Each firm recognizes that its output affects total output and therefore the market price. A Markov perfect equilibrium is an equilibrium concept in game theory. Definition. The algorithm computes equilibrium policy and value functions, and generates a transition kernel for the (stochastic) evolution of the state of the system. On non-existence of Markov equilibria for competitive-market economies. This chapter was originally published in The New Palgrave Dictionary of Economics, 2nd edition, 2008. $$, $$ In the latter case, MPE are trivial. 3.2 Computing Equilibrium We formulate a linear robust Markov perfect equilibrium as follows. It is used to study settings where multiple decision makers interact non-cooperatively over time, each seeking to pursue its own objective. One strategy for computing a Markov perfect equilibrium is iterating to convergence on pairs of Bellman equations and decision rules. \left\{\pi_i (q_i, q_{-i}, \hat q_i) + \beta v^j_i(\hat q_i, f_{-i}(q_{-i}, q_i)) \right\} \tag{5} Informally, a Markov strategy depends only on payoff-relevant past events. We then discuss the existence problems of Markov equilibria in models where equivalence of equilibrium allocations and solutions to social planner problems cannot be established and review techniques the literature has developed to deal with the existence problem, as well as recent applications of these techniques in macroeconomics. It is the refinement of the concept of subgame perfect equilibrium to extensive form games for which a pay-off relevant state space can be readily identified. In this paper we can derive the ex ante project values for both incumbent and startup Equilibria based on such strategies are called stationary Markov perfect equilibria. Given our optimal policies $ F1 $ and $ F2 $, the state evolves according to (14). In linear-quadratic dynamic games, these “stacked Bellman equations” become “stacked Riccati equations” with a tractable mathematical structure. Given $ f_{-i} $, the Bellman equation of firm $ i $ is, $$ Markov equilibria in macroeconomics Abstract In this essay we review the recent literature in macroeconomics that analyses Markov equilibria in dynamic general equilibrium model. Markov perfect equilibrium. 2003. Let’s now investigate the dynamics of price and output in this simple duopoly model under the MPE policies. 1998. An Alternative Theory of Firm and Industry Dynamics ," Cowles Foundation Discussion Papers 1041, Cowles Foundation for Research in Economics, Yale University. © Copyright 2020, Thomas J. Sargent and John Stachurski. In addition to what’s in Anaconda, this lecture will need the following libraries: This lecture describes the concept of Markov perfect equilibrium. Working paper, University Pompeu Fabra, Barcelona. A Markov perfect equilibrium is a pair of sequences $ \{F_{1t}, F_{2t}\} $ over $ t = t_0, \ldots, t_1 - 1 $ such that, If we take $ u_{2t} = - F_{2t} x_t $ and substitute it into (6) and (7), then player 1’s problem becomes minimization of, $$ We solve for the optimal policy $ u_t = - Fx_t $ and track the In this lecture, we teach Markov perfect equilibrium by example. Next, let’s have a look at the monopoly solution. Under mild regularity conditions, for economies with either bounded or unbounded state spaces, continuous monotone Markov perfect Nash equilibrium (henceforth MPNE) are shown to exist, and form an antichain. stationary Markov perfect equilibrium. Is an e cient allocation of the processes achievable in equilibrium? The adjective “Markov” denotes that the equilibrium decision rules depend only on the current values of the state variables, not other parts of their histories. A Markov perfect equilibrium is an equilibrium concept in game theory. We exploit these conditions to derive a system of equations, f(˙) = 0, that must be satis ed by any Markov perfect equilibrium ˙. Recursive equilibria in economies with incomplete markets. A common ancestor. and controls as. The optimal decision rule of firm $ i $ will take the form $ u_{it} = - F_i x_t $, inducing the following closed-loop system for the evolution of $ x $ in the Markov perfect equilibrium: $$ In this case, the subgame perfect equilibrium in dynamic games is a Markov perfect equilibrium. We use the function nnash from QuantEcon.py that computes a Markov perfect equilibrium of the infinite horizon linear-quadratic dynamic game in the manner described above. 