{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "
\n", " \n", " \"QuantEcon\"\n", " \n", "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Markov Perfect Equilibrium\n", "\n", "\n", "" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Overview\n", "\n", "\n", "\n", "This lecture describes the concept of Markov perfect equilibrium\n", "\n", "Markov perfect equilibrium is a key notion for analyzing economic problems involving dynamic strategic interaction, and a cornerstone of applied game theory\n", "\n", "In this lecture we teach Markov perfect equilibrium by example\n", "\n", "We will focus on settings with\n", "\n", "- two players \n", "- quadratic payoff functions \n", "- linear transition rules for the state \n", "\n", "\n", "Other references include chapter 7 of [[LS18]](https://lectures.quantecon.org/zreferences.html#ljungqvist2012)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Setup" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "hide-output": true }, "outputs": [], "source": [ "using InstantiateFromURL\n", "github_project(\"QuantEcon/quantecon-notebooks-julia\", version = \"0.3.0\")" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "hide-output": false }, "outputs": [], "source": [ "using LinearAlgebra, Statistics, QuantEcon" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Background\n", "\n", "\n", "\n", "Markov perfect equilibrium is a refinement of the concept of Nash equilibrium\n", "\n", "It is used to study settings where multiple decision makers interact non-cooperatively over time, each seeking to pursue its own objective\n", "\n", "The agents in the model face a common state vector, the time path of which is influenced by – and influences – their decisions\n", "\n", "In particular, the transition law for the state that confronts each agent is affected by decision rules of other agents\n", "\n", "Individual payoff maximization requires that each agent solve a dynamic programming problem that includes this transition law\n", "\n", "Markov perfect equilibrium prevails when no agent wishes to revise its policy, taking as given the policies of all other agents\n", "\n", "Well known examples include\n", "\n", "- Choice of price, output, location or capacity for firms in an industry (e.g., [[EP95]](https://lectures.quantecon.org/zreferences.html#ericson1995markov), [[Rya12]](https://lectures.quantecon.org/zreferences.html#ryan2012costs), [[DS10]](https://lectures.quantecon.org/zreferences.html#doraszelski2010computable)) \n", "- Rate of extraction from a shared natural resource, such as a fishery (e.g., [[LM80]](https://lectures.quantecon.org/zreferences.html#levhari1980great), [[VL11]](https://lectures.quantecon.org/zreferences.html#van2011dynamic)) \n", "\n", "\n", "Let’s examine a model of the first type" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Example: A duopoly model\n", "\n", "Two firms are the only producers of a good the demand for which is governed by a linear inverse demand function\n", "\n", "\n", "\n", "$$\n", "p = a_0 - a_1 (q_1 + q_2) \\tag{1}\n", "$$\n", "\n", "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 $\n", "\n", "In [(1)](#equation-game2) and what follows,\n", "\n", "- the time subscript is suppressed when possible to simplify notation \n", "- $ \\hat x $ denotes a next period value of variable $ x $ \n", "\n", "\n", "Each firm recognizes that its output affects total output and therefore the market price\n", "\n", "The one-period payoff function of firm $ i $ is price times quantity minus adjustment costs:\n", "\n", "\n", "\n", "$$\n", "\\pi_i = p q_i - \\gamma (\\hat q_i - q_i)^2, \\quad \\gamma > 0 , \\tag{2}\n", "$$\n", "\n", "Substituting