Multilateral Index Number Systems for International Price Comparisons: Properties, Existence and Uniqueness
MMultilateral Index Number Systems for International Price Comparisons:Properties, Existence and Uniqueness
Gholamreza Hajargasht a, ∗ , D.S. Prasada Rao b, ∗∗ a Swinburne Business School, Swinburne University of Technology b School of Economics, University of Queensland
Abstract
Over the past five decades a number of multilateral index number systems have been proposed forspatial and cross-country price comparisons. These multilateral indexes are usually expressed assolutions to systems of linear or nonlinear equations. In this paper, we provide general theoremsthat can be used to establish necessary and su ffi cient conditions for the existence and uniquenessof the Geary-Khamis, IDB, Neary and Rao indexes as well as potential new systems including twogeneralized systems of index numbers. One of our main results is that the necessary and su ffi cientconditions for existence and uniqueness of solutions can often be stated in terms of graph-theoreticconcepts and a verifiable condition based on observed quantities of commodities. Keywords: purchasing power parities, international prices, nonlinear Perron-Frobenius problem,connected graphs, DAD problem, generalized mean
JEL classification:
E31, C19
1. Introduction
Purchasing power parity (PPP) is a widely used multilateral index for comparing price levelsand macroeconomic aggregates such as gross domestic product (GDP) and its components acrosscountries. PPPs are now regularly compiled as part of the World Bank’s International ComparisonProgram (ICP). The most recently released findings from the ICP are for the year 2011 covering 177countries of the world (World Bank, 2015). The PPPs from the ICP are used in assessing the sizeand distribution of the world economy and rankings of economies. For example, the latest ICP ∗ Corresponding author. address: Swinburne Business School, BA Building, 27 John St, Hawthorn Victoria 3122,Australia, email: [email protected] ∗∗ The authors wish to thank Erwin Diewert for his helpful comments on an earlier version of this paper. We areindebted to Stephane Gaubert for pointing us to the connection between our fourth theorem and the DAD theorem inthe mathematics literature. The authors are thankful to an anonymous referee for the incisive comments and helpfulsuggestions. We wish also to thank Bill Gri ffi ths and Beth Webster whose comments have improved the expositionof the paper. This research is partly supported by ARC Grant DP 170103559. A part of work by Rao was undertakenwhen he was visiting the Institute of Economic Research, Hitotsubashi University in 2017. a r X i v : . [ ec on . T H ] D ec eport indicates that the United States was the world’s largest economy in 2011 followed by China,India and Japan. Results from the ICP also indicate that there has been a significant reduction inglobal inequality based on PPP-converted per capita income data. ICP results are also used forcalculating World Development Indicators, the Human Development Index (HDI), regional andglobal poverty, and for comparing health and education expenditures across countries.The PPPs within the ICP are obtained by aggregating price data collected from countriesusing appropriate multilateral index formulas . A variety of multilateral index numbers havebeen proposed for the purpose of PPP compilation over the last five decades including butnot limited to Gini-Elteto-Koves-Szulc (GEKS); Geary-Khamis (GK); generalized GK; Ikl´e (orIDB); Rao; and the Country-Product-Dummy (CPD). Rao (2013b) and Diewert (2013) describethe methods currently employed within the ICP. Economic theoretic approaches to multilateralsystems have been discussed in Neary (2004), Feenstra et al. (2009), and Feenstra et al. (2013)] whileDiewert (1988, 1999) and Balk (2008, 2009) provide overviews of the axiomatic or test approach tomultilateral index numbers. Hill (2000, 2009) and Hajargasht et al. (2018) discuss spatial chainingmethods based on minimum spanning trees and Rao (2009) o ff ers a collection of papers thatdescribe the state of the art and advances that have been made. The quest for indexes with betterproperties is ongoing and new indexes are being proposed e.g. see Hajargasht & Rao (2010) andRao & Hajargasht (2016) for a new stochastic approach.A common attribute of the multilateral index number systems used in international comparisonsis that the price indexes from these methods are usually solutions to some suitably formulatedsystems of equations. These systems can be linear as is the case with the GK system or nonlinearas is the case with the Rao (1990) system. These systems can be meaningful only if they havesolutions which are positive and unique (up to a factor of proportionality ). Not surprisingly,some e ff orts have already been put into proving the existence of solutions to these systems [e.g.Rao (1971, 1976), Khamis (1972); Balk (1996, 2009); Neary (2004)]. However, the existing resultsdo not often assume the most general conditions and do not cover some indexes.This paper contributes to the literature on existence and uniqueness of multilateral indexesin several ways. (i) It provides several theorems for existence and uniqueness of solutions tomultilateral indexes in their most general forms. We use these theorems to prove viability of manyindexes for some of which the results are new (e.g. for ”equally weighted GK” and arithmeticindex). We also provide both necessary and su ffi cient conditions for existence and uniquenessof IDB and Rao indexes. (ii) The paper brings together and clarifies the mathematical conceptsand tools required for establishing existence and uniqueness of solutions to di ff erent types of Details of the ICP methodology can be found in Rao (2013a) and World Bank (2013). Linear here means that equations are linear in unknowns or known functions of the unknowns. For example, asystem is considered linear when it is linear in 1 / PPP j . A constant multiple of a given set of PPPs leaves price comparisons between countries unchanged. Therefore, it issu ffi cient if PPPs are determined uniquely up to a factor of proportionality. ρ . This general class encompasses most ofthe known systems including the commonly used systems. (iv) Another insight from our resultsis that a compatibility condition is often required when defining an index in the sense that theweights in the equations defining world average prices ( P s) and the weights in equations definingpurchasing power parities ( PPP s) must be compatible. (v) Finally, an important contribution ofthe paper is to show that in general, existence and uniqueness of the indexes are guaranteed if theobserved quantity matrix is connected.The paper is organized as follows: In Section 2 we define basic notations and concepts thatunderpin the multilateral index number systems considered in the paper. Section 3 describeslinear and nonlinear systems including several commonly used index number systems. Section 4states and proves the main theorems on existence and uniqueness of general classes of multilateralindex numbers. These general theorems are in turn used to prove the existence and uniqueness ofmany of the index numbers currently used in international comparisons. Appendix A provides amathematical toolkit (including various connectedness concepts, nonlinear eigenvalue theoremsand their links to each other) that is used to prove the theorems stated in Section 4.
2. Notation and Basic Concepts
Let p ij and q ij represent the price and the quantity of the i -th commodity in the j -th countryrespectively where i = , ..., N and j = , ..., M . We assume that prices are strictly positive andquantities are non-negative . We further assume: (i) for each i , q ij is positive for at least one j and (ii) for each j , q ij is positive for at least one i . As additional notation, we let p and q represent( N × M ) matrices of prices and quantities of all commodities in all the countries where p is strictlypositive and q is non-negative where due to (i) and (ii) each row and column of q has at least onestrictly positive element. Let
PPP j denote the purchasing power parity of currency of country j or the general price level in j -th country relative to a numeraire country. PPP j shows the number of currency units of country j that have the same purchasing power as one unit of currency of a reference country. For example,if PPP for currency of India is equal to INR 15 .
