A mean-field model of optimal investment

Calvia, Alessandro; Federico, Salvatore; Ferrari, Giorgio; Gozzi, Fausto Working Paper A mean-field model of optimal investment Center for Mathematical Economics Working Papers, No. 690 Provided in Cooperation with: Center for Mathematical Economics (IMW), Bielefeld University Suggested Citation: Calvia, Alessandro; Federico, Salvatore; Ferrari, Giorgio; Gozzi, Fausto (2024) : A mean-field model of optimal investment, Center for Mathematical Economics Working Papers, No. 690, Bielefeld University, Center for Mathematical Economics (IMW), Bielefeld, https://nbn-resolving.de/urn:nbn:de:0070-pub-29883840 This Version is available at: https://hdl.handle.net/10419/289847 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ 690 April 2024 A Mean-Field Model of Optimal Investment Alessandro Calvia, Salvatore Federico, Giorgio Ferrari, and Fausto Gozzi Center for Mathematical Economics (IMW) Bielefeld University Universit¨atsstraße 25 D-33615 Bielefeld ·Germany e-mail: [email protected] uni-bielefeld.de/zwe/imw/research/working-papers ISSN: 0931-6558 Unless otherwise noted, this work is licensed under a Creative Commons Attribution 4.0 International (CC BY) license. Further information: https://creativecommons.org/licenses/by/4.0/deed.en https://creativecommons.org/licenses/by/4.0/legalcode.en A MEAN-FIELD MODEL OF OPTIMAL INVESTMENT ALESSANDRO CALVIAa,1, SALVATORE FEDERICOb,2, GIORGIO FERRARIc,3, AND FAUSTO GOZZId,4 Abstract. We establish the existence and uniqueness of the equilibrium for a stochastic mean-field game of optimal investment. The analysis covers both finite and infinite time horizons, and the mean-field interaction of the representative company with a mass of identical and indistinguishable firms is modeled through the time-dependent price at which the produced good is sold. At equilibrium, this price is given in terms of a nonlinear function of the expected (optimally controlled) production capacity of the representative company at each time. The proof of the existence and uniqueness of the mean-field equilibrium relies on a priori estimates and the study of nonlinear integral equations, but employs different techniques for the finite and infinite horizon cases. Additionally, we investigate the deterministic counterpart of the mean-field game under study. Keywords: mean-field games; mean-field equilibrium; forward-backward ODEs; optimal investment; price formation. AMS 2020: 35Q89; 47H10; 49N10; 49N80; 91A07; 91B38; 91B70. JEL classification: C02; C61; C62; C72; D25; D41. 1. Introduction In this paper, we consider a mean-field model of optimal investment with competition à la Cournot, where the price of the good produced by a representative company depends on the aggregate production of the entire economy through a nonlinear inverse demand function. In the absence of interventions, the production capacity of the representative company evolves stochastically over time as a geometric Brownian motion, and its level can be increased through investment, subject to quadratic costs. The representative company discounts profits and costs at a constant rate and aims to maximize total expected profits from production, net of investment costs. Instantaneous profits depend linearly on the company’s production capacity (thus, production occurs at full capacity) and on the time-dependent price of the produced good. The mean-field equilibrium investment and average production processes (b u,bq)are such that expected total net profits are maximized and, assuming an isoelastic inverse demand function, the price is given in terms of a nonlinear function of the average optimally controlled production at each time (see also Achdou et al. [2] and Remark 2.4 below). We are able to prove the existence and uniqueness of the equilibrium pair (b u,bq)when the problem’s time horizon is finite or infinite. Furthermore, we demonstrate that the existence and aUniversity of Parma, Department of Economics and Management, Via J. F. Kennedy 6, 43125 Parma (Italy). bUniversity of Bologna, Department of Mathematics, Piazza di Porta San Donato 5, 40126 Bologna (Italy). cBielefeld University, Center for Mathematical Economics (IMW), Universitätsstrasse 25, 33615 Bielefeld (Germany). dLUISS University, Department of Economics and Finance, Viale Romania 32, 00197 Roma (Italy). 1E-mail: [email protected]. 2E-mail: [email protected]. 3E-mail: [email protected]. 4E-mail: [email protected]. Giorgio Ferrari gratefully acknowledges financial support from Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)– Project-ID 317210226– SFB 1283. This work started during the visit of Alessandro Calvia, Salvatore Federico and Fausto Gozzi at the Center for Mathematical Economics (IMW) at Bielefeld University. Those authors thank IMW and the SFB 1283 for the support and hospitality. Alessandro Calvia, Salvatore Federico and Fausto Gozzi are supported by the Italian Ministry of University and Research (MUR), in the framework of PRIN project 2017FKHBA8 001 (The Time-Space Evolution of Economic Activities: Mathematical Models and Empirical Applications). Alessandro Calvia and Salvatore Federico are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica "Francesco Severi" (INdAM). . 