the inverse demand curve [(1)](#equation-game2) into [(2)](#equation-game1) lets us express the one-period payoff as\n", "\n", "\n", "\n", "$$\n", "\\pi_i(q_i, q_{-i}, \\hat q_i) = a_0 q_i - a_1 q_i^2 - a_1 q_i q_{-i} - \\gamma (\\hat q_i - q_i)^2 , \\tag{3}\n", "$$\n", "\n", "where $ q_{-i} $ denotes the output of the firm other than $ i $\n", "\n", "The objective of the firm is to maximize $ \\sum_{t=0}^\\infty \\beta^t \\pi_{it} $\n", "\n", "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}) $\n", "\n", "An essential aspect of a Markov perfect equilibrium is that each firm takes the decision rule of the other firm as known and given\n", "\n", "Given $ f_{-i} $, the Bellman equation of firm $ i $ is\n", "\n", "\n", "\n", "$$\n", "v_i(q_i, q_{-i}) = \\max_{\\hat q_i}\n", " \\left\\{\\pi_i (q_i, q_{-i}, \\hat q_i) + \\beta v_i(\\hat q_i, f_{-i}(q_{-i}, q_i)) \\right\\} \\tag{4}\n", "$$\n", "\n", "**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,\n", "\n", "- The value function $ v_i $ satisfies the Bellman equation [(4)](#equation-game4) \n", "- The maximizer on the right side of [(4)](#equation-game4) is equal to $ f_i(q_i, q_{-i}) $ \n", "\n", "\n", "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\n", "\n", "“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\n", "\n", "> - 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 " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Computation\n", "\n", "One strategy for computing a Markov perfect equilibrium is iterating to convergence on pairs of Bellman equations and decision rules\n", "\n", "In particular, let $ v_i^j,f_i^j $ be the value function and policy function for firm $ i $ at the $ j $-th iteration\n", "\n", "Imagine constructing the iterates\n", "\n", "\n", "\n", "$$\n", "v_i^{j+1}(q_i, q_{-i}) = \\max_{\\hat q_i}\n", " \\left\\{\\pi_i (q_i, q_{-i}, \\hat q_i) + \\beta v^j_i(\\hat q_i, f_{-i}(q_{-i}, q_i)) \\right\\} \\tag{5}\n", "$$\n", "\n", "These iterations can be challenging to implement computationally\n", "\n", "However, they simplify for the case in which the one-period payoff functions are quadratic and the transition laws are linear — which takes us to our next topic" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Linear Markov perfect equilibria\n", "\n", "\n", "\n", "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\n", "\n", "In linear quadratic dynamic games, these “stacked Bellman equations” become “stacked Riccati equations” with a tractable mathematical structure\n", "\n", "We’ll lay out that structure in a general setup and then apply it to some simple problems" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Coupled linear regulator problems\n", "\n", "We consider a general linear quadratic regulator game with two players\n", "\n", "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\n", "\n", "Player $ i $ takes $ \\{u_{-it}\\} $ as given and minimizes\n", "\n", "\n", "\n", "$$\n", "\\sum_{t=t_0}^{t_1 - 1}\n", "\\beta^{t - t_0}\n", "\\left\\{\n", " x_t' R_i x_t +\n", " u_{it}' Q_i u_{it} +\n", " u_{-it}' S_i u_{-it} +\n", " 2 x_t' W_i u_{it} +\n", " 2 u_{-it}' M_i u_{it}\n", "\\right\\} \\tag{6}\n", "$$\n", "\n", "while the state evolves according to\n", "\n", "\n", "\n", "$$\n", "x_{t+1} = A x_t + B_1 u_{1t} + B_2 u_{2t} \\tag{7}\n", "$$\n", "\n", "Here\n", "\n", "- $ x_t $ is an $ n \\times 1 $ state vector and $ u_{it} $ is a $ k_i \\times 1 $ vector of controls for player $ i $ \n", "- $ R_i $ is $ n \\times n $ \n", "- $ S_i $ is $ k_{-i} \\times k_{-i} $ \n", "- $ Q_i $ is $ k_i \\times k_i $ \n", "- $ W_i $ is $ n \\times k_i $ \n", "- $ M_i $ is $ k_{-i} \\times k_i $ \n", "- $ A $ is $ n \\times n $ \n", "- $ B_i $ is $ n \\times k_i $ " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Computing Equilibrium\n", "\n", "We formulate a linear Markov perfect equilibrium as follows\n", "\n", "Player $ i $ employs linear decision rules $ u_{it} = - F_{it} x_t $, where $ F_{it} $ is a $ k_i \\times n $ matrix\n", "\n", "A Markov perfect equilibrium is a pair of sequences $ \\{F_{1t}, F_{2t}\\} $ over $ t = t_0, \\ldots, t_1 - 1 $ such that\n", "\n", "- $ \\{F_{1t}\\} $ solves player 1’s problem, taking $ \\{F_{2t}\\} $ as given, and \n", "- $ \\{F_{2t}\\} $ solves player 2’s problem, taking $ \\{F_{1t}\\} $ as given \n", "\n", "\n", "If we take $ u_{2t} = - F_{2t} x_t $ and substitute it into [(6)](#equation-orig-1) and [(7)](#equation-orig-0), then player 1’s problem becomes minimization of\n", "\n", "\n", "\n", "$$\n", "\\sum_{t=t_0}^{t_1 - 1}\n", "\\beta^{t - t_0}\n", " \\left\\{\n", " x_t' \\Pi_{1t} x_t +\n", " u_{1t}' Q_1 u_{1t} +\n", " 2 u_{1t}' \\Gamma_{1t} x_t\n", " \\right\\} \\tag{8}\n", "$$\n", "\n", "subject to\n", "\n", "\n", "\n", "$$\n", "x_{t+1} = \\Lambda_{1t} x_t + B_1 u_{1t}, \\tag{9}\n", "$$\n", "\n", "where\n", "\n", "- $ \\Lambda_{it} := A - B_{-i} F_{-it} $ \n", "- $ \\Pi_{it} := R_i + F_{-it}' S_i F_{-it} $ \n", "- $ \\Gamma_{it} := W_i' - M_i' F_{-it} $ \n", "\n", "\n", "This is an LQ dynamic programming problem that can be solved by working backwards\n", "\n", "The policy rule that solves this problem is\n", "\n", "\n", "\n", "$$\n", "F_{1t}\n", "= (Q_1 + \\beta B_1' P_{1t+1} B_1)^{-1}\n", "(\\beta B_1' P_{1t+1} \\Lambda_{1t} + \\Gamma_{1t}) \\tag{10}\n", "$$\n", "\n", "where $ P_{1t} $ solves the matrix Riccati difference equation\n", "\n", "\n", "\n", "$$\n", "P_{1t} =\n", "\\Pi_{1t} -\n", "(\\beta B_1' P_{1t+1} \\Lambda_{1t} + \\Gamma_{1t})' (Q_1 + \\beta B_1' P_{1t+1} B_1)^{-1}\n", "(\\beta B_1' P_{1t+1} \\Lambda_{1t} + \\Gamma_{1t}) +\n", "\\beta \\Lambda_{1t}' P_{1t+1} \\Lambda_{1t} \\tag{11}\n", "$$\n", "\n", "Similarly, the policy that solves player 2’s problem is\n", "\n", "\n", "\n", "$$\n", "F_{2t} =\n", "(Q_2 + \\beta B_2' P_{2t+1} B_2)^{-1}\n", "(\\beta B_2' P_{2t+1} \\Lambda_{2t} + \\Gamma_{2t}) \\tag{12}\n", "$$\n", "\n", "where $ P_{2t} $ solves\n", "\n", "\n", "\n", "$$\n", "P_{2t} =\n", "\\Pi_{2t} - (\\beta B_2' P_{2t+1} \\Lambda_{2t} + \\Gamma_{2t})' (Q_2 + \\beta B_2' P_{2t+1} B_2)^{-1}\n", "(\\beta B_2' P_{2t+1} \\Lambda_{2t} + \\Gamma_{2t}) +\n", "\\beta \\Lambda_{2t}' P_{2t+1} \\Lambda_{2t} \\tag{13}\n", "$$\n", "\n", "Here in all cases $ t = t_0, \\ldots, t_1 - 1 $ and the terminal conditions are $ P_{it_1} = 0 $\n", "\n", "The solution procedure is to use equations [(10)](#equation-orig-3), [(11)](#equation-orig-4), [(12)](#equation-orig-5), and [(13)](#equation-orig-6), and “work backwards” from time $ t_1 - 1 $\n", "\n", "Since we’re working backwards, $ P_{1t+1} $ and $ P_{2t+1} $ are taken as given at each stage\n", "\n", "Moreover, since\n", "\n", "- some terms on the right hand side of [(10)](#equation-orig-3) contain $ F_{2t} $ \n", "- some terms on the right hand side of [(12)](#equation-orig-5) contain $ F_{1t} $ \n", "\n", "\n", "we need to solve these $ k_1 + k_2 $ equations simultaneously" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Key insight\n", "\n", "A key insight is that equations [(10)](#equation-orig-3) and [(12)](#equation-orig-5) are linear in $ F_{1t} $ and $ F_{2t} $\n", "\n", "After these equations have been solved, we can take $ F_{it} $ and solve for $ P_{it} $ in [(11)](#equation-orig-4) and [(13)](#equation-orig-6)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Infinite