50 with respect to one US dollar then 15 .
50 Indianrupees in India have the same purchasing power as one US dollar in the United States. We notehere that
PPP j s would naturally be functions of observed price and quantity data. Multilateral An implication of this is that expenditure on an item in a country is positive if and only if the correspondingquantity is positive. j and k , denoted by P jk , is given by P jk = PPP k PPP j ∀ j and k (1)The set of all binary price comparisons, P jk ( j , k = , , ..., M ), defined in (1) are transitive and thatthe price comparisons are una ff ected when all the PPP j s are multiplied by a non-zero constant.Note that, for multilateral price comparisons to be meaningful it is necessary that PPP j s are strictlypositive and determined uniquely up to a factor of proportionality, so that P jk in (1) is unique. A common feature of the index number systems we study in this paper is that these systemsalso determine international average prices of the commodities included in the comparisons. Let P i denote the world average price of the i -th commodity ( i = , , ..., N ). These P i s are typicallyexpressed as a function of the observed price and quantity data as well as the PPP j s of currencies.Intuitively, the international average price of a commodity is an average of prices of the commodityacross countries. As prices in di ff erent countries are denominated in di ff erent currencies, it isnecessary to convert these prices into a common currency unit prior to averaging them. This isachieved by converting observed prices using PPP s of currencies.
PPP j s are also defined in termsof observed price and quantity data as well as international prices. In Section 3, we describedi ff erent methods for averaging prices such as arithmetic, geometric and harmonic averages. A multilateral index number system is an interrelated system of equations which expresses theunknown purchasing power parities and international prices as functions of price and quantitydata from di ff erent countries. In addition to being functions of observed price and quantity data,each PPP j is typically a function of all the international prices and similarly each P i is a functionof all the unknown PPP j s thus leading to a system of M + N equations in M unknown PPP j s and N unknown P i s. A general multilateral system may be specified as a system of equations of thefollowing form: P i = H i ( PPP , p , q ) ( i = , ....., N ) (2a) PPP j = H j ( P , p , q ) ( j = , ....., M ) (2b)where PPP and P are, respectively, M × N × H i and H j are strictly positive and continuous in all the arguments. Transitivity requires that all the pairwise comparisons are internally consistent and satisfy the condition P jk = P jl × P lk for all j , k and i . See Balk (2008) for more on transitivity. i s are often in the form of some weighted average of p ij / PPP j over j and H j s are in the form ofsome weighted average of p ij / P i . Di ff erent index number systems di ff er in their specification offunctional forms for the equations in (2).
3. Multilateral Index Numbers Systems Used in ICP
In this section, we present a number of multilateral systems with an emphasis on those whichhave been used in the International Comparison Program since its beginning in 1968.
The GK system was first proposed by Geary (1958) as a method for comparing agricultural outputsacross countries and later extended to its general form by Khamis (1972). The index was adoptedas the main aggregation method in the ICP until 1985 (see Kravis et al., 1982, for a discussion ofthe method). The GK system consists of the following system of M + N equations:1 PPP j = N (cid:88) n = p nj q nj (cid:80) Nn = p nj q nj P n p nj ( j = , ....., M ) (3a) P i = M (cid:88) m = q im (cid:80) Mm = q im p im PPP m ( i = , ....., N ) (3b)Here, PPP j is defined as a weighted harmonic mean of price relatives. Similarly, the internationalaverage price, P i , is defined as a quantity-share weighted average of prices across countries afterthey are converted into a common currency using PPP j s. Note that Equation (3) is linear in 1 / PPP j sand P i s. The necessary and su ffi cient conditions for existence and uniqueness of a solution havebeen derived by Rao (1971), Khamis (1972) and Balk (1996). We provide a proof of existence for ageneric linear system so that results for specific systems such as GK can be obtained as corollaries.Cuthbert (1999) proposed the following generalized class of index number systems in order toshow that, with respect to satisfying the additivity property, GK is not unique PPP j = N (cid:88) n = p nj q nj (cid:80) Nn = p nj q nj P n p nj ( j = , ....., M ) (4a) P i = M (cid:88) m = β m q im (cid:80) Mm = β m q im p im PPP m ( i = , ....., N ) (4b)where β j s are known constants. The GK system in (3) is a special case of (4) when β j = β for all j = , , ..., M . Other choices for β j could lead to the Ikl´e system (e.g. Balk, 1996, pp. 207). Cuthbert For a definition and further discussion of additivity see Balk (2008) pages 244-251. Cuthbert (1999) disproved aconjecture made by Rao indicating that the GK system is the only multilateral system satisfying additivity. Sakumaet al. (2009) provided another system that satisfies additivity but di ff ers from the GK index. ffi cient conditions for the existence of a unique positivesolution to this system as a special case.Note that in systems (3) and (4), each PPP j is defined as a weighted harmonic mean of ratios { p nj / P n , n = , ..., N } and each P i is an arithmetic mean of the ratios { p im / PPP m , m = , ..., M } . Amore general system based on generalized means of order ρ can be written as1 PPP j = N (cid:88) n = w nj (cid:16) p nj P n (cid:17) ρ /ρ ( j = , ....., M ) (5a) P i = M (cid:88) m = β j q im (cid:80) Mm = β m q im (cid:16) p im PPP m (cid:17) ρ /ρ ( i = , ....., N ) (5b)where w ij represents the expenditure share of commodity i in country j defined as w ij = p ij q ij (cid:80) Nn = p nj q nj .A common feature of the systems in (3), (4) and (5) is that they share the same type of weights.The PPP j definition uses expenditure share weights whereas the international prices are definedusing quantity share weights. It is easy to show that the GK and generalized GK systems definedabove are special cases of the new system defined in (5). Of particular interest could be the casewith ρ → We now turn to multilateral index number systems that make use of expenditure shares as weightsfor defining both
PPP s and P s. The following two sets of weights, w ij and weights w ∗ ij are used w ij = p ij q ij (cid:80) Nn = p nj q nj w ∗ ij = w ij (cid:80) Mm = w im Here, w ij is the expenditure share of commodity i in country j , whereas w ∗ ij is the expenditureshare of commodity i in country j relative to the total share of this commodity across all thecountries.Rao (1990) defines a system for international price comparisons as follows PPP j = N (cid:89) n = (cid:16) p nj P n (cid:17) w nj ( j = , ....., M ) (6a) P i = M (cid:89) m = (cid:16) p im PPP m (cid:17) w ∗ im ( i = , ....., N ) (6b) Rao (2005) has shown that the solution to this system can be obtained as weighted least squares estimates from thecountry-product dummy (CPD) method. Further details of the CPD method and its links with multilateral systems arediscussed in Hajargasht & Rao (2010) and Rao & Hajargasht (2016).