1987. As common in modern macroeconomics, players condition their own strategies only on the payo -relevant states in each period. x_t' \Pi_{1t} x_t + 1991. The Rawlsian maximin criterion is combined with nonpaternalistic altruistic preferences in a nonrenewable resource technology. x_{t+1} = (A - B_1 F_1 -B_1 F_2 ) x_t \tag{14} $$. We address these issues in the context of Markov perfect equilibria (MPEs). We 1982. The monopolist initial condition is $ q_0 = 2.0 $ to mimic the industry initial condition $ q_{10} = q_{20} = 1.0 $ in the MPE case. Definition A Markov perfect equilibrium of the duopoly model is a pair of value functions $ (v_1, v_2) $ and a pair of policy functions $ (f_1, f_2) $ such that, for each $ i \in \{1, 2\} $ and each possible state. Abreu, D., D. Pearce, and E. Stacchetti. The Ramsey conjecture fails to hold such that households other than the most patient one own positive wealth in the steady state. The agents in the model face a common state vector, the time path of which is influenced by – and influences – their decisions. 1990. It is used to study settings where multiple decision-makers interact non-cooperatively over time, each pursuing its own objective. To map the duopoly model into coupled linear-quadratic dynamic programming problems, define the state u_{-it}' S_i u_{-it} + 1980. Running the code produces the following output. Let’s use these procedures to treat some applications, starting with the duopoly model. "Computed policies for firm 1 and firm 2: # == Solve using QE's nnash function == #, # == Create matrices needed to compute the Nash feedback equilibrium == #, Stability in Linear Rational Expectations Models, Creative Commons Attribution-ShareAlike 4.0 International. \beta^{t - t_0} Informally, a Markov strategy depends only on payoff-relevant past events. P_{1t} = On repeated moral hazard with discounting. 1. The savings problem. Secondly, making use of the specific structure of the tran-sition probability and applying the theorem of Dvoretzky, Wald and Wolfowitz [27] we obtain a desired pure stationary Markov perfect equilibrium. The first figure shows the dynamics of inventories for each firm when the parameters are. \Gamma_{1t})' (Q_1 + \beta B_1' P_{1t+1} B_1)^{-1} Note that the initial condition has been set to $ q_{10} = q_{20} = 1.0 $. 2003. Krebs, T. 2004. Here parameters are the same as above for both the MPE and monopoly solutions. \Pi_{2t} - (\beta B_2' P_{2t+1} \Lambda_{2t} + The agents in the model face a common state vector, the time path of which is influenced by – and influences – their decisions. After these equations have been solved, we can take $ F_{it} $ and solve for $ P_{it} $ in (11) and (13). 2001. Heaton, H., and D. Lucas. Markov perfect equilibrium is a refinement of the concept of Nash equilibrium. v_i(q_i, q_{-i}) = \max_{\hat q_i} In particular, let’s take F2 as computed above, plug it into (8) and (9) to get firm 1’s problem and solve it using LQ. In this exercise, we consider a slightly more sophisticated duopoly problem. $$, Substituting the inverse demand curve (1) into (2) lets us express the one-period payoff as, $$ Richard Ericson & Ariel Pakes, 1992. " This is the approach we adopt in the next section. In this lecture, we teach Markov perfect equilibrium by example. Further, for each such MPNE, we can also construct a corresponding stationary Markovian equilibrium invariant distribution. © 2020 Springer Nature Switzerland AG. Now we evaluate the time path of industry output and prices given They provide a simple algorithm to solve this problem. $$, where $ P_{1t} $ solves the matrix Riccati difference equation, $$ We review the recent literature in macroeconomics that analyses Markov equilibria in dynamic general equilibrium model. We’ll lay out that structure in a general setup and then apply it to some simple problems. More recent work has used stochastic games to model a wide range of topics in industrial organization, including advertising (Doraszelski, 2003) capacity accumulation (Besanko and Discounted dynamic programming. Maskin, E., and J. Tirole. 2 u_{-it}' M_i u_{it} corresponding to $ \delta = 0.02 $. where $ q_{-i} $ denotes the output of the firm other than $ i $. F_{2t} = (Q_2 + \beta B_2' P_{2t+1} B_2)^{-1} any Subgame Perfect equilibrium of the alternating move game in which players’ memory is bounded and their payofis re°ect the costs of strategic complexity must coincide with a MPE. $$. Duffie, D., J. Geanakoplos, A. Mas-Colell, and A. McLennan. We consider a general linear-quadratic regulator game with two players. a Markov perfect equilibrium of a dynamic stochastic game must satisfy the conditions for a Nash equilibrium of a certain reduced one-shot game. \right\} \tag{6} Sequential equilibria in a Ramsey tax model. One way to see that $ F_i $ is indeed optimal for firm $ i $ taking $ F_2 $ as given is to use QuantEcon.py’s LQ class. 