horizon\n", "\n", "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 $\n", "\n", "In practice, we usually fix $ t_1 $ and compute the equilibrium of an infinite horizon game by driving $ t_0 \\rightarrow - \\infty $\n", "\n", "This is the approach we adopt in the next section" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Implementation\n", "\n", "We use the function [nnash](https://github.com/QuantEcon/QuantEcon.jl/blob/master/src/lqnash.jl) from [QuantEcon.jl](http://quantecon.org/julia_index.html) that computes a Markov perfect equilibrium of the infinite horizon linear quadratic dynamic game in the manner described above" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Application\n", "\n", "\n", "\n", "Let’s use these procedures to treat some applications, starting with the duopoly model" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### A duopoly model\n", "\n", "To map the duopoly model into coupled linear-quadratic dynamic programming problems, define the state\n", "and controls as\n", "\n", "$$\n", "x_t :=\n", "\\begin{bmatrix}\n", " 1 \\\\\n", " q_{1t} \\\\\n", " q_{2t}\n", "\\end{bmatrix}\n", "\\quad \\text{and} \\quad\n", "u_{it} :=\n", "q_{i,t+1} - q_{it}, \\quad i=1,2\n", "$$\n", "\n", "If we write\n", "\n", "$$\n", "x_t' R_i x_t + u_{it}' Q_i u_{it}\n", "$$\n", "\n", "where $ Q_1 = Q_2 = \\gamma $,\n", "\n", "$$\n", "R_1 :=\n", "\\begin{bmatrix}\n", " 0 & -\\frac{a_0}{2} & 0 \\\\\n", " -\\frac{a_0}{2} & a_1 & \\frac{a_1}{2} \\\\\n", " 0 & \\frac{a_1}{2} & 0\n", "\\end{bmatrix}\n", "\\quad \\text{and} \\quad\n", "R_2 :=\n", "\\begin{bmatrix}\n", " 0 & 0 & -\\frac{a_0}{2} \\\\\n", " 0 & 0 & \\frac{a_1}{2} \\\\\n", " -\\frac{a_0}{2} & \\frac{a_1}{2} & a_1\n", "\\end{bmatrix}\n", "$$\n", "\n", "then we recover the one-period payoffs in expression [(3)](#equation-game3)\n", "\n", "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\n", "\n", "$$\n", "A :=\n", "\\begin{bmatrix}\n", " 1 & 0 & 0 \\\\\n", " 0 & 1 & 0 \\\\\n", " 0 & 0 & 1\n", "\\end{bmatrix},\n", "\\quad\n", "B_1 :=\n", "\\begin{bmatrix}\n", " 0 \\\\\n", " 1 \\\\\n", " 0\n", "\\end{bmatrix},\n", "\\quad\n", "B_2 :=\n", "\\begin{bmatrix}\n", " 0 \\\\\n", " 0 \\\\\n", " 1\n", "\\end{bmatrix}\n", "$$\n", "\n", "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:\n", "\n", "\n", "\n", "$$\n", "x_{t+1} = (A - B_1 F_1 -B_1 F_2 ) x_t \\tag{14}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Parameters and Solution\n", "\n", "Consider the previously presented duopoly model with parameter values of:\n", "\n", "- $ a_0 = 10 $ \n", "- $ a_1 = 2 $ \n", "- $ \\beta = 0.96 $ \n", "- $ \\gamma = 12 $ \n", "\n", "\n", "From these we compute the infinite horizon MPE using the following code" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "hide-output": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Computed policies for firm 1 and firm 2:\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "F1 = [-0.6684661455442794 0.295124817744414 0.07584666305807419]\n", "F2 = [-0.6684661455442794 0.07584666305807419 0.295124817744414]\n" ] } ], "source": [ "using QuantEcon, LinearAlgebra\n", "\n", "# parameters\n", "a0 = 10.0\n", "a1 = 2.0\n", "β = 0.96\n", "γ = 12.0\n", "\n", "# in LQ form\n", "A = I + zeros(3, 3)\n", "B1 = [0.0, 1.0, 0.0]\n", "B2 = [0.0, 0.0, 1.0]\n", "\n", "R1 = [ 0.0 -a0 / 2.0 0.0;\n", " -a0 / 2.0 a1 a1 / 2.0;\n", " 0.0 a1 / 2.0 0.0]\n", "\n", "R2 = [ 0.0 0.0 -a0 / 2.0;\n", " 0.0 0.0 a1 / 2.0;\n", " -a0 / 2.0 a1 / 2.0 a1]\n", "\n", "Q1 = Q2 = γ\n", "S1 = S2 = W1 = W2 = M1 = M2 = 0.0\n", "\n", "# solve using QE's nnash function\n", "F1, F2, P1, P2 = nnash(A, B1, B2, R1, R2, Q1, Q2, S1, S2, W1, W2, M1, M2,\n", " beta=β)\n", "\n", "# display