A Method for Solving and Estimating Heterogenous Agent Macro Models (by Thomas Winberry)

Transcription

1 A Method for Solving and Estimating Heterogenous Agent Macro Models (by Thomas Winberry) Miguel Angel Cabello Macro Reading Group October, 2018 Miguel Cabello Perturbation and HA Models 1 / 95

2 Outline 1 Motivation 2 Summary of Results 3 Benchmark Model: Khan & Thomas (2008) 4 Computational Method 5 Results KT Model 6 Krusell & Smith model (1998) Miguel Cabello Perturbation and HA Models 2 / 95

3 Motivation 1 Unlike RA models, HA macro models are computationally challenging since the aggregate state vector contains infite-dimensional objects (distribution of agents). Miguel Cabello Perturbation and HA Models 3 / 95

4 Motivation 1 Unlike RA models, HA macro models are computationally challenging since the aggregate state vector contains infite-dimensional objects (distribution of agents). 2 Different numerical algorithms have been developed to overcome this challenge (value function or policy function iteration, gold search, etc). Miguel Cabello Perturbation and HA Models 3 / 95

5 Motivation 1 Unlike RA models, HA macro models are computationally challenging since the aggregate state vector contains infite-dimensional objects (distribution of agents). 2 Different numerical algorithms have been developed to overcome this challenge (value function or policy function iteration, gold search, etc). 3 However, none of them are as general, efficient, or easy-to-apply as the standard perturbation methods (typically employed for soving RA models). Miguel Cabello Perturbation and HA Models 3 / 95

6 Motivation 1 Unlike RA models, HA macro models are computationally challenging since the aggregate state vector contains infite-dimensional objects (distribution of agents). 2 Different numerical algorithms have been developed to overcome this challenge (value function or policy function iteration, gold search, etc). 3 However, none of them are as general, efficient, or easy-to-apply as the standard perturbation methods (typically employed for soving RA models). 4 Thus, is it possible to apply perturbation methods to solve heterogeneous agent models? Miguel Cabello Perturbation and HA Models 3 / 95

7 Motivation 1 Yes, it is! But, how to do this? Miguel Cabello Perturbation and HA Models 4 / 95

8 Motivation 1 Yes, it is! But, how to do this? 2 Two important basis: Miguel Cabello Perturbation and HA Models 4 / 95

9 Motivation 1 Yes, it is! But, how to do this? 2 Two important basis: Approximates the cross-sectional distribution using a parametric family. For instance, Algan, Allais and Haan (2008) use a globally accurate projection technique to solve Krusell-Smith (KS) model (but, this is computationally slow). Miguel Cabello Perturbation and HA Models 4 / 95

10 Motivation 1 Yes, it is! But, how to do this? 2 Two important basis: Approximates the cross-sectional distribution using a parametric family. For instance, Algan, Allais and Haan (2008) use a globally accurate projection technique to solve Krusell-Smith (KS) model (but, this is computationally slow). Usage of globally accurate and locally accurate approximations to solve for the dynamics of HA models. For example, Reiter (2009) employs locally accurate approximation with respect to state vector for solving KS model; however, his method relies on a fine histogram approximation, which requires many parameters to achieve aceptable accuracy. Miguel Cabello Perturbation and HA Models 4 / 95

11 Summary of Results 1 Based on these computation techniques, the developed method employs between seconds to solve the model. 2 Besides, to illustrate the power of the method, this is used to estimate a HA model with full-information Bayesian techniques. 3 Another feature of this method is that it could be applied to a wide range of of HA model. In these slides, it will be discussed the HA model of Khan and Thomas (2008) (KT) and Krussel and Smith (1998) (KS). 4 Finally, the computational method is implemented in Dynare. Miguel Cabello Perturbation and HA Models 5 / 95

12 Benchmark Model: KT Model Households 1 It is assumed a representative infinite-lived household, whose preferences are defined by: E [ t=0 β t ( C 1 σ t 1 1 σ )] χ N1+ψ t 1 + ψ where C t is consumption, N t is labor supplied to the market. β the discount factor, σ the relative risk aversion coefficient, χ governs the disutility of labor, and 1/ψ is the Frish elasticity of labor supply. (1) Miguel Cabello Perturbation and HA Models 6 / 95

13 Benchmark Model: KT Model Households 1 It is assumed a representative infinite-lived household, whose preferences are defined by: E [ t=0 β t ( C 1 σ t 1 1 σ )] χ N1+ψ t 1 + ψ where C t is consumption, N t is labor supplied to the market. β the discount factor, σ the relative risk aversion coefficient, χ governs the disutility of labor, and 1/ψ is the Frish elasticity of labor supply. 2 The representative househod is endowed with a unit of time (N t [0, 1]). The household owns all the firms in the economy. Markets are complete. (1) Miguel Cabello Perturbation and HA Models 6 / 95

14 Benchmark Model: KT Model Firms 1 There exist a continuum of firms with a total unit mass, j [0, 1], which produce output y jt according to: y jt = e zt e ε jt k θ jtl ν jt, θ + ν < 1 (2) where z t is an aggregate productivity shock, ε jt, a idiosyncratic one. k jt the capital input, l jt is labor input. θ the elasticity of output respect to capital, and ν the elasticity of output respect to labor. Miguel Cabello Perturbation and HA Models 7 / 95

15 Benchmark Model: KT Model Firms 1 There exist a continuum of firms with a total unit mass, j [0, 1], which produce output y jt according to: y jt = e zt e ε jt k θ jtl ν jt, θ + ν < 1 (2) where z t is an aggregate productivity shock, ε jt, a idiosyncratic one. k jt the capital input, l jt is labor input. θ the elasticity of output respect to capital, and ν the elasticity of output respect to labor. 2 Law of motions: Aggregate shock, z t, evolves as follows: z t+1 = ρ z z t + η z ω z t+1, with ω z t+1 N(0, 1) (3) Miguel Cabello Perturbation and HA Models 7 / 95