1 2 A. CALVIA, S. FEDERICO, G. FERRARI, AND F. GOZZI uniqueness carry over to the deterministic counterpart of our model, essentially using the same techniques as in the stochastic case. The investment problem under study falls under the category of mean-field games with scalar interaction. Mean-field games, independently introduced by Lasry and Lions [17], and Huang, Caines, and Malhamé [15], represent limit models of non-cooperative symmetric N-player games with mean-field interaction as the number of players Ntends to infinity. An exhaustive review of mean-field models can be found in the two-volume book by Carmona and Delarue [8]. In our context, the consistency condition that the equilibrium price aligns at each time with a decreasing nonlinear function of the expected (optimally controlled) production capacity can be viewed as the limit, as the number Nof identical and indistinguishable companies operating in the market diverges, of the requirement that price inversely depends on the aggregate production of the entire economy, scaled by a factor of 1/N. As discussed in Huang et al. [15], this scaling can be justified by considering situations where "an increasing number of firms join together to serve an increasing number of consumers" (see the discussion in Huang et al. [15], after Equation (2.4) therein). The equilibrium construction for a given time horizon T∈(0,+∞]follows a three-step approach. Firstly, given a deterministic path of the average production q:= (qs)s∈[0,T ], we solve the representative company’s optimal investment problem. Because the production capacity evolves as a geometric Brownian motion and is linearly dependent on the investment process and since the performance criterion has linear dependence on the production capacity and quadratic dependence on the investment costs, the resulting optimal control problem for the representative company is of linear-quadratic type and has an explicit solution. Moreover, the optimal control b u(T,q)for fixed average production trajectory qis deterministic. In the second step, we calculate the expectation of the optimally controlled production capacity process, which is easily computable given the explicit representation of the state process. Finally, in the third step, we impose the consistency condition, requiring that this expectation must match qsfor each time s∈[0, T]. This condition leads to a nonlinear integral equation for the equilibrium average production trajectory bq:= (bqs)s∈[0,T]. By construction the control b u:=b u(T,bq)and the function bqform a mean-field equilibrium. It is important to note that although the company’s optimal control problem for a fixed average production trajectory qis linear-quadratic, due to the nonlinear dependence of the net profit functional on q, the overall mean-field problem is not of linear-quadratic type. In this regard, our results differ from those presented in Bensoussan et al. [3], Delarue and Tchuendom [12], Tchuendom [19], among others, which focus on linear-quadratic mean-field games. While the two-step approach previously outlined works for both the case where T < +∞and the case where T= +∞, different technical arguments are used to prove the existence and uniqueness of the equilibrium in the finite and infinite horizon cases. As a matter of fact, the integral equation uniquely characterizing the equilibrium average production bqdoes not fall into the standard theory of integro-differential equations, as we have an initial-value problem with the integral being backward in time, and thus requires a careful ad hoc analysis. Specifically, the proof of the existence of a solution to it relies on a priori estimates and Schauder’s fixed point theorem when T < +∞, whereas it utilizes a priori estimates and the Frechet-Kolmorogorov Lpcompactness theorem in the case of T= +∞. Uniqueness is then established in both cases through suitable contradiction arguments that exploit the properties of any mean-field equilibrium. Notably, when T= +∞, we can also prove that the time-dependent equilibrium bqmonotonically converges to the stationary average production level. The latter is obtained as the unique constant solution to the integral equation that uniquely determines bq. Given the geometric dynamics of the production capacity, the linear dependence of the profit functional on the controlled state variable, the quadratic costs of investment, and the fact that the mean-field interaction is of scalar type and only involves the expected value of the latter, it is not surprising that the mean-field equilibrium is deterministic and does not depend on the volatility coefficient σappearing in the production capacity’s dynamics. Guided by this observation, we also consider the deterministic counterpart of the previously discussed mean-field game and show that a unique equilibrium exists in this setting as well. This equilibrium can indeed be constructed by following exactly the same arguments employed in the stochastic case. A MEAN-FIELD MODEL OF OPTIMAL INVESTMENT 3 Games of optimal investment in both stochastic and deterministic settings are extensively covered in the Economics literature, and a comprehensive overview of models and results can be found in Vives [20]. Specifically, mean-field problems with Cournot competition have garnered interest in recent literature; see Chan and Sircar [10] for an insightful overview. Graber and Bensoussan [14] study the existence (and, under certain conditions, uniqueness) of a solution to a system of partial differential equations (PDEs) (namely, the Hamilton-Jacobi-Bellman (HJB) and FokkerPlanck equations) associated with a mean-field game involving Bertrand and Cournot competition among a continuum of players. In Graber and Sircar [13], existence and uniqueness of the master equation associated with a mean-field game of controls with absorption are proven, while Chan and Sircar [9] delve into dynamic mean-field games with exhaustible capacities and interactions akin to Cournot and Bertrand competitions. An optimal transport perspective on Cournot-Nash equilibria is explored in Acciaio et al. [1]. Finally, Cao et al. [6] considers stationary discounted and ergodic mean-field games of singular controls motivated by irreversible investment and provide existence and uniqueness results, as well as relations across the two classes of considered problems. To the best of our knowledge, the existence and uniqueness of the (nonstationary) mean-field equilibrium for a mean-field model of optimal investment with an isoelastic demand function, as discussed in this paper, is presented here for the first time. The rest of the paper is organized as follows. The stochastic mean-field game is introduced in Section 2. In Section 3the company’s optimal investment problem is solved for both the cases T < +∞and T= +∞, while existence and uniqueness of the equilibrium is shown in Section 4, again for the finite and infinite time horizon cases. The deterministic version of the mean-field problem is finally considered in Section 5. 