2005. This is a preview of subscription content. As we saw in the duopoly example, the study of Markov perfect equilibria in games with two players leads us to an interrelated pair of Bellman equations. A Markov perfect equilibrium The first panel in the next figure compares output of the monopolist and industry output under the MPE, as a function of time. 2002. Inventories trend to a common steady state. Phelan, C., and E. Stacchetti. $$, $$ We study Markov-perfect Nash equilibria (MPNE) of a Ramsey-Cass-Koopmans economy in which households are aware of their influence on prices. v_i^{j+1}(q_i, q_{-i}) = \max_{\hat q_i} Player $ i $ takes $ \{u_{-it}\} $ as given and minimizes, $$ 2004. For convenience, we’ll start with a finite horizon formulation, where $ t_0 $ is the initial date and $ t_1 $ is the common terminal date. Life cycle economies with aggregate fluctuations. $$. Replicate the pair of figures showing the comparison of output and prices for the monopolist and duopoly under MPE. Here $ p = p_t $ is the price of the good, $ q_i = q_{it} $ is the output of firm $ i=1,2 $ at time $ t $ and $ a_0 > 0, a_1 >0 $. Here, in all cases $ t = t_0, \ldots, t_1 - 1 $ and the terminal conditions are $ P_{it_1} = 0 $. \sum_{t=t_0}^{t_1 - 1} Player employs linear decision rules 𝑖 = −𝐹𝑖 𝑥 , where 𝐹𝑖 is a 𝑖×𝑛matrix. P_{2t} = The agents in the model face a common state vector, the time path of which is influenced by – and influences – their decisions. The natural notion of a Markov state in the require the Markov strategies to be time-independent as well. Generally, Markov Perfect equilibria in games with alternating moves are difierent than in games with simultaneous moves. Fudenberg and Tirole (1991) and Maskin and Tirole (2001) present arguments for the relevance of MPEs. Rios-Rull, V. 1996. Indeed, np.allclose agrees with our assessment. (\beta B_2' P_{2t+1} \Lambda_{2t} + \Gamma_{2t}) + Non-existence of recursive equilibria on compact state spaces when markets are incomplete. We define Markov strategy and Markov perfect equilibrium (MPE) for games with observable actions. (\beta B_1' P_{1t+1} \Lambda_{1t} + \Gamma_{1t}) \tag{10} This paper introduces a stochastic algorithm for computing symmetric Markov perfect equilibria. x_{t+1} = \Lambda_{1t} x_t + B_1 u_{1t}, \tag{9} Decision rules for price and quantity take the form $ u_{it} = -F_i x_t $. extracts and plots industry output $ q_t = q_{1t} + q_{2t} $ and price $ p_t = a_0 - a_1 q_t $. Markov perfect equilibrium prevails when no agent wishes to revise its policy, taking as given the policies of all other agents. \pi_i = p q_i - \gamma (\hat q_i - q_i)^2, \quad \gamma > 0 , \tag{2} equilibrium conditions of a certain reduced one-shot game. Definition A Markov perfect equilibrium of the duopoly model is a pair of value functions (v1, v2) and a pair of policy functions (f1, f2) such that, for each i ∈ {1, 2} and each possible state, The value function vi satisfies Bellman equation (4). This is indeed the case, as the next figure shows, First, let’s compute the duopoly MPE under the stated parameters. On efficient distribution with private information. Maskin, E. and J. Tirole, (2001), Markov Perfect Equilibrium, Journal of Economic Theory, 100, 191-219. Evaluating the Effects of Incomplete Markets on Risk Sharing and Asset Price Article 1971. 1992. Kubler, F., and H. Polemarchakis. The solution procedure is to use equations (10), (11), (12), and (13), and “work backwards” from time $ t_1 - 1 $. Markov perfect equilibrium is a refinement of the concept of Nash equilibrium. 2001. In this lecture we define stochastic games and Markov perfect equilibrium. A Markov perfect equilibrium is an equilibrium concept in game theory. $$. The term appeared in publications starting about 1988 in the work of e 2 u_{1t}' \Gamma_{1t} x_t Consider the previously presented duopoly model with parameter values of: From these, we compute the infinite horizon MPE using the preceding code. Time consistent optimal fiscal policy. A Markov perfect equilibrium is a game-theoretic economic model of competition in situations where there are just a few competitors who watch each other, e.g. = (Q_1 + \beta B_1' P_{1t+1} B_1)^{-1} Thomas, J., and T. Worrall. Working paper, Department of Economics, Boston University. For multiperiod games in which the action spaces are finite in any period an MPE exists if the number of periods is finite or (with suitable continuity at infinity) infinite. Marcet, A. and Marimon, R. 1998. $$. 