16 Benchmark Model: KT Model Firms 1 There exist a continuum of firms with a total unit mass, j [0, 1], which produce output y jt according to: y jt = e zt e ε jt k θ jtl ν jt, θ + ν < 1 (2) where z t is an aggregate productivity shock, ε jt, a idiosyncratic one. k jt the capital input, l jt is labor input. θ the elasticity of output respect to capital, and ν the elasticity of output respect to labor. 2 Law of motions: Aggregate shock, z t, evolves as follows: z t+1 = ρ z z t + η z ω z t+1, with ω z t+1 N(0, 1) (3) Idiosyncratic shock, ε jt, evolves as follows: ε jt+1 = ρ ε ε jt + η ε ω ε t+1, with ω ε t+1 N(0, 1) (4) Miguel Cabello Perturbation and HA Models 7 / 95

17 Benchmark Model: KT Model Firms: Decision and Timing 1 Each period, firm j inherits its capital stock from previous periods investments. Miguel Cabello Perturbation and HA Models 8 / 95

18 Benchmark Model: KT Model Firms: Decision and Timing 1 Each period, firm j inherits its capital stock from previous periods investments. 2 After observing the two productivity shocks, each firm hires labor from a perfect competitive labor market, and produces output. Miguel Cabello Perturbation and HA Models 8 / 95

19 Benchmark Model: KT Model Firms: Decision and Timing 1 Each period, firm j inherits its capital stock from previous periods investments. 2 After observing the two productivity shocks, each firm hires labor from a perfect competitive labor market, and produces output. 3 After production, the firm invests in capital for the next period. Gross investment, i jt, yields: k jt+1 = (1 δ)k jt + i jt (5) where δ is the depreciation rate of capital (which is assumed to be homogeneous among the firms). Miguel Cabello Perturbation and HA Models 8 / 95

20 Benchmark Model: KT Model Firms: Investment Decision 1 If i jt k jt [ a, a], the firm does not pay any fixed adjustment cost. Miguel Cabello Perturbation and HA Models 9 / 95

21 Benchmark Model: KT Model Firms: Investment Decision 1 If i jt k jt [ a, a], the firm does not pay any fixed adjustment cost. 2 On the contrary, when i jt k jt / [ a, a], the firm must pay a fixed adjustment cost, ξ jt, measured in labor units. Miguel Cabello Perturbation and HA Models 9 / 95

22 Benchmark Model: KT Model Firms: Investment Decision 1 If i jt k jt [ a, a], the firm does not pay any fixed adjustment cost. 2 On the contrary, when i jt k jt / [ a, a], the firm must pay a fixed adjustment cost, ξ jt, measured in labor units. 3 Hence, the parameter a governs a region around zero investment, within which firms do not incur the fixed cost. Miguel Cabello Perturbation and HA Models 9 / 95

23 Benchmark Model: KT Model Firms: Investment Decision 1 If i jt k jt [ a, a], the firm does not pay any fixed adjustment cost. 2 On the contrary, when i jt k jt / [ a, a], the firm must pay a fixed adjustment cost, ξ jt, measured in labor units. 3 Hence, the parameter a governs a region around zero investment, within which firms do not incur the fixed cost. 4 ξ jt is uniformly distributed in the interval [0, ξ], and is i.i.d over firms and time. Miguel Cabello Perturbation and HA Models 9 / 95

24 Benchmark Model: KT Model Firms: Optimization Problem 1 Following KT, the Bellman equation for the firm is: v(ε, k, ξ; s) = λ(s) max {e z e ε k θ l ν w(s)l} l + max{v a (ε, k; s) ξλ(s)w(s), v n (ε, k; s)} (6) where s is the aggregate state vector, λ(s) = C(s) σ. Miguel Cabello Perturbation and HA Models 10 / 95

25 Benchmark Model: KT Model Firms: Optimization Problem 1 Following KT, the Bellman equation for the firm is: v(ε, k, ξ; s) = λ(s) max {e z e ε k θ l ν w(s)l} l + max{v a (ε, k; s) ξλ(s)w(s), v n (ε, k; s)} (6) where s is the aggregate state vector, λ(s) = C(s) σ. 2 Besides: v a [ (ε, k; s) = max λ(s)(k (1 δ)k) + β E[ ˆv(ε, k ; s (z, s)) ε, k; s] ] k R (7) v n (ε, k; s) = max k A [ λ(s)(k (1 δ)k) + β E[ ˆv(ε, k ; s (z, s))] ε, k; s ] with A = [(1 δ a)k, (1 δ + a)k], ˆv(ε, k; s) = E ξ [v(ε, k, ξ; s)]. (8) Miguel Cabello Perturbation and HA Models 10 / 95

26 Benchmark Model: KT Model Firms: Optimization Problem 1 Let k a (ε, k; s) denotes the capital choice from (7), and k n (ε, k; s) the optimal choice from (8). Miguel Cabello Perturbation and HA Models 11 / 95

27 Benchmark Model: KT Model Firms: Optimization Problem 1 Let k a (ε, k; s) denotes the capital choice from (7), and k n (ε, k; s) the optimal choice from (8). 2 The firm will pay the fixed cost iff: v a (ε, k; s) ξλ(s)w(s) v n (ε, k; s) (9) Miguel Cabello Perturbation and HA Models 11 / 95

28 Benchmark Model: KT Model Firms: Optimization Problem 1 Let k a (ε, k; s) denotes the capital choice from (7), and k n (ε, k; s) the optimal choice from (8). 2 The firm will pay the fixed cost iff: v a (ε, k; s) ξλ(s)w(s) v n (ε, k; s) (9) 3 Hence, there exists a unique value of the fixed cost ξ which makes the firm indifferent between the two options: ξ(ε, k; s) = va (ε, k; s) v n (ε, k; s) λ(s)w(s) (10) Miguel Cabello Perturbation and HA Models 11 / 95

29 Benchmark Model: KT Model Firms: Optimization Problem 1 Let k a (ε, k; s) denotes the capital choice from (7), and k n (ε, k; s) the optimal choice from (8). 2 The firm will pay the fixed cost iff: v a (ε, k; s) ξλ(s)w(s) v n (ε, k; s) (9) 3 Hence, there exists a unique value of the fixed cost ξ which makes the firm indifferent between the two options: ξ(ε, k; s) = va (ε, k; s) v n (ε, k; s) λ(s)w(s) (10) 4 To prevent values that could be outside the support of ξ, let define: ˆξ(ε, k; s) = min{max{0, ξ(ε, k; s)}, ξ} (11) Miguel Cabello Perturbation and HA Models 11 / 95