1.1. Notation. In this section we collect the main notation used in this work. •Throughout the paper the set Ndenotes the set of natural integers without the zero element, i.e., N={1,2, . . . }, while Rdenotes the set of real numbers. Whenever T= +∞, the notation [0, T]indicates the interval [0,+∞). •For any p≥1, any measure space (E, E, µ), and any interval I⊆R, we indicate by Lp((E, E, µ); I)the set of all functions with values in Ithat are p-integrable with respect to µ. If E⊆R, we take L(E)as the Lebesgue σ-algebra and as µthe Lebesgue measure, which is denoted by Leb, and we simply write Lp(E;I). When computing integrals with respect to this measure, we simply write dxinstead of Leb(dx). •For any η > 0and E, I ⊆R, the notation Lp η(E;I)indicates the set of all functions f:E→Isuch that t7→ e−ηtf(t)is p-integrable with respect to the Lebesgue measure. •Given two intervals I, J ⊆R, we denote by C(I;J)the set of all continuous functions from Ito J, endowed with the usual sup-norm. The notation Cp(I;J),p∈N∪ {+∞}, denotes the set of functions from Ito Jthat are continuously differentiable ptimes. 2. The stochastic model Let (Ω,F,F:= (Fs)s≥0,P)be a complete filtered probability space, with Fsatisfying the usual assumptions, supporting an F-adapted standard Brownian motion Band an independent F0-measurable random variable ξ. Throughout the paper, T∈(0,+∞)∪ {+∞} denotes a finite or infinite time horizon. In what follows, whenever T= +∞the notation [0, T]indicates the interval [0,+∞). We consider a real-valued F-adapted process X= (Xs)s∈[0,T], satisfying the stochastic differential equation (SDE) dXs=−δXsds+σXsdBs+usds, s ∈(0, T], X0=ξ , (2.1) where δ, σ > 0are given coefficients, and the control process u:= (us)s∈[0,T]is chosen in either of the following two classes of admissible controls: if T < +∞, UT:=u= (us)s∈[0,T ]s.t. u: Ω ×[0, T]→[0,+∞)is (Fs)s∈[0,T ]-progressively measurable and 4 A. CALVIA, S. FEDERICO, G. FERRARI, AND F. GOZZI EZT 0 u2 sds<+∞;(2.2) if, instead, T= +∞, U∞:=u= (us)s≥0s.t. u: Ω ×[0,∞)→[0,+∞)is F-progressively measurable, EZt 0 usds<+∞,P-a.s., ∀t≥0,and EZ+∞ 0 e−ρsu2 sds<+∞.(2.3) Whenever necessary, to stress the dependence of the solution to (2.1) on the initial condition ξand on the control u, we denote it by Xξ,u. We observe that the solution to SDE (2.1) can be written explicitly (cf. [16, Problem 5.6.15]), for each u∈ UT, as Xξ,u s=Ysξ+Zs 0 Ys Yr urdr, s ∈[0, T ],(2.4) where Ys:= eσBs−δ+σ2 2s, s ∈[0, T].(2.5) In the mean-field game studied in this paper, the process Xdescribes the private state of a representative player. In particular, it models the evolution of her/his production capacity, which depreciates at a rate δand can be increased by choosing the investment rate u∈ UT. Note that the statistical distribution of the state of the other players in the economy does not influence the production capacity level of the representative player. Throughout the paper, we work under the following assumption. Assumption 2.1. The initial condition ξof SDE (2.1)is an F0-measurable, positive, and integrable random variable. More precisely, ξ > 0,P-a.s., and 0<E[ξ]<+∞. The assumption above ensures that, for any u∈ UT,Xξ,u s≥0,P-a.s., for all s∈[0, T]. Thus, the production capacity level of the representative agent is never negative. Moreover, it grants us the following result, which will be used later on. Lemma 2.2. Under Assumption 2.1 and for any u∈ UT, the unique solution to SDE (2.1)has finite first moment, given by E[Xξ,u s] = E[ξ]e−δs +EZs 0 e−δ(s−r)urdr, s ∈[0, T].(2.6) Proof. Fix ξand uas above. Using the expression of Xξ,ugiven in (2.4) we have that E[Xξ,u s] = E[Ysξ] + EZs 0 Ys Yr urdr, s ∈[0, T ]. Since the random variable Ysξis non-negative and Ysis independent of F0, for all s∈[0, T], we can directly compute the first summand E[Ysξ] = E[ξ]E[Ys] = E[ξ]e−δs. For the second summand, observe that the integrand is non-negative and that, for each fixed s∈[0, T], the random variable Ys Yris independent of Fr, for all r∈[0, s]. Therefore, applying the Fubini-Tonelli theorem, EZs 0 Ys Yr urdr=Zs 0 EYs Yr urdr=Zs 0 EYs YrE[ur] dr=EZs 0 e−δ(s−r)urdr, which is finite thanks to the assumptions on u. Therefore, we get (2.6). □ In our mean-field game we assume that every player aims at maximizing the discounted net profit functional JT,q(ξ, u):=EZT 0 e−ρs Xξ,u sq−β s−1 2u2 sds,(2.7) A MEAN-FIELD MODEL OF OPTIMAL INVESTMENT 5 where ρ > 0is a discount factor, β > 0is a fixed parameter, and q= (qs)s∈[0,T]is a given deterministic measurable function. At equilibrium, the function qwill identify with the average production capacity of the whole population of agents. This is formalized in the next definition. Definition 2.3. Fix a random variable ξunder Assumption 2.1. A pair (b u,bq), where b u∈ UTand bq: [0, T]→(0,+∞)is a measurable function, is an equilibrium of the mean-field game if (i) JT,bq(ξ, b u)≥JT,bq(ξ, u), for all u∈ UT; (ii) bqs=E[Xξ,b u s], for all s∈[0, T ]. In the next sections we show that there exists a unique equilibrium by adopting a classic fixed point approach. The first step is to solve an optimization problem for a given measurable function q, representing the evolution of the average production capacity level of the agents in the economy. We show that there exists a unique explicit and deterministic optimal control b u∈ UT, depending on q, which provides us with the best response of the representative agent to the distribution of the states of the other agents in the