A Model with Collateral Constraints

Transcription

1 A Model with Collateral Constraints Jesús Fernández-Villaverde University of Pennsylvania December 2, 2012 Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

2 Motivation Kiyotaki and Moore, Main ideas: 1 Feedback loop between financial constraints and economic activity. 2 Dual role of assets as factors of production and as collateral (fire sale Shleifer and Vishny, 1992). Simple model: 1 Discrete time. 2 Perfect foresight. 3 Little heterogeneity. Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

3 Preferences Continuum of infinitely lived, risk-neutral agents: 1 Mass 1 of farmers: E 0 β t x t t=0 2 Mass m of gatherers: E 0 β t xt t=0 Assumption A: β < β. Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

4 Goods and Markets Two goods: 1 Durable asset (land): does not depreciate, fixed supply K. 2 Nondurable commodity (fruit): x t and x t. Fruit is the numeraire. Competitive spot market for land in each period t: price of 1 unit of land q t. Credit market: one unit of fruit at period t is exchanged for R t units of fruit at period t + 1. Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

5 Technology for Farmers Linear production function that uses land k t to produce fruit: y t+1 = (a + c) k t Two parts of output: 1 a: tradable output. 2 c: non-tradable output (basically to induce some current consumption). Assumption B: non-tradable output is big enough ( ) 1 c > β 1 a Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

6 Budget Constraint for the Farmer Farmers buy (net) land k t k t 1 at price q t. Farmers borrow a quantity b t at interest rate R t. Farmers consume x t at cost x t ck t 1 (total consumption less non-tradable output). Farmers sell output ak t 1. Therefore: q t (k t k t 1 ) + R t b t=1 + x t ck t 1 = ak t 1 + b t Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

7 Borrowing Constraint Hart and Moore, 1994 Farmer labor input is necessary and lot-specific once production has started. Farmer labor cannot be precommitted. Hence: outside value = q t+1 k t < (a + c) k t = inside value Under renegotiation after a default, the farmer can never get less than q t+1 k t. A farmer can, then borrow a quantity b t such that (secure debt): R t b t q t+1 k t Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

8 Technology for Gatherers Concave production function that uses land k t to produce fruit: y t+1 = G (k t 1). Assumption C: to avoid corner solutions: ) G (0) > ar t > G ( K m that is, marginal productivity of gatherers is such that, in equilibrium both farmers and gatherers hold some land (easy because we will see below that R t is constant). Budget constraint: q t (kt kt 1) + R t bt=1 + xt = G (kt 1) + bt No specific skill in production: no borrowing constraint. Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

9 Equilibrium I Definition An equilibrium is an allocation {k t, kt, x t, xt } t=0, debt {b t, bt } t=0, and prices {q t, R t } t=0 such that: 1 Given prices {q t, R t } t=0, farmers solve their problem: max E 0 {k t,x t,b t } β t x t t=0 t=0 s.t. q t (k t k t 1 ) + R t b t 1 + x t ck t 1 = ak t 1 + b t R t b t q t+1 k t Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

10 Equilibrium II Definition 2. Given prices {q t, R t } t=0, gatherers solve their problem: max E 0 {kt,xt,bt } β t x t t=0 t=0 s.t. q t (k t k t 1) + R t b t 1 + x t = G (k t 1) + b t 3. Markets clear: x t + mx t = (a + c) k t 1 + mg (kt 1) k t + mkt = K b t + mbt = 0 Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

11 Characterizing the Equilibrium I Since they are never constrained, gatherers satisfy Euler equation: 1 = β R t+1 R = R t = 1 β Also, by non-arbitrage, gatherers are indifferent at the margin about buying one unit more of land q t + 1 R ( G (k t ) + q t+1 ) = 0 or, rearranging terms, equate the rate of return of land to its user cost u t : ( G (kt ) = R q t 1 ) R q t+1 = Ru t Hence, all gatherers have the same amount of land. Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

12 Characterizing the Equilibrium II R t = 1 β together with assumption A farmers are always constrained. Decision rule of farmers (close to the steady state, role of assumption B): 1 Borrow the maximum amount possible: b t = q t+1k t R 2 Consume just their non-tradable fruit: x t = ck t 1 Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