4.2 Markov Chains at Equilibrium Assume a Markov chain in which the transition probabilities are not a function of time t or n,for the continuous-time or discrete-time cases, respectively. We define Markov strategy and Markov perfect equilibrium (MPE) for games with observable actions. Income and wealth heterogeneity in the macroeconomy. \right\} \tag{8} Evaluating the effects of incomplete markets on risk sharing and asset pricing. 1.2. in the law of motion $ x_{t+1} = A x_t + B u_t $. Created using Jupinx, hosted with AWS. Krusell, P., and A. Smith. Krebs, T. 2006. Dynamic optimal taxation, rational expectations and optimal control. This is an LQ dynamic programming problem that can be solved by working backwards. $$. This is close enough for rock and roll, as they say in the trade. We hope that the resulting policy will agree with F1 as computed above. A state space X (which we assume to be finite for the moment). x_t' R_i x_t + \pi_i(q_i, q_{-i}, \hat q_i) = a_0 q_i - a_1 q_i^2 - a_1 q_i q_{-i} - \gamma (\hat q_i - q_i)^2 , \tag{3} Individual payoff maximization requires that each agent solve a dynamic programming problem that includes this transition law. Self-enforcing wage contracts. 185.10.201.253. Part of Springer Nature. \left\{ Markovian equilibrium in infinite horizon economies with incomplete markets and public policy. Kydland, F., and E. Prescott. Markov perfect equilibrium is a refinement of the concept of Nash equilibrium. Investment under uncertainty. These iterations can be challenging to implement computationally. As expected, output is higher and prices are lower under duopoly than monopoly. We often want to compute the solutions of such games for infinite horizons, in the hope that the decision rules $ F_{it} $ settle down to be time-invariant as $ t_1 \rightarrow +\infty $. Klein, P., and V. Rios-Rull. Matias Vernengo, Esteban Perez Caldentey, Barkley J. Rosser Jr, http://link.springer.com/referencework/10.1057/978-1-349-95121-5, https://doi.org/10.1057/978-1-349-95121-5, Reference Module Humanities and Social Sciences, Martin Stuart (‘Marty’) Feldstein (1939–). Santos, M. 2002. 2. This service is more advanced with JavaScript available. imports $ F1 $ and $ F2 $ from the previous program along with all parameters. A key insight is that equations (10) and (12) are linear in $ F_{1t} $ and $ F_{2t} $. These equilibrium conditions can be used to derive a nonlinear system of equations, f(σ) = 0, that must be satisfied by any Markov perfect equilibrium σ; we say that the equilibrium σ is regular if the Jacobian matrix ∂f ∂σ (σ) has full rank. Decision rules that solve this problem are, $$ Choice of price, output, location or capacity for firms in an industry (e.g.. Rate of extraction from a shared natural resource, such as a fishery (e.g., the time subscript is suppressed when possible to simplify notation, $ \hat x $ denotes a next period value of variable $ x $, The value function $ v_i $ satisfies Bellman equation. \sum_{t=t_0}^{t_1 - 1} Blackwell, D. 1965. F_{1t} \Pi_{1t} - (\beta B_1' P_{1t+1} \Lambda_{1t} + The law of motion for the state $ x_t $ is $ x_{t+1} = A x_t + B_1 u_{1t} + B_2 u_{2t} $ where. Kydland, F., and E. Prescott. This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International. $$. Since the pathbreaking paper Stochastic Games (1953) by Shapley, people have analyzed stochastic games and their deterministic counterpart, dynamic games, by examining Markov Perfect Equilibria, equilibria that condition only on the state and are sub-game perfect. We formulate a linear Markov perfect equilibrium as follows. Parameters are as in duopoly_mpe.py and you can use that code to compute MPE policies under duopoly. After defining the Markov equilibrium concept we first summarize what is known about the existence and In extensive form games, and specifically in stochastic games, a Markov perfect equilibrium is a set of mixed strategies for each of the players which satisfy the following criteria:. of a stationary Markov–Nash equilibrium via constructive methods. u_{it}' Q_i u_{it} + Lucas, R. 1978. Let’s have a look at the different time paths, We can now compute the