30 Benchmark Model: KT Model Firms: Characterization of ˆv(ε, k; s) = E ξ [v(ε, k, ξ; s)] 1 Since the extensive margin decision is characterized by the cutoff (10), it is possible to compute analytically ˆv(ε, k; s): ˆv(ε, k; s) = λ(s) max {e z e ε k θ l ν w(s)l} + v a (ε, k; s)p(i/k / [ a, a]) l λ(s)w(s) E[ξ1(ξ ˆξ(ε, k; s))] + v n (ε, k; s)(1 P(i/k / [ a, a])) Miguel Cabello Perturbation and HA Models 12 / 95

31 Benchmark Model: KT Model Firms: Characterization of ˆv(ε, k; s) = E ξ [v(ε, k, ξ; s)] 1 Since the extensive margin decision is characterized by the cutoff (10), it is possible to compute analytically ˆv(ε, k; s): ˆv(ε, k; s) = λ(s) max {e z e ε k θ l ν w(s)l} + v a (ε, k; s)p(i/k / [ a, a]) l λ(s)w(s) E[ξ1(ξ ˆξ(ε, k; s))] + v n (ε, k; s)(1 P(i/k / [ a, a])) 2 Note that: P(i/k / [ a, a]) = E[ξ1(ξ ˆξ(ε, k; s))] = ˆξ(ε,k;s) 0 ˆξ(ε,k;s) 0 df (ξ) = ˆξ(ε, k; s) ξ ξdf (ξ) = ˆξ(ε, k; s) 2 2ξ (12) (13) Miguel Cabello Perturbation and HA Models 12 / 95

32 Benchmark Model: KT Model Firms: Characterization of ˆv(ε, k; s) = E ξ [v(ε, k, ξ; s)] 1 Thus: ˆv(ε, k; s) = λ(s) max l + ˆξ(ε, k; s) ξ +v n (ε, k; s) {e z e ε k θ l ν w(s)l} ( v a (ε, k; s) λ(s)w(s) ˆξ(ε, ) k; s) 2 ( 1 ˆξ(ε, ) k; s) ξ (14) Miguel Cabello Perturbation and HA Models 13 / 95

33 Benchmark Model: KT Model Recursive Competitive Equilibrium (RCE) 1 The aggregate vector s contains the current draw of the aggregate productivity shock, z, and the distribution (density) of firms over (ε, k)- space, g(ε, k). Miguel Cabello Perturbation and HA Models 14 / 95

34 Benchmark Model: KT Model Recursive Competitive Equilibrium (RCE) 1 The aggregate vector s contains the current draw of the aggregate productivity shock, z, and the distribution (density) of firms over (ε, k)- space, g(ε, k). 2 The RCE for KT model consists in a set of functions: (i) ˆv, l, k a, k n, ˆξ depending on (ε, k; s), (ii) λ, w depending on s, and (iii) s (z ( ) ; s) = z ; g (z, g) such that: (Firm opt.) Taking λ, w and s as given: l, k a, k n, ˆξ solve the firm s optimization problem (6). (Household opt.) For all s: λ(s) = C(s) σ, with C(s) [ = ( E ε,k e z e ε k θ l(ε, k; s) ν + (1 δ)k k a (ε, k; s) ˆξ(ε,k;s) ξ k n (ε, k; s) 1 ˆξ(ε,k;s) ξ [ ] ) w(s) satisfies E ε,k l(ε, k; s) + ˆξ(ε,k;s) 2 1/ψ = 2ξ ( w(s)λ(s) χ )] Miguel Cabello Perturbation and HA Models 14 / 95

35 Benchmark Model: KT Model Recursive Competitive Equilibrium (RCE) - Cont. 3 (Law of motion for distribution) For all (ε, k ): ( [ 1 ρ εε + η εω ε = ε ) ( ˆξ(ε,k;s) 1 k a ) (ε, k; z, s) = k.. ξ ( ) g (ε, k ; z, s) = ˆξ(ε,k;s) ξ 1 (k n )] (ε, k; z, s) = k p(ω ε)g(ε, k; s)dω εdεdk) (15) where p(.) is the p.d.f of idiosyncratic productivity shock. Miguel Cabello Perturbation and HA Models 15 / 95

36 Benchmark Model: KT Model Recursive Competitive Equilibrium (RCE) - Cont. 3 (Law of motion for distribution) For all (ε, k ): ( [ 1 ρ εε + η εω ε = ε ) ( ˆξ(ε,k;s) 1 k a ) (ε, k; z, s) = k.. ξ ( ) g (ε, k ; z, s) = ˆξ(ε,k;s) ξ 1 (k n )] (ε, k; z, s) = k p(ω ε)g(ε, k; s)dω εdεdk) (15) where p(.) is the p.d.f of idiosyncratic productivity shock. (Law of motion for aggregate shocks) z = ρ z z + η z ω z (16) Miguel Cabello Perturbation and HA Models 15 / 95

37 Computational Method 1 Winberry s method involves three main steps: 1. Approximation of Infinite-dimensional objects. 2. Computation of Stationary Equilibrium. 3. Local accurate approximation around the stationary equilibrium. Miguel Cabello Perturbation and HA Models 16 / 95

38 Computational Method 1 Winberry s method involves three main steps: 1. Approximation of Infinite-dimensional objects. 2. Computation of Stationary Equilibrium. 3. Local accurate approximation around the stationary equilibrium. 2 Note that steps 2 and 3 are the typical ones when a RA model is solved using perturbation techniques. Miguel Cabello Perturbation and HA Models 16 / 95