13 Characterizing the Equilibrium III Using 1. and 2.: farmers buy as much land as possible: q t (k t k t 1 ) + Rb t 1 + x t ck t 1 = ak t 1 + b t q t (k t k t 1 ) + q t k t 1 = ak t 1 + q t+1k t R (Rq t q t+1 ) k t = Rak t 1 1 k t = q t 1 R q ak t 1 t+1 k t = 1 u t ak t 1 = 1 u t ((a + q t ) k t 1 Rb t 1 ) Last equation: farmer leverages his net wealth (a + q t ) k t 1 Rb t 1 given the down payment u t (which is also the user cost of land!) Decision rules are linear: distribution of land among farmers is irrelevant. Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

14 Steady State I First, Second, Third, k = 1 u ak u = a u = q 1 R q = (1 β ) q q = a 1 β k = G 1 (Ru) = G 1 (Ra) k = K mk Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

15 Steady State II Fourth, b = qk R = b = b m β 1 β ak Fifth, x = ck x = a m k + G (k ) Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

16 Comparison with Social Planner Optimal allocation of land: G ( k sp) = a + c k sp = G 1 (a + c) (same marginal productivity in both sectors). In the market allocation: Thus: Intuition. k = G 1 (a) k > k sp Consumption: it would depend on the social planner s objective function. Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

17 Computation of the Equilibrium I Given some initial k t and k t, we find: u t = 1 R G (k t ) k t+1 = 1 u t ak t k t+1 = 1 m ( K kt+1 ) By imposing a transversality condition to run out bubbles, we can find {q t } t=0 that satisfies: u t = q t 1 R q t+1 Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

18 Computation of the Equilibrium II To close the model: x t b t = q t+1k t R b t = b t m x t = ck t 1 = a m k t 1 + G (k t 1) Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

19 A Shock the Economy Think about the case where, unanticipatedly, farmers produce at time t y t = (a + c) k t 1 Farmers are poorer. Farmers demand less land: q t goes down, u t goes down, land moves from farmers to gatherers fall in production. But a lower q t means farmers can borrow less (they are leveraged, and hence their net wealth goes down more than proportionally to the shock) and get even less land. Comparison with social planner s response. Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

20 Feedback Loop Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

21 An Extended Model Two modifications: 1 Opportunities to invest arrive randomly. 2 Trees in addition to land. We will explore them later. Main idea. Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

22 Computation Basic model where: y t = (a + ε t + c) k t 1 where Dynare. ε t N (0, σ) Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

23 Main Idea We can think about equilibrium conditions as a system of functional equations of the form: E t H (d) = 0 for an unknown decision rule d. Perturbation solves the problem by specifying: d n (x, θ) = n θ i (x x 0 ) i i=0 We use implicit-function theorems to find coeffi cients θ i s. Inherently local approximation. However, often good global properties. Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

24 Motivation Many complicated mathematical problems have: 1 either a particular case 2 or a related problem. that is easy to solve. Often, we can use the solution of the simpler problem as a building block of the general solution. Very successful in physics. Sometimes perturbation is known as asymptotic methods. Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

25 Applications to Economics Judd and Guu (1993) showed how to apply it to economic problems. Recently, perturbation methods have been gaining much popularity. In particular, second- and third-order approximations are easy to compute and notably improve accuracy. Perturbation theory is the generalization of the well-known linearization strategy. Hence, we can use much of what we already know about linearization. Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

26 References General: 1 A First Look at Perturbation Theory by James G. Simmonds and James E. Mann Jr. 2 Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory by Carl M. Bender, Steven A. Orszag. Economics: 1 Perturbation Methods for General Dynamic Stochastic Models by Hehui Jin and Kenneth Judd. 2 Perturbation Methods with Nonlinear Changes of Variables by Kenneth Judd. 3 A gentle introduction: Solving Dynamic General Equilibrium Models Using a Second-Order Approximation to the Policy Function by Martín Uribe and Stephanie Schmitt-Grohe. Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