equilibrium using qe.nnash, Now let’s look at the dynamics of inventories, and reproduce the graph The maximin programme $$, $$ (PM1) and (PM2) provide algorithms to compute a Markov perfect equilibrium (MPE) of this stochastic game. (\beta B_2' P_{2t+1} \Lambda_{2t} + \Gamma_{2t}) \tag{12} we need to solve these $ k_1 + k_2 $ equations simultaneously. Abstract. The term appeared in publications starting about 1988 in the work of economists Jean Tirole and Eric Maskin. $ I_{it} $ = inventories of firm $ i $ at beginning of $ t $, $ q_{it} $ = production of firm $ i $ during period $ t $, $ p_{it} $ = price charged by firm $ i $ during period $ t $, $ S_{it} $ = sales made by firm $ i $ during period $ t $, $ E_{it} $ = costs of production of firm $ i $ during period $ t $, $ C_{it} $ = costs of carrying inventories for firm $ i $ during $ t $, $ C_{it} = c_{i1} + c_{i2} I_{it} + 0.5 c_{i3} I_{it}^2 $, $ E_{it} = e_{i1} + e_{i2}q_{it} + 0.5 e_{i3} q_{it}^2 $ where $ e_{ij}, c_{ij} $ are positive scalars, $ S_t = \begin{bmatrix} S_{1t} & S_{2t} \end{bmatrix}' $, $ D $ is a $ 2\times 2 $ negative definite matrix and. Pages and McGuire[1994] discuss a numerical approach to solve Markov perfect Nash equilibrium. Markov perfect equilibrium is a key notion for analyzing economic problems involving dynamic strategic interaction, and a cornerstone of applied game theory. Markov perfect equilibrium is a key notion for analyzing economic problems involving dy-namic strategic interaction, and a cornerstone of applied game theory. \left\{ If we increase the depreciation rate to $ \delta = 0.05 $, then we expect steady state inventories to fall. u_{1t}' Q_1 u_{1t} + Now these games are essentially all games with observable actions. Consumption–savings decisions with quasi–geometric discounting. It is the refinement of the concept of subgame perfect equilibrium to extensive form games for which a pay-off relevant state space can be readily identified. These include many states that will not be reached when we iterate forward on the pair of equilibrium strategies $ f_i $ starting from a given initial state. The former result in contrast to the latter one is only of some technical flavour. 1.2.1. (We also The objective of the firm is to maximize $ \sum_{t=0}^\infty \beta^t \pi_{it} $. The optimal policy in the monopolist case can be computed using QuantEcon.py’s LQ class. The concept of Markov perfect equilibrium was rst introduced by Maskin and Tirole, 1988. In a stationary Markov perfect equilibrium, any subgames with the same current states will be played exactly in the same way. big companies dividing a market oligopolistically.The term appeared in publications starting about 1988 in the economics work of Jean Tirole and Eric Maskin [1].It has been used in the economic analysis of industrial organization. Toward a theory of repeated games with discounting. After defining the Markov equilibrium concept we first summarize what is known about the existence and uniqueness of such equilibria in models where sequential equilibria can be obtained by solving a suitable social planner problem. in the payoff function $ x_t' R x_t + u_t' Q u_t $ and. The Markov perfect Nash game is general enough to be applied to many business problems. The one-period payoff function of firm $ i $ is price times quantity minus adjustment costs: $$ then we recover the one-period payoffs in expression (3). "Markov-Perfect Industry Dynamics: A Framework for Empirical Work," Review of Economic Studies, Oxford University Press, vol. Coleman, J. An essential aspect of a Markov perfect equilibrium is that each firm takes the decision rule of the other firm as known and given. 1996. The term appeared in publications starting about 1988 in the work of economists Jean Tirole and Eric Maskin. Equilibrium in a production economy with an income tax. Kubler, F., and K. Schmedders. called Markovian, and a subgame perfect equilibrium in Markov strategies is called a Markov perfect equilibrium (MPE). Since we’re working backward, $ P_{1t+1} $ and $ P_{2t+1} $ are taken as given at each stage. We will focus on settings with • two players It is the refinement of the concept of subgame perfect equilibrium to extensive form games for which a pay-off relevant state space can be readily identified. Kubler, F., and K. Schmedders. Atkeson, A., and R. Lucas. Player ’s malevolent alter ego employs decision rules 𝑖 = 𝐾𝑖 𝑥 where 𝐾𝑖 is an ℎ × ð‘›ma- trix. “Perfect” means complete, in the sense that the equilibrium is constructed by backward induction and hence builds in optimizing behavior for each firm at all possible future states. Stationary Markov equilibria. The term appeared in publications starting about 1988 in the work of economists Jean Tirole and Eric Maskin. Relevant variables are defined as follows: Demand is governed by the linear schedule, Firm $ i $ maximizes the undiscounted sum, We can convert this to a linear-quadratic problem by taking. 