, k ( t ) , e c ( t )) (cid:3) dt . Dividing by τ − t and letting ̺ → 0, we get L c ϕ ( t , k ) + φ ( t , k , e c (0)) ≥ − ǫ . Since ǫ is arbitrary, this showsthat sup c { L c ϕ ( t , k ) + φ ( t , k , c ) } ≥ 0. Hence, V i ( t, k ) is a viscosity sub-solution.(ii) We next prove that V i ( t, k ) is a viscosity super-solution. We choose ϕ ( t k ) ∈ C ([0 , ∞ ) × R + ) and ( t , k ) such that ( t , k ) is a local minima of V i ( t, k ) − ̺ ( t, k ). We must show thatmax n sup c ∈ U h L c ϕ ( t , k ) + φ ( t , k , c ) i , M i V i ( t , k ) − V i ( t , k ) o ≤ . (A.10)By Lemma 4.2, M i V i ( t , k ) ≤ V i ( t , k ) always holds. Hence it suffices to show thatsup c ∈ U h L c ϕ ( t , k ) + φ ( t , k , c ) i ≤ . To this end, fix c ∈ U and consider the control ( ξ, e c ) such that τ = ∞ , that is, there isno switching at all. Then by Lemma 4.2, we have V i ( t , k ) ≥ Z rt φ ( t, k t ,k ( t, e c ( t )) , e c ( t )) dt + V i ( r, k ( r )) . Since ( t , k ) minimizes V i ( t, k ) − ϕ ( t, k ) locally, we have V i ( t , k ) − ϕ ( t , k ) ≤ V i ( r, k ( r )) − ̺ ( r, k ( r )). Hence Z rt φ ( t, k t ,k ( t, e c ( t )) , e c ( t )) dt ≤ V i ( t , k ) − V i ( r, k ( r )) ≤ ϕ ( t , k ) − ϕ ( r, k ( r )) = − Z rt L c ϕ ( t, k ) dt, or R rt (cid:2) L c ϕ ( t, k ) + φ ( t, k t ,k ( t, e c ( t )) , e c ( t )) (cid:3) dt ≤ . Dividing by r − t and letting r → t , weget (A.10). (cid:3) Proof of Theorem 4.3 . We argue by contradiction under assumptions (B1), (B2), and (B3). Step 1 . The switching costs η ij ( i, j ∈ I , i = j ) satisfy η ij ( t ) + η jl ( t ) > η il ( t ) for l ∈ I , l = i, l = j and uniformly for all t . Let α i ( t ) = min j = i η ji ( t ) and β i ( t, k ) := Ce − ρt (1+ k )+ α i ( t )for t ∈ [0 , ∞ ) and k ∈ R + , where C is a positive constant to be determined later.sup c L β i ( t, k ) = − Cρe − ρt (1 + k ) + Ce − ρt sup c µ i ( t, k, c ) ≤ − Cρe − ρt (1 + k ) + C e − ρ ′ t (1 + k ) ≤ − Cρ + C . C such that − Cρ + C < − 1. Thus sup c L β i ( t, k ) < − t, k ) ∈ [0 , ∞ ) × R + and i ∈ I . Let ν i ( t ) := ( M i β i − β i )( t , k ) = max j = i ( α j ( t ) − η ij ( t )) − α i ( t ) . We have ν i ( t ) < t and i ∈ I . To see this, fix i ∈ I and let l ∈ I such that max j = i ( α j ( t ) − η ij ( t )) = α l ( t ) − η il ( t ). Set i such that α i ( t ) = min j = i η ji ( t ) = η ii ( t ). Then ν i ( t ) = α l ( t ) − η il ( t ) − η ii ( t ) < min j = l η jl ( t ) − η il ( t ) ≤ t . Let ν = max i ∈ I ,t ≥ t ν i , then ν < t, k ) ∈ [0 , ∞ ) × R + and i ∈ I , ( M i β i − β i )( t, k ) ≤ ν < Step 2 . For any ǫ > 0, consider a function W ǫi ( t, k ) := (1 − ǫ ) W i ( t, k ) + ǫβ i ( t, k ), for( t, k ) ∈ [0 , ∞ ) × R . We show that W ǫi ( t, k ) is a strict viscosity super-solution of (4.16) in thefollowing sense. For any ϕ ( t, k ) ∈ C ([0 , ∞ ) × R + ) and any ( t , k ) such that( t , k ) = arg min ( t,k ) (cid:2) W