policies\n", "println(\"Computed policies for firm 1 and firm 2:\")\n", "println(\"F1 = $F1\")\n", "println(\"F2 = $F2\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Running the code produces the following output\n", "\n", "One way to see that $ F_i $ is indeed optimal for firm $ i $ taking $ F_2 $ as given is to use [QuantEcon.jl](http://quantecon.org/julia_index.html)’s LQ type\n", "\n", "In particular, let’s take F2 as computed above, plug it into [(8)](#equation-eq-mpe-p1p) and [(9)](#equation-eq-mpe-p1d) to get firm 1’s problem and solve it using LQ\n", "\n", "We hope that the resulting policy will agree with F1 as computed above" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "hide-output": false }, "outputs": [ { "data": { "text/plain": [ "1×3 Array{Float64,2}:\n", " -0.668466 0.295125 0.0758467" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Λ1 = A - (B2 * F2)\n", "lq1 = QuantEcon.LQ(Q1, R1, Λ1, B1, bet=β)\n", "P1_ih, F1_ih, d = stationary_values(lq1)\n", "F1_ih" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This is close enough for rock and roll, as they say in the trade\n", "\n", "Indeed, isapprox agrees with our assessment" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "hide-output": false }, "outputs": [ { "data": { "text/plain": [ "true" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "isapprox(F1, F1_ih, atol=1e-7)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Dynamics\n", "\n", "Let’s now investigate the dynamics of price and output in this simple duopoly model under the MPE policies\n", "\n", "Given our optimal policies $ F1 $ and $ F2 $, the state evolves according to [(14)](#equation-eq-mpe-cle)\n", "\n", "The following program\n", "\n", "- imports $ F1 $ and $ F2 $ from the previous program along with all parameters \n", "- computes the evolution of $ x_t $ using [(14)](#equation-eq-mpe-cle) \n", "- extracts and plots industry output $ q_t = q_{1t} + q_{2t} $ and price $ p_t = a_0 - a_1 q_t $ " ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "hide-output": false }, "outputs": [ { "data": { "image/png": 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" }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "using Plots\n", "gr(fmt=:png);\n", "\n", "AF = A - B1 * F1 - B2 * F2\n", "n = 20\n", "x = zeros(3, n)\n", "x[:, 1] = [1 1 1]\n", "for t in 1:n-1\n", " x[:, t+1] = AF * x[:, t]\n", "end\n", "q1 = x[2, :]\n", "q2 = x[3, :]\n", "q = q1 + q2 # total output, MPE\n", "p = a0 .- a1 * q # price, MPE\n", "\n", "plt = plot(q, color=:blue, lw=2, alpha=0.75, label=\"total output\")\n", "plot!(plt, p, color=:green, lw=2, alpha=0.75, label=\"price\")\n", "plot!(plt, title=\"Output and prices, duopoly MPE\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Note that the initial condition has been set to $ q_{10} = q_{20} = 1.0 $\n", "\n", "To gain some perspective we can compare this to what happens in the monopoly case\n", "\n", "The first panel in the next figure compares output of the monopolist and industry output under the MPE, as a function of time\n", "\n", "The second panel shows analogous curves for price\n", "\n", "\n", "\n", "\n", "\n", " \n", "Here parameters are the same as above for both the MPE and monopoly solutions\n", "\n", "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\n", "\n", "As expected, output is higher and prices are lower under duopoly than monopoly" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Exercises" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Exercise 1\n", "\n", "Replicate the [pair of figures](#mpe-vs-monopolist) showing the comparison of output and prices for the monopolist and duopoly under