{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Insurance and Incentives\n", "\n", "*By Sebastian Graves and Thomas Sargent*\n", "\n", "This notebook computes optimal contracts for the three examples that lead off chapter 21 of \n", "**Recursive Macroeconomic Theory, Fourth edition** by Lars Ljungqvist and Thomas Sargent.\n", "\n", "The examples illustrate different sorts of tradeoffs between insurance and incentives that emerge under different\n", "limits on enforcement and information. \n", "\n", "In each of the three economies, a planner or money-lender designs an efficient contract to supply insurance to a risk-averse consumer who receives an exogenous random stream of a non-storable endowment.\n", "\n", "The only way that the consumer to smooth consumption across states and time is to interact with the planner. \n", "\n", "The three models differ in the constraints that they impose on the planner. \n", "\n", "These constraints express the planner's limited ability either to enforce a contract or to observe the consumer's endowment\n", "\n", "Each of the examples uses a version of what we have nicknamed **dynamic programming squared**\n", "\n", "In a dynamic programming squared problem, a value function from one Bellman equation is an argument of another Bellman equation. \n", "\n", "In the examples below, a planner or money lender's value function will have as an argument the value of a villager\n", "that satisfies a Bellman equation\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Three models of a villager and a money lender\n", "\n", "Imagine a village with a large number of ex ante\n", "identical households. Each household has preferences over\n", "consumption streams that are ordered by\n", "$$E_{-1}\\sum_{t=0}^\\infty \\beta^t u(c_t),$$\n", "where $u(c)$ is an increasing, strictly concave, and twice\n", "continuously differentiable function,\n", " $\\beta \\in (0,1)$ is a discount factor, and $E_{-1}$ is the mathematical expectation\n", " not conditioning on any information available at time $0$ or later. \n", " \n", "Each household\n", "receives a stochastic endowment stream $\\{y_t\\}_{t=0}^\\infty$,\n", "where for each $t \\geq 0$, $y_t$ is independently and\n", "identically distributed according to the discrete\n", "probability distribution ${\\rm Prob} (y_t = \\overline y_s) = \\Pi_s,$\n", "where $s \\in \\{1, 2, \\ldots ,S\\}\\equiv {\\bf S}$ and\n", "$\\overline y_{s+1}>\\overline y_s$. \n", "\n", "The consumption\n", "good is not storable. \n", "\n", "At time $t \\geq 1$, the\n", "household has received a history of endowments\n", "$h_t = (y_t, y_{t-1}, \\ldots, y_0).$\n", "\n", "Endowment processes are distributed independently and identically\n", " both across time and\n", "across households.\n", "\n", "\n", "##### Competitive equilibrium\n", "\n", "In this setting, if there were a competitive equilibrium with\n", "complete markets, at date\n", "$0$ households would trade history- and date-contingent claims. \n", "\n", "Since households are ex ante\n", "identical, each household would consume the per capita\n", "endowment in every period, and its lifetime utility would be\n", "\n", "$$v_{\\rm pool} = \\sum_{t=0}^\\infty\n", "\\beta^t \\, u\\!\\left(\\sum_{s=1}^S \\Pi_s \\overline y_s\\right) =\n", " {1 \\over 1-\\beta}\\, u\\!\\left(\\sum_{s=1}^S \\Pi_s \\overline y_s\\right) .\n", "$$\n", " \n", " Households would thus insure away all\n", "risks from their individual endowment processes.\n", "\n", "But the\n", " incentive constraints that we are about to specify make\n", "this allocation unattainable. \n", "\n", "For each specification of incentive\n", "constraints, we shall solve a planning problem for an efficient\n", "allocation that respects those constraints.