39 Computational Method 1 Winberry s method involves three main steps: 1. Approximation of Infinite-dimensional objects. 2. Computation of Stationary Equilibrium. 3. Local accurate approximation around the stationary equilibrium. 2 Note that steps 2 and 3 are the typical ones when a RA model is solved using perturbation techniques. 3 In step 1, the two infinite-dimensional objects to approximate are: (i) firm s value function ˆv(ε, k; s), and (ii) cross-section distribution g(ε, k). Miguel Cabello Perturbation and HA Models 16 / 95

40 Approximation of infinite-dimensional objects Cross-section distribution 1 Following Algan, Allais and Haan (2008), Winberry approximates g(ε, k) using the following parametric family: g(ε, k) g 0 exp{g1 1 (ε m1) 1 + g1 2 (log(k) m1)... 2 n g i [ ]... + g j i (ε m1) 1 i j (log(k) m1) 2 j m j i } (17) i=2 j=0 where n g is the degree of approximation, {g 0, g1 1, g 1 2, {g j i }i j=0 ng i=2 } are parameters, and {m1 1, m2 1, {mj i }i j=0 ng i=2 } are centralized moments of the distribution. Miguel Cabello Perturbation and HA Models 17 / 95

41 Approximation of infinite-dimensional objects Cross-section distribution 1 Following Algan, Allais and Haan (2008), Winberry approximates g(ε, k) using the following parametric family: g(ε, k) g 0 exp{g1 1 (ε m1) 1 + g1 2 (log(k) m1)... 2 n g i [ ]... + g j i (ε m1) 1 i j (log(k) m1) 2 j m j i } (17) i=2 j=0 where n g is the degree of approximation, {g 0, g1 1, g 1 2, {g j i }i j=0 ng i=2 } are parameters, and {m1 1, m2 1, {mj i }i j=0 ng i=2 } are centralized moments of the distribution. 2 When n g = 2, the approximate function lies in the family of multivariate Normal distributions. Miguel Cabello Perturbation and HA Models 17 / 95

42 Approximation of infinite-dimensional objects Cross-section distribution { 1 The parameter vector g = g 0... gn ng g }, and the moment vector { } m = m mn ng g have to be consistent with each other. That is, moments should be implied by the parameters. Thus: m1 1 = εg(ε, k)dεdk m1 2 = log(k)g(ε, k)dεdk (18) m j i = (ε m1) 1 i j (log(k) m1) 2 j g(ε, k)dεdk for i = 1,..., n g, j = 0,..., i Miguel Cabello Perturbation and HA Models 18 / 95

43 Approximation of infinite-dimensional objects Cross-section distribution { 1 The parameter vector g = g 0... gn ng g }, and the moment vector { } m = m mn ng g have to be consistent with each other. That is, moments should be implied by the parameters. Thus: m1 1 = εg(ε, k)dεdk m1 2 = log(k)g(ε, k)dεdk (18) m j i = (ε m1) 1 i j (log(k) m1) 2 j g(ε, k)dεdk for i = 1,..., n g, j = 0,..., i 2 Plugging in the approximate functional form (17) in system (18) results in a non-linear system on m and g. But, Algan, Allais and Haan (2008) develop a robust method for solving g given a vector m. Thus, m completely characterizes the approximated density. Miguel Cabello Perturbation and HA Models 18 / 95

44 Approximation of infinite-dimensional objects Cross-section distribution 1 Using the law of motions 3, 4 and 15, and the approximated density g(ε, k; m), one can get: m1 1 (z, m) = m 2 1 (z, m) = m j i (z, m) = where log(k ) = ˆξ(ε,k;s) κ j = ˆξ(ε,k;s) ξ ( ξ log(k a (ε, k; z, m)) + [ log(k a (ε, k; z, m)) m1 2 (ρ ε ε + ω ε)p(ω ε)g(ε, k; m)dω εdεdk log(k )p(ω ε)g(ε, k; m)dω εdεdk (19) (ρ ε ε + ω ε m 1 1 ) i j κ j p(ω ε)g(ε, k; m)dω εdεdk ) 1 ˆξ(ε,k;s) ξ log(k n (ε, k; z, m)) ) [log(k n (ε, k; z, m)) m1 2 ] j + (1 ˆξ(ε,k;s) ξ ] j Miguel Cabello Perturbation and HA Models 19 / 95

45 Approximation of infinite-dimensional objects Cross-section distribution 1 System (19) represents a mapping from the current aggregate state (z, m) into the next period moments m (z, m). Miguel Cabello Perturbation and HA Models 20 / 95

46 Approximation of infinite-dimensional objects Cross-section distribution 1 System (19) represents a mapping from the current aggregate state (z, m) into the next period moments m (z, m). 2 Besides, one can iterate system (19) in order to find the steady-state values of moment vector, m. Miguel Cabello Perturbation and HA Models 20 / 95

47 Approximation of infinite-dimensional objects Cross-section distribution 1 System (19) represents a mapping from the current aggregate state (z, m) into the next period moments m (z, m). 2 Besides, one can iterate system (19) in order to find the steady-state values of moment vector, m. 3 Even though, theoretically, there is no guarantee for the convergence of a non-linear system like (19), Winberry shows that, in practice, convergence happens. Miguel Cabello Perturbation and HA Models 20 / 95

48 Approximation of infinite-dimensional objects Firm Value Function 1 Winberry approximates firm s value function with respect to individual states (ε, k) using orthogonal polynomials. Hence: n ε n k ˆv(ε, k; z, m) ϑ ij (z, m)t i (ε)t j (k) (20) i=1 j=1 where n ε and n k are the degree of approximation of individual states ε and k, respectively. T i (ε) and T j (k) are Chebyshev polynomials, and ϑ ij (z, m) are coefficients on those polynomials. Miguel Cabello Perturbation and HA Models 21 / 95

49 Approximation of infinite-dimensional objects Firm Value Function 1 With this approximation, we can obtain a numerical approximation of Bellman equation (6) using collocation. Let define a set of grid points { ε i nε i=1, k j n k j=1 }. Miguel Cabello Perturbation and HA Models 22 / 95