27 Asymptotic Expansion y t = y (s t, ε t ; σ) k,0,0 = y (s, 0; 0) +y s (s, 0; 0) (s t s) + y ε (s, 0; 0) ε t + y σ (s, 0; 0) σ y ss (s, 0; 0) (s t s) y sε (s, 0; 0) (s t s) ε t y sσ (s, 0; 0) (s t s) σ y εs (s, 0; 0) z t (k t k) y εε (s, 0; 0) ε 2 t y εσ (s, 0; 0) ε t σ y σs (s, 0; 0) σ (k t k) y σε (s, 0; 0) σε t y σ 2 (s, 0; 0) σ Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

28 The General Case Most of previous argument can be easily generalized. The set of equilibrium conditions of many DSGE models can be written as (note recursive notation) E t H(y, y, x, x ) = 0, where y t is a n y 1 vector of controls and x t = (s t, ε t ) is a n x 1 vector of states. Define n = n x + n y. Then H maps R n y R n y R n x R n x into R n. Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

29 Partitioning the State Vector The state vector x t can be partitioned as x = [x 1 ; x 2 ] t. x 1 = s t is a (n x n ɛ ) 1 vector of endogenous state variables. x 2 = ε t is a n ɛ 1 vector of exogenous state variables. Why do we want to partition the state vector? Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

30 Exogenous Stochastic Process Process with 3 parts: x 2 = Λx 2 + ση ɛ ɛ 1 The deterministic component Λx 2 : 1 Λ is a n ɛ n ɛ matrix, with all eigenvalues with modulus less than one. 2 More general: x 2 = Γ(x 2) + ση ɛ ɛ, where Γ is a non-linear function satisfying that all eigenvalues of its first derivative evaluated at the non-stochastic steady state lie within the unit circle. 2 The scaled innovation η ɛ ɛ where: 1 η ɛ is a known n ɛ n ɛ matrix. 2 ɛ is a n ɛ 1 i.i.d innovation with bounded support, zero mean, and variance/covariance matrix I. 3 The perturbation parameter σ. We can accommodate very general structures of x 2 through changes in the definition of the state space: i.e. stochastic volatility. Note we do not impose gaussianity. Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

31 The Perturbation Parameter The scalar σ 0 is the perturbation parameter. If we set σ = 0 we have a deterministic model. Important: there is only ONE perturbation parameter. The matrix η ɛ takes account of relative sizes of different shocks. Why bounded support? Samuelson (1970) and Jin and Judd (2002). Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

32 Solution of the Model The solution to the model is of the form: y = g(x; σ) x = h(x; σ) + σηɛ where g maps R n x R + into R n y and h maps R n x R + into R n x. The matrix η is of order n x n ɛ and is given by: [ ] η = η ɛ Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

33 Perturbation We wish to find a perturbation approximation of the functions g and h around the non-stochastic steady state, x t = x and σ = 0. We define the non-stochastic steady state as vectors ( x, ȳ) such that: H(ȳ, ȳ, x, x) = 0. Note that ȳ = g( x; 0) and x = h( x; 0). This is because, if σ = 0, then E t H = H. Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

34 Plugging-in the Proposed Solution Substituting the proposed solution, we define: F (x; σ) E t H(g(x; σ), g(h(x; σ) + ησɛ, σ), x, h(x; σ) + ησɛ ) = 0 Since F (x; σ) = 0 for any values of x and σ, the derivatives of any order of F must also be equal to zero. Formally: F x k σj (x; σ) = 0 x, σ, j, k, where F x k σj (x, σ) denotes the derivative of F with respect to x taken k times and with respect to σ taken j times. Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

35 First-Order Approximation We are looking for approximations to g and h around (x, σ) = ( x, 0) of the form: As explained earlier, g(x; σ) = g( x; 0) + g x ( x; 0)(x x) + g σ ( x; 0)σ h(x; σ) = h( x; 0) + h x ( x; 0)(x x) + h σ ( x; 0)σ g( x; 0) = ȳ h( x; 0) = x The remaining four unknown coeffi cients of the first-order approximation to g and h are found by using the fact that: F x ( x; 0) = 0 F σ ( x; 0) = 0 Before doing so, I need to introduce the tensor notation. Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

