A mean field game model of firm--level innovation
AA MEAN FIELD GAME MODEL OF FIRM–LEVEL INNOVATION
MATT BARKER ∗ , PIERRE DEGOND † , RALF MARTIN ‡ , AND
MIRABELLE MU ˆULS
Abstract.
Knowledge spillovers occur when a firm researches a new technology and that tech-nology is adapted or adopted by another firm, resulting in a social value of the technology that islarger than the initially predicted private value. As a result, firms systematically under–invest inresearch compared with the socially optimal investment strategy. Understanding the level of under–investment, as well as policies to correct it, is an area of active economic research. In this paper,we develop a new model of spillovers, taking inspiration from the available microeconomic data. Weprove existence and uniqueness of solutions to the model, and we conduct some initial simulationsto understand how indirect spillovers contribute to the productivity of a sector.
AMS subject classifications.
Key words. mean field games, knowledge spillovers, innovation model
1. Introduction.
When a business invests in research and development (R&D),such strategy only takes into account how a potential innovation may increase theinvesting company’s private value. However, other businesses may utilise innovationsmade by the original investing company to increase their own profits. This is known ineconomic literature as the knowledge spillover effect. By only considering its privatereturn, businesses systematically undervalue their own innovations and hence under–invest in R&D, compared with the socially optimal investment level. To counteractthe under–investment, governments introduce R&D subsidy policies for certain sectorsof the economy. In order to effectively allocate such subsidies, it is therefore importantto understand the extent of under–investment and how it varies between sectors.To understand the spillover effect we develop a mean field game (MFG) model offirms distributed heterogeneously between sectors and according to their productivitylevel, taking into account their microscopic behaviour. From a microeconomic per-spective, the size of knowledge spillovers can be inferred from the network of patentcitations [15]. When an industrial technology is developed, it often gets patented. Aspart of the patent any previous technology that has been used must be cited. Thisresults in a network of patent citations, where each citation can be used as a proxyfor a spillover from one technology to another, so spillover sizes can be evaluated [11].In the model we develop, sectors are connected by a graph that is informed by andcan be calibrated to the microeconomic network of patent citation data.A first model of knowledge spillovers, by Cohen and Levinthal [10], consideredthe stock of knowledge of a firm to depend on the amount of investment in R&Dof that firm and the total amount of investment by all other firms, through a meanfield–type interaction. Only an initial analysis of the model was conducted in [10]. Alater model, acknowledged in Section 13.2 of [1], started from a macroscopic perspec- ∗ Science and Solutions for a Changing Planet DTP, and the Department of Mathematics, ImperialCollege London, London, UK ([email protected]). This work was supported by the NaturalEnvironment Research Council [grant number NE/L002515/1] † Department of Mathematics, Imperial College London, London, UK ([email protected]).PD acknowledges support by the Engineering and Physical Sciences Research Council (EPSRC)under grants no. EP/M006883/1 and by the Royal Society and the Wolfson Foundation through aRoyal Society Wolfson Research Merit Award no. WM130048. PD’s affiliation from 01/01/2021 is”Institut de Math´ematiques de Toulouse, CNRS & Universit´e Paul Sabatier, 31062 TOULOUSE,France. MB is supported by the Natural Environment Research Council ‡ Imperial Business School, Imperial College London, London, UK ([email protected],[email protected]). 1 a r X i v : . [ m a t h . O C ] J a n M. BARKER, P. DEGOND, R. MARTIN, M. MU ˆULS tive, hence only the aggregate knowledge of the entire economy was considered andspillovers were assumed to increase the aggregate uniformly. This does not explainhow spillovers heterogeneously affect firms. Similar models have also been used tostudy entrepreneurship and intellectual property rights, such as in [2]. There hasbeen particularly extensive research of knowledge spillovers in cross–country models.In such models, a country’s own output is aggregated and the knowledge level increasesat a rate that depends on the leading country’s knowledge level. However, simplifyingassumptions are made that may affect their accuracy, such as in [12], where interac-tions take place in a discrete time setting, or [3, 16] where the interactions betweenfirms was described only through the evolution of aggregate quantities. In this paper,we use an MFG model to both increase the complexity of the description of firms,and to link firm–level evolution directly to microeconomic data for spillover sizes.There have been several other papers focussing on MFG–type models of knowledgespillovers, see [7, 21]. The Boltzmann model studied in both papers doeas not con-sider how innovation among firms evolves, nor did it incorporate the microeconomicdata related to patent citations in its formulation. As a result, the model studied inthis paper can give greater insight into firm–level dynamics.In this paper, we analyse a stationary MFG model describing the spillover effect.The MFG model describes the long–term behaviour of firms with full anticipationof the future. MFGs were described mathematically by Lasry and Lions [18, 19],and simultaneously by Huang, Caines and Malham´e [17] and they build on the workof Aumann and related authors on anonymous games [5, 22]. The novelties of thesystem we develop are, first, that the distribution dependence enters into the driftterm rather than in the cost functional and, second, that we are considering morethan one population of agents. Therefore our MFG model can be classed as a multi–population MFG with a non–separable Hamiltonian. There has been some work inboth multi–population MFGs (see [9]) and MFG models with non–separable Hamil-tonians (see [4, 14]). However, we are aware of no literature for models that displayboth characteristics, so although our model is one dimensional, its interest reachesbeyond this setting. As a result of the novelty of our model, the techniques we useto prove existence and uniqueness are also novel. However, they rely heavily on theability to write a stationary Fokker–Planck equation in the form of an exponential.This characteristic has previously been used in [6] to prove existence and uniquenessin MFG and BRS models in a slightly different framework.The paper is organised as follows. In Section 2, we develop the spillover model bydescribing firm behaviour at a microscopic level and formally deriving the mean fieldlimit. In section 3, we describe the MFG problem and prove existence of solutions toit. We also show uniqueness of such solutions holds, provided the coupling strengthbetween sectors is small enough. In Section 4, we provide some deeper insights intothe effects of the modelling parameters, through numerical simulations. The first sim-ulations show how parameters describing effects unrelated to spillovers (for examplethe discount factor, the noise level and the labour efficiency) change the MFG model.Our second group of simulations demonstrate the effect of the spillover network onthe model. The spillover network is a sector–level network that aggregates the patentcitation network. We show that the effect of a spillover on any sector is a result of allpaths to that sector in the associated network, and not just the immediate connectionsbetween sectors, which is contrary to the current economic state of the art. Finally,in Section 5, we briefly discuss future research prospects for the model, including howwe intend to apply the model to economic questions relevant to R&D subsidy policy.
MEAN FIELD GAME OF INNOVATION
2. Model development.2.1. The microscopic model.Firms.
Assume there are L sectors within the economy, and in sector (cid:96) there are N (cid:96) firms. We assume firm i in group (cid:96) has s (cid:96),(cid:96) (cid:48) i,j links with firm j in group (cid:96) (cid:48) , where s (cid:96),(cid:96) (cid:48) i,j is a random variable, taking a value s ∈ N with probability N (cid:96) (cid:48) p ( (cid:96), (cid:96) (cid:48) , s ). The i th firm in sector (cid:96) has a productivity level Z (cid:96),i ∈ Ω = (0 , ¯ z ), which increases as aresult of employing labour h (cid:96),i or due to knowledge spillovers from firms that theyare linked with. The productivity dynamics are also affected by noise with strength σ ∈ (0 , ∞ ). As a result, Z (cid:96),i evolves according to the following SDE dZ (cid:96),i ( t ) = ( h (cid:96),i ( t )) γ + 1 N L (cid:88) (cid:96) (cid:48) =1 N (cid:96) (cid:48) (cid:88) j =1 s (cid:96),(cid:96) (cid:48) i,j Z (cid:96) (cid:48) ,j ( t ) dt + σdB (cid:96),i ( t )(2.1a) L ( Z (cid:96),i (0)) = m (cid:96) , (2.1b)where N = (cid:80) L(cid:96) =1 N (cid:96) , B (cid:96),i is an independent Brownian motion with reflection atboundaries 0 and ¯ z , and γ ∈ (0 ,
1) represents the inefficiency in converting one unitof labour to one unit of knowledge. In the initial condition (2.1b), L ( Z (cid:96),i (0)) denotesthe law of the random variable Z (cid:96),i (0) and m (cid:96) is an initial distribution, which may bedifferent for each sector. We assume firms produce a quantity of differentiated goodat a rate q (cid:96),i according to the production function(2.2) q (cid:96),i = Z (cid:96),i . Each firm sells their product at a market–determined price r (cid:96),i and maximises theirprofit subject to the other firms’ decisions. Each agent’s profit functional is given by(2.3) J (cid:96),i ( h ) = E (cid:20)(cid:90) ∞ ( r (cid:96),i ( t ) q (cid:96),i ( t ) − wh (cid:96),i ( t )) e − ρt dt (cid:21) , where h = (cid:16) ( h (cid:96),i ) N (cid:96) i =1 (cid:17) L(cid:96) =1 . The wage, w , and the discount rate, ρ , are given constants. Consumers.
Assume there is a representative consumer with preferences givenby Q = N (cid:80) L(cid:96) =1 (cid:80) N (cid:96) i =1 q α(cid:96),i , the Dixit–Stiglitz constant elasticity of substitution (CES)form, and with average income Y . The value α ∈ (0 ,
1) is related to the elasticityof substitution. The demand for each variety can be found by maximising Q un-der the budget constraint that average expenditure is equal to average income, i.e. N (cid:80) L(cid:96) =1 (cid:80) N (cid:96) i =1 r (cid:96),i q (cid:96),i = Y . This gives(2.4) q (cid:96),i = Br α − (cid:96),i , B = Y R α − α , R = (cid:32) N L (cid:88) (cid:96) =1 N (cid:96) (cid:88) i =1 r αα − (cid:96),i (cid:33) α − α . For the purposes of the firm–level optimisation problem, we assume R is fixed, inthat it can’t be changed by any individual firm — this becomes true as N (cid:96) → ∞ foreach (cid:96) . For the mathematical analysis we assume B to be a fixed constant, whichsimplifies matters and enhances the model’s relevant features. Later in the numericalsimulations B will be determined as the solution to a fixed point problem, whichendogenises the price formation through the interaction between firms and consumers. M. BARKER, P. DEGOND, R. MARTIN, M. MU ˆULS
Firm profits revisited.