50 Approximation of infinite-dimensional objects Firm Value Function 1 With this approximation, we can obtain a numerical approximation of Bellman equation (6) using collocation. Let define a set of grid points { ε i nε i=1, k j n k j=1 }. 2 First, equation (7) it is approximated: v a (ε i, k j ; z, m) λ(z, m)(k a (ε i, k j ; z, m) (1 δ)k j )... [ ( + β E z z Eω ε ˆv(ρε ε + η ε ω ε, k a (ε i, k j ; z, m); z, m (z, m)) )] (21) Miguel Cabello Perturbation and HA Models 22 / 95

51 Approximation of infinite-dimensional objects Firm Value Function 1 With this approximation, we can obtain a numerical approximation of Bellman equation (6) using collocation. Let define a set of grid points { ε i nε i=1, k j n k j=1 }. 2 First, equation (7) it is approximated: v a (ε i, k j ; z, m) λ(z, m)(k a (ε i, k j ; z, m) (1 δ)k j )... [ ( + β E z z Eω ε ˆv(ρε ε + η ε ω ε, k a (ε i, k j ; z, m); z, m (z, m)) )] (21) 3 Now, equation (8): v n (ε i, k j ; z, m) λ(z, m)(k n (ε i, k j ; z, m) (1 δ)k j )... [ ( + β E z z Eω ε ˆv(ρε ε + η ε ω ε, k n (ε i, k j ; z, m); z, m (z, m)) )] (22) Miguel Cabello Perturbation and HA Models 22 / 95

52 Approximation of infinite-dimensional objects Firm Value Function 1 Thus, the numerical approximation of equation (14) is: ˆv(ε i, k j ; z, m) = λ(z, m) max {e z e ε i kj θ l ν w(z, m)l}... l + ˆξ(ε ( i, k j ; z, m) ṽ a (ε i, k j ; z, m) λ(z, m)w(z, m) ˆξ(ε ) i, k j ; z, m) ξ 2 ( +ṽ n (ε i, k j ; z, m) 1 ˆξ(ε ) i, k j ; z, m) (23) ξ where ṽ a (.) and ṽ n (.) are the right-hand side of equations (21, 22), respectively. Miguel Cabello Perturbation and HA Models 23 / 95

53 Approximation of infinite-dimensional objects Approximated Equilibrium Conditions 1 The last step of first stage is to approximate the equilibrium conditions. Winberry shows that these approximated eq. conditions may be written as a system of 2n ε n k + n g n g + 1 equations. Miguel Cabello Perturbation and HA Models 24 / 95

54 Approximation of infinite-dimensional objects Approximated Equilibrium Conditions 1 The last step of first stage is to approximate the equilibrium conditions. Winberry shows that these approximated eq. conditions may be written as a system of 2n ε n k + n g n g + 1 equations. 2 Let {τi ε, ωε i }mε i=1 denote the weights and nodes of the one-dimensional Gauss-Hermite quadrature used to approximate: E ω ε ( ˆv(ρ ε ε + η ε ω ε, k q (ε i, k j ; z, m); z, m (z, m))) for q = a, n. Miguel Cabello Perturbation and HA Models 24 / 95

55 Approximation of infinite-dimensional objects Approximated Equilibrium Conditions 1 The last step of first stage is to approximate the equilibrium conditions. Winberry shows that these approximated eq. conditions may be written as a system of 2n ε n k + n g n g + 1 equations. 2 Let {τi ε, ωε i }mε i=1 denote the weights and nodes of the one-dimensional Gauss-Hermite quadrature used to approximate: E ω ε ( ˆv(ρ ε ε + η ε ω ε, k q (ε i, k j ; z, m); z, m (z, m))) for q = a, n. 3 Hence: E ω ε ( ˆv(ρε ε + η ε ω ε, k q (ε i, k j ; z, m); z, m (z, m)) ) =... m ε τ ε n ε n k o o=1 p =1 r =1... where ϑ p r = ϑ p r (z, m (z, m)). ϑ p r T p (ρ εε i + η ε ω ε o)t r (k q (ε i, k j )) (24) Miguel Cabello Perturbation and HA Models 24 / 95

56 Approximation of infinite-dimensional objects Approximated Equilibrium Conditions 1 Thus, equation (23) can be written as follows: n ε n k E ϑ pr T p (ε i )T r (k j ) λ(e z e ε i kj θ l(ε i, k j ) ν wl(ε i, k j ) + (1 δ)k j )... p=1 r=1 + ˆξ(ε i, k j ) ξ β ˆξ(ε i, k j ) ξ β ( 1 ˆξ(ε i, k j ) ξ (λ(k a (ε i, k j ) w ˆξ(ε ( i, k j ) )) + 1 ˆξ(ε ) i, k j ) λk n (ε i, k j )... 2 ξ m ε τ ε n ε n k o ϑ p r T p (ρ εε i + η ε ωo)t ε r (k a (ε i, k j ))... o=1 p =1 r =1 ) mε n ε τo ε n k ϑ p r T p (ρ εε i + η ε ωo)t ε r (k n (ε i, k j )) = 0 o=1 p =1 r =1 (25) for the collocation nodes i = 1,..., n ε and j = 1,..., n k. Miguel Cabello Perturbation and HA Models 25 / 95

57 Approximation of infinite-dimensional objects Approximated Equilibrium Conditions 1 The optimal labor demand is given by the FOC: νe z e ε k θ l ν 1 w(s) = 0 Its numerical version, for the grid {ε i, k j }, is: l(ε i, k j ) = ( νe z e ε i k θ j w ) 1 1 ν (26) Miguel Cabello Perturbation and HA Models 26 / 95

58 Approximation of infinite-dimensional objects Approximated Equilibrium Conditions 1 The optimal capital decision in case the firm assumes the fix cost is given by the FOC: λ(s) + β k E[ ˆv(ε, k ; s (z, s)) ε, k; s] = 0 Its numerical version, for the grid {ε i, k j }, is: m ε E λ β τ ε n ε n k o o=1 p =1 r =1 where T r (ka (ε i, k j )) = ϑ p r T p (ρ εε i + η ε ωo)t ε r (ka (ε i, k j )) = 0 k a T r (k a (ε i, k j )) (27) Miguel Cabello Perturbation and HA Models 27 / 95