6n the system first proposed by Ikl´e (1972), and later simplified and clarified by Dikhanov(1997) and Balk (1996) , expenditure share weights are used along with harmonic averages asshown in (7).1 PPP j = N (cid:88) n = (cid:16) w nj P n p nj (cid:17) ( j = , ....., M ) (7a)1 P i = M (cid:88) m = (cid:16) w ∗ im PPP m p im (cid:17) ( i = , ....., N ) (7b)Note that, in the Rao system in (6), PPP s and P s are defined as geometric means of deflatednational prices while in the Ikl´e-Balk-Dikhanov (IDB) system in (7), harmonic means of theconverted national prices are used in a similar manner. Hajargasht & Rao (2010) proposed asimilar system of equations but using arithmetic means: PPP j = N (cid:88) n = (cid:16) w nj p nj P n (cid:17) ( j = , ....., M ) (8a) P i = M (cid:88) m = (cid:16) w ∗ im p im PPP m (cid:17) ( i = , ....., N ) (8b)Hill (2000) proposed the ”equally weighted GK system” (EWGK) defined below and found that ithas better properties than the GK system1 PPP j = N (cid:88) n = w nj P n p nj ( j = , ....., M ) P i = M (cid:88) m = w ∗ im p im PPP m ( i = , ....., N )where the PPP j equations are the same as those in the GK and Ikl´e indexes and the P i equations arethe same as those in the arithmetic index. A new general index number system which encompassesthe systems described in (6), (7) and (8) can be defined as: PPP j = N (cid:88) n = w nj (cid:16) p nj P n (cid:17) ρ /ρ ( j = , ....., M ) (9a) P i = M (cid:88) m = w ∗ im (cid:16) p im PPP m (cid:17) ρ /ρ ( i = , ....., N ) (9b) We follow Diewert (2013) and refer to this system as the Ikl´e-Dikhanov-Balk (IDB) system. ff erent values for ρ leads to di ff erent indexes. For example ρ = ρ = − ρ = . Theorem-5 proved in this papercan be used to establish the existence and uniqueness of all of these systems and more. The Neary (2004) and Rao (1976) systems bring economic theory and the notion of cost of livingindexes into the definition of
PPP s. The Konus cost of living index is defined as the ratio ofminimum expenditure required to attain a certain level of utility at two di ff erent sets of prices .The Neary (2004) system uses the Konus index to define PPP s as follows:1
PPP j = N (cid:88) n = p nj q ∗ n ( P , q j ) (cid:80) Nn = p nj q nj P n p nj ( j = , ....., M ) (10a) P i = M (cid:88) m = q im (cid:80) Mm = q ∗ i ( P , q m ) p im PPP m ( i = , ....., N ) (10b) q ∗ i ( P , q j ) in (10a) and (10b) are optimal (cost-minimizing) quantities obtained as the solution tothe following cost minimization problem where U ( . ) is a well-behaved utility function argmin ( q ∗ ,..., q ∗ N ) N (cid:88) n = P n q ∗ n subject to U ( q ∗ , .., q ∗ N ) ≥ U ( q j ) (10c)This problem is solved for each country separately. The main di ff erence between the Nearysystem and the GK system is in the use of unobserved quantities, q ∗ ij . In the Neary system wehave quantities q ∗ ij s which depend on the vector of international average prices P . This calls for adi ff erent theorem for establishing existence and uniqueness.The system proposed by Rao (1976) is similar to the Neary (2004) system except that theinternational average prices are defined di ff erently: P i = M (cid:88) m = q im (cid:80) Mm = q im p im PPP m ( i = , ....., N ) (11)Rao (1976) studied existence of solutions to this system by using a version of a nonlinear eigenvaluetheorem but his proof is incomplete. It might be possible to associate a stochastic model to this system and therefore estimate ρ or statistically testbetween these indexes. Diewert (1976) discusses the Konus index in detail. The notation q ∗ i ( P , q j ) could be replaced by q ∗ ij to indicate that these quantities depend upon the utility obtained bycountry i with quantity vector q j . Further details can be found in Neary (2004) and Rao (1976). In fact, he could not prove that λ ∗ = . Existence Theorems for Multilateral Index Number Systems As we mentioned before, multilateral index number systems can be classified into two groups:The first group are essentially linear systems in terms of
PPP j s, or P i s, or in terms of some functionsof PPP j s and P i s whereas the second group are nonlinear. In this section we provide theoremsthat can be applied to both cases. For stating and proving of these theorems we need a variety ofmathematical concepts and tools that are summarized in Appendix A. The following theorem proves existence and uniqueness for the linear class of systems in its mostgeneral form.
Theorem 1.
Consider the following general system of M + N equations:f j ( PPP j ) = N (cid:88) n = a nj g n ( P n ) ( j = , ....., M ) (12a) g i ( P i ) = M (cid:88) m = b im f m ( PPP m ) ( i = , ....., N ) (12b) where f j ( . ) and g i ( . ) can be any bijective functions from R + → R + ; a ij and b ij are non-negative weights.Under the following assumption:(T.1) a ij and b ij can be written such that a ij = d ij (cid:80) Nn = c nj and b ij = c ij (cid:80) Mm = d im with q ij > ⇔ d ij > andq ij > ⇔ c ij > .A necessary and su ffi cient condition for the existence of a unique positive solution (up to a positive scalarfactor) is connectedness of the quantity matrix q . Before o ff ering a proof for this theorem, we examine some aspects of the theorem. • Note that for existence of a non-trivial solution, a ij and b ij in equation (12) cannot be completelyindependent and a compatibility condition is needed. One such condition is given by Assumption( T.1 ) above. Khamis & Rao (1989) considered an interesting system in which conditions T . • As we discussed in defining PPP indexes, each
PPP j is often set as some kind of average of p ij s(over i ) deflated by international prices P i s and each P i as some kind of average of p ij s (over j )deflated by PPP j s. With appropriate definitions for d ij and c ij , f j and g i we can easily cover such9ndexes. As an example, consider the GK system (3) i.e.1 PPP j = N (cid:88) n = p nj q nj (cid:80) Nn = p nj q nj P n p nj ( j = , ....., M ) P i = M (cid:88) m = q im (cid:80) Mm = q im p im PPP m ( i = , ....., N )It is easy to see that by defining d ij = q ij , c ij = p ij q ij , f j ( x ) = / x and g i ( x ) = x in Theorem-1 weobtain this system.To prove the theorem, we establish the following two lemmas where we use the notation f j = f j ( PPP j ) and g i = g i ( P i ). If we obtain solutions for f j and g i , then we can obtain solutions for PPP j and P i by invoking the bijective nature of these functions. Lemma 1.