MPE\n", "\n", "Parameters are as in duopoly_mpe.jl and you can use that code to compute MPE policies under duopoly\n", "\n", "The optimal policy in the monopolist case can be computed using [QuantEcon.jl](http://quantecon.org/julia_index.html)’s LQ type" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Exercise 2\n", "\n", "In this exercise we consider a slightly more sophisticated duopoly problem\n", "\n", "It takes the form of infinite horizon linear quadratic game proposed by Judd [[Jud90]](https://lectures.quantecon.org/zreferences.html#judd1990)\n", "\n", "Two firms set prices and quantities of two goods interrelated through their demand curves\n", "\n", "Relevant variables are defined as follows:\n", "\n", "- $ I_{it} $ = inventories of firm $ i $ at beginning of $ t $ \n", "- $ q_{it} $ = production of firm $ i $ during period $ t $ \n", "- $ p_{it} $ = price charged by firm $ i $ during period $ t $ \n", "- $ S_{it} $ = sales made by firm $ i $ during period $ t $ \n", "- $ E_{it} $ = costs of production of firm $ i $ during period $ t $ \n", "- $ C_{it} $ = costs of carrying inventories for firm $ i $ during $ t $ \n", "\n", "\n", "The firms’ cost functions are\n", "\n", "- $ C_{it} = c_{i1} + c_{i2} I_{it} + 0.5 c_{i3} I_{it}^2 $ \n", "- $ E_{it} = e_{i1} + e_{i2}q_{it} + 0.5 e_{i3} q_{it}^2 $ where $ e_{ij}, c_{ij} $ are positive scalars \n", "\n", "\n", "Inventories obey the laws of motion\n", "\n", "$$\n", "I_{i,t+1} = (1 - \\delta) I_{it} + q_{it} - S_{it}\n", "$$\n", "\n", "Demand is governed by the linear schedule\n", "\n", "$$\n", "S_t = D p_{it} + b\n", "$$\n", "\n", "where\n", "\n", "- $ S_t = \\begin{bmatrix} S_{1t} & S_{2t} \\end{bmatrix}' $ \n", "- $ D $ is a $ 2\\times 2 $ negative definite matrix and \n", "- $ b $ is a vector of constants \n", "\n", "\n", "Firm $ i $ maximizes the undiscounted sum\n", "\n", "$$\n", "\\lim_{T \\to \\infty}\\ {1 \\over T}\\ \\sum^T_{t=0}\\ \\left( p_{it} S_{it} - E_{it} - C_{it} \\right)\n", "$$\n", "\n", "We can convert this to a linear quadratic problem by taking\n", "\n", "$$\n", "u_{it} =\n", "\\begin{bmatrix}\n", " p_{it} \\\\\n", " q_{it}\n", "\\end{bmatrix}\n", "\\quad \\text{and} \\quad\n", "x_t =\n", "\\begin{bmatrix}\n", " I_{1t} \\\\\n", " I_{2t} \\\\\n", " 1\n", "\\end{bmatrix}\n", "$$\n", "\n", "Decision rules for price and quantity take the form $ u_{it} = -F_i x_t $\n", "\n", "The Markov perfect equilibrium of Judd’s model can be computed by filling in the matrices appropriately\n", "\n", "The exercise is to calculate these matrices and compute the following figures\n", "\n", "The first figure shows the dynamics of inventories for each firm when the parameters are" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "hide-output": false }, "outputs": [ { "data": { "text/plain": [ "3-element Array{Int64,1}:\n", " 10\n", " 10\n", " 3" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "δ = 0.02\n", "D = [ -1 0.5;\n", " 0.5 -1]\n", "b = [25, 25]\n", "c1 = c2 = [1, -2, 1]\n", "e1 = e2 = [10, 10, 3]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", " \n", "Inventories trend to a common steady state\n", "\n", "If we increase the depreciation rate to $ \\delta = 0.05 $, then we expect steady state inventories to fall\n", "\n", "This is indeed the case, as the next figure shows\n", "\n", "" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Solutions" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Exercise 1\n", "\n", "First let’s compute the duopoly MPE under the stated parameters" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "hide-output": false }, "outputs": [ { "data": { "text/plain": [ "([-0.6684661455442794 0.295124817744414 