\n", "\n", "\n", " Following a tradition started by\n", "Edward Green (1987) [*Lending and the Smoothing of Uninsurable\n", " Income*, in Edward C. Prescott and Neil Wallace, editors, **Contractual Arrangements for\n", " Intertemporal Trade**, Minnesota Studies in Macroeconomics series, Vol.\n", " 1, Minneapolis: University of Minnesota Press, pp. 3--25], we assume that a *moneylender* or *planner* is\n", "the only person in the village who has access to\n", "a risk-free loan market outside the village.\n", "\n", "The moneylender can borrow or lend at a constant one-period\n", "risk-free gross interest rate $R=\\beta^{-1}$.\n", "\n", "Households cannot borrow or lend with each other,\n", "and can trade only with the moneylender. \n", "\n", "Furthermore,\n", "we assume that the moneylender is committed to honor his\n", "promises.\n", "\n", "We will study three distinct environments in which there are three alternative types of incentive constraints.\n", "\n", "\n", "**Enviroment a.** Both the money lender and the household observe the household's history of endowments at each time $t$.\n", "Although the moneylender can commit to honor a\n", "contract, households cannot commit and at any time are\n", " free to walk away from an arrangement\n", "with the moneylender\n", "and live in perpetual autarky thereafter. They must be induced not to do so\n", "by the structure of\n", "the contract.\n", "This is a model of *one-sided commitment* in which the\n", "contract must be *self-enforcing*. That is, it must be structured to induce the household to prefer to\n", "conform to it.\n", "\n", "**Environment b.** Households *can* make commitments and enter\n", "into enduring and binding contracts with the moneylender,\n", "but they have private\n", "information about their own incomes. The moneylender\n", "can see neither their income nor their consumption. Instead,\n", " exchanges between the moneylender and a household must\n", "be based on the household's own reports about income\n", "realizations. An incentive-compatible contract induces\n", "a household to report its income truthfully.\n", "\n", "**Environment c.** The environment is the same as b except that now households have access to a storage technology that\n", "cannot be observed by the moneylender.\n", "Households can store nonnegative amounts of goods at a risk-free\n", "gross return of $R$ equal to the interest rate that\n", "the moneylender faces in the outside credit market.\n", "Since the moneylender can both borrow and lend at the interest\n", "rate $R$ outside of the village,\n", "the private storage technology does not change the economy's\n", "aggregate resource constraint, but it does affect the set of\n", "incentive-compatible contracts between the moneylender and the\n", "households.\n", "\n", "\n", "#### Preview\n", "\n", "\n", "When we compute efficient allocations for each of these three\n", "environments, we find that the dynamics of the implied\n", "consumption allocations differ dramatically. \n", "\n", " \n", " We shall see that the dynamics\n", "of consumption outcomes evidently differ substantially across the\n", "three environments, increasing monotonically and then flattening out in environment a,\n", "stochastically heading south in environment b, and stochastically heading north in\n", "environment c.\n", "These sample path properties will reflect how the optimal contracts cope with the three different frictions that we have put into the environment. \n", "\n", "Chapter 21 of RMT4 explains why sample paths of consumption differ\n", "so much across these three settings." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Three computed contracts\n", "\n", "\n", "For all three environments discussed, consumers have a utility function:\n", "\n", "$$u(c) = - \\gamma^{-1} \\exp(-\\gamma c)$$\n", "\n", "We set $\\gamma = 0.7$, and the discount factor, $\\beta$ to 0.8. \n", "\n", "The consumers receive an iid endowment that can take any integer in the range $[\\bar y_1,...,\\bar y_{5}] = [6,...,10]$. \n", "\n", "The probability of each realisation is $\\Pi_s = \\frac{1-\\lambda}{1-\\lambda^{5}}\\lambda^{s-1}$ with $\\lambda = 0.4$.\n", "\n", "As mentioned above, an interesting benchmark case is a complete markets environment.\n", "\n", "Because all households are *ex ante* identical, in a complete markets economy each household would consume the per capita endowment in every period, and its lifetime utility would be:\n", "\n", "$$v_{pool} = \\frac{1}{1-\\beta} u \\left( \\sum_{s=1}^S \\Pi_s \\bar y_s \\right) = \\frac{u(c_{pool})}{1-\\beta}$$\n", "\n", "Later we will compare the consumption paths for each environment to that which would occur in the complete markets environment. \n", "\n", "In each environment, we compute allocations for the situation in which the planner or money lender just breaks even.\n", "\n", "## Environment a\n", "\n", "The first environment is one in which the planner is able to commit, but households are not.\n", "\n", "At any time households are free to walk away from an arrangement with the planner, and live in perpetual autarky thereafter.\n", "\n", "RMT4 shows how this problem can be written in a recursive form. \n", "\n", "Equations 21.3.4 to 21.3.8 in RMT4 express the planners's problem as:\n", "\n", "\\begin{align}\n", "\t&P(v) = \\max_{c_s,w_s} \\sum_{s=1}^S \\Pi_s \\left[ (\\bar y_s - c_s) + \\beta P(w_s) \\right] \\\\\n", "\t&\\text{subject to} \\\\\n", "\t&\\sum_{s=1}^S \\Pi_s \\left[ u(c_s) + \\beta w_s \\right] \\geq v \\\\\n", "\t&u(c_s) + \\beta w_s \\geq u(\\bar y_s) + \\beta v_{aut} \\text{ , s = 1,...,S} \\\\\n", "\t&c_s \\in [c_{min},c_{max}] \\\\\n", "\t&w_s \\in [v_{aut},\\bar v]\n", "\\end{align}\n", "\n", "where $w_s$ is the promised value with which the consumer will enter the next period, given that $y = \\bar y_s$ this period. \n", "\n", "The first constraint is a promise keeping constraint, while the second set of constraints are participation constraints. $[c_{min},c_{max}]$ is a bounded set, while $\\bar v$ just needs to be a very large number. \n", "\n", "\n", "The value of autarky to the households is:\n", "\n", "$$v_{aut} = \\frac{1}{1-\\beta} \\sum_{s=1}^S \\Pi_s u(\\bar y_s)$$\n", "\n", "Below we solve the moneylender's problem in this environment by approximating $P(v)$ using Chebyshev polynomials." ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "from scipy.optimize import minimize, fsolve\n", "from scipy.interpolate import UnivariateSpline\n", "import matplotlib.pyplot as plt\n", "import numpy.polynomial.chebyshev as cheb\n", "%matplotlib inline" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "# Parameter values\n", "gamma = 0.7\n", "beta = 0.8\n", "lamb = 0.4\n", "S = 5\n", "y_grid = np.linspace(6,5+S,S)\n", "prob_grid = np.zeros(S)\n", "for i in range(S):\n", " prob_grid[i] = (1 - lamb)/(1-lamb**S)*lamb**(i)\n", " \n", "# Utility function\n", "u = lambda c: -gamma**(-1)*np.exp(-gamma*c)\n", "u_inv = lambda u: np.log(-gamma*u)/(-gamma)\n", "\n", "# Calculate complete markets consumption \n", "c_pool = np.dot(prob_grid,y_grid)\n", "\n", "# Calculate value of autarky\n", "v_aut = 1/(1-beta)*np.dot(prob_grid, u(y_grid))" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "# Functions used in each environment\n", "\n", "# Nodes and basis matrix for Chebyshev approximation\n", "\n", "def