59 Approximation of infinite-dimensional objects Approximated Equilibrium Conditions 1 The optimal capital decision in case the firm does not assume the fix cost is: (1 δ + a)k j, if k a (ε i, k j ) > (1 δ + a)k j k n (ε i, k j ) = k a (ε i, k j ), if k a (ε i, k j ) [(1 δ a)k j, (1 δ + a)k j ] (1 δ a)k j, if k a (ε i, k j ) < (1 δ a)k j (28) Miguel Cabello Perturbation and HA Models 28 / 95

60 Approximation of infinite-dimensional objects Approximated Equilibrium Conditions 1 The numerical version of optimal threshold ξ(ε i, k j ; z, m) is: ξ(ε i, k j ) = 1 wλ [ λ(ka (ε i, k j ) k n (ε i, k j ))... m ε τ ε n ε n k o o=1 p =1 r =1 +β ϑ p r T p (ρ εε i + η ε ωo) ε (T r (k a (ε i, k j )) T r (k n (ε i, k j ))) (29) Miguel Cabello Perturbation and HA Models 29 / 95

61 Approximation of infinite-dimensional objects Approximated Equilibrium Conditions 1 The numerical version of optimal threshold ξ(ε i, k j ; z, m) is: ξ(ε i, k j ) = 1 wλ [ λ(ka (ε i, k j ) k n (ε i, k j ))... m ε τ ε n ε n k o o=1 p =1 r =1 +β ϑ p r T p (ρ εε i + η ε ωo) ε (T r (k a (ε i, k j )) T r (k n (ε i, k j ))) 2 And, the numerical version of the bounded threshold is defined: (29) ˆξ(ε i, k j ) = min{max{0, ξ(ε i, k j )}, ξ} (30) Miguel Cabello Perturbation and HA Models 29 / 95

62 Approximation of infinite-dimensional objects Approximated Equilibrium Conditions 1 Now, let {τ g i, ε i, k i } mg i=1 denote the weights and nodes of the twodimensional Gauss-Legendre quadrature used to approximate the integral with respect to the distribution. Miguel Cabello Perturbation and HA Models 30 / 95

63 Approximation of infinite-dimensional objects Approximated Equilibrium Conditions 1 Now, let {τ g i, ε i, k i } mg i=1 denote the weights and nodes of the twodimensional Gauss-Legendre quadrature used to approximate the integral with respect to the distribution. 2 Then, household optimization condition may be written as follows: λ ( mg τ g h h=1 ˆξ(ε h, k h ) k a (ε h, k h ) ξ [ e z e ε h k θ hl(ε h, k h ) + (1 δ)k h... ( 1 ˆξ(ε h, k h ) ξ ) k n (ε h, k h ) ] g(ε h, k h m)) σ = 0 (31) Miguel Cabello Perturbation and HA Models 30 / 95

64 Approximation of infinite-dimensional objects Approximated Equilibrium Conditions 1 And, the numerical version of optimal labor supply choice is: with ( ) 1 m g [ wλ ψ τ g h χ h=1 l(ε h, k h ) + ˆξ(ε h, k h ) 2 ] g(ε h, k h m) = 0 (32) 2ξ g(ε i, k j m) = g 0 exp{g 1 1 (ε m 1 1) + g 2 1 (log(k) m 2 1) n g i ε i ε=2 j k =0 g j k i ε [ (ε i m 1 1) iε j k (log(k j ) m 2 1) j k m j k iε ]} Miguel Cabello Perturbation and HA Models 31 / 95

65 Approximation of infinite-dimensional objects Approximated Equilibrium Conditions 1 The approximated law of motion for the distribution, based on equation (19), will be: mg mε 0 = m 1 1 τ g τ ε h o (ρεε h + η εω ε o )g(ε h, k h m) h=1 o=1 mg mε [ ˆξ(ε 0 = m 2 1 τ g τ ε h, k h ) h o log(k a (ε h, k h )) + ξ h=1 o=1 mg mε 0 = m j i τ g h τ ε o h=1 o=1 ( ˆξ(ε h, k h ) 1 ξ [ (ρ εε h + ω ε o m1 1 )i j κ j ] g(ε h, k h m) ) log(k n (ε h, k h )) ] g(ε h, k h m) (33) where κ j = ˆξ(ε h, k h ) ξ [ log(k a (ε h, k h )) m 2 1 ] j + ( ˆξ(ε h, k h ) 1 ξ ) [ ] log(k n (ε h, k h )) m 2 j 1 Miguel Cabello Perturbation and HA Models 32 / 95

66 Approximation of infinite-dimensional objects Approximated Equilibrium Conditions 1 And, the approximated moment consistency system, based on equation (18), will be: m g 0 = m1 1 τ g h ε hg(ε h, k h m) h=1 m g 0 = m1 2 τ g h log(k h)g(ε h, k h m) (34) 0 = m j i h=1 m g τ g h h=1 [ (ε h m1) 1 i j ( log(k h ) m1 2 ) j ] g(ε h, k h m) for i = 2,..., n g ; j = 0,..., i Miguel Cabello Perturbation and HA Models 33 / 95

67 Approximation of infinite-dimensional objects Approximated Equilibrium Conditions 1 And, the approximated moment consistency system, based on equation (18), will be: m g 0 = m1 1 τ g h ε hg(ε h, k h m) h=1 m g 0 = m1 2 τ g h log(k h)g(ε h, k h m) (34) 0 = m j i h=1 m g τ g h h=1 [ (ε h m1) 1 i j ( log(k h ) m1 2 ) j ] g(ε h, k h m) for i = 2,..., n g ; j = 0,..., i 2 Finally, the law of motion for the aggregate productivity shock is: E[z ρ z z] = 0 (35) Miguel Cabello Perturbation and HA Models 33 / 95

68 Approximation of infinite-dimensional objects Approximated Equilibrium Conditions 1 Therefore, it is straightforward to see that equations (23), (27), (31), (32), (33), (34) and (35) defines a system of 2n ε n k + n g n g + 1 equations. Miguel Cabello Perturbation and HA Models 34 / 95