36 Tensors General trick from physics. An n th -rank tensor in a m-dimensional space is an operator that has n indices and m n components and obeys certain transformation rules. [H y ] i α is the (i, α) element of the derivative of H with respect to y: 1 The derivative of H with respect to y is an n n y matrix. 2 Thus, [H y ] i α is the element of this matrix located at the intersection of the i-th row and α-th column. 3 Thus, [H y ] i α[g x ] α β [h x ] β j = n y α=1 n x β=1 Hi g α h β y α. x β x j [H y y ]i αγ: 1 H y y is a three dimensional array with n rows, n y columns, and n y pages. 2 Then [H y y ]i αγ denotes the element of H y y located at the intersection of row i, column α and page γ. Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

37 Solving the System I g x and h x can be found as the solution to the system: [F x ( x; 0)] i j = [H y ] i α[g x ] α β [h x ] β j + [H y ] i α[g x ] α j + [H x ] i β [h x ] β j + [H x ] i j = 0 i = 1,..., n; j, β = 1,..., n x ; α = 1,..., n y Note that the derivatives of H evaluated at (y, y, x, x ) = (ȳ, ȳ, x, x) are known. Then, we have a system of n n x quadratic equations in the n n x unknowns given by the elements of g x and h x. We can solve with a standard quadratic matrix equation solver. Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

38 Solving the System II g σ and h σ are the solution to the n equations: Then: [F σ ( x; 0)] i = E t {[H y ] i α[g x ] α β [h σ] β + [H y ] i α[g x ] α β [η]β φ [ɛ ] φ + [H y ] i α[g σ ] α +[H y ] i α[g σ ] α + [H x ] i β [h σ] β + [H x ] i β [η]β φ [ɛ ] φ } i = 1,..., n; α = 1,..., n y ; β = 1,..., n x ; φ = 1,..., n ɛ. [F σ ( x; 0)] i = [H y ] i α[g x ] α β [h σ] β + [H y ] i α[g σ ] α + [H y ] i α[g σ ] α + [f x ] i β [h σ] β = 0; i = 1,..., n; α = 1,..., n y ; β = 1,..., n x ; φ = 1,..., n ɛ. Certainty equivalence: linear and homogeneous equation in g σ and h σ. Thus, if a unique solution exists, it satisfies: h σ = 0 Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

39 Second-Order Approximation I The second-order approximations to g around (x; σ) = ( x; 0) is [g(x; σ)] i = [g( x; 0)] i + [g x ( x; 0)] i a[(x x)] a + [g σ ( x; 0)] i [σ] [g xx ( x; 0)] i ab[(x x)] a [(x x)] b [g x σ( x; 0)] i a[(x x)] a [σ] [g σx ( x; 0)] i a[(x x)] a [σ] [g σσ( x; 0)] i [σ][σ] where i = 1,..., n y, a, b = 1,..., n x, and j = 1,..., n x. Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

40 Second-Order Approximation II The second-order approximations to h around (x; σ) = ( x; 0) is [h(x; σ)] j = [h( x; 0)] j + [h x ( x; 0)] j a[(x x)] a + [h σ ( x; 0)] j [σ] [h xx ( x; 0)] j ab [(x x)] a[(x x)] b [h x σ( x; 0)] j a[(x x)] a [σ] [h σx ( x; 0)] j a[(x x)] a [σ] [h σσ( x; 0)] j [σ][σ], where i = 1,..., n y, a, b = 1,..., n x, and j = 1,..., n x. Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

41 Second-order Approximation III The unknowns of these expansions are [g xx ] i ab, [g x σ] i a, [g σx ] i a, [g σσ ] i, [h xx ] j ab, [h x σ] j a, [h σx ] j a, [h σσ ] j. These coeffi cients can be identified by taking the derivative of F (x; σ) with respect to x and σ twice and evaluating them at (x; σ) = ( x; 0). By the arguments provided earlier, these derivatives must be zero. Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