economy, summarized by the average q. The second step is to determine the optimally controlled dynamics of the production capacity level of the representative agent. Thanks to (2.6), we can compute explicitly E[Xξ,b u s],s∈[0, T]. Since the optimal control b udepends on q, the fixed point argument follows from condition (ii) in Definition 2.3. We show that proving that there exists a unique equilibrium reduces to finding the unique solution of an integro-differential equation. Remark 2.4. The profit function appearing in (2.7)– i.e., the function (x, p)7→ xp−β– is related to the isoelastic demand obtained from Spence-Dixit-Stiglitz preferences and can be motivated as follows.1 Assume that each player in our economy is a firm indexed by its productivity (or size) x > 0. We can consider the firms’ production capacities as a proxy for this index and, thus, this setting is coherent with the model introduced above. Each firm produces a single good and faces the demand function π(x) P−γ ,with γ > 1, where π(x)is the price set by a firm with productivity level x, and P=RRπ(x)1−γµ(dx)1/(1−γ)is a price index, which is computed according to the statistical distribution µof the firms productivity in the economy. Here we are assuming that the price set by two firms having an equal productivity level is the same. In other words, the price policy of each firm is exclusively determined by its productivity. Each firm (i.e., fix x > 0) aims at maximizing the function π7→ ππ P−γ−1 xπ P−γ. It is easy to see that the maximum profit is 1 γ−1γ γ−1−γ xγ−1Pγ,(2.8) which is attained setting the price π(x) = γ (γ−1)x. This entails that P=γ (γ−1) RRxγ−1µ(dx)1/(1−γ), and hence, substituting this expression into (2.8), we get that the profit is 1 γ−1xγ−1p−γ,with p=ZR xγ−1µ(dx)1 γ−1.(2.9) Therefore, setting β=γ= 2, we get the profit function (x, p)7→ xp−βappearing in our model. Note, however, that in the discounted net profit functional introduced in (2.7)we do not restrict the choice of the parameter βto the one dictated by the economic application outlined above. Instead, 1See also the discussion in Achdou et al. [2, p. 7, Footnote 5] concerning the the model in Luttmer [18]. 6 A. CALVIA, S. FEDERICO, G. FERRARI, AND F. GOZZI we consider any possible value β > 0, as the mathematical results that we prove can be stated in this more general framework. The discussion above highlights that it is possible to consider more general forms of interaction between players in the profit function. In this paper we consider the average of the production capacities of the firms in the economy, but in (2.9)the geometric average appears. More generally, one can consider p=FZR f(x)µ(dx), for some F, f :R→[0,+∞)strictly increasing, see, e.g., [6]. However, as will be clear later on, the analysis carried out in this work heavily relies on the fact that the interaction between players is through the average of the production capacities. Indeed, thanks to this feature, we will find an explicit characterization of the solution to the mean-field game introduced above. 3. The optimization problem In this section we consider the optimization problem associated to the mean-field game described in Section 2. We show that, for each possible choice (in a suitable class of measurable functions) of the function qappearing in the discounted net profit functional (2.7), the corresponding maximization problem has a unique solution and we compute explicitly the associated optimal control. We divide our analysis into two subsections, the first devoted to the finite time horizon case, the second one to the infinite time horizon case. 3.1. The finite time horizon case. Let us consider the finite time horizon case, i.e., fix T < +∞. For each fixed q: [0, T]→(0,+∞)such that q−β∈L1((0, T]; (0,+∞)), we consider the problem of maximizing the functional JT,q(ξ, u):=EZT 0 e−ρs Xξ,u sq−β s−1 2u2 sds,(3.1) where ξsatisfies Assumption 2.1,uis chosen in the class of admissible controls introduced in (2.2), and Xξ,uis the unique solution of SDE (2.1). We also introduce the value function corresponding to the optimization problem, namely, VT,q(ξ):= sup u∈UT JT,q(ξ, u), ξ ∈L1((Ω,F0,P); (0,+∞)).(3.2) In the next proposition we are going to show important properties of the functional JT,q. Proposition 3.1. Fix a random variable ξsatisfying Assumption 2.1,u∈ UT, and q: [0, T]→ (0,+∞)such that q−β∈L1((0, T]; (0,+∞)). Then, the functional JT,q defined in (3.1)is finite and verifies JT,q(ξ, u) = E[ξ]z(T,q) 0+EZT 0 e−ρs z(T,q) sus−1 2u2 sds,(3.3) where z(T,q): [0, T]→[0,+∞)is the deterministic function given by z(T,q) s:=ZT s e−(ρ+δ)(r−s)q−β rdr, s ∈[0, T ].(3.4) Moreover, if ξ, ξ′both verify Assumption 2.1 and are such that E[ξ] = E[ξ′], then JT,q(ξ;u) = JT,q(ξ′;u), for any u∈ UT. Proof. Fix ξ∈L1((Ω,F0,P); (0,+∞)), and u∈ UT. Since the assumptions of Lemma 2.2 are verified, we can use (2.6) and apply the Fubini-Tonelli theorem (note that the integrand is nonnegative) to get EZT 0 e−ρsXξ,u sq−β sds=E[ξ]ZT 0 e−ρse−δsq−β sds+EZT 0Zs 0 e−ρse−δ(s−r)q−β surdrds. Clearly, the first summand of the last equality is finite, thanks to the assumptions on ξand q. Moreover, E[ξ]ZT 0 e−ρse−δsq−β sds=E[ξ]z(T,q) 0. A MEAN-FIELD MODEL OF OPTIMAL INVESTMENT 7 Also the second summand is finite, thanks to the assumptions on qand u. Exchanging the order of the two time integrals we get EZT 0Zs 0 e−ρse−δ(s−r)q−β surdrds=EZT 0 eδrurZT r e−(ρ+δ)sq−β sdsdr =EZT 0 e−ρrurZT r e−(ρ+δ)(s−r)q−β sdsdr=EZT 0 e−ρrz(T,q) rurdr. It easily follows that JT,q is finite and that Equation (3.3) holds, which also entails the last statement of the Proposition. □ Remark 3.2. The last statement in Proposition 3.1 implies that the value function VT,q depends on the initial condition ξonly through its average. More precisely, if ξ, ξ′both verify Assumption 2.1 and are such that E[ξ] = E[ξ′], then VT,q(ξ) = VT,q(ξ′). Thanks to the Proposition above we can find the optimal