Consider the following system of equations defined in terms of ( f , g ) f j = N (cid:88) n = d nj (cid:80) Nn = c nj g n ( j = , ....., M ) (13a) g i = M (cid:88) m = c im (cid:80) Mm = d im f m ( i = , ....., N ) (13b) then a necessary and su ffi cient condition for the existence of a unique positive f ∗ = ( f ∗ , ..., f ∗ M ) (cid:48) and g ∗ = ( g ∗ , ..., g ∗ N ) (cid:48) (up to a positive scalar factor) is irreducibility of matrices B , C or D defined below.Proof of Necessity: Through direct substitution, we first express the system (13) in matrix form · · · c N (cid:80) n = c n · · · c MN (cid:80) n = c nM ... ... ... · · · c N N (cid:80) n = c n · · · c NMN (cid:80) n = c nM d M (cid:80) m = d m · · · d N M (cid:80) m = d Nm · · · ... ... ... ... d MM (cid:80) m = d m · · · d NMM (cid:80) m = d Nm · · · g M (cid:80) m = d m ...... g N M (cid:80) m = d Nm f N (cid:80) n = c n ...... f M N (cid:80) n = c nM = g M (cid:80) m = d m ...... g N M (cid:80) m = d Nm f N (cid:80) m = c m ...... f M N (cid:80) m = c mM or more stacked form X X = X X where C , D , X and X can be identified from the10bove expression and B = .Note that by defining F = DC , we can focus on the subsystem involving only f = ( f , ..., f M ) (cid:48) : FX = X ⇒ N (cid:80) n = d n c n M (cid:80) m = d nmN (cid:80) n = c n · · · · · · N (cid:80) n = d n c nMM (cid:80) m = d nmN (cid:80) n = c nM ... ...... ... N (cid:80) n = d nM c n M (cid:80) m = d nmN (cid:80) n = c n · · · · · · N (cid:80) n = d nM c nMM (cid:80) m = d nmN (cid:80) n = c nM f N (cid:80) n = c n ...... f M N (cid:80) n = c nM = f N (cid:80) n = c n ...... f M N (cid:80) n = c nM (14)Alternatively, we can define E = CD and focus on a subsystem involving only g = ( g , ..., g N ) (cid:48) .We can write each of these linear systems as AX = X where matrix A (i.e. F , E or B ) is a non-negativematrix with columns that sum to one. To prove the necessity part of the theorem, we note thatthe matrix A (cid:48) , transpose of A , is a stochastic matrix (each column of matrix A (cid:48) sums to 1)and therefore it has a dominant eigenvalue equal to one with corresponding eigenvector equalto (1 , ...,
1) (see e.g. Gantmacher & Brenner, 2005, pp 100). Now, for AX = X to have a uniquepositive solution, matrix A must have a dominant eigenvalue equal to one with a correspondingpositive eigenvector. But this (i.e. both A and A (cid:48) having the same dominant eigenvalue withpositive eigenvectors) is possible only if A is irreducible (see corollary in page 96 of Gantmacher& Brenner, 2005).To prove su ffi ciency , we use the Perron-Frobenius theorem. According to this theorem if matrix A is irreducible then AX = λ X has a unique positive solution with λ >
0. Furthermore, since eachcolumn of A sums to one, we must have λ =
1. Note also that properties of the q matrix ensurethat (cid:80) Mj = d ij > (cid:80) Ni = c ij > f ∗ = ( f ∗ , ..., f ∗ M ) (cid:48) and g ∗ = ( g ∗ , ..., g ∗ N ) (cid:48) are well-defined. Lemma 2.
A necessary and su ffi cient condition for matrix F in (14), (and therefore E or B ), to be irreducibleis the connectedness of the set of countries based on the quantity matrix q . A matrix is said to be stochastic if it is non-negative and each row sums to one. Note that connectedness is necessary for uniqueness of the solutions. Without connectedness there can be solutionsbut they are not unique since the system of equations for some J ⊂ , ..., M can be divided into at least two independentsubsystems with independent solutions f ∗ J = { f ∗ j | j ∈ J } and f ∗ J c = { f ∗ j | j ∈ J c } with γ f ∗ J and γ f ∗ J c for any γ > γ > roof of Su ffi ciency : If q is connected, then for any non-empty J ⊂ { , , ..., M } , there exists at leastone k ∈ J and i ∈ { , ...., N } such that q ik > j (cid:60) J such that q ij >
0. Note that thisimplies d ik > , c ik > , d ij > , c ij >
0. The su ffi ciency is proved if we show that { F } kj (i.e. element k , j of the matrix F ) is positive. Note that { F } kj = N (cid:80) n = d nk c nj (cid:46) M (cid:80) m = d nmN (cid:80) n = c nj It can be seen that this is positive, if and only if, there is at least one i ∈ , ...., N such that both d ik > c ij >
0, but connectedness as argued in the previous paragraph guarantees existenceof such an i . Proof of Necessity : If matrix F is irreducible, then for any J ⊂ { , , ..., M } (cid:44) ∅ there exists at least one k ∈ J and j (cid:60) J such that { F } kj > i ∈ , ..., N such that d ik > c ij > q ik > q ij > Proof of Theorem-1 : From Lemma-1, irreducibility of matrices F ( E or B ) is necessary and su ffi cientfor the existence of a unique positive solution for the system (12) in Theorem-1. Lemma-2 showsthat connectedness of the quantity matrix q is necessary and su ffi cient for irreducibility of thematrices involved thus establishing Theorem-1.We can restate the necessary and su ffi cient conditions stated in Theorem-1 on the quantitymatrix q as: unique positive solutions to multilateral systems of the form (12) exist if and only ifthe adjacent graph G q is connected (see Appendix A for further information). f ( . ) and g ( . )Theorem-1 stated above is quite general and functions f and g are (otherwise unrestricted) positivebijective functions. In this section, we introduce a simple axiom that underpins meaningfulinternational price comparisons and show that under this axiom the functions, f and g , must havea particular form. In the Lemmas and Theorem-1 proved above, we have shown that if there isa positive f ∗ = ( f ∗ , ..., f ∗ M ) (cid:48) and g ∗ = ( g ∗ , ..., g ∗ N ) (cid:48) that solves the system of equations then δ f ∗ and δ g ∗ for every δ > f j and g i are invertible functions,therefore there exist vectors PPP and P that solve system (12). However, in general if PPP j s and P i s are solutions then f − j { δ f j ( PPP j ) } and g − i { δ g i ( P i ) } are also solutions.In order to narrow the class of functions f ( . ) and g ( . ), we invoke the following axiom which statesthat if the unit of measurement of the reference currency is multiplied by γ then the internationalaverage price of the commodity must be divided by γ . For example, if the reference currency is1 US dollar and if the international average price of wheat per tonne is US $125; then when thereference currency is changed to a 100 US dollar unit of currency then the international average12rice would be 1.25 units of reference currency (US $100 unit). Axiom of Change of Reference Currency Unit : This axiom simply states that if
PPP ∗ and P ∗ aresolutions to the multilateral system, then γ PPP ∗ and (1 /γ ) P ∗ should also be solutions for every γ > f ( . ) and g ( . ) that can be used in internationalcomparisons. Theorem-2 below establishes the functional forms for f and g that satisfy the axiomof units of reference currency unit. Theorem 2.