Now, the profit functional (2.3) can be rewritten as J (cid:96),i ( h ) = E (cid:104)(cid:82) ∞ (cid:16) ( Z (cid:96),i ( t )) α B α − − wh (cid:96),i ( t ) (cid:17) e − ρt dt (cid:105) , using the consumer behaviour (2.4)and the production function (2.2). When there are large numbers of firms in each sector,the microscopic model developed in Section 2.1 can become intractable. Instead, weassume the number of firms in each sector, N (cid:96) , goes to infinity while N (cid:96) N → A (cid:96) forsome A (cid:96) ∈ (0 , (cid:96) . In order toderive the limiting mean field model, we first define the empirical distributions foreach sector (cid:96) = 1 , . . . , L by m N (cid:96) (cid:96) = N (cid:96) (cid:80) N (cid:96) i =1 δ Z (cid:96),i , where δ Z (cid:96),i is a Dirac delta at thepoint Z (cid:96),i . We can then rewrite the dynamics (2.1) using m N (cid:96) (cid:96) as dZ (cid:96),i ( t ) = ( h (cid:96),i ( t )) γ + L (cid:88) (cid:96) (cid:48) =1 ( N (cid:96) (cid:48) ) N (cid:90) Ω z (cid:48) dm N (cid:96) (cid:48) (cid:96) (cid:48) ( z (cid:48) , t ) 1 N (cid:96) (cid:48) N (cid:96) (cid:48) (cid:88) j =1 s (cid:96),(cid:96) (cid:48) i,j dt + σdB (cid:96),i ( t ) L ( Z (cid:96),i (0)) = m (cid:96) , Assuming m N (cid:96) (cid:96) has a limit, m (cid:96) , as N (cid:96) → ∞ then, in the limiting model, a representa-tive firm in sector (cid:96) evolves according to the SDE dZ h,m(cid:96) ( t ) = (cid:32) ( h (cid:96) ( t )) γ + L (cid:88) (cid:96) (cid:48) =1 A (cid:96) (cid:48) p ( (cid:96), (cid:96) (cid:48) ) (cid:90) Ω z (cid:48) dm (cid:96) (cid:48) ( z (cid:48) , t ) (cid:33) dt + σdB (cid:96) ( t )(2.5a) L (cid:16) Z h,m(cid:96) (0) (cid:17) = m (cid:96) , (2.5b)where, by the law of large numbers, p ( (cid:96), (cid:96) (cid:48) ) = (cid:80) ∞ s =0 p ( (cid:96), (cid:96) (cid:48) , s ). The correspondingprofit functional is(2.6) J (cid:96) ( h ; m ) = E (cid:90) ∞ (cid:16) Z h,m(cid:96) ( t ) (cid:17) α B α − − wh (cid:96) ( t ) e − ρt dt . If all firms act in the same way as the representative firm, then the distribution of firmswith respect to productivity level is given by a system of L Fokker–Planck equations ∂ t m (cid:96) = − ∂ z (cid:34)(cid:32) ( h (cid:96) ) γ + L (cid:88) (cid:96) (cid:48) =1 A (cid:96) (cid:48) p ( (cid:96), (cid:96) (cid:48) ) (cid:90) Ω z (cid:48) dm (cid:96) (cid:48) ( z (cid:48) , t ) (cid:33) m (cid:96) (cid:35) + σ ∂ zz m (cid:96) (2.7a) − (cid:32) ( h (cid:96) ) γ + L (cid:88) (cid:96) (cid:48) =1 A (cid:96) (cid:48) p ( (cid:96), (cid:96) (cid:48) ) (cid:90) Ω z (cid:48) dm (cid:96) (cid:48) ( z (cid:48) , t ) (cid:33) m (cid:96) + σ ∂ z m (cid:96) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z =0 , ¯ z = 0(2.7b) m (cid:96) ( z,
0) = m (cid:96) ( z ) . (2.7c)
3. The MFG model.3.1. Problem formulation.
The MFG problem is related to the search forNash equilibria in the optimisation of the profit functional (2.6) while agents evolveaccording to the dynamics (2.5).
Definition
The MFG problem is to find a pair ( h ∗ , m ∗ ) , where h ∗ = ( h ∗ (cid:96) ) L(cid:96) =1 is a sequence of controls and m ∗ = ( m ∗ (cid:96) ) L(cid:96) =1 is a sequence of probability distributions MEAN FIELD GAME OF INNOVATION on ¯Ω , such that for any other sequence of controls h and every (cid:96)J (cid:96) ( h ∗ (cid:96) , m ∗ ) ≥ J (cid:96) ( h (cid:96) , m ∗ )(3.1a) and m ∗ (cid:96) = L (cid:16) Z h ∗ ,m ∗ (cid:96) (cid:17) . (3.1b) Such a distribution is called an MFG equilibrium.
To find an MFG equilibrium we first describe the Hamilton–Jacobi–Bellman(HJB) PDE related to the optimisation part of the problem (3.1a). Then we couplethe HJB PDE to the Fokker–Planck PDE (2.7) to solve the consistency part (3.1b).We start by defining L Hamiltonians H (cid:96) : Ω × (cid:0) H (Ω) (cid:1) L × R → R , for (cid:96) = 1 , . . . L as(3.2) H (cid:96) ( z, m, λ ) = sup h ≥ (cid:32) h γ + L (cid:88) (cid:96) (cid:48) =1 A (cid:96) (cid:48) p ( (cid:96), (cid:96) (cid:48) ) (cid:90) Ω z (cid:48) m (cid:96) (cid:48) ( z (cid:48) ) dz (cid:48) (cid:33) λ + z α B α − − wh = (1 − γ ) (cid:16) γw (cid:17) γ − γ max(0 , λ ) − γ + λ L (cid:88) (cid:96) (cid:48) =1 A (cid:96) (cid:48) p ( (cid:96), (cid:96) (cid:48) ) (cid:90) Ω z (cid:48) m (cid:96) (cid:48) ( z (cid:48) ) dz (cid:48) + z α B α − , where z ∈ Ω is productivity, m = ( m (cid:96) ) L(cid:96) =1 is a distribution of firms in each sector and λ is an adjoint variable. The optimal control is given by h ∗ (cid:96) = (cid:0) γw max(0 , λ ) (cid:1) − γ , for (cid:96) = 1 , . . . , L . Then we define the running profit V (cid:96) ( z, t ), for (cid:96) = 1 , . . . , L , by(3.3) V (cid:96) ( z, t ) = sup h (cid:96) E (cid:20) (cid:90) ∞ t (cid:18) ( Z (cid:96) ( s )) α B α − − wh (cid:96) ( z ) (cid:19) e − ρ ( s − t ) ds (cid:12)(cid:12)(cid:12)(cid:12) Z (cid:96) ( t ) = z (cid:21) , where Z (cid:96) ( s ) follows (2.5). If we let the equilibrium distribution be given by m (cid:96) (for (cid:96) = 1 , . . . , L ), then the MFG PDE system is stationary and given by V (cid:96) ∈ H (Ω)(3.4a) m (cid:96) ∈ H (Ω)(3.4b) − σ V (cid:48)(cid:48) (cid:96) + ρV (cid:96) − H (cid:96) ( z, m, V (cid:48) (cid:96) ) = 0(3.4c) − σ m (cid:48)(cid:48) (cid:96) + ( ∂ λ H (cid:96) ( z, m, V (cid:48) (cid:96) ) m (cid:96) ) (cid:48) = 0(3.4d) V (cid:48) (cid:96) | z =0 , ¯ z = 0(3.4e) − σ m (cid:48) (cid:96) + ∂ λ H (cid:96) ( z, m, m (cid:96) (cid:12)(cid:12)(cid:12)(cid:12) z =0 , ¯ z = 0(3.4f) (cid:90) Ω m (cid:96) ( z ) dz = 1 . (3.4g)It can be shown, using either the dynamic programming principle (c.f [25]) or thestochastic maximum principle (c.f. [8]), that V (cid:96) ( z ), as defined by (3.3), satisfies theHJB equation (3.4c), (3.4e). The Fokker–Planck system (3.4d), (3.4f), (3.4g) comesfrom the distribution in the previous section (2.7) and the consistency condition (3.1b). Definition
A solution to the innovation MFG model (3.4) is defined to bea tuple ( m, V ) = ( m , . . . , m L , V , . . . , V L ) such that m (cid:96) : Ω → (0 , ∞ ) , V (cid:96) : Ω → R satisfy (3.4) in the weak sense for each (cid:96) = 1 , . . . , L . M. BARKER, P. DEGOND, R. MARTIN, M. MU ˆULS
Theorem
There exists a solution ( m, V ) ∈ (cid:2) C (Ω) ∩ C (cid:0) ¯Ω (cid:1)(cid:3) L to (3.4) .Furthermore, if (cid:80) L(cid:96) (cid:48) =1 A (cid:96) (cid:48) p ( (cid:96), (cid:96) (cid:48) ) is small enough for every (cid:96) = 1 , . . . L , then the solu-tion is unique.Proof outline. As noted in the introduction, this proof is based on the proof ofexistence and uniqueness in [6]. The proof presented here has some technical differ-ences compared with the one in [6], hence it is reproduced in full. However, it followsa similar framework and so we do not claim the proof to be new. First, for k ∈ [0 , ∞ )we introduce an auxiliary system of PDEs defined by V k ∈ H (Ω)(3.5a) − σ (cid:0) V k (cid:1) (cid:48)(cid:48) + ρV k − H k (cid:16) z, (cid:0) V k (cid:1) (cid:48) (cid:17) = 0(3.5b) (cid:0) V k (cid:1) (cid:48) (cid:12)(cid:12)(cid:12) z =0 , ¯ z = 0 , (3.5c) m k ∈ H (Ω)(3.6a) − σ (cid:0) m k (cid:1) (cid:48)(cid:48) + (cid:18)(cid:20)(cid:16) γw max(0 , (cid:0) V k (cid:1) (cid:48) ) (cid:17) γ − γ + k (cid:21) m k (cid:19) (cid:48) = 0(3.6b) − σ (cid:0) m k (cid:1) (cid:48) + km k (cid:12)(cid:12)(cid:12)(cid:12) z =0 , ¯ z = 0(3.6c) (cid:90) Ω m k ( z ) dz = 1 , (3.6d)where H k ( z, λ ) = (1 − γ ) (cid:0) γw (cid:1) γ − γ (max (0 , λ )) − γ + kλ + z α B α − . We use a modifiedversion of upper and lower solutions (c.f. [23]) to prove existence and uniquenessof a weak solution V k to (3.5) for any k ∈ [0 , ∞ ), and use elliptic regularity the-ory to show V k ∈ C (Ω) ∩ C (cid:0) ¯Ω (cid:1) . Next we define m k = (cid:107) ¯ m k (cid:107) ¯ m k , where ¯ m k = e σ (cid:18) kz + (cid:82) z ( γw max ( , ( V k ) (cid:48) )) γ − γ dy (cid:19) and (cid:13)(cid:13) ¯ m k (cid:13)(cid:13) = (cid:82) Ω ¯ m k dz , for k ∈ [0 , ∞ ). We provethat m k ∈ C (Ω) ∩ C (cid:0) ¯Ω (cid:1) and that m k is the unique solution of (3.6). Finally, we de-fine a map Φ : [0 , ∞ ) L → [0 , ∞ ) L by Φ (cid:96) ( k ) = (cid:80) L(cid:96) (cid:48) =1 A (cid:96) (cid:48) p ( (cid:96), (cid:96) (cid:48) ) (cid:82) Ω zm k (cid:96) (cid:48) ( z ) dz for (cid:96) =1 , . . . , L , and using the Brouwer fixed point theorem we prove there exists ¯ k ∈ [0 , ∞ ) L such that Φ (cid:0) ¯ k (cid:1) = ¯ k . We use the contraction mapping theorem to prove uniquenessunder certain smallness assumptions for the data. Then it follows, by replacing ¯ k (cid:96) with Φ (cid:0) ¯ k (cid:96) (cid:1) in (3.5) and (3.6), that (cid:16) m ¯ k , V ¯ k (cid:17) = (cid:16) m ¯ k , . . . , m ¯ k L , V ¯ k , . . . , V ¯ k L (cid:17) is a(unique) solution to (3.4) with the required regularity. Solutions to the auxiliary HJB PDE.