ǫi − ϕ (cid:3) ( t, k ) = arg min ( t,k ) (cid:2) (1 − ǫ ) W i − ( ϕ − ǫβ i ) (cid:3) ( t, k ) ,W i is a viscosity super-solution, thenmax n sup c ∈ U h L c ( ϕ − ǫβ i )( t , k ) + φ ( t , k , c ) i , M i W i ( t , k ) − W i ( t , k ) o ≤ . Since sup c ∈ U h L c ϕ ( t , k ) + φ ( t , k , c ) i − ǫ sup c ∈ U L c β i ( t , k )= sup c ∈ U h L c ( ϕ − ǫβ i )( t , k ) + φ ( t , k , c ) i ≤ , we have sup c ∈ U h L c ϕ ( t , k ) + φ ( t , k , c ) i ≤ ǫ sup c L c β i ( t , k ) ≤ − ǫ. In addition, by the result in Step 1, M i W ǫi ( t , k ) − W ǫi ( t , k ) = max j = i (cid:2) ((1 − ǫ ) W j + ǫβ j )( t , k ) − η ij (cid:3) − ((1 − ǫ ) W i + ǫβ i )( t , k ) ≤ (1 − ǫ )( M i W i − W i )( t , k ) + ǫ ( M i β i − β i )( t , k ) ≤ ǫν ( < . Hence, let δ = min(1 , − v ) > n sup c ∈ U h L c ( ϕ )( t , k ) + φ ( t , k , c ) i , M i W ǫi ( t , k ) − W ǫi ( t , k ) o ≤ − ǫδ. (A.11) Step 3 . To show the comparison principle, it suffices to show that for all ǫ ∈ (0 , j ∈ I sup ( t,k ) ∈ Q (cid:2) U j ( t, k ) − W ǫj ( t, k ) (cid:3) ≤ , since letting ǫ → ǫ ∈ (0 , 1) and i ∈ I such that M ǫ := max j ∈ I sup ( t,k ) ∈ Q (cid:2) U j ( t, k ) − W ǫj ( t, k ) (cid:3) = sup ( t,k ) ∈ Q (cid:2) U i ( t, k ) − W ǫi ( t, k ) (cid:3) > . (A.12)58rom the linear growth condition (4.11), we have U j ( t, k ) − W ǫj ( t, k ) goes to −∞ as t and/or k go to infinity. Hence U j ( t, k ) − W ǫj ( t, k ) attains its maximum M ǫ . For n = 1 , , · · · , and( t, k ) , ( s, h ) ∈ Q , define ϕ n,ǫ ( t, k, s, h ) := n t − s ) + n k − h ) + ǫ (cid:2) β i ( t, k ) + β i ( s, h ) (cid:3) ,H n,ǫ ( t, k, s, h ) := U i ( t, k ) − (1 − ǫ ) W i ( s, h ) − ϕ n,ǫ ( t, k, s, h )and M n,ǫ = sup ( t,k,s,h ) ∈ Q × Q H n,ǫ ( t, k, s, h ) . The linear growth condition of U i and W i implies that 0 < M n,ǫ < ∞ for all n , and hencethere exists ( t n , k n , s n , h n ) ∈ Q × Q such that M n,ǫ = H n,ǫ ( t n , k n , s n , h n ) = sup ( t,k,s,h ) ∈ Q H n,ǫ ( t, k, s, h ) . Note that, for all n ≥ M ǫ = sup ( t,k ) ∈ Q (cid:2) U i ( t, k ) − W ǫi ( t, k ) (cid:3) = sup ( t,k ) ∈ Q H n,ǫ ( t, k, t, k ) ≤ sup ( t,k,s,h ) ∈ Q H n,ǫ ( t, k, s, h ) = M n,ǫ = H n,ǫ ( t n , k n , s n , h n ) ≤ U i ( t n , k n ) − (1 − ǫ ) W i ( s n , h n ) − ǫ (cid:2) β i ( t n , k n ) + β i ( s n , h n ) (cid:3) . (A.13)The bounded sequence { ( t n , k n , s n , h n ) } n ≥ converges, up to a subsequence, to some ( e t, e k, e s, e h ) ∈ Q . Moreover, since the sequence { U i ( t n , k n ) − (1 − ǫ ) W i ( s n , h n ) − ǫ (cid:2) β i ( t n , k n )+ β i ( s n , h n ) (cid:3) } n ≥ is bounded, we see from (A.13) that the sequence { n ( t n − s n ) + n ( k n − h n ) } n ≥ is alsobounded, which implies that lim n →∞ n ( t n − s n ) + n ( k n − h n ) = 0 up to a subsequence.Hence e t = e s , e k = e h . Sincelim n →∞ n U i ( t n , k n ) − (1 − ǫ ) W i ( s n , h n ) − ǫ (cid:2) β i ( t n , k n ) + β i ( s