The axiom of change of reference currency unit is both necessary and su ffi cient for the functionsf j ( . ) and g i ( . ) to be of the form f j ( x ) = α j x ρ and g i ( x ) = β i x − ρ for any α j > , β i > and ρ ∈ R. Proof of Necessity : Note that having both [
PPP , P ] and [ γ PPP , (1 /γ ) P ] as solutions for any γ > f j ( PPP j ) , g i ( P i )] and [ f j ( γ PPP j ) , g i (1 /γ ) P i )] as solutions to equations in(12). On the other hand, according to Theorem-1 if [ f j ( PPP j ) , g i ( P i )] is a solution then [ δ f j ( PPP j ) , δ g i ( P i )]is also a solution for every δ >
0. Therefore, for every γ > δ > f j ( γ PPP j ) = δ f j ( PPP j ) j = , , ..., Mg i ( 1 γ P i ) = δ g i ( P i ) i = , , ..., N Since we are assuming this to be true for all possible solutions [
PPP , P ] > , we apply theseconditions to the case where PPP j = P i = f j ( γ ) = δ f j (1) ( j = , , ..., M ) g i ( 1 γ ) = δ g i (1) ( i = , , ..., N )Then the equations can be rewritten as: f j ( γ PPP j ) = f j ( γ ) f j (1) f j ( PPP j ) for j = , , ..., Mg i ( 1 γ P i ) = g i (1 /γ ) g i (1) g i ( P i ) for i = , , ..., N where f j ( γ ) and g i (1 /γ ) are continuous functions defined over R + . In general, each equation inthe above system is a special form of the 4th Pexiders functional equation (see e.g. Acz´el, 1966;Diewert, 2011). The non-trivial solution to this system of functional equations takes the form f j ( x ) = α j x ρ j and g i ( x ) = β i x ρ i for any ρ j , ρ i ∈ R and f j (1) = α j and g i (1) = β i . But note that Here we exclude trivial situations where either f j ( x ) = g i ( y ) = = f ( γ ) f (1) = .... = f M ( γ ) f M (1) = g (1 /γ ) g (1) = .... = g N (1 /γ ) g N (1) ⇒ γ ρ , = .... = γ ρ , M = γ − ρ , = .... = γ − ρ , N But since this is true for every γ >
0; we must have ρ , = .... = ρ , M = − ρ , = .... = − ρ , N = ρ . Proof of Su ffi ciency : The proof is trivial. We can use Theorem-1 to establish the existence conditions for the GK system and its generalizations.
Corollary 1 : A necessary and su ffi cient condition for existence and uniqueness of the GK index (3),generalized GK (4), EWGK and the system based on generalized means (5) is connectedness ofquantity matrix, q . Proof : In system (12), defining f j ( x ) = x , g ( x ) = x , c ij = p ij q ij and d ij = q ij leads to (3). Existence anduniqueness of generalized GK index (4) can be proved by defining f j ( x ) = x , g ( x ) = x , c ij = β j p ij q ij and d ij = β j q ij .To prove the result for system (5), define f j ( x ) = x ρ , g ( x ) = x ρ , c ij = β j p ij q ij and d ij = β j q ij .To prove the results for EWGK, define f j ( x ) = x , g ( x ) = x , c ij = w ij and d ij = w ij / p ij . Theorem-1 does not cover all multilateral systems of interest. Theorem-3 below extends Theorem-1to the case where the d ij s and c ij s are functions of P . As we saw in Section 3, Rao (1976) and Neary(2004) o ff er examples of such indexes. This specification makes the systems more complex andconditions for their existence and uniqueness more di ffi cult to establish. Theorem 3.
Consider the following general system of equations PPP j = N (cid:88) n = d nj ( P , p , q ) (cid:80) Nn = c nj ( P , p , q ) P n ( j = , ....., M ) (15a) P i = M (cid:88) m = c im ( P , p , q ) (cid:80) Mm = d im ( P , p , q ) 1 PPP m ( i = , ....., N ) (15b) Let d ij ( P , p , q ) ≥ and c ij ( P , p , q ) ≥ be continuous homogeneous functions of the same degree withrespect to P then(i) there is at least one non-negative solution with some positive elements (up to a positive scalarmultiple). The theorem can be written in terms of functions of P i s and PPP j s with conditions stated in Theorem-2 but to avoidcumbersome notation here we focus on f j ( x ) = x and g i ( x ) = / x . ii) there is at least one positive solution (up to a positive scalar multiple) if d ij ( P , p , q ) and c ij ( P , p , q ) are such that the vector function G defined below satisfies monotonicity and strong connectedness.(iii) there is a unique positive solution (up to a positive scalar multiple) if d ij ( P , p , q ) and c ij ( P , p , q ) are such that the vector function G satisfies monotonicity and indecomposibility.Proof : Substitute PPP j from (15a) into (15b) and define G as G i = M (cid:88) m = c im ( P , p , q ) N (cid:80) n = d nm ( P , p , q ) (cid:80) Nn = c nm ( P , p , q ) P nM (cid:80) m = d im ( P , p , q ) ( i = , ....., N )It is easy to see that G satisfies conditions (1) and (2) of the nonlinear Perron-Frobenius theorem and therefore the system G ( P ) = λ P has a non-negative solution (see e.g. Nikaido, 2016, pp 151 fora simple proof). If G also satisfies (3) and (4), there is at least one positive solution for the systemand if G satisfies indecomposibility the positive solution is unique.The next step is to establish that the eigenvalue associated with the solution is equal to one(i.e. λ ∗ = λ ∗ P ∗ i = M (cid:80) m = c im ( P ∗ , p , q ) N (cid:80) n = d nm ( P ∗ , p , q ) (cid:80) Nn = c nm ( P ∗ , p , q ) P ∗ nM (cid:80) m = d im ( P ∗ , p , q ) ( i = , ....., N )or λ ∗ (cid:18) M (cid:88) m = d im ( P ∗ , p , q ) (cid:19) P ∗ i = M (cid:88) m = c im ( P ∗ , p , q ) N (cid:88) n = d nm ( P ∗ , p , q ) (cid:80) Nn = c nm ( P ∗ , p , q ) P ∗ n ( i = , ....., N )Summing the equations over i we have λ ∗ N (cid:88) i = (cid:18) M (cid:88) m = d im ( P ∗ , p , q ) P ∗ i (cid:19) = N (cid:88) i = (cid:18) M (cid:88) m = c im ( P ∗ , p , q ) N (cid:88) n = d nm ( P ∗ , p , q ) (cid:80) Nn = c nm ( P ∗ , p , q ) P ∗ n (cid:19) = M (cid:88) m = (cid:18) N (cid:80) i = c im ( P ∗ , p , q ) (cid:80) Nn = c nm ( P ∗ , p , q ) N (cid:88) n = d nm ( P ∗ , p , q ) P ∗ n (cid:19) = M (cid:88) m = (cid:18) N (cid:88) n = d nm ( P ∗ , p , q ) P ∗ n (cid:19) see Appendix A for detailed discussion of eigenvalue theorems. Assuming that for each j there is at least one i for which c ij > i there is at least one j for which d ij > λ ∗ = q ij > ⇒ d ij > ⇒ c ij > G is indecomposable if q is connected but without knowing the exact form of c ij and d ij s or placing stringent conditions on these functions, we cannot prove monotonicity of G . Corollary 2 : The Neary system (10), has at least one non-negative solution. It has a unique positivesolution if q is connected and G satisfies monotonicity. Proof : In System (15) define c ij = p ij q ij and d ij = q ∗ ij where it is evident that the c ij are homogenousof degree zero and the d ij are homogenous of degree zero with respect to P since the q ∗ ij are derivedthrough (10c) . We now focus on a variant of Theorem-1 where the system cannot be turned into a linear systemin terms of g i and f j . This theorem can be used to establish existence and uniqueness of solutionsfor systems that are not covered by system (12) such as the multilateral index numbers proposedin Rao (1990), IDB, and the arithmetic index (8) described in Section 3. Theorem 4.