control for our optimization problem and this, as a byproduct, allows us to explicitly compute the value function. Theorem 3.3. Fix a random variable ξsatisfying Assumption 2.1 and q: [0, T]→(0,+∞)such that q−β∈L1((0, T]; (0,+∞)). Then, b u(T,q):=z(T,q)∈ UT, where z(T,q)is the function defined in (3.4), is an optimal control for problem (3.2), which is deterministic and independent of ξ. Moreover, b u(T,q)is essentially unique, i.e., if u(T,q)∈ UTis an optimal control for problem (3.2) different from b u(T,q), then u(T,q) s=bu(T,q) s,for P⊗Leb-a.e. (ω, s)∈Ω×[0, T]. Finally, the value of the optimization problem admits the explicit expression VT,q(ξ) = E[ξ]z(T,q) 0+1 2ZT 0 e−ρs(z(T,q) s)2ds, (3.5) and the optimally controlled state process Xξ,b u(T,q)is given by Xξ,b u(T,q) s=Ysξ+Zs 0 Ys Yr z(T,q) rdr, s ∈[0, T ],(3.6) where Yis the process defined in (2.5). Proof. Fix ξ∈L1((Ω,F0,P); (0,+∞)). From (3.3) it immediately follows that VT,q(ξ) = E[ξ]z(T,q) 0+ sup u∈UT EZT 0 e−ρs z(T,q) sus−1 2u2 sds.(3.7) Hence, if we can find b u(T,q)∈ UTthat maximizes the integrand z(T,q) sus−1 2u2 s, for P⊗Leb-a.e. (ω, s)∈Ω×[0, T], then b u(T,q)it must be optimal. Clearly, the integrand is maximized in the sense above if we take b u(T,q)=z(T,q). Its admissibility is a consequence of the fact that z(T,q)is bounded. Therefore, b u(T,q)is optimal, and clearly essentially unique in the sense specified above. Substituting its definition in (3.7) we immediately get (3.5) and (3.6). □ 3.2. The infinite time horizon case. We analyze now the infinite time horizon case, i.e., we set T= +∞. For each fixed q: [0,+∞)→(0,+∞)such that q−β∈L1((0,+∞); (0,+∞)), we consider the problem of maximizing the functional J∞,q(ξ, u):=EZ+∞ 0 e−ρs Xξ,u sq−β s−1 2u2 sds,(3.8) where ξsatisfies Assumption 2.1,uis chosen in the class of admissible controls defined in (2.3), and Xξ,uis the unique solution of SDE (2.1). We also introduce the value function corresponding to the optimization problem, namely, V∞,q(ξ):= sup u∈U∞ J∞,q(ξ, u), ξ ∈L1((Ω,F0,P); (0,+∞)).(3.9) The next result is analogous to Proposition 3.1. The proof proceeds along the same lines and, thus, we omit it. 14 A. CALVIA, S. FEDERICO, G. FERRARI, AND F. GOZZI We observe that, given the bounds in (4.22), the family Zis equibounded. We want to show that it is also equi-integrable in L1 ρ+2δ. For any s≥0and h > 0we have that z(n) s+h−z(n) s=1[0,n−h](s)Zs+h sz(n) r′dr+1[n−h,n](s)Zn sz(n) r′dr. Let K:=x−β ρ+δ−δβ . Using the fact that 0<z(n) s′≤Keδ(1+β)s, for all s≥0, and all n≥1, we get that, for any h > 0small enough, ∥z(n) ·+h−z(n) ·∥1,ρ+2δ=Z+∞ 0 e−(ρ+2δ)s|z(n) s+h−z(n) s|ds ≤Zn−h 0 e−(ρ+2δ)sZs+h sz(n) r′drds+Zn n−h e−(ρ+2δ)sZn sz(n) r′dr ≤K(eδ(1+β)h−1) δ(1 + β)Zn−h 0 e−(ρ+δ−δβ)sds +Keδ(1+β)n δ(1 + β)Zn n−h e−(ρ+2δ)sds−K δ(1 + β)Zn n−h e−(ρ+δ−δβ)sds | {z } ≤0 ≤K(eδ(1+β)h−1) δ(1 + β)(ρ+δ−δβ)[1 −e−(ρ+δ−δβ)(n−h) | {z } ≤0 ] + Ke−(ρ+δ−δβ)n δ(1 + β)(ρ+ 2δ)[e(ρ+2δ)h−1] ≤K(eδ(1+β)h−1) δ(1 + β)(ρ+δ−δβ)+K δ(1 + β)(ρ+ 2δ)[e(ρ+2δ)h−1],∀n≥1. Therefore, we obtain that lim h→0+sup n≥1 ∥z(n) ·+h−z(n) ·∥1,ρ+2δ= 0, i.e., that the family Zis equi-integrable in L1 ρ+2δ. By the Frechet-Kolmogorov theorem, it follows that Zis relatively compact, so we can extract a subsequence {z(n)}n≥1⊂ Z, still labeled (with an abuse of notation) by {z(n)}n≥1, such that z(n)−→ z(∞)with respect to ∥·∥1,ρ+2δ, as n→ ∞. Then, we can extract a sub-subsequence, still labeled (with an abuse of notation) by {z(n)}n≥1, such that z(n) s−→ z(∞) s, for almost every s≥0, as n→ ∞. Clearly, the limit z(∞)belongs to L1 ρ+2δand, since z(n)∈ Cx,∞, for any n≥1, we have that αx,∞ s≤z(∞) s≤αx,∞ s, for almost every s≥0, where αx,∞and αx,∞are the functions defined in (4.20) and (4.21). Now we use (4.7) to get that also the derivative {z(n)′}of the last subsequence {z(n)}converges a.e. to the function w(∞) s:= e(ρ+2δ)sZ+∞ s e−(ρ+δ−δβ)uz(∞) u−βdu, s ≥0, which belongs to L1 ρ+2δ. By standard computations, it follows that {z(n)′} ⊂ L1 ρ+2δand that 0<z(n) s′≤x−β ρ+δ−δβ eδ(1+β)s=:gs,∀s≥0,∀n≥1, with g∈L1 ρ+2δ. Therefore, the sequence of derivatives {z(n)′}converges in L1 ρ+2δto w(∞). We deduce that the subsequence {z(n)}converges in the weighted Sobolev space W1,1 ρ+2δ(cf. [4, Chapter 8, Remark 4]). By completeness of this space, we get that w(∞) s= (z(∞) s)′, for a.e. s≥0. Moreover, using the same argument as in the proof of [4, Theorem 8.2], we get that there exists a function z(∞)∈C([0,+∞); (0,+∞)) such that z(∞) s=z(∞) s, for almost every s≥0, and z(∞) s=x+Zs 0 (z(∞) u)′du=x+Zs 0 w(∞) udu, ∀s≥0. A MEAN-FIELD MODEL OF OPTIMAL INVESTMENT 15 Substituting the expression of w(∞)in the previous equality we get that z(∞) s=x+Zs 0 e(ρ+2δ)rZ+∞ r e−(ρ+δ−δβ)uz(∞) u−βdudr s ≥0, that is, z(∞)∈ Cx,∞and solves (4.7). Uniqueness. As we observed at the beginning of the proof, any solution zto (4.7) is (at least) twice differentiable. Differentiating (4.7) with respect to the time variable and denoting by ′and ′′ firstand second-order derivatives with respect to this variable, we have that zverifies (z′′ s= (ρ+ 2δ)z′ s−eδ(1+β)sz−β s, s ≥0, z0=x . (4.25) The equation above is a second-order ODE with locally Lipschitz coefficients over (s, z)∈[0,+∞)× (0,+∞), to which we can associate the following initial value problem      z′′ s= (ρ+ 2δ)z′ s−eδ(1+β)sz−β s, s ≥0, z0=x , z′ 0=ζ > 0. (4.26) Note that we only need to consider ζ > 0, since solutions to (4.7) are such that their first derivative is strictly positive for all s≥0. For each fixed x > 0and ζ > 0, (4.26) has a unique solution zx,ζ on the maximal interval [0, τ∗(x, ζ)), with τ∗(x, ζ)≤+∞. Since any solution zto (4.7) also verifies (4.25), it also satisfies (4.26) for some ζ > 0. Therefore, if we show that for each given x > 0, there exists a unique ζ > 0such that (4.26) has a unique global solution zx,ζ ∈ Cx,∞, then necessarily ζ=ζand zx,ζ must also be the unique solution to (4.7). The idea is, thus, to study, for each fixed x > 0, the dependence of