Theorem
There exists a unique solution V k ∈ C ,τ (cid:0) ¯Ω (cid:1) to the auxiliaryHJB PDE (3.5) for any k ∈ [0 , ∞ ) and some τ ∈ (0 , , where C ,τ (cid:0) ¯Ω (cid:1) is the setof C functions on ¯Ω whose second derivative is H¨older continuous with exponent τ .Furthermore, ≤ V k ≤ ¯ z α ρB α − Proof.
The existence part of the proof uses the theory of upper and lower solu-tions, specifically Theorem 4.3. in [20], and follows along similar lines to the proofof Proposition 3.12 in [6]. This shows that a solution V k ∈ W ,p (Ω) to the auxiliaryHJB PDE exists, for some p ≥
1, provided the following hold true:
MEAN FIELD GAME OF INNOVATION
71. There exist constants ¯ V ≤ ¯ V such that ρ ¯ V − z α B α − ≤ ≤ ρ ¯ V − z α B α − , forevery z ∈ ¯Ω.2. There exist constants a k ∈ R and b k > (cid:12)(cid:12)(cid:12)(cid:12) ρu − (1 − γ ) (cid:16) γw (cid:17) γ − γ (max(0 , λ )) − γ − kλ − z α B α − (cid:12)(cid:12)(cid:12)(cid:12) ≤ a k + b k | λ | p , for every z ∈ Ω, u ∈ (cid:2) ¯ V, ¯ V (cid:3) and every λ ∈ R .If these two properties hold, then ¯ V ≤ V k ≤ ¯ V . The first assertion is true by taking¯ V = 0 and ¯ V = ¯ z α ρB α − , which also gives the required bounds for V k . The secondassertion is true with b k = k + (1 − γ ) (cid:0) γw (cid:1) γ − γ , a k = z α B α − + b k , and p = − γ , as then (cid:12)(cid:12)(cid:12)(cid:12) ρu − (1 − γ ) (cid:16) γw (cid:17) γ − γ (max(0 , λ )) − γ − kλ − z α B α − (cid:12)(cid:12)(cid:12)(cid:12) ≤ ρ | u | + (cid:18) k + (1 − γ ) (cid:16) γw (cid:17) γ − γ (cid:19) max (cid:16) , | λ | − γ (cid:17) + ¯ z α B α − ≤ a k + b k | λ | p . Now, since Ω is bounded and p > V k ∈ H (Ω). To show V k ∈ C ,τ (cid:0) ¯Ω (cid:1) , take anysolution V k to (3.5) and define f = 2 σ (cid:18)(cid:18) σ − ρ (cid:19) V k + k (cid:0) V k (cid:1) (cid:48) + (1 − γ ) (cid:16) γw (cid:17) γ − γ (cid:16) max(0 , (cid:0) V k (cid:1) (cid:48) (cid:17) − γ + z α B α − (cid:19) . Then V k is a solution of − u (cid:48)(cid:48) + u = f , where f ∈ L (Ω). So, from the elliptic regularityresult of Proposition 7.2. p.404 in [24], V k ∈ H (Ω). Therefore (cid:0) V k (cid:1) (cid:48) ∈ H (Ω), andso f ∈ H (Ω) because α ∈ (0 , V k ∈ H (Ω). Then, by the Sobolev inequality (c.f. Theorem 6p.270 in [13]) V k ∈ C ,τ (cid:0) ¯Ω (cid:1) .To prove uniqueness we use the strong maximum principle and Hopf’s lemma, asstated in [13] Section 6.4.2. pp. 330–333. Suppose, for some k ∈ [0 , ∞ ), there aretwo solutions V , V ∈ C (Ω) ∩ C ( ¯Ω) to (3.5) and V (cid:54) = V . If we define u = V − V ,then u must attain its maximum at some point z ∗ ∈ ¯Ω. Suppose at this point u > z ∗ ∈ Ω. Since thisis the maximal point, u (cid:48) ( z ∗ ) = 0, so V (cid:48) ( z ∗ ) = V (cid:48) ( z ∗ ). Hence, there exists an open,connected and bounded region U such that U ⊂ Ω, z ∗ ∈ U and − σ u (cid:48)(cid:48) = − ρu + ku (cid:48) + (1 − γ ) (cid:16) γw (cid:17) γ − γ (cid:104) max (0 , V (cid:48) ) − γ − max (0 , V (cid:48) ) − γ (cid:105) ≤ , for every z ∈ U . So, by the strong maximum principle, u is constant in U . Inparticular, using (3.5b), u ( z ∗ ) = 0. But this is a contradiction. The only other caseis z ∗ ∈ ∂ Ω and u ( z ) < u ( z ∗ ) for every z ∈ Ω. Then, ∂u∂ν (cid:12)(cid:12) z ∗ > ∂u∂ν = ∂V ∂ν − ∂V ∂ν = 0. This again leads to a contradiction. Therefore V = V and solutions to (3.5) are unique for every k ∈ [0 , ∞ ). Proposition
Fix k, k , k ∈ [0 , ∞ ) . Then, the unique classical solution tothe auxiliary HJB PDE (3.5) , as found in Theorem 3.4, satisfies the following prop-erties: V k is an increasing function on ¯Ω i.e. (cid:0) V k (cid:1) (cid:48) ≥ M. BARKER, P. DEGOND, R. MARTIN, M. MU ˆULS (cid:0) V k (cid:1) (cid:48) > for all z ∈ Ω3. (cid:107) (cid:0) V k (cid:1) (cid:48) (cid:107) ∞ = sup z ∈ Ω (cid:0) V k (cid:1) (cid:48) ( z ) ≤ (cid:104) ¯ z α (1 − γ ) B α − (cid:105) − γ (cid:16) wγ (cid:17) γ (cid:107) V k − V k (cid:107) ∞ ≤ ρ (cid:104) ¯ z α (1 − γ ) B α − (cid:105) − γ (cid:16) wγ (cid:17) γ | k − k | V k is strictly increasing with respect to k (cid:107) (cid:0) V k (cid:1) (cid:48) − (cid:0) V k (cid:1) (cid:48) (cid:107) ∞ ≤ zσ (cid:104) ¯ z α (1 − γ ) B α − (cid:105) − γ (cid:16) wγ (cid:17) γ | k − k | (cid:0) V k (cid:1) (cid:48)(cid:48) (0) > > (cid:0) V k (cid:1) (cid:48)(cid:48) (¯ z ) .Proof. Property (1): Suppose, for a contradiction, there exists z ∈ ¯Ω such that (cid:0) V k (cid:1) (cid:48) ( z ) <
0. First, by the boundary condition (3.5c), z ∈ Ω. So, by the boundaryconditions and continuity of (cid:0) V k (cid:1) (cid:48) , there exists z , z ∈ ¯Ω with z < z , (cid:0) V k (cid:1) (cid:48) ( z ) = (cid:0) V k (cid:1) (cid:48) ( z ) = 0 and (cid:0) V k (cid:1) (cid:48) ( z ) ≤ z ∈ ( z , z ). Suppose that z , z ∈ Ω. Then (cid:0) V k (cid:1) (cid:48)(cid:48) ( z ) ≤ ≤ (cid:0) V k (cid:1) (cid:48)(cid:48) ( z ) by construction of z , z and differentiability of (cid:0) V k (cid:1) (cid:48) .Furthermore, V k ( z ) > V k ( z ) because (cid:0) V k (cid:1) (cid:48) < z , z ). So, using (3.5b)0 = − σ (cid:16)(cid:0) V k (cid:1) (cid:48)(cid:48) ( z ) − (cid:0) V k (cid:1) (cid:48)(cid:48) ( z ) (cid:17) + ρ (cid:0) V k ( z ) − V k ( z ) (cid:1) − B α − ( z α − z α ) < . This is a contradiction, so z = 0 or z = ¯ z . Assume z = 0, we will again provea contradiction (the other two cases of z = ¯ z and both z = 0 , z = ¯ z follow alongsimilar arguments so their proofs are omitted). Since (cid:0) V k (cid:1) (cid:48) (0) = (cid:0) V k (cid:1) (cid:48) ( z ) = 0 and (cid:0) V k (cid:1) (cid:48) ( z ) < z ∈ (0 , z ) then, by continuity of (cid:0) V k (cid:1) (cid:48)(cid:48) , we can find (cid:15) , δ ∈ (0 , z ) such that (cid:0) V k (cid:1) (cid:48)(cid:48) ( z ) ≤ z ∈ (0 , (cid:15) ] and (cid:0) V k (cid:1) (cid:48)(cid:48) ( z ) ≥ z ∈ [ z − δ , z ). Furthermore, V k is strictly decreasing on ( z , z ). So, using these twofacts and continuity of (cid:0) V k (cid:1) (cid:48) there exists δ ∈ (0 , δ ] and (cid:15) ∈ (0 , (cid:15) ] such that1. (cid:0) V k (cid:1) (cid:48) ( (cid:15) ) = (cid:0) V k (cid:1) (cid:48) ( z − δ ) = min (cid:16)(cid:0) V k (cid:1) (cid:48) ( (cid:15) ) , (cid:0) V k (cid:1) (cid:48) ( z − δ ) (cid:17) V k ( (cid:15) ) > V k ( z − δ )3. (cid:0) V k (cid:1) (cid:48)(cid:48) ( (cid:15) ) ≤ ≤ (cid:0) V k (cid:1) (cid:48)(cid:48) ( z − δ ).Then − σ (cid:16)(cid:0) V k (cid:1) (cid:48)(cid:48) ( z − δ ) − (cid:0) V k (cid:1) (cid:48)(cid:48) ( (cid:15) ) (cid:17) + ρ (cid:0) V k ( z − δ ) − V k ( (cid:15) ) (cid:1) − ( z − δ ) α − (cid:15) α B α − < V k is a solution to (3.5). Therefore, (cid:0) V k (cid:1) (cid:48) ≥ (cid:0) V k (cid:1) (cid:48) ≥
0. Now suppose, for acontradiction, there exists z ∗ ∈ Ω such that (cid:0) V k (cid:1) (cid:48) ( z ∗ ) = 0. Then (cid:0) V k (cid:1) (cid:48)(cid:48) ( z ∗ ) = 0,since it is a minimum of (cid:0) V k (cid:1) (cid:48) . So, by (3.5b), V k ( z ∗ ) = z α ρB α − and, since (cid:0) V k (cid:1) (cid:48) ( z ∗ ) < ddz (cid:16) z α ρB α − (cid:17) , there exists z , z ∈ Ω with z < z ∗ < z such that1. (cid:0) V k (cid:1) (cid:48) ( z ) = (cid:0) V k (cid:1) (cid:48) ( z )2. V k ( z ) > z α ρB α − and V k ( z ) < z α ρB α − (cid:0) V k (cid:1) (cid:48)(cid:48) ( z ) ≤ ≤ (cid:0) V k (cid:1) (cid:48)(cid:48) ( z ).Then − σ (cid:16)(cid:0) V k (cid:1) (cid:48)(cid:48) ( z ) − (cid:0) V k (cid:1) (cid:48)(cid:48) ( z ) (cid:17) + ρ (cid:0) V k ( z ) − V k ( z ) (cid:1) − B α − ( z α − z α ) < (cid:0) V k (cid:1) (cid:48) ( z ) > z ∈ Ω.Property (3): Since (cid:0) V k (cid:1) (cid:48) is continuous on ¯Ω, (cid:0) V k (cid:1) (cid:48) ≥ (cid:0) V k (cid:1) (cid:48) (0) = (cid:0) V k (cid:1) (cid:48) (¯ z ) = 0, then (cid:0) V k (cid:1) (cid:48) must have a maximum that it attains at some point z ∗ ∈ Ω.Furthermore, since (cid:0) V k (cid:1) (cid:48) is continuously differentiable in Ω, then (cid:0) V k (cid:1) (cid:48)(cid:48) ( z ∗ ) = 0. So, MEAN FIELD GAME OF INNOVATION V k found in Theorem 3.40 ≤ (cid:0) V k (cid:1) (cid:48) ( z ) ≤ (cid:0) V k (cid:1) (cid:48) ( z ∗ ) = (cid:34) w γ − γ (1 − γ ) γ γ − γ (cid:18) ρV k ( z ∗ ) − k (cid:0) V k (cid:1) (cid:48) ( z ∗ ) − ( z ∗ ) α B α − (cid:19)(cid:35) − γ ≤ (cid:20) ¯ z α (1 − γ ) B α − (cid:21) − γ (cid:18) wγ (cid:19) γ . Property (4): Take k , k ∈ [0 , ∞ ) such that k < k . First we show V k − V k ≥