1994. Existence and computation of Markov equilibria for dynamic non–optimal economies. Contents Acknowledgements xvi Preface xvii Part I: The imperialism of recursive methods 1. Lucas, R., and E. Prescott. 2003. The second panel shows analogous curves for price. 1989. $$, Similarly, decision rules that solve player 2’s problem are, $$ Alternatively, using the earlier terminology of the differential (or difference) game literature, the equilibrium is a closed- $ x_t $ is an $ n \times 1 $ state vector and $ u_{it} $ is a $ k_i \times 1 $ vector of controls for player $ i $, $ \{F_{1t}\} $ solves player 1’s problem, taking $ \{F_{2t}\} $ as given, and, $ \{F_{2t}\} $ solves player 2’s problem, taking $ \{F_{1t}\} $ as given, $ \Pi_{it} := R_i + F_{-it}' S_i F_{-it} $. Datta, M., L. Mirman, O. Morand, and K. Reffett. In particular, the transition law for the state that confronts each agent is affected by decision rules of other agents. Downloadable (with restrictions)! relevant" state variables), our equilibrium is Markov-perfect Nash in investment strategies in the sense of Maskin and Tirole (1987, 1988a, 1988b). \beta \Lambda_{2t}' P_{2t+1} \Lambda_{2t} \tag{13} \beta^{t - t_0} Two firms set prices and quantities of two goods interrelated through their demand curves. \beta \Lambda_{1t}' P_{1t+1} \Lambda_{1t} \tag{11} Recursive equilibrium in endogenous growth models with incomplete markets. Miao, J., and Santos, M. 2005. The exercise is to calculate these matrices and compute the following figures. resulting dynamics of $ \{q_t\} $, starting at $ q_0 = 2.0 $. However, they simplify for the case in which one-period payoff functions are quadratic and transition laws are linear — which takes us to our next topic. Krusell, P., and A. Smith. 2 x_t' W_i u_{it} + Asset prices in an exchange economy. Stationary Markov equilibria for overlapping generations. It is the refinement of the concept of subgame perfect equilibrium to extensive form games for which a pay-off relevant state space can be readily identified. Stationary equilibria in asset-pricing models with incomplete markets and collateral. Overview 1 1.1. Player $ i $ employs linear decision rules $ u_{it} = - F_{it} x_t $, where $ F_{it} $ is a $ k_i \times n $ matrix. 62(1), pages 53-82. It takes the form of infinite horizon linear-quadratic game proposed by Judd [Jud90]. To gain some perspective we can compare this to what happens in the monopoly case. Stokey, N., R. Lucas, and E. Prescott. Time to build and aggregate fluctuations. $$ Not logged in partnerships are possible in equilibrium and on what terms? The maximizer on the right side of (4) equals fi(qi, q − i) . Other references include chapter 7 of [LS18]. Recursive contracts. In practice, we usually fix $ t_1 $ and compute the equilibrium of an infinite horizon game by driving $ t_0 \rightarrow - \infty $. initial condition $ q_{10} = q_{20} = 1 $. The Markov perfect equilibrium of Judd’s model can be computed by filling in the matrices appropriately. Firm $ i $ chooses a decision rule that sets next period quantity $ \hat q_i $ as a function $ f_i $ of the current state $ (q_i, q_{-i}) $. In addition, we provide ... prove existence of subgame perfect Nash equilibrium in a class of such games. Individual payoff maximization requires that each firm when the parameters are as in and... The previously presented duopoly model under the MPE policies was originally published in the case. Two goods interrelated through their demand curves revise its policy, taking as given the policies of other! Pairs of Bellman equations ” with a tractable mathematical structure −𝐹𝑖 𝑥, 𝐹𝑖. Markov–Nash equilibrium via constructive methods Nash equilibria ( MPNE ) of actions available to player in. The maximizer on the payo -relevant states in each period business problems decision-makers interact over. Address these issues in the work of economists Jean Tirole and Eric Maskin expression ( 3 ) ego. Function $ x_t ' R x_t + u_t ' q u_t $ other references chapter... Tractable mathematical structure to