0.07584666305807419], [-0.6684661455442794 0.07584666305807419 0.295124817744414], [-100.74013291681779 -13.28370101134053 2.435873888234418; -13.283701011340526 5.441368457863284 1.9305445296892967; 2.4358738882344166 1.9305445296892967 -0.18944247543524873], [-100.74013291681771 2.435873888234418 -13.283701011340526; 2.435873888234417 -0.18944247543524873 1.9305445296892967; -13.283701011340526 1.9305445296892967 5.441368457863284])" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# parameters\n", "a0 = 10.0\n", "a1 = 2.0\n", "β = 0.96\n", "γ = 12.0\n", "\n", "# in LQ form\n", "A = I + zeros(3, 3)\n", "B1 = [0.0, 1.0, 0.0]\n", "B2 = [0.0, 0.0, 1.0]\n", "\n", "R1 = [ 0.0 -a0 / 2.0 0.0;\n", " -a0 / 2.0 a1 a1 / 2.0;\n", " 0.0 a1 / 2.0 0.0]\n", "\n", "R2 = [ 0.0 0.0 -a0 / 2.0;\n", " 0.0 0.0 a1 / 2.0;\n", " -a0 / 2.0 a1 / 2.0 a1]\n", "\n", "Q1 = Q2 = γ\n", "S1 = S2 = W1 = W2 = M1 = M2 = 0.0\n", "\n", "# solve using QE's nnash function\n", "F1, F2, P1, P2 = nnash(A, B1, B2, R1, R2, Q1, Q2, S1, S2, W1, W2, M1, M2,\n", " beta=β)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we evaluate the time path of industry output and prices given\n", "initial condition $ q_{10} = q_{20} = 1 $" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "hide-output": true }, "outputs": [], "source": [ "AF = A - B1 * F1 - B2 * F2\n", "n = 20\n", "x = zeros(3, n)\n", "x[:, 1] = [1 1 1]\n", "for t in 1:(n-1)\n", " x[:, t+1] = AF * x[:, t]\n", "end\n", "q1 = x[2, :]\n", "q2 = x[3, :]\n", "q = q1 + q2 # Total output, MPE\n", "p = a0 .- a1 * q # Price, MPE" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Next let’s have a look at the monopoly solution\n", "\n", "For the state and control we take\n", "\n", "$$\n", "x_t = q_t - \\bar q\n", "\\quad \\text{and} \\quad\n", "u_t = q_{t+1} - q_t\n", "$$\n", "\n", "To convert to an LQ problem we set\n", "\n", "$$\n", "R = a_1\n", "\\quad \\text{and} \\quad\n", "Q = \\gamma\n", "$$\n", "\n", "in the payoff function $ x_t' R x_t + u_t' Q u_t $ and\n", "\n", "$$\n", "A = B = 1\n", "$$\n", "\n", "in the law of motion $ x_{t+1} = A x_t + B u_t $\n", "\n", "We solve for the optimal policy $ u_t = - Fx_t $ and track the\n", "resulting dynamics of $ \\{q_t\\} $, starting at $ q_0 = 2.0 $" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "hide-output": true }, "outputs": [], "source": [ "R = a1\n", "Q = γ\n", "A = B = 1\n", "lq_alt = QuantEcon.LQ(Q, R, A, B, bet=β)\n", "P, F, d = stationary_values(lq_alt)\n", "q̄ = a0 / (2.0 * a1)\n", "qm = zeros(n)\n", "qm[1] = 2\n", "x0 = qm[1]-q̄\n", "x = x0\n", "for i in 2:n\n", " x = A * x - B * F[1] * x\n", " qm[i] = float(x) + q̄\n", "end\n", "pm = a0 .- a1 * qm" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let’s have a look at the different time paths" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "hide-output": false }, "outputs": [ { "data": { "image/png": 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" }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "plt_q = plot(qm, color=:blue, lw=2, alpha=0.75, label=\"monopolist output\")\n", "plot!(plt_q, q, color=:green, lw=2, alpha=0.75, label=\"MPE total output\")\n", "plot!(plt_q, xlabel=\"time\", ylabel=\"output\", ylim=(2,4),legend=:topright)\n", "\n", "plt_p = plot(pm, color=:blue, lw=2, alpha=0.75, label=\"monopolist price\")\n", "plot!(plt_p, p, color=:green, lw=2, alpha=0.75, label=\"MPE price\")\n", "plot!