n , h n ) (cid:3)o = U i ( e t, e k ) − W ǫi ( e t, e k ) ≤ M ǫ . (A.14)Together with (A.13), we have M ǫ = U i ( e t, e k ) − W ǫi ( e t, e k ) . Using (A.13) and (A.14) and letting n → ∞ , we obtain thatlim n →∞ M n,ǫ = lim n →∞ H n,ǫ ( t n , k n , s n , h n ) = M ǫ = sup ( t,k ) ∈ Q (cid:2) U i ( t, k ) − W ǫi ( t, k ) (cid:3) . (A.15) Step 4 . By definition of ( t n , k n , s n , h n ),( t n , k n ) is a local maximum of ( t, k ) → U i ( t, k ) − ϕ n,ǫ ( t, k, s n , h n ) on Q, ( s n , h n ) is a local minimum of ( s, h ) → (1 − ǫ ) W i ( s, h ) + ϕ n,ǫ ( t n , k n , s, h ) on Q. i ( t, k ) is a viscosity sub-solution. By definition, for ϕ n,ǫ ( t, k, s n , h n ) ∈ C , ( Q ), we have, at( t n , k n )max n sup c ∈ U (cid:2) L ct,k ϕ n,ǫ ( t n , k n , s n , h n ) + φ ( t n , k n , c ) (cid:3) , M i U i ( t n , k n ) − U i ( t n , k n ) o ≥ , (A.16)where L ct,k ϕ n,ǫ ( t n , k n , s n , h n ) = n ( t n − s n ) + n ( k n − h n ) µ i ( t n , k n , c ) + ǫ L c β i ( t n , k n ) . Similarly, W i ( s, h ) is a viscosity super-solution. By definition, for − ϕ n,ǫ ( t n , k n , s, h ) ∈ C , ( Q ), we havethat, at ( s n , h n ),max n sup c ∈ U (cid:2) L ct,k ( − ϕ n,ǫ )( t n , k n , s n , h n ) + φ ( t n , k n , c ) (cid:3) , (cid:2) M i W i ( s n , h n ) − W i ( s n , h n ) (cid:3)o ≤ , (A.17)where L ct,k ( − ϕ n,ǫ )( t n , k n , s n , h n ) = n ( t n − s n ) + h ( k n − h n ) µ i ( t n , k n , c ) − ǫ L c β i ( s n , h n ) . Sincesup c ∈ U (cid:2) L ct,k ϕ n,ǫ ( t n , k n , s n , h n ) + φ ( t n , k n , c ) (cid:3) − sup c ∈ U (cid:2) L ct,k ( − ϕ n,ǫ )( t n , k n , s n , h n ) + φ ( t n , k n , c ) (cid:3) ≤ ǫ (cid:2) L β i ( t n , k n ) + L β i ( s n , h n ) (cid:3) → ǫ L β i ( e t, e k ) < − ǫ < . This suggests that there exists an N > n > N ,sup c ∈ U (cid:2) L ct,k ϕ n,ǫ ( t n , k n , s n , h n ) + φ ( t n , k n , c ) (cid:3) < sup c ∈ U (cid:2) L ct,k ( − ϕ n,ǫ )( t n , k n , s n , h n ) + φ ( t n , k n , c ) (cid:3) ≤ . Therefore, inequality (A.16) implies that, for n ≥ N , M i U i ( t n , k n ) − U i ( t n , k n ) ≥ . (A.18)From the argument in Step 2, W ǫi ( t, k ) is a strict viscosity super-solution, and hence inequal-ity (A.11) implies that, for all n ≥ N , M i W ǫi ( s n , h n ) − W ǫi ( s n , h n ) ≤ − ǫδ < . (A.19)Combining (A.18) and (A.19) yields that M i W ǫi ( s n , h n ) − W ǫi ( s n , h n ) ≤ − ǫδ < ≤ M i U i ( t n , k n ) − U i ( t n , k n )or U i ( t n , k n ) − W ǫi ( s n , h n ) + ǫδ < M i U i ( t n , k n ) − M i W ǫi ( s n , h n )Letting n → ∞ yields thatlim n →∞ (cid:2) U i ( t n , k n ) − W ǫi ( s n , h n ) (cid:3) + ǫδ ≤ lim n →∞ (cid:2) max j = i ( U i ( t n , k n ) − η ij ) − max j = i ( W ǫi ( s n , h n ) − η ij ) (cid:3) = lim n →∞ (cid:2) max j = i ( U i ( t n , k n ) − η ij ) − max j = i ( W ǫi ( t n , k n ) − η ij ) (cid:3) ≤ lim n →∞ max j = i (cid:2) U j ( t n , k n ) − W ǫj ( t n , k n ) (cid:3) . That is, M ǫ + ǫδ ≤ M ǫ . Contradiction! Therefore, U i ( t, k ) ≤ W i ( t, k ) for all ( t, k ) ∈ Q and i ∈ I . (cid:3) .4 Proof of Theorem 4.8 Proof of Theorem 4.8 . Step 1 . Lemma 4.1 implies that there exists ρ ′ > C > ± C e − ρ ′ ( t − t ) (1 + || x || ) are super- and subsolutions of (4.13). Then define that V i ( t, x ) = lim inf ǫ → + , ( s,y ) → ( t,x ) V ǫi ( s, y ) , V i ( t, x ) = lim sup ǫ → + , ( s,y ) → ( t,x ) V ǫi ( s, y ) . Observe that, for any i ∈ I , V ǫj ( t, k ) − η ij ( t ) ≤ V ǫi ( t, k ) for all ( t, x ) ∈ [ t , ∞ ) × R p and any j = i . When ǫ → + , it follws that V ( t, x ) = V ( t, x ) = · · · = V I ( t, x ) , V ( t, x ) = V ( t, x ) = · · · = V I ( t, x ) . Denote by V ( t, x ) and V ( t, x ), respectively, the common values of V i ( t, x ) and V i ( t, x ).By construction, V ( t, x ) and V ( t, x ) are, respectively, lower and upper semicontinuous and V ( t, x ) ≤ V ( t, x ). Step 2 . We claim that V ( t, x ) and V ( t, x ) are, respectively,a super- and a subsolution of(4.33) in the viscosity sense.Let ϕ ∈ C ( Q ) and ( t, k ) is a strict maximum point for V − ϕ in a closed neighborhood B of ( t, k ). Then there exists ǫ n → + and for all i , a sequence ( t in , k in ) n ≥ in B and suchthat ( V ǫ n i − ϕ )( t in , k in ) = max ( t,k ) ∈ B ( V ǫ n i − ϕ )( t, k ), and( t in , k in ) → ( t, k ) , V ǫ n i ( t in , k in ) → V i ( t, k ) = V ( t, k ) , as n → ∞ . (A.20)Since ( V ǫ , . . . , V ǫI ) is a viscosity subsolution of (4.13), for all i ∈ I ,sup c ∈ U h ϕ∂t ( t in , k in ) + µ i ( t in , k in , c ) ϕ∂k + φ ( t in , k in , c ) i ≤ . Let n → ∞ in the above, and by (A.20), V is a viscosity subsolution of (4.33).Let ϕ ∈ C ( Q ) and ( t, k ) is a strict minimum point for V − ϕ in a closed neighborhood B of ( t, k ). As above, there exists ǫ n → + and ( t in , k in ) n ≥ in B such that ( V ǫ n i − ϕ )( t in , k in ) =min ( t,k ) ∈ B ( V ǫ n i − ϕ )( t, k ), and( t in , k in ) → ( t, k ) , V ǫ n i ( t in , k in ) → V i ( t, k ) = V ( t, k ) , as n → ∞ . (A.21)Choose i n ∈ I such that ( V ǫ n i n − ϕ )( t i n n , k i n n ) = min j ∈ I min ( t,k ) ∈ B ( V ǫ n i − ϕ )( t, k ) and set( t n , k n ) := ( t i n n , k i n n ). Then V ǫ n i n ( t n , k n ) ≤ V ǫ n j ( t n , k n ) < V ǫ n j ( t n , k n ) + η ǫ n ( i n , j ), for all j = i n .Since ( V ǫ , . . . , V ǫI ) is a viscosity subsolution of (4.13), for all i ∈ I ,sup c ∈ U h ϕ∂t ( t in , k in ) + µ i ( t in , k in , c ) ϕ∂k + φ ( t in , k in , c ) i ≥ . i n → i or i n = i for all n large enough. Let n → ∞ in the equalityabove and use (A.21), we obtainsup c ∈ U h ϕ∂t ( t, k ) + µ i ( t, k, c ) ϕ∂k + φ ( t, k, c ) i ≥ . Hence, V is a viscosity supersolution of (4.33). Step 3 . Then the comparison theorem imply that V = V . Denote by e V ( t, k ) the commonvalue of V and V , we have V ǫi → e V locally uniformly in [ t , ∞ ) × R p , for all i ∈ I . Hence e V = V in (4.34). (cid:3) References 1. Abdel-Kader, K. (2013) What are structural policies? Finance and Development . 