Cheb_basis(order,lb,ub):\n", " # Calculate roots of Chebyshev polynomial\n", " k = np.linspace(order, 1, order)\n", " roots = np.cos((2*k - 1)*np.pi/(2*order))\n", " # Scale to approximation space\n", " s = lb + (roots - -1)/2*(ub-lb)\n", " # Create basis matrix\n", " Phi = cheb.chebvander(roots, order-1)\n", " return s, Phi\n", "\n", "# Value Function Iteration\n", "\n", "def Bellman_Iterations(s, Phi, P_fun, x_store, coeff, tolc=1e-6, bnds=None, cons=(), max_iters=100):\n", " global x, c\n", " c = coeff\n", " order = Phi.shape[1]\n", " iters = 0\n", " diff = 1\n", " \n", " while diff > tolc:\n", " # 1. Maximization, given value function guess\n", " P_iter = np.zeros(order)\n", " for i in range(order):\n", " x = s[i]\n", " res = minimize(P_fun, x_store[i], method = 'SLSQP', bounds = bnds, constraints=cons, tol=1e-15)\n", " x_store[i] = res.x\n", " P_iter[i] = -P_fun(res.x)\n", " # 2. Bellman updating of Value Function coefficients\n", " c1 = np.linalg.solve(Phi, P_iter)\n", " # 3. Compute distance and update\n", " diff = max(abs(c1 - c))\n", " print(diff)\n", " c = np.copy(c1)\n", " iters = iters + 1\n", " \n", " if iters >= max_iters:\n", " print('Convergence failed after {} iterations'.format(iters))\n", " break\n", " \n", " if diff < tolc:\n", " print('Convergence achieved after {} iterations'.format(iters))\n", "\n", " return c" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "0.02046457448057115\n", "0.01420417271952934\n", "0.007919316517300268\n", "0.004362488108637486\n", "0.0024521591210679983\n", "0.0014181717553074513\n", "0.0008450151371991454\n", "0.0005180673399127755\n", "0.0003258997519962614\n", "0.0002096164890845742\n", "0.00013734153609978872\n", "9.134953200051754e-05\n", "6.149551030254496e-05\n", "4.179812741511579e-05\n", "2.8630491596959295e-05\n", "1.973504389185532e-05\n", "1.367544312058655e-05\n", "9.5197594243146e-06\n", "6.6539162798529006e-06\n", "4.66802753196216e-06\n", "3.286358093546049e-06\n", "2.323138710569328e-06\n", "1.666813717604576e-06\n", "1.1999303339838008e-06\n", "8.667152331387484e-07\n", "Convergence achieved after 25 iterations\n" ] } ], "source": [ "# Value Function Approximation\n", "# Set bounds and approximation order\n", "v_min = v_aut\n", "v_max = -0.065\n", "c_min = 0\n", "c_max = 50\n", "order = 70\n", "\n", "# Calculate nodes and basis matrix\n", "s, Phi = Cheb_basis(order, v_min, v_max)\n", "\n", "# Bounds for Maximisation\n", "lb = np.concatenate([np.ones(S)*c_min, np.ones(S)*v_min], axis=0)\n", "ub = np.concatenate([np.ones(S)*c_max, np.ones(S)*v_max], axis=0)\n", "\n", "# Initialize Value Function coefficients and goess for c,w\n", "y = (c_pool - u_inv(s*(1-beta)))/(1-beta)\n", "c = np.linalg.solve(Phi, y)\n", "x_init = np.concatenate([np.ones(S)*c_min, np.ones(S)*v_min], axis=0)\n", "\n", "# Function to minimize and constraints\n", "def P_fun(x):\n", " scale = -1 + 2*(x[S:2*S] - v_min)/(v_max - v_min)\n", " P = np.dot(cheb.chebvander(scale,order-1),c)\n", " P_fun = - prob_grid.dot((y_grid - x[0:S]) + beta*P)\n", " return P_fun\n", "\n", "def cons12(y):\n", " global x\n", " return prob_grid.dot(u(y[0:S]) + beta*y[S:2*S]) - x\n", "\n", "cons1 = ({'type': 'ineq', 'fun': lambda y: u(y[0:S]) + beta*y[S:2*S] - u(y_grid) - beta*v_aut},\n", " {'type': 'ineq', 'fun': cons12})\n", "\n", "bnds1 = np.concatenate([lb.reshape(2*S, 1), ub.reshape(2*S, 1)], axis = 1)\n", "\n", "# Bellman Iterations\n", "NBell = 5\n", "tolc = 1e-6\n", "diff = 1\n", "iters = 1\n", "x_store = {}\n", "for i in range(order):\n", " x_store[i] = x_init\n", " \n", "c = Bellman_Iterations(s, Phi, P_fun, x_store, c, bnds=bnds1, cons=cons1)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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