0, then we show V k − V k ≤ (cid:107) ( V k ) (cid:48) (cid:107) ∞ ρ ( k − k ), and we can conclude usingProperty (3). Let u = V k − V k and assume, for a contradiction, there exists z ∈ ¯Ωsuch that u ( z ) <
0. Then, u attains a minimum at z ∗ ∈ ¯Ω and u ( z ∗ ) <
0. Firstsuppose z ∗ ∈ Ω, then u (cid:48) ( z ∗ ) = 0 and from (3.5b) − σ u (cid:48)(cid:48) ( z ∗ ) = − ρu ( z ∗ ) + ( k − k ) (cid:0) V k (cid:1) (cid:48) ( z ∗ ) > , since (cid:0) V k (cid:1) (cid:48) ≥ u <
0. Then, by continuity of u (cid:48)(cid:48) , there exists an open bounded,connected Ω (cid:48) ⊂ Ω such that z ∗ ∈ Ω (cid:48) and u (cid:48)(cid:48) < z ∈ Ω (cid:48) . So, by the strongmaximum principle, u is constant in Ω (cid:48) . In particular, u (cid:48)(cid:48) = 0, which contradicts u (cid:48)(cid:48) < z ∈ Ω (cid:48) . So, z ∗ ∈ ∂ Ω and u ( z ) < u ( z ∗ ) for all z ∈ Ω. However, fromHopf’s lemma u (cid:48) ( z ∗ ) (cid:54) = 0, which contradicts (3.5c). So, we conclude that u ≥ u = V k − V k − (cid:15) , with (cid:15) = (cid:107) ( V k ) (cid:48) (cid:107) ∞ ρ ( k − k ) < ∞ . We assume, for acontradiction, there exists z ∈ ¯Ω such that u ( z ) >
0. Then u attains a maximumat z ∗ ∈ ¯Ω and u ( z ∗ ) >
0. First suppose z ∗ ∈ Ω, then u (cid:48) ( z ∗ ) = 0 and from (3.5b) − σ u (cid:48)(cid:48) ( z ∗ ) = − ρu ( z ∗ ) + ( k − k ) (cid:0) V k (cid:1) (cid:48) ( z ∗ ) − ρ(cid:15) < , since u > ρ(cid:15) ≥ ( k − k ) (cid:0) V k (cid:1) (cid:48) ( z ∗ ). Then, by continuity of u (cid:48)(cid:48) , there existsan open bounded, connected Ω (cid:48) ⊂ Ω such that z ∗ ∈ Ω (cid:48) and u (cid:48)(cid:48) > z ∈ Ω (cid:48) .So, by the strong maximum principle, u is constant in Ω (cid:48) . In particular, u (cid:48)(cid:48) = 0,which contradicts u (cid:48)(cid:48) > z ∈ Ω (cid:48) . So, z ∗ ∈ ∂ Ω and u ( z ) > u ( z ∗ ) for all z ∈ Ω. However, from Hopf’s lemma u (cid:48) ( z ∗ ) (cid:54) = 0, which contradicts (3.5c). So, we canconclude that u ≤ V k is increasing with respectto k . Now suppose, for a contradiction, there exists z ∗ ∈ ¯Ω such that k < k but V k ( z ∗ ) = V k ( z ∗ ). First, assume z ∗ ∈ Ω and define u = V k − V k . Then, u ( z ∗ ) = 0, u (cid:48) ( z ∗ ) = 0 and u (cid:48)(cid:48) ( z ∗ ) ≥
0, since z ∗ is a minimum of u . Furthermore, fromProperty (2), (cid:0) V k (cid:1) (cid:48) ( z ∗ ) >
0. Therefore, using (3.5b), we get the contradiction0 = − σ u (cid:48)(cid:48) + ( k − k ) (cid:0) V k (cid:1) (cid:48) < . Hence, z ∗ ∈ ∂ Ω, so u (cid:48) ( z ∗ ) = 0 ,using (3.5c). But, u (cid:48) ( z ∗ ) (cid:54) = 0 by Hopf’s lemma, whichis a contradiction. So, V k is strictly increasing with respect to k .Property (6): Fix k , k ∈ [0 , ∞ ). Let u = V k − V k . Then, u satisfies σ u (cid:48)(cid:48) = ρu − k u (cid:48) +( k − k ) (cid:0) V k (cid:1) (cid:48) − (1 − γ ) (cid:16) γw (cid:17) γ − γ (cid:18)(cid:16)(cid:0) V k (cid:1) (cid:48) (cid:17) − γ − (cid:16)(cid:0) V k (cid:1) (cid:48) (cid:17) − γ (cid:19) . M. BARKER, P. DEGOND, R. MARTIN, M. MU ˆULS
Suppose for z ∈ Ω, u (cid:48) ( z ) ≥
0. Then, since u (cid:48) (0) = 0, there exists z ∈ [0 , z ] such that u (cid:48) ( y ) ≥ y ∈ [ z , z ] and u (cid:48) ( z ) = 0. Therefore0 ≤ u (cid:48) ( z ) = (cid:90) zz u (cid:48)(cid:48) ( y ) dy ≤ zσ (cid:16) ρ || u || ∞ + | k − k | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:0) V k (cid:1) (cid:48) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ (cid:17) ≤ zσ (cid:20) ¯ z α (1 − γ ) B α − (cid:21) − γ (cid:18) wγ (cid:19) γ | k − k | . We can similarly show that u (cid:48) ( z ) ≥ − zσ (cid:104) ¯ z α (1 − γ ) B α − (cid:105) − γ (cid:16) wγ (cid:17) γ | k − k | if u (cid:48) ( z ) ≤ (cid:0) V k (cid:1) (cid:48) is Lipschitz continuous with respect to k with the required constant.Property (7): First, we will show (cid:0) V k (cid:1) (cid:48)(cid:48) (0) > (cid:0) V k (cid:1) (cid:48)(cid:48) (¯ z ) <
0. Both steps use a similar method. Note that (cid:0) V k (cid:1) (cid:48)(cid:48) (0) ≥ (cid:0) V k (cid:1) (cid:48)(cid:48) (¯ z ) ≤ (cid:0) V k (cid:1) (cid:48) (0) = (cid:0) V k (cid:1) (cid:48) (¯ z ) = 0 and (cid:0) V k (cid:1) ( z ) > z ∈ (0 , ¯ z ). So suppose,for a contradiction, that (cid:0) V k (cid:1) (cid:48)(cid:48) (0) = 0. Then, since V k ∈ C ,τ (cid:0) ¯Ω (cid:1) , we can usecontinuity of V k , (cid:0) V k (cid:1) (cid:48) , (cid:0) V k (cid:1) (cid:48)(cid:48) and (3.5b), (3.5c) to show V k (0) = 0. We can alsouse continuity of (cid:0) V k (cid:1) (cid:48)(cid:48) to show that for every C > (cid:15) > z ∈ (0 , (cid:15) ) = ⇒ (cid:0) V k (cid:1) (cid:48)(cid:48) ( z ) < C . Therefore, for any z ∈ (0 , (cid:15) )(3.7) V k ( z ) = (cid:90) z (cid:90) y (cid:0) V k (cid:1) (cid:48)(cid:48) ( y (cid:48) ) dy (cid:48) dy − z (cid:0) V k (cid:1) (cid:48) (0) − V k (0) < C z . Since (cid:0) V k (cid:1) (cid:48) > (cid:15) > (cid:0) V k (cid:1) (cid:48) increases on (0 , (cid:15) ). There-fore, (cid:0) V k (cid:1) (cid:48)(cid:48) ≥ , (cid:15) ). Take C = ρB α − and (cid:15) = min( (cid:15) , (cid:15) , V k ( z ) < z ρB α − ≤ z α ρB α − , for all z ∈ (0 , (cid:15) ). But, by rearranging (3.5b) and using (cid:0) V k (cid:1) (cid:48) , (cid:0) V k (cid:1) (cid:48)(cid:48) ≥
0, we get V k ( z ) ≥ z α ρB α − , which contradicts (3.8). Hence, (cid:0) V k (cid:1) (cid:48)(cid:48) (0) >
0. Now suppose, for acontradiction, that (cid:0) V k (cid:1) (cid:48)(cid:48) (¯ z ) = 0. Then, since V k ∈ C ,τ (cid:0) ¯Ω (cid:1) , we can use continuityof V k , (cid:0) V k (cid:1) (cid:48) , (cid:0) V k (cid:1) (cid:48)(cid:48) and (3.5b), (3.5c) to show V k (¯ z ) = ¯ z α ρB α − . We can also usecontinuity of (cid:0) V k (cid:1) (cid:48)(cid:48) to show that for every C > (cid:15) > z ∈ (¯ z − (cid:15) , ¯ z ) = ⇒ (cid:0) V k (cid:1) (cid:48)(cid:48) ( z ) > − C . Therefore, for any z ∈ (¯ z − (cid:15) , ¯ z )(3.9) V k ( z ) = (cid:90) ¯ zz (cid:90) ¯ zy (cid:0) V k (cid:1) (cid:48)(cid:48) ( y ) dy + V k (¯ z ) − (¯ z − z ) (cid:0) V k (cid:1) (cid:48) (¯ z ) > ¯ z α ρB α − − C z − z ) . Since (cid:0) V k (cid:1) (cid:48) > (cid:15) > (cid:0) V k (cid:1) (cid:48) decreases on (¯ z − (cid:15) , ¯ z ).Therefore, (cid:0) V k (cid:1) (cid:48)(cid:48) ≤ z − (cid:15) , ¯ z ). Take C > kCρ + C C − γ ρ (1 − γ ) (cid:16) γw (cid:17) γ − γ ≤ α ¯ z α − ρB α − , and (cid:15) = min( (cid:15) , (cid:15) , V k ( z ) > ¯ z α ρB α − − C z − z ) ≥ ¯ z α ρB α − − C z − z ) , MEAN FIELD GAME OF INNOVATION z ∈ (¯ z − (cid:15), ¯ z ). But, from (3.5b) and using (cid:0) V k (cid:1) (cid:48) ≤ C (¯ z − z ), (cid:0) V k (cid:1) (cid:48)(cid:48) ≤
0, we get V k ( z ) = 1 ρ (cid:18) σ (cid:0) V k (cid:1) (cid:48)(cid:48) ( z ) + k (cid:0) V k (cid:1) (cid:48) ( z ) + (1 − γ ) (cid:16) γw (cid:17) γ − γ (cid:16)(cid:0) V k (cid:1) (cid:48) (cid:17) − γ + z α B α − (cid:19) ≤ kCρ (¯ z − z ) + C − γ ρ (1 − γ ) (cid:16) γw (cid:17) γ − γ (¯ z − z ) − γ + z α ρB α − ≤ z α ρB α − + (cid:32) kCρ + C − γ ρ (1 − γ ) (cid:16) γw (cid:17) γ − γ (cid:33) (¯ z − z ) . So, using (3.10), we get α ¯ z α − ρB α − (¯ z − z ) < (cid:18) kCρ + C − γ ρ (1 − γ ) (cid:0) γw (cid:1) γ − γ (cid:19) (¯ z − z ), whichcontradicts the definition of C . Hence, (cid:0) V k (cid:1) (cid:48)(cid:48) (¯ z ) < The auxiliary Fokker–Planck equation.