50,46-47.2. Acemoglu, D. (2009). Introduction to Modern Economic Growth . Princeton, NY:Princeton University Press.3. Acemoglu, D. and Guerrieri, V. (2008). Capital deepening and non–balanced economicgrowth. Journal of Political Economy , 116, 467-498.4. Aghion, P. and Howitt, P. (1992). A model of growth through creative destruction. Econometrica , 60, 323-351.5. Bardi, M. and Capuzzo-Dolcetta, I. (1997). Optimal Control and Viscosity Solutionsof Hamilton-Jacobi-Bellman Equations . Berlin, Germany: Birkh¨auser.6. Crandall, M. G., Ishii, H., and Lions, P.-L. (1992). A user’s guide to viscosity solutions, Bulletin of the American Mathematical Society , 27, 1-67.7. Crandall, M. G. and Lions, P.-L. (1984). Viscosity solutions of Hamilton-Jacobi equa-tions. Transactions of the American Mathematical Society , 277, 1-42.8. Fleming, W. H. and Soner, H. M. (2006). Controlled Markov Processes and ViscositySolutions . New York: Springer.9. Herrendorf, B., Rogerson, R., and Valentinyi, A. (2014). Growth and structural trans-formation. In Handbook of Economic Growth , Volume 2, edited by P. Aghion and S.N. Durlauf, 855-941. Oxford: North Holland.10. Hirschman, A. O. (1958). The Strategy of Economic Development . New Haven, CT:Yale University Press. 621. Ju, J., Lin, Y., and Wang, Y. (2015). Endowment structures, industrial dynamics, andeconomic growth. Journal of Monetary Economics , 76, 244-263.12. Krugman, P. (1994). The myth of Asia’s miracle. Foreign Affairs , 73 (6), 62-78.13. Kuznets, S. (1966). Modern Economic Growth . New Haven, CT: Yale University Press.14. Kuznets, S. (1973). Modern economic growth: Findings and reflections. AmericanEconomic Review , 63, 247-258.15. Lewis, W. A. (1954). Economic development with unlimited supplies of labor , Manch-ester School, 22, 139-191.16. Lin, J. Y. (1989). An economic theory of institutional change: Induced and imposedchange. Cato Journal , 9 (1), 1-33.17. Lin, J. Y. (2011). New structural economics: A framework for rethinking development. World Bank Research Observer , 26 (2), 193-221.18. Marx, K. and Engels, F. (1848). Manifesto of the Communist Party. Marx/EngelsSelected Works , Vol. 1, Progress Publishers, Moscow, 1969, pp. 98-137.19. Murphy, K. M., Shleifer, A., and Vishny, R. W. (1989). Industrialization and the bigpush. Journal of Political Economy , 97 (5), 1003-1026.20. North, D. C. (1981). Structure and Change in Economic History . New York: W. W.Norton and Company.21. Ranis, G. and Fei, J. C. H. (1961). A theory of economic development. AmericanEconomic Review , 51, 533-565.22. Romer, P. M. (1990). Endogenous technological change. Journal of Political Economy ,98, 71-102.23. Weber, M. (1904, 1905).