69 Approximation of infinite-dimensional objects Approximated Equilibrium Conditions 1 Therefore, it is straightforward to see that equations (23), (27), (31), (32), (33), (34) and (35) defines a system of 2n ε n k + n g n g + 1 equations. 2 All these equation can be defined as a mapping f (y, y, x, x; η) such that: E[f (y, y, x, x; η)] = 0 (36) { } where y = ϑ, k a, g, λ, w is the control variables vector and x = { } z, m, the state variables vector. η represents the perturbation parameter, and k a denotes the target capital stock along the collocation grid. Miguel Cabello Perturbation and HA Models 34 / 95

70 Computation of Stationary Equilibrium 1 It can be noted that equation (36) has the canonical form studied by Schmitt-Grohe and Uribe (2004). Miguel Cabello Perturbation and HA Models 35 / 95

71 Computation of Stationary Equilibrium 1 It can be noted that equation (36) has the canonical form studied by Schmitt-Grohe and Uribe (2004). 2 Then, stationary equilibrium can be computed from: f (y, y, x, x ; 0) = 0 Miguel Cabello Perturbation and HA Models 35 / 95

72 Computation of Stationary Equilibrium 1 It can be noted that equation (36) has the canonical form studied by Schmitt-Grohe and Uribe (2004). 2 Then, stationary equilibrium can be computed from: f (y, y, x, x ; 0) = 0 3 In principle, the above non-linear system can be solved using some root-finding algorithm. However, in practice, such system is quite large that numerical solvers fail to converge usually. Miguel Cabello Perturbation and HA Models 35 / 95

73 Computation of Stationary Equilibrium 1 It can be noted that equation (36) has the canonical form studied by Schmitt-Grohe and Uribe (2004). 2 Then, stationary equilibrium can be computed from: f (y, y, x, x ; 0) = 0 3 In principle, the above non-linear system can be solved using some root-finding algorithm. However, in practice, such system is quite large that numerical solvers fail to converge usually. 4 Winberry s method for solving the stationary equilibrium is similar to the developed by Hopenhayn and Rogerson (1993). Miguel Cabello Perturbation and HA Models 35 / 95

74 Computation of Stationary Equilibrium Algorithm 1 Winberry s algorithm solves for w which clears labor market. Labor demand can be computed in the following way: 1 Guess a value for equilibrium wage, w 0. Miguel Cabello Perturbation and HA Models 36 / 95

75 Computation of Stationary Equilibrium Algorithm 1 Winberry s algorithm solves for w which clears labor market. Labor demand can be computed in the following way: 1 Guess a value for equilibrium wage, w 0. 2 Given w 0, compute firm s value function ϑ 0 by iterating on Bellan equation (23). Miguel Cabello Perturbation and HA Models 36 / 95

76 Computation of Stationary Equilibrium Algorithm 1 Winberry s algorithm solves for w which clears labor market. Labor demand can be computed in the following way: 1 Guess a value for equilibrium wage, w 0. 2 Given w 0, compute firm s value function ϑ 0 by iterating on Bellan equation (23). 3 Using firm s decision rules, compute invariant distribuion m 0 by iterating on (33). Miguel Cabello Perturbation and HA Models 36 / 95

77 Computation of Stationary Equilibrium Algorithm 1 Winberry s algorithm solves for w which clears labor market. Labor demand can be computed in the following way: 1 Guess a value for equilibrium wage, w 0. 2 Given w 0, compute firm s value function ϑ 0 by iterating on Bellan equation (23). 3 Using firm s decision rules, compute invariant distribuion m 0 by iterating on (33). 4 Finally, aggregate individual firms labor demand: ( L D = l(ε, k) + ˆξ(ε, ) k) 2 g(ε, k)dεdk 2ξ Miguel Cabello Perturbation and HA Models 36 / 95

78 Computation of Stationary Equilibrium Algorithm 1 Winberry s algorithm solves for w which clears labor market. Labor demand can be computed in the following way: 1 Guess a value for equilibrium wage, w 0. 2 Given w 0, compute firm s value function ϑ 0 by iterating on Bellan equation (23). 3 Using firm s decision rules, compute invariant distribuion m 0 by iterating on (33). 4 Finally, aggregate individual firms labor demand: ( L D = l(ε, k) + ˆξ(ε, ) k) 2 g(ε, k)dεdk 2ξ 2 Labor supply can be computed from: L S = where λ 0 may be computed using C 0 = Y 0 I 0 ( ) w 0 λ 0 1/ψ χ Miguel Cabello Perturbation and HA Models 36 / 95

79 Local accurate approximation around the stationary equilibrium 1 Once steady state vector, (y, x ), is computed, it may be applied a Taylor expansion around such value in order to compute the aggregate Dynamics. Miguel Cabello Perturbation and HA Models 37 / 95

80 Local accurate approximation around the stationary equilibrium 1 Once steady state vector, (y, x ), is computed, it may be applied a Taylor expansion around such value in order to compute the aggregate Dynamics. 2 Like Schmitt-Grohe and Uribe (2004), it is assumed a solution of the form: where φ = (1, 0 ng 1). y = g(x; η) x = h(x; η) + η φω z Miguel Cabello Perturbation and HA Models 37 / 95

81 Local accurate approximation around the stationary equilibrium 1 Once steady state vector, (y, x ), is computed, it may be applied a Taylor expansion around such value in order to compute the aggregate Dynamics. 2 Like Schmitt-Grohe and Uribe (2004), it is assumed a solution of the form: where φ = (1, 0 ng 1). y = g(x; η) x = h(x; η) + η φω z 3 Then, a first order Taylor approximation around steady-stat yields: g(x; 1) g x (x ; 0)(x x ) + g η (x ; 0) h(x; 1) h x (x ; 0)(x x ) + h η (x ; 0)(1 0) + φω z (37) Miguel Cabello Perturbation and HA Models 37 / 95

82 Results KT model Figure 1: Stationary Distribution of Firms (Winberry (2018)) Miguel Cabello Perturbation and HA Models 38 / 95

83 Results KT model Table 1: Aggregate Variable in Steady State (Winberry (2018)) Variable n g = 1 n g = 2 n g = 3 n g = 4 n g = 6 Histogram Output Consumption Capital Wage Marg. Utility Miguel Cabello Perturbation and HA Models 39 / 95