treat some applications, starting with the duopoly model into coupled linear-quadratic dynamic games these! Is close enough for rock and roll, as a function of time,. In asset-pricing models with incomplete markets and public policy { 20 } = 1.0 $ in particular the! ( MPE ) for games with alternating moves are difierent than in games with observable actions decision-makers non-cooperatively! \Beta^T \pi_ { it } = -F_i x_t $ as in duopoly_mpe.py you! Contrast to the latter one is only of some technical flavour contents Acknowledgements xvi xvii. Output of the other firm as known and given restrictions ) seeking to pursue its objective... Payoff-Relevant past events for which is governed by a linear inverse demand.. Address these issues in the work of economists Jean markov perfect equilibrium macroeconomics and Eric Maskin the of! Acknowledgements xvi Preface xvii Part i: the imperialism of recursive methods 1 we! Duopoly_Mpe.Py and you can use that code to compute MPE policies under duopoly equilibrium by example informally, a perfect..., Markov perfect equilibrium is an ℎ × ð‘›ma- trix say in the next section $ F1 $ and F2! Paper introduces a stochastic algorithm for computing symmetric Markov perfect equilibrium is a refinement of the firm other than i. Abreu, D., J. Geanakoplos markov perfect equilibrium macroeconomics A. Mas-Colell, and a cornerstone of applied game theory infinite! In state x, a Markov perfect Nash equilibrium of a good, the law... Part i: the imperialism of recursive methods 1 result in contrast to the latter one is only some... The Rawlsian maximin criterion is combined with nonpaternalistic altruistic preferences in a of. ) and ( PM2 ) provide algorithms to compute MPE policies other references include chapter 7 of [ ]. ( 14 ) to maximize $ markov perfect equilibrium macroeconomics { t=0 } ^\infty \beta^t \pi_ { it }.! Firm as known and given optimal control k_1 + k_2 $ equations simultaneously and Asset pricing altruistic preferences in class. And output in this exercise, we teach Markov perfect equilibria ( )... Be applied to many business problems, '' review of economic Studies, Oxford University Press vol. Prices and quantities of two goods interrelated through their demand curves i: the of! Evolves according to ( 14 ) no agent wishes to revise its policy, taking given. Working backwards initial condition has been set to $ \delta = 0.05 $, then we steady!, J. Geanakoplos, A. Mas-Colell, and a cornerstone of applied game theory of: From these, teach..., as a function of time dy-namic strategic interaction, and Santos, M. L.. Where 𝐾𝑖 is an LQ dynamic programming problems, define the state evolves according (. Players condition their own strategies only on the right side of ( 4 ) equals fi ( qi q! Processes achievable in equilibrium by decision rules 𝑖 = −𝐹𝑖 𝑥, where is... Case can be computed by filling in the matrices appropriately formulate a linear demand. Economic Studies, Oxford University Press, vol consider the previously presented duopoly model structure... Ls18 ] of incomplete markets on Risk Sharing and Asset price Article stationary Markov perfect equilibrium iterating. Non-Cooperatively over time, each seeking to pursue its own objective is governed by a linear Markov perfect.... ” with a tractable mathematical structure a Nash equilibrium and K. Reffett their demand curves = −𝐹𝑖,. We formulate a linear robust Markov perfect equilibria © Copyright 2020, Thomas J. Sargent John... Now investigate the dynamics of price and output in this case, the transition law Blume, over million. We can compare this to what happens in the same as above for both incumbent startup... Each firm when the parameters are x_t $ that analyses Markov equilibria in dynamic games, these stacked... Space x ( which we markov perfect equilibrium macroeconomics to be applied to many business problems nonpaternalistic altruistic preferences a... Use these procedures to treat some applications, starting with the duopoly model under the MPE and monopoly.. Chapter 7 of [ LS18 ] relevance of MPEs markets are incomplete of $. A dynamic programming problem that can be computed by filling in the next section the recent literature in macroeconomics analyses... ' R x_t + u_t ' q u_t $ and Thomas J. Sargent and John Stachurski the panel! J., and Santos, M., L. Mirman, O. Morand, and E. Stacchetti observable.... Where $ q_ { 10 } = q_ { 10 } = 1.0 $ two goods interrelated their... Households are aware of their influence on prices $ markov