the solution zx,ζ to (4.26) on ζ > 0. Let us fix x > 0. Standard results (see, e.g., [11, Theorem 7,5, Chapter 1]) ensure that the solution zx,ζ to (4.26) and its firstand second-order derivatives with respect to the time variable are differentiable with respect to ζand that the order of differentiation can be exchanged. Therefore, defining bzs(ζ):=∂ ∂ζ zx,ζ s, for all s≥0, we get from (4.26) that bz(ζ)solves the initial value problem     bz′′ s(ζ)=(ρ+ 2δ)bz′ s(ζ) + βeδ(1+β)szs(ζ)−β−1bzs(ζ), s ≥0, bz0(ζ) = 0 , bz′ 0(ζ) = 1 , (4.27) where zs(ζ):=zx,ζ s. Let s∗(ζ):= inf{s≥0: bz′ s(ζ)≤0}>0. By definition z′ s(ζ)>0, for all s∈[0, s∗(ζ)), which implies bzs(ζ)>0,∀s∈[0, s∗(ζ)). Therefore, using (4.27) we get that bz′′ s(ζ)>0, for all s∈[0, s∗(ζ)). It follows that bz′ s(ζ)≥1,∀s∈[0, s∗(ζ)), which implies that s∗(ζ) = +∞and, in turn, that bzs(ζ)≥s, for all s≥0. Using again (4.27), we then get (bz′′ s(ζ)≥(ρ+ 2δ)bz′ s(ζ), s ≥0, bz′ 0(ζ) = 1 ,(4.28) which, implies, by comparison, bz′ s(ζ)≥e(ρ+2δ)s,∀s≥0, and, consequently, bzs(ζ)≥1 ρ+ 2δ[e(ρ+2δ)s−1],∀s≥0. Thus, for every a<b, we have 16 A. CALVIA, S. FEDERICO, G. FERRARI, AND F. GOZZI zs(b)−zs(a)≥Zb a ˆzs(ζ) dζ≥(b−a)1 ρ+ 2δ[e(ρ+2δ)s−1],∀s≥0. We know that, for some ζ > 0, there exists a global solution z(ζ)∈ Cx,∞to (4.26), since any solution zto (4.7) also verifies this equation. Using the estimate above and (4.22), we get, for all b > a :=ζ, zs(b)≥zs(ζ)+(b−ζ)[e(ρ+2δ)s−1] ≥x+ (b−ζ)1 ρ+ 2δ[e(ρ+2δ)s−1],∀s≥0. This implies that z(b)would grow exponentially at a rate at least ρ+ 2δ > δ(1 + β), which contradicts the fact that solutions in (4.22) need to grow at a rate at most δ(1 + β). Similarly, we get, for all a < ζ =:b, zs(a)≤zs(ζ)−(ζ−a)[e(ρ+2δ)s−1] ≤αx,∞ s−(ζ−a)1 ρ+ 2δ[e(ρ+2δ)s−1],∀s≥0. This implies that z(a)becomes negative in finite time, which contradicts the fact that solutions in (4.22) are positive. By arbitrariness of b > ζ and a < ζ, we conclude. Convergence. As recalled at the beginning of the proof, uniqueness of the solution to (4.7) implies uniqueness of the solution to (4.6), which is also twice continuously differentiable. Differentiating (4.6), we get the second-order ODE y′′ s=ρy′ s+ (ρ+δ)δys−y−β s, s ≥0.(4.29) Hence, any solution to (4.6) is also a solution to (4.29). We observe that y∞is the unique solution to the algebraic equation (ρ+δ)δy −y−β= 0.(4.30) Hence, the function ys=y∞, for all s≥0, is the unique stationary solution to (4.6). Now, we show monotonicity and convergence of the solution to (4.6) when x=y∞. We prove the case x>y∞, as the other one can be established by similar arguments. Let ybe the unique solution to (4.6) and fix x>y∞. We prove, first, that ys> y∞, for all s≥0. Let us define s0:= inf{s≥0: ys=y∞}, and assume, by contradiction, that s0<+∞. We have two cases: (i) y′ s0<0; (ii) y′ s0= 0. Consider the first case. By the flow property, the function eys:=ys−s0,s≥s0, solves (4.6) with initial condition equal to y∞and it satisfies the a priori bounds given in (4.23) (with x=y∞). However, this solution is different from the constant one, which is the unique solution to (4.6) with initial condition y∞and satisfying the aforementioned bounds. Hence we have a contradiction. The second case leads to a contradiction as well. Indeed, we would have two different solutions in the interval [0, s0]to the Cauchy problem (4.29), with terminal conditions ys0=y∞and y′ s0= 0, which clearly admits unique solution, that is, ys=y∞, for all s≥0. Hence, we have proved that ys> y∞, for all s≥0. This implies, by strict monotonicity of the map y7→ (ρ+δ)δy −y−βand (4.30), that (ρ+δ)δys−y−β s>(ρ+δ)δy∞−y−β ∞= 0,∀s≥0,(4.31) and hence, from (4.29), that y′′ s> ρy′ s,∀s≥0.(4.32) Now we show that y′ s<0, for all s≥0. Let s1:= inf{s≥0: y′ s≥0}. A MEAN-FIELD MODEL OF OPTIMAL INVESTMENT 17 Assume, by contradiction, that s1<+∞. Then, from (4.32) we get that y′′ s1>0. It follows that there exist η, ε > 0such that y′ s1+ε=η > 0. Using (4.32) again, we get (y′′ s> ρy′ s,∀s≥s1+ε, y′ s1+ε=η > 0.(4.33) Therefore, solving the corresponding Cauchy problem and using the comparison, we get y′ s≥ηeρ[s−(s1+ε)] >0, s ≥s1+ε. (4.34) Next, differentiating (4.29), we get that y∈C3([0,+∞); (0,+∞)) and that it satisfies y′′′ s=ρy′′ s+ (ρ+δ)δy′ s+βy−β−1 sy′ s, s ≥0.(4.35) Since, by (4.34), we know that y′ s>0for all s≥s1+ε, we deduce that      y′′′ s> ρy′′ s+ (ρ+δ)δy′ s,∀s≥s1+ε, y′ s1+ε=η, y′′ s1+ε=κ, with η > 0and κ > ηρ > 0, by (4.33). Solving the corresponding Cauchy problem and using the comparison, we get y′ s>ηδ +κ ρ+ 2δe(ρ+δ)[s−(s1+ε)] +η(ρ+δ)−κ ρ+ 2δe−δ[s−(s1+ε)],∀s≥s1+ε . This implies that ygrows, in the long run, at least with rate ρ+δwhich is strictly bigger than δβ, as prescribed by the admissibility condition in (4.23). The contradiction follows, and hence we proved that y′ s<0, for all s≥0. We are now going to show that y′′ s>0, for all s≥0. Since we proved that y′ s<0, for all s≥0, we deduce from (4.35) that y′′′ s< ρy′′ s,∀s≥0.(4.36) Assume by contradiction that y′′ s2≤0for some s2≥0. Then, from (4.36) we get that y′′′ s2<0. It follows that there exist ϑ < 0,τ > 0such that y′′ s2+τ=ϑ < 0. Using again (4.36), we get (y′′′ s< ρy′′ s,∀s≥s2+τ, y′′ s2+τ=ϑ. Therefore, y′′ s≤ϑeρs <0, for all s≥s2+τ. Considering that y′ s<0, for all s≥0, this implies that the graph of s7→ yslies below a straight line with strictly negative slope, contradicting the fact that ys> y∞, for all s≥0, as previously established. So, we have proved that, for all s≥0, ys> y∞, y′ s<0, y′′ s>0. This implies that there exists ¯y:= lim s→∞ ys≥y∞and that lim s→∞ y′ s= 0.However, it cannot be that ¯y > y∞, as it would imply, using (4.29), that lim s→∞ y′′ s>0, contradicting the fact that y′ s→0, as s→+∞. Therefore, ¯y=y∞.□ We are now ready to establish existence and uniqueness of an equilibrium for the mean-field game. Proposition 4.13. For each fixed random variable ξsatisfying Assumption 2.1, there exists a unique equilibrium (b u,bq)for the mean-field game with infinite time horizon, among all equilibria verifying condition (4.3). More precisely, b u=z(∞,bq), where z(∞,q)is the function defined in (3.10), and bqis the unique solution to (4.6), with initial condition x=E[ξ]. 