42 Solving the System I We use F xx ( x; 0) to identify g xx ( x; 0) and h xx ( x; 0): [F xx ( x; 0)] i jk = [H y y ]i αγ[g x ] γ δ [h x ] δ k + [H y y ] i αγ[g x ] γ k + [H y x ]i αδ [h x ] δ k + [H y x ] i αk ) [g x ] α β [h x ] β j +[H y ] i α[g xx ] α βδ [h x ] k δ [h x ] β j + [H y ] i α[g x ] α β [h xx ] β jk ( ) + [H yy ] i αγ[g x ] γ δ [h x ] k δ + [H yy ] i αγ[g x ] γ k + [H yx ]i αδ [h x ] k δ + [H yx ] i αk [g x ] j α +[H y ] i α[g xx ] jk α ( ) + [H x y ]i βγ [g x ] γ δ [h x ] k δ + [H x y ] i βγ [g x ] γ k + [H x x ]i βδ [h x ] k δ + [H x x ] i βk [h x ] β j +[H x ] i β [h xx ] β jk +[H xy ] i jγ[g x ] γ δ [h x ] δ k + [H xy ] i jγ[g x ] γ k + [H xx ]i jδ [h x ] δ k + [H xx ] i jk = 0; i = 1,... n, j, k, β, δ = 1,... n x ; α, γ = 1,... n y. Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

43 Solving the System II We know the derivatives of H. We also know the first derivatives of g and h evaluated at (y, y, x, x ) = (ȳ, ȳ, x, x). Hence, the above expression represents a system of n n x n x linear equations in then n n x n x unknowns elements of g xx and h xx. Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

44 Solving the System III Similarly, g σσ and h σσ can be obtained by solving: [F σσ ( x; 0)] i = [H y ] i α[g x ] α β [h σσ] β +[H y y ]i αγ[g x ] γ δ [η]δ ξ [g x ] α β [η]β φ [I ]φ ξ +[H y x ]i αδ [η]δ ξ [g x ] α β [η]β φ [I ]φ ξ +[H y ] i α[g xx ] α βδ [η]δ ξ [η]β φ [I ]φ ξ + [H y ]i α[g σσ ] α +[H y ] i α[g σσ ] α + [H x ] i β [h σσ] β +[H x y ]i βγ [g x ] γ δ [η]δ ξ [η]β φ [I ]φ ξ +[H x x ]i βδ [η]δ ξ [η]β φ [I ]φ ξ = 0; i = 1,..., n; α, γ = 1,..., n y ; β, δ = 1,..., n x ; φ, ξ = 1,..., n ɛ a system of n linear equations in the n unknowns given by the elements of g σσ and h σσ. Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

45 Cross Derivatives The cross derivatives g x σ and h x σ are zero when evaluated at ( x, 0). Why? Write the system F σx ( x; 0) = 0 taking into account that all terms containing either g σ or h σ are zero at ( x, 0). Then: [F σx ( x; 0)] i j = [H y ] i α[g x ] α β [h σx ] β j + [H y ] i α[g σx ] α γ[h x ] γ j + [H y ] i α[g σx ] α j + [H x ] i β [h σx ] β j = 0; i = 1,... n; α = 1,..., n y ; β, γ, j = 1,..., n x. This is a system of n n x equations in the n n x unknowns given by the elements of g σx and h σx. The system is homogeneous in the unknowns. Thus, if a unique solution exists, it is given by: g σx = 0 h σx = 0 Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

46 Structure of the Solution The perturbation solution of the model satisfies: g σ ( x; 0) = 0 h σ ( x; 0) = 0 g x σ ( x; 0) = 0 h x σ ( x; 0) = 0 Standard deviation only appears in: 1 A constant term given by 1 2 g σσσ 2 for the control vector y t. 2 The first n x n ɛ elements of 1 2 h σσσ 2. Correction for risk. Quadratic terms in endogenous state vector x 1. Those terms capture non-linear behavior. Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47

47 Higher-Order Approximations We can iterate this procedure as many times as we want. We can obtain n-th order approximations. Problems: 1 Existence of higher order derivatives (Santos, 1992). 2 Numerical instabilities. 3 Computational costs. Jesús Fernández-Villaverde (PENN) Collateral Constraints December 2, / 47