84 Results KT model Figure 2: Impulse Responses of Aggregates: First Order Perturbation (Winberry (2018)) Miguel Cabello Perturbation and HA Models 40 / 95

85 Krusell & Smith model Households Continuum of households i [0, 1], with preferences over consumption c it : [ ] E β t c1 σ it 1 1 σ t=0 (38) 1 Each household supplies ε it efficiency units of labor inelastically, and this follows a 2-state Markov process ε it {0, 1} with transition probabilities π(ε ε). 2 When ε it = 1, households receive w t (1 τ); when ε it = 0, households receive unemployment benefit bw t (financed with tax labor). 3 a it+1 a (borrowing constraint). Miguel Cabello Perturbation and HA Models 41 / 95

86 Krusell & Smith model Firms Representative firm, produces output Y t according to: with z t aggregate prod. shock. Aggregate TFP evolces according to: Y t = e zt K α t L 1 α t (39) z t+1 = ρ z z t + σ z ω t+1 (40) Miguel Cabello Perturbation and HA Models 42 / 95

87 Krusell & Smith model RCE The aggregate state for this economy s = (z, g). functions a (ε, a; s), r(s), w(s), g (s) such that: 1 HH: taking r(s), w(s) and g (s) as given, a (.) satisfies: The RCE is a list of c(ε, a; s) σ βe[(1 + r(s ))c(ε, a ; s ) σ ε, z, g] with equality if a (ε, a; s) > a. 2 Firms and Market Clearing: r(s) = αe z K α 1 L 1 α δ w(s) = (1 α)e z K α L α K = adg(ε, a) ε 3 Evolution of distribution g (ε, A a ) = ε π(ε ε) 1{a ( ε, a; s) A a }g( ε, da) Miguel Cabello Perturbation and HA Models 43 / 95

88 Krusell & Smith model Approximating Infinite Dimensional Objects: Distribution Mass at Constraint: ˆm εt+1 = 1 [ (1 ˆm εt )π(ε ε)π( ε) π(ε)... + ε ε ˆm εt π(ε ε)π( ε)1{a t( ε, a) = a} 1{a t( ε, a) = a}g εt (a)da... ] Distribution away from Constraint: { n g g εt (a) gεt 0 exp gεt(a 1 mεt) 1 + gεt i i=2 [ ] } (a mεt) 1 i mεt i Miguel Cabello Perturbation and HA Models 44 / 95

89 Krusell & Smith model Approximating Infinite Dimensional Objects: Distribution Relation between g εt and m εt : mεt 1 = ag εt (a)da mεt i = (a mεt) 1 i g εt (a)da, i = 2,..., n g Evolution of moments: [ m 1 εt+1 = 1 π(ε) (1 ˆm εt )π(ε ε)π( ε) ε [ m i εt+1 = 1 (1 ˆm εt )π(ε ε)π( ε) π(ε) ε a t ( ε, a)g εt (a)da + ε (a t ( ε, a) m1 εt+1 )i g εt (a)da + ε ˆm εt π(ε ε)π( ε)a t ( ε, a) ] ˆm εt π(ε ε)π( ε)(a t ( ε, a) m1 εt+1 )i ] Miguel Cabello Perturbation and HA Models 45 / 95

90 Krusell & Smith model Approximating Infinite Dimensional Objects: Distribution Integrals are solved using Gauss-Legendre quadrature, with nodes {a j } mg j=1 and weights {τ j } mg j=1. So: m g a ( ε, a)g εt (a)da = τ j a ( ε, a j )g εt (a j ) j=1 (a ( ε, a) mεt+1) 1 i g εt (a)da = τ j [a ( ε, a j ) mεt+1] 1 i g εt (a j ) m g j=1 With the same nodes and weights, aggregate capital is computed: K t = ε m g π(ε) τ j a j g εt (a j ) j=1 Miguel Cabello Perturbation and HA Models 46 / 95

91 Krusell & Smith model Approximating Infinite Dimensional Objects: Household Decision Rules Calling ψ t (ε, a) = E t [β(1 + r t+1 )(c t+1 (ε, a t(ε, a))) σ ], then: a t(ε, a) = max{a, w t ((1 τ)ε + b(1 ε)) + (1 + r t )a ψ t (ε, a) 1/σ } c t (ε, a) = w t ((1 τ)ε + b(1 ε)) + (1 + r t )a a t(ε, a) ψ t (ε, a) is approximated by: { nψ } ˆψ t exp θ εit T i (ξ(a)) i=1 thus: â t(ε, a) = max{a, w t ((1 τ)ε + b(1 ε)) + (1 + r t )a ˆψ t (ε, a) 1/σ } ĉ t (ε, a) = w t ((1 τ)ε + b(1 ε)) + (1 + r t )a â t(ε, a) when borrowing constraint is not binding: { nψ } ] = E t [β(1 + r t+1 ) ε π(ε ε)ĉ t (ε, â t(ε, a j )) σ exp θ εit T i (ξ(a)) i=1 Miguel Cabello Perturbation and HA Models 47 / 95

92 Krusell & Smith model Solving for Steady State The approximated equilibrium can be written in the form: E[f (y, y, x, x ; η)] = 0 Then, the steady-state can be found by: Algorithm: f (y, y, x, x ; η) = 0 1 Compute factor prices r(k) and w(k) 2 Solve for the parameters of conditional expectation θ. 3 Using implied decision rules, solve for invariant m and g. 4 Update aggregate capital K = ε π(ε) m g j=1 τ ja (ε, a j )g ε (a j ). 5 Solve for a zero of K K. Miguel Cabello Perturbation and HA Models 48 / 95

93 Results KT model Figure 3: Stationary Distribution of Households (Winberry (2016)) Miguel Cabello Perturbation and HA Models 49 / 95

94 Results KT model Figure 4: Decision Rules in Steady State (Winberry (2016)) Miguel Cabello Perturbation and HA Models 50 / 95

95 Results KT model Figure 5: Impulse Response to Aggregate TFP shock (Winberry (2016)) Miguel Cabello Perturbation and HA Models 51 / 95