Definition
Fix k ∈ [0 , ∞ ) and let V k ∈ C (Ω) ∩ C (cid:0) ¯Ω (cid:1) denote the uniquesolution to (3.5) . Then, we define the function m k : Ω → (0 , ∞ ) by (3.11a) ¯ m k = e σ (cid:18) kz + (cid:82) z ( γw ( V k ) (cid:48) ) γ − γ dy (cid:19) (3.11b) (cid:13)(cid:13) ¯ m k (cid:13)(cid:13) = (cid:90) Ω ¯ m k dz (3.11c) m k = 1 (cid:107) ¯ m k (cid:107) ¯ m k . Proposition
For every k ∈ [0 , ∞ ) , m k ∈ C (Ω) ∩ C (cid:0) ¯Ω (cid:1) where m k isdefined by (3.6) .Proof. First, note that m k is well defined because (cid:0) V k (cid:1) (cid:48) ≥ (cid:0) V k (cid:1) (cid:48) is uni-formly bounded. Hence, there exists C ∈ (1 , ∞ ) such that ¯ m k ( z ) ∈ [1 , C ] and (cid:12)(cid:12)(cid:12)(cid:12) ¯ m k (cid:12)(cid:12)(cid:12)(cid:12) ∈ [¯ z, C ¯ z ], so m k ( z ) ∈ (cid:2) C ¯ z , C ¯ z (cid:3) . Furthermore, m k ∈ C (cid:0) ¯Ω (cid:1) because V k ∈ C (cid:0) ¯Ω (cid:1) . Now, if m k ∈ C (Ω) ∩ C (cid:0) ¯Ω (cid:1) , then its derivatives would be(3.12a) (cid:0) m k (cid:1) (cid:48) = 2 σ (cid:18) k + (cid:16) γw (cid:0) V k (cid:1) (cid:48) (cid:17) γ − γ (cid:19) m k (3.12b) (cid:0) m k (cid:1) (cid:48)(cid:48) = 2 σ (cid:18) k + (cid:16) γw (cid:0) V k (cid:1) (cid:48) (cid:17) γ − γ (cid:19) (cid:0) m k (cid:1) (cid:48) + 2 γ σ w (1 − γ ) (cid:16) γw (cid:0) V k (cid:1) (cid:48) (cid:17) γ − − γ (cid:0) V k (cid:1) (cid:48)(cid:48) m k . But, since V k ∈ C (cid:0) ¯Ω (cid:1) and m k ∈ C (cid:0) ¯Ω (cid:1) , then σ (cid:18) k + (cid:16) γw (cid:0) V k (cid:1) (cid:48) (cid:17) γ − γ (cid:19) m k is well–defined and continuous for all z ∈ ¯Ω. Hence, m k ∈ C (cid:0) ¯Ω (cid:1) . Then, (cid:0) m k (cid:1) (cid:48) , (cid:0) V k (cid:1) and (cid:0) V k (cid:1) (cid:48)(cid:48) are continuous in Ω and from Proposition 3.5 (cid:0) V k (cid:1) (cid:48) > (cid:0) m k (cid:1) (cid:48)(cid:48) is well–defined in Ω, (cid:0) m k (cid:1) (cid:48)(cid:48) ∈ C (Ω) and m k ∈ C (Ω) ∩ C (cid:0) ¯Ω (cid:1) .2 M. BARKER, P. DEGOND, R. MARTIN, M. MU ˆULS
Theorem
There exists a unique solution m k ∈ C (Ω) ∩ C (cid:0) ¯Ω (cid:1) to the aux-iliary Fokker–Planck PDE (3.6) for any k ∈ [0 , ∞ ) .Proof. Take m k defined in Definition 3.6. Then, m k ∈ C (Ω) ∩ C (cid:0) ¯Ω (cid:1) by Propo-sition 3.7. Furthermore, from (3.12), m k satisfies (3.6b), (3.6c). Finally, by con-struction, m k satisfies (3.6d). Therefore, a solution to the auxiliary Fokker–Planckequation (3.6) exists, it is given by m k , and m k ∈ C (Ω) ∩ C (cid:0) ¯Ω (cid:1) . To prove unique-ness we follow the same proof in [6]. For brevity we only outline the argument here.First, with ¯ m k defined as in (3.11), we can use regularity of ¯ m k from Proposition 3.7to show (3.6) is equivalent to m k , m k ¯ m k ∈ H (Ω)(3.13a) (cid:32) ¯ m k (cid:18) m k ¯ m k (cid:19) (cid:48) (cid:33) (cid:48) = 0(3.13b) ¯ m k (cid:18) m k ¯ m k (cid:19) (cid:48) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z =0 , ¯ z = 0 , (cid:90) Ω m k dz = 1 . (3.13c)Then, by multiplying (3.13b) by m k ¯ m k , integrating over Ω and using integration byparts, the system (3.13) is equivalent to(3.14) m k ∈ H (Ω) , there exists Z > m k = 1 Z ¯ m k , (cid:90) Ω m k dz = 1 . From the previous results in this section, we have shown there exists a unique solutionto (3.14) given by m k from Definition 3.6. Hence, existence and uniqueness of theauxiliary Fokker–Planck PDE follows from the equivalence between (3.6) and (3.14). The fixed point problem.
Definition
Fix k = ( k (cid:96) ) L(cid:96) =1 ∈ [0 , ∞ ) L . For (cid:96) = 1 , . . . , L , let V k (cid:96) be theunique solution to the auxiliary HJB PDE (3.5) with constant k (cid:96) , and let m k (cid:96) be theunique solution to the auxiliary Fokker–Planck PDE (3.6) with constant k (cid:96) . Then wedefine the function Φ : [0 , ∞ ) L → [0 , ∞ ) L by Φ (cid:96) ( k ) = L (cid:88) (cid:96) (cid:48) =1 A (cid:96) (cid:48) p ( (cid:96), (cid:96) (cid:48) ) (cid:90) ¯ z zm k (cid:96) (cid:48) ( z ) dz , (cid:96) = 1 , . . . , L . Proposition
The function Φ defined in Definition 3.9 is bounded. Fur-thermore, defining P as the L × L matrix with entries P (cid:96),(cid:96) (cid:48) = p ( (cid:96), (cid:96) (cid:48) ) and A as thecolumn vector ( A , . . . , A L ) T , then ≤ (cid:107) Φ( k ) (cid:107) ≤ ¯ z (cid:107) P A (cid:107) , where the 1–norm (cid:107) · (cid:107) is defined as (cid:107) x (cid:107) = (cid:80) L(cid:96) =1 | x (cid:96) | for any x ∈ R L .Remark ζ = ¯ z (cid:107) P A (cid:107) and consideronly the restriction of Φ to [0 , ζ ] L , which we will still denote by Φ for convenience. Proof.