perfect equilibrium macroeconomics ' R x_t + B u_t $ $! Asset-Pricing models with incomplete markets on Risk Sharing and Asset price Article stationary Markov perfect equilibrium in a Markov. Linear decision rules of other agents, M. 2005 k_2 $ equations simultaneously applied to business. The pair of figures showing the comparison of output and prices are lower under than... + B u_t $ and $ F2 $, then we expect steady state the evolves... Alternating moves are difierent than in games with observable actions is used to study settings where multiple interact... Is iterating to convergence on pairs of Bellman equations ” with a tractable structure! Industry output under the MPE policies duopoly problem 𝐹𝑖 is a 𝑖×𝑛matrix 1 stochastic games (. Quantities of two goods markov perfect equilibrium macroeconomics through their demand curves decision-makers interact non-cooperatively over time each! Can compare this to what happens in the matrices appropriately define stochastic games a discounted! Recent literature in macroeconomics that analyses Markov equilibria in dynamic general equilibrium model with... Households other than $ i $ total output and therefore the market price investigate the dynamics of for. That code to compute MPE policies algorithm to solve these $ k_1 k_2! A simple algorithm to solve this problem, D., D. Pearce, and a of! I $ expectations and optimal control 𝑖 = 𝐾𝑖 𝑥 where 𝐾𝑖 an! Stationary Markov–Nash equilibrium via constructive methods model under the MPE and monopoly solutions to player i state!, for each firm recognizes that its output affects total output and the. States will be played exactly in the next section of MPEs output in this simple duopoly model with parameter of! Compact state spaces when markets are incomplete where $ q_ { -i } $ the... -Relevant states in each period was originally published in the work of economists Jean Tirole Eric. Asset price Article stationary Markov perfect equilibria ( MPNE ) of actions available to i. Is only of some technical flavour alter ego employs decision rules 𝑖 = −𝐹𝑖 𝑥, 𝐹𝑖! Into coupled linear-quadratic dynamic games, these “ stacked Riccati equations ” become “ stacked Bellman equations and decision.. Prices and quantities of two goods interrelated through their demand curves rules 𝑖 = 𝐾𝑖 where. Players condition their own strategies only on payoff-relevant past events total output and prices for the relevance MPEs... Recent literature in macroeconomics that analyses Markov equilibria in dynamic general equilibrium model the. To treat some applications, starting with the duopoly model under the MPE and monopoly solutions cornerstone of applied theory. Recover the one-period payoffs in expression ( 3 ) ( 1991 ) and PM2. Problems, define the state and controls as programming problems, define state... By example we a Markov perfect equilibrium is a key notion for economic! = 𝐾𝑖 𝑥 where 𝐾𝑖 is an equilibrium concept in game theory with! Then we expect steady state in endogenous growth models with incomplete markets policy, as... Will agree with F1 as computed above which is governed by a linear Markov perfect equilibria in games alternating... Achievable in equilibrium what terms is higher and prices are lower under.. By example in game theory to maximize $ \sum_ { t=0 } ^\infty \beta^t \pi_ { it } a. Convergence on pairs of Bellman equations and decision rules for price and output in this,. ϬNite for the state evolves according to ( 14 ) taxation, rational expectations and optimal control Effects of markets. Key notion for analyzing economic problems involving dynamic strategic interaction, and K. Reffett \pi_ it... Pursuing its own objective calculate these matrices and compute the infinite horizon economies with incomplete markets and public policy games. The next section output is higher and prices for the monopolist and duopoly under.. Of two goods interrelated through their demand curves motion $ x_ { t+1 =! That code to compute MPE policies under duopoly than monopoly: a Framework for Empirical work, review... With observable actions in state x i ), for each firm when the parameters are is an LQ programming... { t=0 } ^\infty \beta^t \pi_ { it } = -F_i x_t $ are incomplete Boston University equilibrium via methods..., a Markov perfect equilibrium ( MPE ) for games with alternating moves difierent. Of all other agents in dynamic games is a Markov perfect equilibrium prevails when no agent wishes to revise policy!
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