18 A. CALVIA, S. FEDERICO, G. FERRARI, AND F. GOZZI Proof. Let bqbe the unique solution to (4.6), with x=E[ξ]. We need to check that condition (4.3) is satisfied. We have the following three cases. Case x=y∞.From Theorem 4.12, we know that bqs=y∞, for all s≥0. Therefore, z(∞,bq) s=K:=y−β ∞ ρ+δ,∀s≥0,(4.37) and hence bqverifies condition (4.3). Case x>y∞.From Theorem 4.12, we know that bqis monotone decreasing and converges to y∞. This implies that, 0< y∞<bqs≤x, for all s≥0, and that z(∞,bq)is positive and bounded above by the constant Kdefined in (4.37). Therefore, bqsatisfies condition (4.3). Case x<y∞.From Theorem 4.12, we know that bqis monotone increasing and converges to y∞. This implies that, 0< x ≤bqs< y∞, for all s≥0, and that z(∞,bq)is positive and bounded above by x−β ρ+δ. Therefore, bqsatisfies condition (4.3). Applying Theorem 4.1-(iv) we get the result. □ Remark 4.14. Also in this case the results of this section can be easily extended to the case where we consider an initial time time t > 0. Clearly, we need to adapt (in an obvious way) the definition of equilibrium given in Definition 2.3 to include the initial time t. The statement of Theorem 4.1 is also adapted accordingly. The integro-differential equation (4.6)becomes    d dsys=−δys+ZT s e−(ρ+δ)(u−s)y−β udu, s ∈[t, T], yt=x , (4.38) while equation (4.7)becomes      d dszs= e(ρ+2δ)sZT s e−(ρ+δ−δβ)uz−β udu, s ∈[t, T], zt= eδtx . (4.39) The sets in which we look for solutions to (4.39)(now dependent on t) are Ct,x :={f∈C([t, T]; (0,+∞)): eδtαx s≤fs≤eδtαx s,∀s∈[t, T]},(4.40) in the finite time horizon case, and Ct,x,∞:={f∈C([t, +∞); (0,+∞)): eδtαx,∞ s≤fs≤eδtαx,∞ s,∀s≥t},(4.41) in the infinite time horizon case. The statements of Theorems 4.9 and 4.12, and of Propositions 4.10 and 4.13, remain the same (except for minor modifications). The bounds given in (4.14)become e−δ(s−t)x≤ys≤e−δ(s−t)αx s, s ∈[t, T],(4.42) while those given in (4.23)become e−δ(s−t)x≤ys≤e−δ(s−t)x−x−β δ(1 + β)(ρ+δ−δβ)+eδβ(s−t)x−β δ(1 + β)(ρ+δ−δβ), s ≥t. (4.43) 5. The deterministic model In this section we discuss the deterministic version of the mean-field game problem introduced in Section 2. The structure of the problem will allow us to exploit the results given in Sections 3 and 4, to get the unique equilibrium of the mean-field game in an explicit form. Let us consider the ordinary differential equation (ODE)    d dsXs=−δXs+us, s ∈(0, T], X0=x , (5.1) A MEAN-FIELD MODEL OF OPTIMAL INVESTMENT 19 where δ > 0is a given coefficient and the control u:= (us)s∈[0,T ]is chosen in either of the following two classes of admissible controls: if T < +∞, UT:=u: [0, T]→[0,+∞)measurable and s.t. ZT 0 u2 sds < +∞;(5.2) if, instead, T= +∞, U∞:=u: [0,∞)→[0,+∞)measurable and s.t. Zs 0 urdr < +∞,∀s≥0, and Z+∞ 0 e−ρsu2 sds < +∞.(5.3) Note that, for any x∈Rand any u∈ UT, Equation (5.1) has a unique solution Xx,u, given by Xx,u s= e−δsx+Zs 0 e−δ(s−r)urdr, s ∈[0, T ].(5.4) As in the stochastic case, Xdescribes the evolution of the production capacity of a representative agent, which depreciates at a rate δand can be increased by choosing the investment rate u∈ UT. Next, we consider the discounted net profit functional JT,q(x, u):=ZT 0 e−ρs Xx,u sq−β s−1 2u2 sds, (5.5) where ρ > 0is a discount factor, β > 0is a fixed parameter, and q= (qs)s∈[0,T]is a given deterministic measurable function. We introduce the following assumption. Assumption 5.1. The initial distribution of the agents in the economy ν0has a density m0with respect to the Lebesgue measure on R, is supported on (0,+∞), and has finite first moment. More precisely, there exists m0∈L1((0,+∞); (0,+∞)) such that ν0(A) = ZA m0(x) dx, ∀A∈L(R),Z+∞ 0 m0(x) dx= 1,Z+∞ 0 x m0(x) dx < +∞. Since the representative agent is chosen randomly by picking her/his initial production capacity level xaccording to the initial distribution ν0, the assumption above ensures that, for any u∈ UT, Xx,u s≥0, for all s∈[0, T]. Thus, the production capacity level of the representative agent is never negative. To introduce the definition of equilibrium for the deterministic mean-field game, we need to consider the so-called continuity equation (also known as Liouville equation, cf. [5]) associated to ODE (5.1)    ∂ ∂sp(s, x) + ∂ ∂x (p(s, x)[−δx +us]) = 0,(s, x)∈(0, T ]×(0,+∞), p(0, x) = p0(x), x ∈(0,+∞), (5.6) where u= (us)s∈[0,T]∈ UTand p0is a given initial probability density function. This equation describes the evolution of p0under the flow determined by ODE (5.1). Solutions to (5.6) are understood in the weak sense, according to the following definition. Definition 5.2. A function p∈L1([0, T]×(0,+∞); (0,+∞)) is a weak solution to (5.6)if, for any function φ∈C∞([0, T]×(0,+∞)) with compact support, and any t > 0, we have Z+∞ 0 φ(t, x)p(t, x) dx(5.7) =Z+∞ 0 φ(0, x)p0(x) dx+Zt 0Z+∞ 0∂ ∂sφ(s, x)+[−δx +us]∂ ∂xφ(s, x)p(s, x) dxds . 20 A. CALVIA, S. FEDERICO, G. FERRARI, AND F. GOZZI Lemma 5.3. For any u∈ UTand any initial probability density p0, supported on (0,+∞)and with finite first moment, the continuity equation (5.6)has a unique solution, given by pp0,u(s, x)=eδsp0eδsx−Zs 0 eδrurdr,(s, x)∈[0, T]×(0,+∞).(5.8) Moreover, for each s∈[0, T], the probability density pp0,u(s, ·)has finite first moment, given by Z+∞ 0 x pp0,u(s, x) dx= e−δs Z+∞ 0 x p0(x) dx+Zs 0 e−δ(s−r)urdr s ∈[0, T].