Take (cid:96) = 1 , . . . , L . Then Φ (cid:96) ( k ) = (cid:80) L(cid:96) (cid:48) =1 A (cid:96) (cid:48) p ( (cid:96), (cid:96) (cid:48) ) (cid:82) ¯ z zm k (cid:96) (cid:48) ( z ) dz ≥ p ( (cid:96), (cid:96) (cid:48) ) ≥ m k (cid:96) (cid:48) ≥
0. Similarly, since m k (cid:96) (cid:48) is a probability distribution, MEAN FIELD GAME OF INNOVATION (cid:96) ( k ) ≤ ¯ z (cid:80) L(cid:96) (cid:48) =1 A (cid:96) (cid:48) p ( (cid:96), (cid:96) (cid:48) ) (cid:82) ¯ z m k (cid:96) (cid:48) ( z ) dz = ¯ z (cid:80) L(cid:96) (cid:48) =1 A (cid:96) (cid:48) p ( (cid:96), (cid:96) (cid:48) ). Therefore,0 ≤ L (cid:88) (cid:96) =1 Φ (cid:96) ( k ) ≤ ¯ z L (cid:88) (cid:96) =1 L (cid:88) (cid:96) (cid:48) =1 A (cid:96) (cid:48) p ( (cid:96), (cid:96) (cid:48) ) = ¯ z (cid:107) P A (cid:107) . Theorem
The function
Φ : [0 , ζ ] L → [0 , ζ ] L defined in Definition 3.9 is Lip-schitz in the 1–norm on R L . The Lipschitz constant is given by ¯ C max (cid:96) =1 ,...,L A (cid:96) P (cid:96) ,where P (cid:96) = (cid:80) L(cid:96) (cid:48) =1 p ( (cid:96) (cid:48) , (cid:96) ) and ¯ C depends on (cid:107) P A (cid:107) , but not explicitly on P or A .Proof. First, fix k ∈ [0 , ζ ]. From Property (5) of Proposition 3.5, the continuityof V k , (cid:0) V k (cid:1) (cid:48) , (cid:0) V k (cid:1) (cid:48)(cid:48) with respect to z in ¯Ω, and equations (3.5b), (3.5c), we find (cid:0) V k (cid:1) (cid:48)(cid:48) (0) = 2 ρσ V k (0) ≥ ρσ V (0) = (cid:0) V (cid:1) (cid:48)(cid:48) (0) > , with the middle inequality an equality if and only if k = 0. Similarly, (cid:0) V k (cid:1) (cid:48)(cid:48) (¯ z ) ≤ (cid:0) V ζ (cid:1) (cid:48)(cid:48) (¯ z ) < k = 0. Moreover, (cid:0) V k (cid:1) (cid:48)(cid:48) is continuous with respect to k due to (3.5b) and continuity of V k , (cid:0) V k (cid:1) (cid:48) withrespect to k , which was proven in Proposition 3.5. Therefore, there exists (cid:15) , (cid:15) ∈ (0 , C , C >
0, independent of k , such that (cid:0) V k (cid:1) (cid:48) ( z ) = (cid:90) z (cid:0) V k (cid:1) (cid:48)(cid:48) ( y ) dy ≥ (cid:90) z (cid:0) V (cid:1) (cid:48)(cid:48) ( y ) dy ≥ C z , if z ∈ [0 , (cid:15) ] (cid:0) V k (cid:1) (cid:48) ( z ) = − (cid:90) ¯ zz (cid:0) V k (cid:1) (cid:48)(cid:48) ( y ) dy ≥ − (cid:90) ¯ zz (cid:0) V ζ (cid:1) (cid:48)(cid:48) ( y ) dy ≥ C z , if z ∈ [¯ z − (cid:15) , ¯ z ] . Furthermore, by continuity of (cid:0) V k (cid:1) (cid:48) with respect to k and compactness of [0 , ζ ], thereexists C > k ∈ [0 ,ζ ] (cid:0) V k (cid:1) (cid:48) ( z ) ≥ C if z ∈ [ (cid:15) , ¯ z − (cid:15) ]. Note that C j for j = 1 , , k ∈ [0 , ζ ]. Therefore, if γ ≤ , for any k , k ∈ [0 , ζ ]:(3.15) (cid:90) ¯ z (cid:20) min (cid:16)(cid:0) V k (cid:1) (cid:48) ( z ) , (cid:0) V k (cid:1) (cid:48) ( z ) (cid:17) (cid:21) γ − − γ dz ≤ (cid:90) (cid:15) ( C z ) γ − − γ dz + (cid:90) ¯ z − (cid:15) (cid:15) C γ − − γ dz + (cid:90) ¯ z ¯ z − (cid:15) ( C (¯ z − z )) γ − − γ dz ≤ − γγ ( C γ − − γ + C γ − − γ ) + C γ − − γ ¯ z , while, using Proposition 3.5, if γ ≥ (3.16) (cid:90) ¯ z (cid:20) max (cid:16)(cid:0) V k (cid:1) (cid:48) ( z ) , (cid:0) V k (cid:1) (cid:48) ( z ) (cid:17)(cid:21) γ − − γ dz ≤ ¯ z (cid:18) ¯ z α (1 − γ ) B α − (cid:19) γ − (cid:18) wγ (cid:19) γ (2 γ − − γ . Now, with the definition of ¯ m k in (3.11), for any k , k ∈ [0 , ζ ] we have | ¯ m k − ¯ m k | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e σ (cid:18) k z + (cid:82) z ( γw ( V k ) (cid:48) ( y ) ) γ − γ dy (cid:19) − e σ (cid:18) k z + (cid:82) z ( γw ( V k ) (cid:48) ( y ) ) γ − γ dy (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ (cid:90) k z + (cid:82) z ( γw ( V k ) (cid:48) ( y ) ) γ − γ dyk z + (cid:82) z ( γw ( V k ) (cid:48) ( y ) ) γ − γ dy e σ u du (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . M. BARKER, P. DEGOND, R. MARTIN, M. MU ˆULS
Then, using the uniform bound on (cid:0) V k (cid:1) (cid:48) ( y ) with respect to k given by Proposition 3.5,we get | ¯ m k − ¯ m k | ≤ C σ (cid:12)(cid:12)(cid:12)(cid:12) ( k − k ) z + (cid:90) z (cid:20)(cid:16) γw (cid:0) V k (cid:1) (cid:48) ( y ) (cid:17) γ − γ − (cid:16) γw (cid:0) V k (cid:1) (cid:48) ( y ) (cid:17) γ − γ (cid:21) dy (cid:12)(cid:12)(cid:12)(cid:12) ≤ C σ (cid:32) | k − k | z + (cid:16) γw (cid:17) γ − γ (cid:90) ¯ z (cid:90) ( V k ) (cid:48) ( y ) ( V k ) (cid:48) ( y ) γ − γ u γ − − γ du dy (cid:33) , where ¯ C = e zσ (cid:16) ζ + (cid:104) γ ¯ zα (1 − γ ) wBα − (cid:105) γ (cid:17) . Then, using Proposition 3.5 and either (3.15)or (3.16), we get(3.17) | ¯ m k − ¯ m k | ≤ C σ (cid:32) | k − k | z + (cid:16) γw (cid:17) γ − γ γ − γ || (cid:0) V k (cid:1) (cid:48) − (cid:0) V k (cid:1) (cid:48) || ∞ (cid:90) z max (cid:104) (cid:16)(cid:0) V k (cid:1) (cid:48) ( y ) (cid:17) γ − − γ , (cid:16)(cid:0) V k (cid:1) (cid:48) ( y ) (cid:17) γ − − γ (cid:105) dy (cid:33) ≤ C σ (cid:0) z + ¯ C (cid:1) | k − k | , where ¯ C = zσ (cid:16) ¯ z (1 − γ ) B α − (cid:17) − γ (cid:0) γw (cid:1) γ − γ (cid:18) C γ − − γ + C γ − − γ + γ − γ C γ − − γ ¯ z (cid:19) , if γ < .While ¯ C = γ − γ z σ (cid:16) w ¯ z α γ (1 − γ ) B α − (cid:17) γ , if γ ≥ . Note that for any k ∈ [0 , ζ ], (cid:13)(cid:13) ¯ m k (cid:13)(cid:13) satisfies(3.18) (cid:12)(cid:12)(cid:12)(cid:12) ¯ m k (cid:12)(cid:12)(cid:12)(cid:12) = (cid:90) ¯ z e σ (cid:20) kz + (cid:82) z ( γw ( V k ) (cid:48) ( y ) ) γ − γ dy (cid:21) dz ≥ , as (cid:0) V k (cid:1) (cid:48) ≥
0. So, for any k , k ∈ [0 , ζ ], using (3.17) and (3.18), we have(3.19) (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) ¯ z z ( m k − m k ) dz (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:107) ¯ m k (cid:107) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ¯ z z ( ¯ m k − ¯ m k ) dz (cid:12)(cid:12)(cid:12)(cid:12) + (cid:90) ¯ z z ¯ m k (cid:107) ¯ m k (cid:107) (cid:107) ¯ m k (cid:107) dz (cid:12)(cid:12)(cid:12) (cid:13)(cid:13) ¯ m k (cid:13)(cid:13) − (cid:12)(cid:12) ¯ m k (cid:13)(cid:13) (cid:12)(cid:12)(cid:12) ≤ z (cid:90) ¯ z | ¯ m k − ¯ m k | dz ≤ C ¯ zσ (cid:90) ¯ z ( z + ¯ C ) dz | k − k | = 2 ¯ C ¯ z (¯ z + 2 ¯ C ) σ | k − k | := ¯ C | k − k | . Now take k (1) , k (2) ∈ [0 , ζ ] L . Define P (cid:96) = (cid:80) L(cid:96) (cid:48) =1 p ( (cid:96) (cid:48) , (cid:96) ), then recalling the definitionof Φ given in Definition 3.9 and using (3.19)(3.20) (cid:13)(cid:13)(cid:13) Φ( k (1) ) − Φ( k (2) ) (cid:13)(cid:13)(cid:13) = L (cid:88) (cid:96) =1 L (cid:88) (cid:96) (cid:48) =1 A (cid:96) (cid:48) p ( (cid:96), (cid:96) (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ¯ z z ( m k (1) (cid:96) (cid:48) − m k (2) (cid:96) (cid:48) ) dz (cid:12)(cid:12)(cid:12)(cid:12) ≤ ¯ C L (cid:88) (cid:96) (cid:48) =1 A (cid:96) (cid:48) P (cid:96) (cid:48) (cid:12)(cid:12)(cid:12) k (1) (cid:96) (cid:48) − k (2) (cid:96) (cid:48) (cid:12)(cid:12)(cid:12) ≤ ¯ C max (cid:96) =1 ,...,L A (cid:96) P (cid:96) (cid:13)(cid:13)(cid:13) k (1) − k (2) (cid:13)(cid:13)(cid:13) , which concludes the proof. MEAN FIELD GAME OF INNOVATION Theorem
For any given data, there exists a solution to the innovationMFG (3.4) . Furthermore, if (cid:107)
P A (cid:107) is fixed, this solution is unique provided A (cid:96) P (cid:96) < C for every (cid:96) = 1 , . . . , L .Proof. From Proposition 3.10 and Theorem 3.12, the function Φ : [0 , ζ ] L → [0 , ζ ] L is a continuous function from a convex compact subset of R L to itself. Therefore, byBrouwer’s fixed point theorem, Φ has a fixed point. Furthermore, Theorem 3.12shows that Φ is a Lipschitz function in (cid:107) · (cid:107) . The Lipschitz constant is given by¯ C max (cid:96) =1 ,...,L A (cid:96) P (cid:96) , where ¯ C depends on (cid:107) P A (cid:107) but not directly on P (cid:96) or A (cid:96) .Therefore, for fixed (cid:107) P A (cid:107) , Φ is a contraction map provided A (cid:96) P (cid:96) < C for every (cid:96) = 1 , . . . , L , and in this case the fixed point is unique.Theorems 3.4 and 3.8 proved existence and uniqueness of solutions to equa-tions (3.5) and (3.6) respectively for any k ∈ [0 , ζ ]. Now, if k ∗ is a fixed pointof Φ then ( m ∗ , V ∗ ) := (cid:0) m k ∗ (cid:96) , V k ∗ (cid:96) (cid:1) L(cid:96) =1 is a solution to (3.4), which can be seenby replacing k ∗ (cid:96) with Φ (cid:96) ( k ∗ ) in (3.5), (3.6) for every (cid:96) = 1 , . . . , L . Conversely, if( m ∗ , V ∗ ) is a solution to (3.5), (3.6), then clearly, by defining k ∗ co–ordinate wise as k ∗ (cid:96) = (cid:80) L(cid:96) (cid:48) =1 A (cid:96) (cid:48) p ( (cid:96), (cid:96) (cid:48) ) (cid:82) ¯ z zm (cid:96) (cid:48) ( z ) dz , k ∗ ∈ [0 , ζ ] L is a fixed point of Φ. Furthermore,by uniqueness of (3.5), (3.6), (cid:0) m k ∗ , V k ∗ (cid:1) = ( m ∗ , V ∗ ). So, existence and uniquenessof solutions to the innovation MFG (3.4) is equivalent to existence and uniqueness offixed points of Φ. Hence, there exists a solution to the innovation MFG. Furthermore,this solution is unique, provided A (cid:96) P (cid:96) < C for every (cid:96) = 1 , . . . , L . Remark A (cid:96) P (cid:96) < C holdsfor every (cid:96) = 1 , . . . , L provided L is large enough. This is because (cid:80) L(cid:96) =1 A (cid:96) = 1. So,for fixed (cid:107) P A (cid:107) , when L is sufficiently large we can take A (cid:96) to be sufficiently smallso that A (cid:96) P (cid:96) < C