(plt_p, xlabel=\"time\", ylabel=\"price\",legend=:topright)\n", "\n", "plot(plt_q, plt_p, layout=(2,1), size=(700,600))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Exercise 2\n", "\n", "We treat the case $ \\delta = 0.02 $" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "hide-output": false }, "outputs": [ { "data": { "text/plain": [ "0.98" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "δ = 0.02\n", "D = [-1 0.5;\n", " 0.5 -1]\n", "b = [25, 25]\n", "c1 = c2 = [1, -2, 1]\n", "e1 = e2 = [10, 10, 3]\n", "δ_1 = 1-δ" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Recalling that the control and state are\n", "\n", "$$\n", "u_{it} =\n", "\\begin{bmatrix}\n", " p_{it} \\\\\n", " q_{it}\n", "\\end{bmatrix}\n", "\\quad \\text{and} \\quad\n", "x_t =\n", "\\begin{bmatrix}\n", " I_{1t} \\\\\n", " I_{2t} \\\\\n", " 1\n", "\\end{bmatrix}\n", "$$\n", "\n", "we set up the matrices as follows:" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "hide-output": false }, "outputs": [ { "data": { "text/plain": [ "2×2 Array{Float64,2}:\n", " 0.0 0.0 \n", " 0.0 0.25" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# create matrices needed to compute the Nash feedback equilibrium\n", "A = [δ_1 0 -δ_1 * b[1];\n", " 0 δ_1 -δ_1 * b[2];\n", " 0 0 1]\n", "\n", "B1 = δ_1 * [1 -D[1, 1];\n", " 0 -D[2, 1];\n", " 0 0]\n", "B2 = δ_1 * [0 -D[1, 2];\n", " 1 -D[2, 2];\n", " 0 0]\n", "\n", "R1 = -[0.5 * c1[3] 0 0.5 * c1[2];\n", " 0 0 0;\n", " 0.5 * c1[2] 0 c1[1]]\n", "\n", "R2 = -[0 0 0;\n", " 0 0.5 * c2[3] 0.5*c2[2];\n", " 0 0.5 * c2[2] c2[1]]\n", "\n", "Q1 = [-0.5*e1[3] 0;\n", " 0 D[1, 1]]\n", "Q2 = [-0.5*e2[3] 0;\n", " 0 D[2, 2]]\n", "\n", "S1 = zeros(2, 2)\n", "S2 = copy(S1)\n", "\n", "W1 = [ 0.0 0.0;\n", " 0.0 0.0;\n", " -0.5 * e1[2] b[1] / 2.0]\n", "W2 = [ 0.0 0.0;\n", " 0.0 0.0;\n", " -0.5 * e2[2] b[2] / 2.0]\n", "\n", "M1 = [0.0 0.0;\n", " 0.0 D[1, 2] / 2.0]\n", "M2 = copy(M1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can now compute the equilibrium using `qe.nnash`" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "hide-output": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "Firm 1's feedback rule:\n", "\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "[0.24366658220856505 0.02723606266195122 -6.827882926030329; 0.3923707338756386 0.13969645088599783 -37.734107288592014]\n", "\n", "Firm 2's feedback rule:\n", "\n", "[0.027236062661951208 0.243666582208565 -6.827882926030323; 0.1396964508859978 0.39237073387563864 -37.73410728859201]\n" ] } ], "source": [ "F1, F2, P1, P2 = nnash(A, B1, B2, R1, R2, Q1, Q2, S1, S2, W1, W2, M1, M2)\n", "\n", "println(\"\\nFirm 1's feedback rule:\\n\")\n", "println(F1)\n", "\n", "println(\"\\nFirm 2's feedback rule:\\n\")\n", "println(F2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now let’s look at the dynamics of inventories, and reproduce the graph\n", "corresponding to $ \\delta = 0.02 $" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "hide-output": false }, "outputs": [ { "data": { "image/png": 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" }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "AF = A - B1 * F1 - B2 * F2\n", "n = 25\n", "x = zeros(3, n)\n", "x[:, 1] = [2 0 1]\n", "for t in 1:(n-1)\n", " x[:, t+1] = AF * x[:, t]\n", "end\n", "I1 = x[1, :]\n", "I2 = x[2, :]\n", "\n", "plot(I1, color=:blue, lw=2, alpha=0.75, label=\"inventories, firm 1\")\n", "plot!(I2, color=:green, lw=2, alpha=0.75, label=\"inventories, firm 2\")\n", "plot!(title=\"delta = 0.02\")" ] } ], "metadata": { "download_nb": 1, "download_nb_path": "https://lectures.quantecon.org/", "filename": "markov_perf.rst", "filename_with_path": "multi_agent_models/markov_perf", "kernelspec": { "display_name": "Julia 1.2.0", "language": "julia", "name": "julia-1.2" }, "language_info": { "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", "version": "1.2.0" }, "title": "Markov Perfect Equilibrium" }, "nbformat": 4, "nbformat_minor": 2 }