Suppose f j and g i are known non-negative, bijective functions defined over non-negativevalues andf j ( PPP j ) = N (cid:88) n = a nj c n g n ( P n ) ( j = , ....., M ) (16a) g i ( P i ) = M (cid:88) m = a im d m f m ( PPP m ) ( i = , ....., N ) (16b) with a ij ≥ , c i > and d j > , then compatibility and connectedness conditions (as defined in AppendixA) are necessary and su ffi cient for having a unique positive solution. It may be di ffi cult to show monotonicity for Neary (2004) as it includes q ∗ and q in the definitions. The Rao (1976) system satisfies monotonicity but the weights are not defined in the form specified in Theorem-3and it is not possible to prove λ ∗ = The existence of IDB system has been proved by Balk (1996) and Diewert (2013). roof : Substituting (16.b) into (16.a), we obtain f j = N (cid:88) n = a nj c nM (cid:80) m = a nm d m f m ( j = , ...., M )This is exactly the DAD Problem discussed in Appendix A where compatibility and connectednessconditions are necessary and su ffi cient for existence of a unique solution.In this general form, we cannot relate the solution of the system to connectedness of thecountries. However, there is a more interesting special case of this system where we can show thatthe connectedness of q is both necessary and su ffi cient for existence of a unique positive solution.This system covers the systems defined in equations (6), (7), (8) and (9) as special cases. Theorem 5.
Consider the following system of equations:f j = N (cid:88) n = d nj (cid:80) Nn = d nj e nj g n ( j = , ....., M ) (17a) g i = M (cid:88) m = d im (cid:80) Mj = m d im e im f m ( i = , ....., N ) (17b) where d ij > ⇔ e ij > ⇔ q ij > . Then connectedness of countries through quantity matrix q is bothnecessary and su ffi cient for uniqueness of the solutions (up to a positive scalar multiple).Proof : Note first that this system is a special case of (16) by defining a ij = d ij e ij ( (cid:80) Mm = d im )( (cid:80) Nn = d nj ) , c i = (cid:80) Mj = d ij and d j = (cid:80) Ni = d ij . Su ffi ciency : Define D = { d ij | i = , ..., N , j = , ..., M } , A = { a ij | i = , ..., N , j = , ..., M } and let I ⊂ { , ..., N } , J ⊂ { , ..., M } with A I c J c = where I c is complement of I and A IJ = { a ij | i ∈ I and j ∈ J } .Since d ij > ⇔ e ij > ⇔ q ij >
0, connectedness of q is equivalent to connectedness of A and D .To show that compatibility is automatically satisfied note that (cid:88) i ∈ I c c i = (cid:88) i ∈ I c (cid:88) j ∈ J c ∪ J d ij = (cid:88) i ∈ I c (cid:88) j ∈ J d ij since (cid:88) i ∈ I c (cid:88) j ∈ J c d ij = (cid:88) j ∈ J d j = (cid:88) i ∈ I c ∪ I (cid:88) j ∈ J d ij = (cid:88) i ∈ I c (cid:88) j ∈ J d ij + (cid:88) i ∈ I (cid:88) j ∈ J d ij ≥ (cid:88) i ∈ I c (cid:88) j ∈ J d ij (19)(18) and (19) ⇔ (cid:88) j ∈ J d j ≥ (cid:88) i ∈ I c c i
17t is also obvious that the inequality is strict if and only if A IJ (cid:44) (or D IJ (cid:44) ) which provescompatibility. To illustrate the above argument, consider the following decomposition of D afterappropriate re-ordering of rows and columns:Since D I c J c = , the sum of the elements in D I c J i.e. (cid:80) i ∈ I c (cid:80) j ∈ J d ij is equal to (cid:80) i ∈ I c c i , but (cid:80) j ∈ J d j issum of the elements in D I c J and D IJ . Necessity : According to the DAD theorem, a unique positive solution to (17) ⇔ compatibility andconnectedness of A = { a ij | i = , ..., N , j = , ..., M } ⇔ connectedness of D ⇔ connectedness of q dueto the assumption d ij > ⇔ q ij > γ > PPP ∗ , P ∗ ] and [ γ PPP ∗ , (1 /γ ) P ∗ ] are solutions to thesystem (16), is imposed then using arguments similar to those in Theorem-3 above, the functionalforms used in the system must be set as f j ( x ) = α j x ρ and g i ( x ) = β i x ρ for any α j > β i > ρ ∈ R . Corollary 3 : If matrix q is connected, systems (6), (7), (8) and (9) have unique positive solutions.We consider system (A.1) which encompasses all the others. Note that if in (17) we define f j ( x ) = g i ( x ) = x ρ and d ij = w ij , e ij = p ρ ij , c i = (cid:80) Mj = w ij and d j = (cid:80) Ni = w ij =
5. Conclusion
This paper provides general theorems for establishing existence and uniqueness of positivesolutions to multilateral index number systems that make use of the twin concepts of internationalaverage prices and purchasing power parities in the context of international comparisons of pricesand real incomes. The main result indicates that connectedness of the matrix of quantities orequivalently connectedness of the associated quantity-adjacent graph is in general both necessaryand su ffi cient for existence of a unique positive solution. The theorems proved in the paperare general and powerful enough to prove the existence and uniqueness of solutions not onlyto the currently used system of index numbers but also systems that may come into vogue inthe future. While simple connectedness guarantees the existence of solutions and therefore theviability of most of the multilateral index number systems, the strength of connectedness couldhave implications for the reliability of the indexes which are being studied in our other works.18 eferences Acz´el, J. (1966).