(5.9) Proof. Fix u∈ UTand p0as above. The fact that pp0,uis a solution to (5.6) can be verified via standard computations (see, e.g., [5, Example 2.5.2] or [7, Lemma 4.15]). Uniqueness follows from an argument similar to the one used in the proof of [7, Lemma 4.16]. Finally, using the change of variables y= eδsx−Rs 0eδrurdr, we get that, for each s∈[0, T], Z+∞ 0 x pp0,u(s, x) dx=Z+∞ 0 xeδsp0eδsx−Zs 0 eδrurdrdx =Z+∞ 0 e−δs y+Zs 0 eδrurdrp0(y) dy= e−δs Z+∞ 0 y p0(x) dy+Zs 0 e−δ(s−r)urdr , which is finite thanks to the assumptions on p0and u. Thus, pp0,u(s, ·)has finite first moment, for any s∈[0, T ], which satisfies (5.9). □ We are now ready to state the definition of equilibrium for our deterministic mean-field game. Definition 5.4. Fix an initial distribution ν0under Assumption 5.1 and let m0be its density with respect to the Lebesgue measure. A pair (b u,bq), where b u∈ UTand bq: [0, T]→(0,+∞)is a measurable function, is an equilibrium of the deterministic mean-field game if (i) JT,bq(x, b u)≥JT,bq(x, u), for all u∈ UTand ν0-a.e. x > 0; (ii) bqs=Z+∞ 0 x mm0,b u(s, x) dx, for all s∈[0, T ], where mm0,b uis the unique solution of the continuity equation (5.6), given by (5.8), with initial condition m0and control b u. Also in this case, we can adopt a fixed point argument to show that there exists a unique equilibrium, by solving, first, the problem of maximizing (5.5) for a given measurable function q, then determining the optimally controlled dynamics of the production capacity level of the representative agent and, finally, setting the fixed point argument from condition (ii) in Definition 5.4. 5.1. The optimization problem. Let us consider the optimization problem VT,q(x):= sup u∈UT JT,q(x, u), x > 0,(5.10) where JT,q is the discounted net profit functional defined in (5.5), for any given and fixed qmeasurable and deterministic function. In the finite time horizon case, i.e. T < +∞, the following result holds, which is analogous to Theorem 3.3. Theorem 5.5. Fix x > 0and q: [0, T]→(0,+∞)such that q−β∈L1((0, T]; (0,+∞)). Then, b u(T,q):=z(T,q)∈ UT, where z(T,q)is the function defined in (3.4), is an optimal control for problem (5.10). Moreover, b u(T,q)is essentially unique, i.e., if u(T,q)∈ UTis an optimal control for problem (5.10) different from b u(T,q), then u(T,q) s=bu(T,q) s,for Leb-a.e. s∈[0, T]. Finally, the value function of the optimization problem admits the explicit expression VT,q(x) = xz(T,q) 0+1 2ZT 0 e−ρs(z(T,q) s)2ds, (5.11) A MEAN-FIELD MODEL OF OPTIMAL INVESTMENT 21 and the optimally controlled state Xx,b u(T,q)is given by Xx,b u(T,q) s= e−δsx+Zs 0 e−δ(s−r)z(T,q) rdr , s ∈[0, T].(5.12) Proof. Replicating the argument of the proof of Proposition 3.1, we get that JT,q(x, u) = xz(T,q) 0+ZT 0 e−ρs z(T,q) sus−1 2u2 sds . Then, the result follows using the same reasoning as in the proof of Theorem 3.3.□ In the infinite time horizon case, i.e. T= +∞, we have the following statement, which is the deterministic counterpart of Theorem 3.6. We state it without proof. Theorem 5.6. Fix x > 0and consider a function q: [0,+∞)→(0,+∞)such that q−β∈ L1 ρ+δ((0,+∞); (0,+∞)) and such that the function z(∞,q), defined in (3.10)is bounded on [0,+∞). Then, b u(∞,q):=z(∞,q)∈ U∞is an optimal control for problem (5.10). Moreover, b u(∞,q)is essentially unique, i.e., if u(∞,q)∈ U∞is an optimal control for problem (5.10)different from b u(∞,q), then u(∞,q) s=bu(∞,q) s,for Leb-a.e. s≥0. Finally, the value function of the optimization problem admits the explicit expression V∞,q(x) = xz(∞,q) 0+1 2Z+∞ 0 e−ρs(z(∞,q) s)2ds, (5.13) and the optimally controlled state Xx,b u(∞,q)is given by Xx,b u(∞,q) s= e−δsx+Zs 0 e−δ(s−r)z(∞,q) rdr , s ≥0.(5.14) 5.2. Existence and uniqueness of equilibria. Also in the deterministic mean-field game previously introduced, the search for an equilibrium boils down to finding a fixed point of a suitable map. According to Definition 5.4, this map is (qs)s∈[0,T ]7→ R+∞ 0x mm0,b u(T,q)(s, x) dxs∈[0,T ], where b u(T,q)is the optimal control for the optimization problem (5.10) – whose expression is given either in Theorem 5.5, in the case T < +∞, or in Theorem 5.6, in the case T= +∞– and mm0,b u(T,q)is the unique solution to (5.6), with initial probability density m0and control b u(T,q). The next result, which is analogous to Theorem 4.1, shows that also in the deterministic case the fixed point map is the solution map of the same integral equation studied in Section 4. We omit its proof, which is a straightforward adaptation of the proof of Theorem 4.1. Theorem 5.7. Let us fix an initial distribution ν0satisfying Assumption 5.1, and let m0be its density with respect to the Lebesgue measure. Consider the deterministic mean-field game in the finite time horizon case, i.e., T < +∞. Then, (i) If there exists an equilibrium (b u,bq)in the sense of Definition 5.4, such that bq−β∈L1((0, T]; (0,+∞)),(5.15) then bqis a solution to the integral equation ys= e−δs Z+∞ 0 y m0(y) dy+Zs 0 e−δ(s−r)ZT r e−(ρ+δ)(u−r)y−β ududr, s ∈[0, T ].(5.16) (ii) Vice versa, if there exist a unique solution bqto (5.16)satisfying (5.15), then there exists a unique equilibrium (b u,bq) = (z(T,bq),bq)of the mean-field game among all equilibria (e u,eq) such that eqverifies (5.15), where z(T,q)is the function defined in (3.4). Consider, instead, the deterministic mean-field game in the infinite time horizon case, i.e., T= +∞. Then, 22 A. CALVIA, S. FEDERICO, G. FERRARI, AND F. GOZZI (iii) If there exists an equilibrium (b u,bq)in the sense of Definition 5.4, such that bq−β∈L1 ρ+δ((0,+∞); (0,+∞)) and z(∞,bq)is bounded on [0,+∞),(5.17) where z(∞,q)is the function defined in (3.10), then bqis a solution to the integral equation ys= e−δs Z+∞ 0 y m0(y) dy+Zs 0 e−δ(s−r)Z+∞ r e−(ρ+δ)(u−r)y−β ududr, s ≥0.(5.18) (iv) Vice versa, if there exist a unique solution bqto (5.18)satisfying (5.17), then there exists a unique equilibrium (b u,bq)=(z(∞,bq),bq)of the mean-field game among all equilibria (e u,eq) such that eqverifies (5.17). Finally, we get the following result, which establishes existence and uniqueness of an equilibrium also for the deterministic mean-field game. We omit its proof, since it follows the same lines of the proofs of Propositions 4.10 and 4.13, except for minor adaptations. Proposition 5.8. 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