4. Numerical simulations.4.1. Consumers.
In the previous analysis, we assumed that consumers play apassive role in the model. In particular, the constant B has been fixed. However,in doing so we have not modelled the active nature of consumers in determining theprice index R . To include this when implementing our numerical methods we returnto (2.4) and we normalise economic output to Y = 1. Then, by rearranging (2.4) andusing the production function q (cid:96),i = Z (cid:96),i , we get B = (cid:104) N (cid:80) L(cid:96) =1 (cid:80) N (cid:96) i =1 Z α(cid:96),i (cid:105) α − . So, asthe number of firms in each sector goes to infinity, B = (cid:104)(cid:80) L(cid:96) =1 A (cid:96) (cid:82) Ω z α m (cid:96) ( z ) dz (cid:105) α − .Note that this now needs to be solved as a fixed point, as m (cid:96) itself depends on B . We computed simulations with synthetic data, using the nu-merical method outlined in Appendix A. From an economics perspective it is impor-tant to understand how the model affects the sector–level productivity. The purposeof the simulations is to provide initial insights into the role of the modelling parametersand of the network configuration.
Parameter effects.
The MFG depends on the parameters σ , w , α , γ and ρ .Recall that σ > w > α ∈ (0 ,
1) is a parameter in the consumer optimisationproblem which ensures convexity, and γ ∈ (0 ,
1) is the returns to labour i.e. theinefficiency in converting one unit of labour to one unit of knowledge, it also ensuresconvexity of the firm–level optimisation problem.6
M. BARKER, P. DEGOND, R. MARTIN, M. MU ˆULS
In order to separate the parameter effects from any effects caused by the sectornetwork, we ran simulations with just a single sector. We fixed ¯ z = 2, A = A = 1 and P = 0 .
1, where ¯ z is the maximum productivity level, A is the proportion of firms insector 1 and P is the strength of connection from sector 1 to itself. For baseline values, (a) Plot of firm distribution with varying α (b) Plot of average productivity against α (c) Plot of firm distribution with varying γ (d) Plot of average productivity against γ (e) Plot of firm distribution with varying ρ (f) Plot of average productivity against ρ Fig. 4.1: Simulations of MFG with varying α , γ , and ρ MEAN FIELD GAME OF INNOVATION σ = 1, w = 1, ρ = 1, γ = 0 . α = 0 .
5. For each simulation, we varied oneparameter while keeping all others at the baseline level. Figures 4.1a and 4.1b showthat the relationship between α and the distribution of firms is a complex one. There issome α ∗ ∈ (0 ,
1) where the average productivity reaches a maximum, while on (0 , α ∗ ]average productivity is monotonically increasing, and on [ α ∗ ,
1) average productivity ismonotonically decreasing. Note that, for fixed productivity level and firm distribution,a firm’s revenue is r (cid:96) q (cid:96) = Z α(cid:96) (cid:104) N (cid:80) L(cid:96) (cid:48) =1 A (cid:96) (cid:48) (cid:82) Ω z α m (cid:96) (cid:48) ( z ) dz (cid:105) − , which consists of aterm that increases with respect to α multiplied by a term that decreases with respectto α . This results in a competing effect between α and a firm’s revenue, which inturn affects a firm’s return on investment, and therefore its level of investment inlabour. Since labour investment has an increasing effect on average productivity,the competing terms in the revenue equation directly correspond to the behaviourexhibited in figure 4.1b.Figures 4.1c and 4.1d shows the effect of γ on the sector–level productivity. Fig-ure 4.1d shows that as γ increases, the average productivity decreases. Since γ re-lates to the inefficiency of converting one unit of labour to one unit of productivework, it seems counter–intuitive at first that average productivity would be a de-creasing function of γ . Recall that the optimal level of employment is given by h ∗ = (cid:0) γw max(0 , V (cid:48) ) (cid:1) − γ , which increases productivity at a rate ( h ∗ ) γ . Then, h ∗ is increasing with respect to γ for fixed V (cid:48) if and only if V (cid:48) ≥ wγ e γ − γ and ( h ∗ ) γ is in-creasing if and only if V (cid:48) ≥ wγ e γ − . Hence, the effect of γ on the average productivitydepends on V (cid:48) and how it changes with respect to γ .The effects of ρ and σ on the average productivity, shown in Figures 4.1e, 4.1fand 4.2a, 4.2b respectively, show the same trend: average productivity decreases aseach parameter increases. The size of ρ is the extent to which a firm discounts futureprofits. As ρ increases, firms care less about the future state of the system and sothey are less willing to invest in labour; it is an investment whose effect is only on thefuture value of productivity. This results in reduced average productivity in the longrun, which can be seen in Figure 4.1f. As σ increases, the randomness in productivityevolution of each firm increases. So, the impact of labour on productivity decreaseswith increasing σ , and this is reflected in Figure 4.2b.Finally, Figures 4.2c and 4.2d shows that average productivity also decreases withincreasing wage, w . The wage rate increases the cost of labour. So, we can directly seethat as the wage increases, the optimal level of employment, and hence the averageproductivity, decreases. Spillover size effects.
The sector–level network, encoded by the vertex weights A (cid:96) for sector (cid:96) , and the edge weights p ( (cid:96), (cid:96) (cid:48) ) for a transfer of knowledge from sector (cid:96) (cid:48) tosector (cid:96) , is called the spillover network as it describes how knowledge and productivityspills over from one sector to another. A path in the spillover network is called aspillover path, or just spillover if there is no ambiguity. A path of length 1 fromsector (cid:96) (cid:48) to sector (cid:96) is called a direct spillover, a path of length 2 or greater fromsector (cid:96) (cid:48) to sector (cid:96) is called an indirect spillover, and in both cases sector (cid:96) is calledthe receiving sector and sector (cid:96) (cid:48) is the originating sector.In almost all economic literature, only direct spillovers have been modelled andwe are aware of no models that pay attention to the effect indirect spillovers have oneconomic productivity. In this subsection, we begin investigating how the productiv-ity of a sector is affected by the structure of the spillover network, and in particularthe effect of indirect spillovers on productivity. To undertake this investigation, we8 M. BARKER, P. DEGOND, R. MARTIN, M. MU ˆULS (a) Plot of firm distribution with varying σ (b) Plot of average productivity against σ (c) Plot of firm density against productivitywith w = 1 , ,
10 (d) Plot of average productivity against valueof w Fig. 4.2: Simulations of MFG with varying σ and w conducted three types of simulations. The first simulations were to model the six net-works in Figure 4.3, to provide initial insight into how indirect spillover paths affectthe distribution of firms. In the second simulations, we randomly generated spillovernetworks in models with three sectors and used the collected data to hypothesise arelationship between the average productivity of a sector and the size of spillovers(direct and indirect) it received. In the final simulations, we tested our hypothesison more randomly generated spillover networks, this time for models with 10 sec-tors, which more closely resembles the number of sectors in the real economy. Weshowed that the hypothesis developed accurately describes the relationship betweenthe spillover network and the average productivity of firms, moreover there was a20% reduction in error when direct and indirect spillovers were taken into account,compared with when only direct spillovers were considered. Therefore, our conclusionfrom this preliminary investigation is that indirect spillovers have a significant effecton economic productivity in our model and they should not be ignored.The networks in Figure 4.3 provide insight into how indirect spillovers affectthe distribution of firms, in comparison to direct spillovers. In network 1, sector C has one direct spillover, in network 2 it has one direct spillover and one indirect MEAN FIELD GAME OF INNOVATION C between network 2 and network 1 will show the effect ofan indirect spillover compared with having no spillover, and the difference betweennetworks 3 and 2 will show the effect of an indirect spillover compared with a directspillover. The differences in density of sector C are plotted in Figure 4.4a. From theplots, it can be seen that the density of firms is larger at high productivity levelsin network 2 compared with network 1 and the density is lower at low productivitylevels. This means that the indirect spillover from sector A to sector C has a positiveeffect on sector C, skewing the distribution towards higher productivity levels. Thesame behaviour can be seen when we compare sector C in network 3 to network 2,however the effect is an order of magnitude larger. Therefore, although an indirectspillover path has some positive effect compared with no path at all, the effect is lessstrong than a direct spillover path.In Figure 4.4b, sector D of networks four to six were modelled. For sector D:in network 4 there is one indirect spillover with path length 2; network 5 has oneindirect spillover with path length 2 and one with path length 3; finally network 6has an infinite number of indirect spillovers, one for every path length. We haveplotted the difference in density of sector D between network 5 and network 4 andbetween networks 6 and 5. The difference between network 5 and network 4 showsthe effect of an indirect spillover of length 3, while the difference between network 6and network 5 shows the effect of indirect spillovers of all lengths greater than 3. Forthe difference between network 5 and network 4, the same qualitative result as thedifference between network 2 and network 1, in Figure 4.4a, is observed. This suggeststhat having spillover paths of greater length do have positive impacts on productivity,but with reduced impact for increased path lengths. Interestingly, sector D in network6 is less productive than sector D in sector 5. Further investigation showed that if B is fixed, rather than the solution of a fixed point problem, then the effect that morepaths result in greater productivity returns, see figure 4.4c. The reason for this is notimmediately obvious and warrants further study. Since the observed change is verysmall, it can’t be ruled out that this result is an artefact from simplifications in themodel.In the second set of simulations, we took a closer look at how the spillover networkstructure affects the average productivity within each sector. Recall that if, given anetwork, we know the value of the fixed point, k ∗ , of the function Φ defined in Defini-tion 3.9. Then the average productivity in sector (cid:96) is (cid:82) Ω zm (cid:96) ( z ) dz = (cid:82) Ω zm k ∗ (cid:96) ( z ) dz .So, to understand the relationship between average productivity and the network,we first need to understand the relationship between (cid:82) Ω zm k (cid:96) ( z ) dz and k (cid:96) , for any k (cid:96) ≥