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Appendix A. Mathematical Toolkit for Existence Theorems
Appendix A.1. Connectedness, Graphs and Irreducibility
In this section, we present the notions of connectedness across countries, (strong) connectednessof graphs, and irreducibility and their relationships to each other.
Connectedness of Matrix q : The quantity matrix q = { q ij } , an N × M matrix of quantities where q ij is the quantity of i -th commodity consumed in j -th country, is said to be connected if the set of allcountries cannot be split into two or more disjoint subsets such that there are no commodities thatare commonly consumed across the two groups.More formally, a non-negative matrix of quantities q is said to be connected if for every nonemptyproper subset of countries J ⊂ { , , ..., M } (cid:44) ∅ there exists at least one country j ∈ J , one country l (cid:60) J and one commodity k ∈ , ...., N such that both q kj > q kl >
0. An equivalentmathematical definition of connectedness is that for every non-empty proper subset I ⊂ { , , ..., N } and J ⊂ { , , ..., M } , with I c and J c as their complements, q I c J c = q IJ (cid:44) q IJ = { q ij , i ∈ I and j ∈ J } .Connectedness of matrix q is critical in international comparisons. If q is disconnected, then20ountries can be divided into two groups with no common items of consumption. In such a case,there is no basis for making price comparisons. Thus, connectedness of q seems necessary. Inthis paper, we show that this condition is both necessary and su ffi cient for uniqueness of severalclasses of multilateral index number systems. It is possible to give this notion of connectedness agraph theoretic interpretation. Quantity-Adjacent Graph : Let G q represent a graph associated with a given quantity matrix q with countries as vertices of the graph. Two vertices j and k are connected by an edge if thereexists a commodity i ∈ { , , ..., N } such that q ij > q ik > Connected Graph : A graph, G q is said to be connected if for any pair of countries j and k , thereexists a sequence of countries { j , ..., j k } ∈ { , , ..., } such that each consecutive pair of countries inthe sequence are connected by an edge.It has been established that the graph G q associated with a quantity matrix q is connected if andonly if the quantity matrix q is connected . Connected Matrices and Adjacent Graphs-An illustration : The notions of connectedness of thematrix of quantities and the connectedness of the graphs associated with quantity matrices arecentral to the existence theorems proved in the paper. We illustrate these notions using threeexamples where we consider four countries and four commodities. The quantity matrix, q , in thiscase is 4 × q =
10 5 100 256 3 75 3580 25 250 1258 6 35 40
In this case, it is easy to see that the matrix q is connected. The adjacent graph is a complete graphwhere each country is directly connected with every other country. This is a case where there iscomplete connectedness.The second example is where the first country consumes all four commodities whereas theother three countries consume only one or two of the commodities. The quantity matrix and theadjacent graphs in this case are of the following form.It is easy to check that the matrix q here is connected. We can see that Country 1 is connected tothe rest of the countries. However, there are no direct links between Countries 2, 3 and 4. In thiscase the graph is connected with no cycles this means between any two countries there is only21 =
10 5 0 06 0 75 080 25 0 08 0 0 40 one chained path connecting the two countries. For example, Countries 3 and 2 are connectedthrough Country 1. This type of graph is referred to as a spanning tree. Here there is connectivityonly through Country 1 but it is still su ffi cient for connectedness of the matrix q .Finally, we consider an example where the four countries are divided into two separate groups,Countries 1 & 2 and Countries 3 & 4. These groups have no commodities commonly consumed.Therefore, the matrix q is not a connected matrix and it consists of two sets of matrices of lowerdimensions (two) each of which are connected but the full matrix is not connected. q =
10 5 0 06 3 0 00 0 250 1250 0 35 40
In this example, countries 1 & 2 have commodities (1 and 2) commonly consumed whereascountries 3 & 4 consume only commodities 3 and 4. In this case, the adjacent graph is disconnected.Countries 3 &4 have no connections with Countries 1&2. In this case there is no basis formultilateral comparisons involving all four countries.There are well-known algorithms such as the Breadth-First Search (BFS) which can be usedfor checking connectedness of a graph (and as a result q ). There are also very simple su ffi cientconditions that guarantee connectedness: ( i ) there is one commodity that every country consumes;( ii ) there is one country that consumes all commodities. These conditions have been used by e.g.Balk (2008) in his proofs of existence.We now introduce several other definitions that are closely related to the connectedness of q andthe existence of multilateral index numbers. Irreducibility of a Square Matrix : A non-negative M × M matrix A = { a ij } is said to be irreducibleif and only if for any proper subset J ⊂ { , , ..., M } (cid:44) ∅ there exists at least one j ∈ J and i (cid:60) J suchthat a ij > Strong Connectedness of the Directed Graph Associated with Matrix A : Let the directed graph(digraph) associated to matrix A , Θ ( A ) be a directed graph (digraph) with vertices of 1 , ..., M andan edge from i to j if and only if a ij >
0. This digraph is said to be strongly connected if for22ny pair of vertices of Θ ( A ), there is a path that connects them. It is a well-known result thatirreducibility of a matrix and strong connectedness of its digraph are equivalent. The followingdefinition generalizes the strong connectedness to nonlinear functions. Strong Connectedness of Digraph Associated with a Function G : For a function G ( x ) : R M ++ → R M ++ ,let Θ ( G ) be the digraph with vertices of 1 , ..., M and an edge from i to j if and only if lim x j →∞ G i ( x { j } ) = ∞ (where vector x { j } refers to a vector that has all elements equal to one except for its j th element).Then G is strongly connected if for any pair of vertices, there is a path that connects them. It iseasy to see that when G ( x ) is linear (i.e. G ( x ) = Ax ), this is equivalent to strong connectedness ofmatrix A . As an example, consider the following nonlinear function G ( x ) = min (cid:110) √ x x , √ x x (cid:111) max (cid:110) √ x x , √ x x (cid:111) max { x , x } The digraph associated to G is strongly connected as the following figure (constructed based onthe above definition) shows Indecomposibility of a Function : For any pair of vectors x and y with x ≥ y ≥ define anonempty proper subset Ω = { j | x j > y j } ⊂ , ..., M . Function G is indecomposable if there exists k (cid:60) Ω such that G k ( x , ... x M ) (cid:44) G k ( y , ... y M ). This concept of indecomposibility is equivalent to theirreducibility of a matrix if G is linear. Indecomposibility is generally a stronger condition thanstrong connectedness. Appendix A.2. Nonlinear Perron-Frobenius Theorems