0. Then, we also need to understand the relationship between k ∗ (cid:96) and the L × L matrix S with entries defined by S (cid:96),(cid:96) (cid:48) = A (cid:96) (cid:48) p ( (cid:96), (cid:96) (cid:48) ), because k ∗ = S (cid:18)(cid:90) Ω zm k ∗ (cid:96) ( z ) dz (cid:19) L(cid:96) =1 . In Figure 4.5, we have plotted (cid:82) Ω zm k ( z ) dz against k . The relationship appears toapproximately follow(4.1) (cid:90) Ω zm k ( z ) dz = ¯ z − b k b + b , for some b , b , b >
0, as can be seen by the second line in Figure 4.5. To understandthe relationship between the fixed point of Φ and the matrix S , we considered networks0 M. BARKER, P. DEGOND, R. MARTIN, M. MU ˆULS
AB C (a) Network 1
AB C (b) Network 2
AB C (c) Network 3
AB CD (d) Network 4
AB CD (e) Network 5
AB CD (f) Network 6
Fig. 4.3: Sector–level networks for simulations in Figures 4.4a and 4.4bof three vertices, with A (cid:96) = 1 / (cid:96) . We created a random network between thevertices by choosing a connection probability p , and making a directed edge betweenvertices with probability p . We then weighted each directed edge with a randomweight, chosen from a uniform distribution on [0 , k ∗ (cid:96) , the (cid:96) th co–ordinate of the fixed point of Φ, against the sum of directspillover strengths (cid:80) L(cid:96) (cid:48) =1 S (cid:96),(cid:96) (cid:48) . In the simulations with a high connection probability,Figure 4.6a, there is a strong linear relationship between k ∗ (cid:96) and (cid:80) L(cid:96) (cid:48) =1 S (cid:96),(cid:96) (cid:48) . However,with low connection probabilities, Figure 4.6b, the simulations tend to follow one oftwo weaker linear relationships with the row sum.To understand the relationships further, we can look at the equation that k ∗ (cid:96) ∈ [0 , ∞ ) implicitly satisfies: k ∗ (cid:96) = (cid:80) L(cid:96) (cid:48) =1 S (cid:96),(cid:96) (cid:48) (cid:82) Ω zm k ∗ (cid:96) (cid:48) dz , where m k (cid:96) is defined by (3.11).So, if sector (cid:96) receives no spillovers then k ∗ (cid:96) = 0. If it has only direct spillovers, thenit is only connected to sectors with no spillovers. So, by defining f ( k ) = (cid:82) Ω zm k dz (4.2) k ∗ (cid:96) = f (0) L (cid:88) (cid:96) (cid:48) =1 S (cid:96),(cid:96) (cid:48) . We can see this linear relationship between k ∗ (cid:96) and (cid:80) L(cid:96) (cid:48) =1 S (cid:96),(cid:96) (cid:48) in Figures 4.6c and 4.6d,where we have taken the simulated points in Figure 4.6b, and split the data into thosepoints which have only direct spillovers and those that have indirect spillovers as well.In Figure 4.6c, where sectors with only direct spillover paths are considered, the linearrelationship described by (4.2) can be clearly seen.To understand how the value of k ∗ (cid:96) depends on the matrix S in the case of indirectspillovers, we can return to the definition of the spillover size and f ( k ). If we assumethat f is approximately linear for sectors with indirect spillovers, i.e. f ( k ) = f + f k , MEAN FIELD GAME OF INNOVATION (a) Difference in density of sector C betweennetworks 2 and 1, and networks 3 and 2 (b) Difference in density of sector D betweennetworks 5 and 4, and networks 6 and 5(c) Difference in density of sector D betweennetworks 5 and 4, and networks 6 and 5, withfixed B = 1 Fig. 4.4: Simulations of MFG comparing distribution of firms in sectors C and Dwith respect to productivity in networks 1, 2 and 3, and networks 4, 5 and 6 fromFigure 4.3then(4.3) k ∗ (cid:96) = ( S ( f + f k ∗ )) (cid:96) , where is the vector of length L with ones in every entry. Using the identity ( I + f S ) − = (cid:80) ∞ n =0 f n S n , we can rearrange (4.3)(4.4) k ∗ (cid:96) = f ∞ (cid:88) n =0 f n (cid:0) S n +1 (cid:1) (cid:96) , which gives a way to estimate the value k ∗ (cid:96) directly from the initial data. Therefore,combining estimates (4.1) and (4.4), we can estimate the value of average productivity2 M. BARKER, P. DEGOND, R. MARTIN, M. MU ˆULS
Fig. 4.5: Plot of average productivity against size of k in auxiliary Fokker–Planckequation (3.6) and plot of y = ¯ z − (¯ z − . k +1 for comparison (a) Probability of directed edge = 0 . . .
2, andlength of longest path to sector = 1 (d) Probability of directed edge = 0 .
2, andlength of longest path to sector > Fig. 4.6: Relationship between k ∗ (cid:96) and sum of direct spillovers (cid:80) L(cid:96) (cid:48) =1 S (cid:96),(cid:96) (cid:48) MEAN FIELD GAME OF INNOVATION S by(4.5) (cid:90) Ω zm (cid:96) ( z ) dz = ¯ z − b (cid:16) f (cid:80) L(cid:96) (cid:48) =1 (cid:80) ∞ n =0 f n ( S n +1 ) (cid:96),(cid:96) (cid:48) (cid:17) b + b . The relationship suggests that the average productivity depends on S n for every n i.e. on indirect spillovers of every path length. Moreover, if f is small enough, theeffect of a spillover path is decreasing by an order of magnitude for every increase inpath length, which agrees with our initial simulations of networks 1–6.In order to verify the hypothesis, in the final simulations we ran a regression to es-timate the parameters f , f , b , b , b and provide evidence that approximation (4.5)is accurate. We performed 1000 simulations on networks of ten vertices, with connec-tion probability chosen randomly and uniformly distributed in [0 , , A (cid:96) also ran-domly chosen. We ran a nonlinear regression, of the form (4.4), on sectors withindirect spillovers, to obtain optimal values of f and f . Then, using the optimalvalues of f and f we ran a second nonlinear regression, of the form (4.5), to find theoptimal values of b , b and b . Table 5.1 gives estimates for the parameters f i and b i .We found that average productivity does behave approximately according to (4.5),with table 5.1 suggesting a statistically significant result. Visually, this can be seenin Figure 5.1b, where we plotted (4.5) using the optimal values of f i and b i . We alsocomputed estimates for the model(4.6) (cid:90) Ω zm (cid:96) ( z ) dz = ¯ z − ¯ b (cid:16) ¯ f (cid:80) L(cid:96) (cid:48) =1 S (cid:96),(cid:96) (cid:48) (cid:17) ¯ b + ¯ b , which assumes average productivity depend on direct spillovers only, and plotted theresult in Figure 5.1b. Comparing plots 5.1a and 5.1b shows that the model (4.5),which includes the effects of indirect spillovers, provides a more accurate estimatefor average productivity than model (4.6), which only accounts for the effect of directspillovers. This is reconfirmed by the 20% reduction in R–squared error when indirectspillover paths are included in the model. Therefore, indirect spillover paths can notbe ignored as a factor determining a sector’s productivity.
5. Conclusion and future research.
We have developed an MFG model offirm–level innovation from a microscopic formulation. The model can be calibrated tofit economic data of spillovers, so its economic validity can be verified. We have beenable to prove existence of solutions and, under a smallness assumption on the data,uniqueness. We have investigated numerically how the modelling parameters and thespillover network affects the sector–level productivity, through the development of asimple algorithm that takes advantage of the structure of the proof of existence anduniqueness.In future work, we hope to compare the MFG model with the socially optimalbehaviour, as described by the mean field optimal control problem. We will also usepatent–level data to calibrate and test the two models for their accuracy. We hope thecomparison between the social optimum and the competitive equilibrium will suggesta method for implementing socially optimal subsidy policies for R&D.
Appendix A. Numerical Methods.
The numerical method we designed tosolve (3.4) is informed by the structure of the proof of existence and uniqueness. The4
M. BARKER, P. DEGOND, R. MARTIN, M. MU ˆULS method of proof relies on the contraction mapping theorem to find a fixed point of themap Φ, defined in Definition 3.9. We are also required to solve a fixed point problemto find the value of the parameter B . In light of this, our numerical method proceedsas follows, after choosing an initial guess k ∈ [0 , ∞ ) L , B ∈ [0 , ∞ ) and tolerances δ , δ .1. Given k i ∈ [0 , ∞ ) L and B i ∈ [0 , ∞ ), solve (3.5b), (3.5c) using the followingmethod, based on a Newton–Raphson method in a Banach space.(a) Define F ( v ) = − σ v (cid:48)(cid:48) + ρv − kv (cid:48) − (1 − γ ) (cid:0) γw (cid:1) γ − γ ( v (cid:48) ) − γ − z α B α − . Wewant to find zeros of F ( v ).(b) We define dF ( v )( u ) = − σ u (cid:48)(cid:48) + ρu − ku (cid:48) − (cid:0) γw v (cid:48) (cid:1) γ − γ u (cid:48) , which is theFr´echet derivative of F .(c) Denote by V k i(cid:96) ,B i the initial guess for the (cid:96) th component of the solutionto (3.5b), (3.5c) with k = k i(cid:96) and B = B i .(d) Given V k i(cid:96) ,B i n , we compute the next iteration, V k i(cid:96) ,B i n +1 , using a Newton–Raphson method: V k i(cid:96) ,B i n +1 = V k i(cid:96) ,B i n − dF (cid:16) V k i(cid:96) ,B i n (cid:17) − (cid:16) F (cid:16) V k i(cid:96) ,B i n (cid:17)(cid:17) .(e) Continue iteratively until (cid:13)(cid:13)(cid:13) F (cid:16) V k i(cid:96) ,B i n (cid:17)(cid:13)(cid:13)(cid:13) ≤ δ and define V i,(cid:96) = V k i(cid:96) ,B i n
2. Given V i , compute the solution to (3.6b) using (3.11) and denote it by m i Variable Coefficient estimate Standard error t stat p value f . × − f .
483 3 . × − b . × − b . × − b . × − (a) Plot of average productivity against righthand side of (4.5), with optimal values for f i , b i (b) Plot of average productivity against righthand side of (4.6), with optimal values for¯ f , ¯ b i Fig. 5.1: Plots of average productivity against (4.5) with models for k (cid:96) given byconsidering direct and indirect spillovers (4.4) or only direct spillovers (4.6) MEAN FIELD GAME OF INNOVATION k i +1 = Φ (cid:0) k i (cid:1) and B i +1 = (cid:104)(cid:80) L(cid:96) =1 A (cid:96) (cid:82) Ω z α m (cid:96) ( z ) dz (cid:105) − α
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