We start with a general multilateral system of M + N equations of the following form: P i = H i ( PPP , p , q ) ( i = , ....., N ) PPP j = H j ( P , p , q ) ( j = , ....., M )where PPP and P are respectively M × N × P i into the equation for PPP j leading to a homogeneous See Gaubert and Gunawardena (2004) for further information on strong connectedness, more examples and itsrelevance to nonlinear eigenvalue theorems. M equations of the form: PPP j = H j ( H i ( PPP , p , q ) , p , q ) (A.1) = G j ( PPP , p , q ) ( j = , ....., M )This means that the set of equations in (A.1) are such that each PPP j is expressed as a function ofobserved price and quantity data as well as all PPP j s. Solving equations in (A.1) is equivalent tosolving the following general system of homogeneous equations: x j = G j ( x , x , ..., x M ) j = , , ..., M or in vector form x = G ( x ) (A.2)To prove the existence of solutions to (A.1) or (A.2), Perron-Frobenius (or eigenvalue) theoremsare the basic tools. There are a host of such theorems which give conditions for existence of asolution under various situations (see e.g. the book by Lemmens & Nussbaum, 2012, for a reviewof eigenvalue theorems). The following theorem is one of the general eigenvalue theorems due toGaubert & Gunawardena (2004) and Morishima (1964). Nonlinear Perron-Frobenius Theorem : Let G j ( x , x , ..., x M ) for j = , ..., M satisfy the followingconditions:1. G ( x ) is a function R M ++ → R M ++
2. Homogeneity: functions G j ( x ) for j = , ..., M are homogeneous of degree one in x
3. Monotonicity: for all x ≥ y , G j ( x ) ≥ G j ( y ) for j = , ...., M
4. Strong connectedness of the digraph associated with G ( x )5. Indecomposibility of function G ( x )and consider the system of equations G j ( x , ....., x M ) = λ x j j = , ...., M (A.3)(i) under conditions (1) and (2), there is at least one non-negative x ∗ (up to a positive scalarmultiple) and λ ∗ that satisfy the equation.(ii) under conditions (1),(2), (3) and (4), there is at least one positive x ∗ (up to a positive scalarmultiple) and λ ∗ that satisfy the equation.(iii) under conditions (1),(2), (3) and (5), there is a unique positive x ∗ (up to a positive scalarmultiple) and λ ∗ that satisfy the equation.A few remarks on the relevance of nonlinear eigenvalue theorems are: • Eigenvalue theorems play an important role in establishing conditions for the existence ofsolutions to the multilateral index number systems examined in this paper. It is su ffi cient if24he two functions, H i and H j , are such that the function G obtained through (A.1) satisfies theconditions stated in the nonlinear eigenvalue theorems. • In the context of multilateral systems, we need to find a solution that satisfies equation (A.1),i.e. x = G ( x ). For this to hold, we need to also establish that λ ∗ = H i and H j which we may refer to as compatibility conditions. • In all eigenvalue theorems, it is assumed that the function G ( x ) is continuous, homogenous andmonotone, conditions which are easy to check but which are often not su ffi cient for a positivesolution. A further assumption such as 4 or 5 above (which could be di ff erent across eigenvaluetheorems) is often required. In this paper, we provide easily verifiable conditions to check suchconditions in terms of the quantity data matrix. • A well-known eigenvalue theorem in mathematical economics proved by Morishima (1964,pp 195-199) and Nikaido (2016, pp 149-161) gives a su ffi cient condition for existence of aunique positive solution to the above problem by replacing assumption (4) above with (5) i.e. indecomposibility of function G ( x ). The linear version of this theorem (i.e. G ( x ) = Ax ) is thewell-known Perron-Frobenius theorem (Gantmacher & Brenner, 2005, Chapter 3) where therequired su ffi cient condition for a unique positive solution is irreducibility of the non-negativematrix A .To prove existence of some of the multilateral systems, we need to appeal to another nonlineareigenvalue theorem known as the DAD theorem (see e.g. Chapter 7 of Lemmens & Nussbaum,2012, or Menon & Schneider (1969)). The DAD theorem entails a triplet { A , c , d } where A = { a ij } is an N × M non-negative matrix, c is an N × d is an M × D = diag { [ δ , ..., δ N ] } and D = diag { [ ∂ , ..., ∂ M ] } with D AD having row sumsequal to d and column sums equal to c . Note that D AD = δ a ∂ · · · δ a M ∂ M ... . . . ...δ N a N ∂ · · · δ N a NM ∂ M Since columns sum to c and rows sums to d , we must have δ M (cid:80) m = a m ∂ m = c ... ...δ N M (cid:80) m = a Nm ∂ m = c N & ∂ N (cid:88) n = a n δ n = d ... ...∂ M N (cid:88) n = a nM δ n = d M M + N equations in M + N unknowns. Substituting δ i s in the set of equationson the right from equations on the left and defining x j = d j /∂ j , it is easy to see that this problemis equivalent to the following eigenvalue problem: DAD Eigenvalue Theorem : Consider the following system of equations: N (cid:88) n = a nj c nM (cid:80) m = a nm d m x m = x j for j = , ...., M or A (cid:48) cA (cid:18) dx (cid:19) = x where in the equation on the right, vector divisions are element by element. Then a necessaryand su ffi cient condition for existence of at least one positive solution is the compatibility condition (defined below). A further condition of connectedness of A provides a necessary and su ffi cientcondition for uniqueness. Compatibility Condition : For every I ⊆ { , ..., N } and J ⊆ { , ..., M } define I c and J c as complements ofthese sets. The compatibility condition implies that for every A I c J c = , the inequality (cid:80) i ∈ I c c i ≤ (cid:80) j ∈ J d j holds, and the inequality is strict if and only if A IJ (cid:44) . Connectedness of A : For every I ⊂ { , ..., N } , J ⊂ { , ..., M } , A I c J c = ⇒ A IJ (cid:44) .To see why compatibility is required, note that, with A I c J c = , after appropriate row and columnpermutations D AD can be written aswhere (cid:80) i ∈ I c c i is the sum of the elements in the horizontal dashed rectangle and (cid:80) j ∈ J d j is the sum ofelements in the red vertical rectangle. Excluding the zeros, the horizontal dashed rectangle is asubset of the red vertical rectangle. Therefore, in order to have a compatible set of equations werequire the compatibility condition. Connectedness is required for uniqueness because otherwisethe system can be decomposed into two unrelated subsystems. Appendix A.3. Relationships between the Concepts
Assuming that the function G satisfies conditions (1) to (3) stated in the nonlinear eigenvaluetheorem, the link between various concepts and eigenvalue theorems can be summarized asfollows: • Irreducibility of a square matrix (equivalently, strong connectedness of its digraph) is a su ffi cientcondition for existence of a unique positive solution to a linear eigenvalue problem (Perron-Frobenius26heorem). • Strong connectedness of the digraph of a function is a su ffi cient condition for existence of atleast one solution to the nonlinear eigenvalue problem. • Indecomposibility is a su ffi cient condition for existence of a unique positive solution for thenonlinear eigenvalue problem. • Indecopmposibility is a stronger condition than strong connectedness but they are both equivalentto irreducibility if the function is linear. • Compatibility and connectedness of matrix A together provide both necessary and su ffiffi