Numerical Simulation of Nonoptimal Dynamic Equilibrium Models∗ Zhigang Feng, Jianjun Miao, Adrian Peralta-Alva, and Manuel S. Santos†

Abstract In this paper we present a recursive method for the numerical simulation of nonoptimal dynamic equilibrium models. This method builds upon a convergent operator over an expanded set of state variables. The fixed point of this operator defines the set of all Markovian equilibria. We study approximation properties of the operator as well as the convergence of the moments of simulated sample paths. We apply our numerical algorithm to various models with heterogeneous agents, incomplete financial markets, endogenous and exogenous borrowing constraints, taxes, and money.

Keywords: Heterogeneous agents, taxes, externalities, financial frictions, competitive equilibrium, computation, simulation. JEL Codes: C6, D5,E2.



An earlier version of this paper was circulated as “Existence and Computation of Markov Equilibria for Dynamic Nonoptimal Economies” by J. Miao and M. Santos, March 2005. We thank Chanont Banternghansa and Jan Auerbach for their computational assistance. † Z. Feng: Department of Banking and Finance, University of Zurich, Plattenstrasse 14, 8032 Zurich Switzerland. J. Miao: Department of Economics, Boston University, 270 Bay State Road, Boston, MA. A. Peralta-Alva: Research Division, Federal Reserve Bank of Saint Louis. M. Santos: Department of Economics, University of Miami, 5250 University Drive, Coral Gables, FL. The usual disclaimer applies.

1

Introduction

In this paper we present a recursive method for the numerical simulation of non-optimal dynamic equilibrium models. Computation of these models is usually a formidable task because of various technical issues that preclude direct application of standard dynamic programming techniques. We address the problem of existence of a Markov equilibrium, and convergence and accuracy properties for both the numerical solution and the simulated moments. We apply our numerical algorithm to various models with heterogeneous agents and real and financial frictions. Computation of these models is critical to advance our understanding in several basic areas of macroeconomics and finance. We simulate a variant of the two-agent model of Kehoe and Levine (2001) to assess the influence of endogenous and exogenous borrowing constraints on the volatility of consumption and asset prices. We simulate various versions of the two-country model of Kehoe and Perri (2002) to assess the influence of borrowing constraints, incomplete markets, preference shocks, and taxes on international risk sharing and investment. We study the overlapping generations (OLG) economy of Kubler and Polemarchakis (2004), and introduce money. We also simulate some simple versions of these models by established algorithms using continuous equilibrium functions. The computed solutions under these latter algorithms present large approximation errors, and fail to mimic the true dynamics. In spite of the large approximation errors, the use of these traditional algorithms may be quite deceptive as they can produce small Euler equation residuals or may do well under some other independent accuracy checks. Standard solution methods search for a continuous equilibrium function over a natural state space. Since the seminal work of Kydland and Prescott (1980), it is known that Markovian solutions conditioning on a natural state space will generally be time-inconsistent. These authors consider a game of optimal taxation with a representative household, but similar technical problems are observed for competitive economies with market frictions because Pareto-optimality may fail to hold. Kydland and Prescott rewrite their model in a recursive form by appending a set of Lagrange multipliers to the original state space so as 1

to characterize the exact solution. Unfortunately, only restrictive conditions presupposing the uniqueness of equilibria may insure the continuity of the transition laws governing the enlarged state space. As seen later, Kydland and Prescott’s methods are not directly suited for the computation of decentralized economies with heterogeneous agents and market frictions. We certainly lack reliable algorithms for the simulation of these economies. Positive results on existence of a continuous equilibrium over a natural state space rely upon certain monotonicity properties of the equilibrium dynamics [e.g., see Bizer and Judd (1989), Coleman (1991), and Datta, Mirman and Reffett (2002)]. For the canonical onesector growth model with taxes and externalities, monotone dynamics follow from fairly mild restrictions on the primitives. But monotone dynamics are much harder to obtain in multi-sector models with heterogeneous agents and incomplete financial markets. Duffie et al. (1994) search for general representations of stationary equilibria over an expanded state space that includes additional endogenous variables such as asset prices and individual consumptions. Building upon these methods, Kubler and Schmedders (2003) show existence of a Markovian equilibrium for a class of financial economies with collateral requirements. Marcet and Marimon (2010) study a general class of contracting problems with incentive constraints. Following Kydland and Prescott (1980) these authors enlarge the state space with a vector of weights for the utility of each agent, and compute a transition for such weights from the shadow values of the agents’ participation constraints. They assume that equilibrium solutions can be characterized by convex social planning problems. By construction this method cannot capture multiple equilibria, but seems to be more operative for the computation of some dynamic incentive problems written in a Pareto-welfare form. Our work is closest to Kubler and Schmedders (2003). We expand the state space with a different set of variables: The shadow values of investment. This choice of the state space seems better suited for computation. Unlike Kubler and Schmedders (2003), in the numerical implementation we iterate over candidate equilibrium sets – rather than functions – to preserve convergence properties of our algorithm. We can thus compute the set of all

2

competitive equilibria. We also establish several convergence and accuracy properties of the algorithm and the simulated moments. Our algorithm can successfully be applied to various types of models, and it seems particularly useful for models with market distortions and endogenous and exogenous borrowing constraints and for OLG economies. These equilibria cannot generally be rewritten as solutions of regular social planning problems. Both projection and perturbation methods may become rather cumbersome in problems with endogenous and exogenous constraints. Projection methods actually assume that the equilibrium can be represented by a continuous function and such a continuous recursive representation of equilibrium may not exist. Perturbation methods are also not adequate if the ergodic region is quite large: In models with incomplete financial markets we may observe long swings in relative wealth as a result of idiosyncratic and aggregate shocks. The shadow value of investment as an additional state variable is actually considered in the the representative agent formulation of Kydland and Prescott (1980) and in some other related works [e.g., Phelan and Stacchetti (2001), and Marcet and Marimon (2010)]. Abreu, Pierce, and Stacchetti (1990) introduce continuation utility values to find a recursive representation of sequential equilibria for dynamic games. This additional state vector – continuation utilities – will not be adequate for the computation of our economies. Using continuation utility values, Miao (2006) provides a recursive characterization of sequential competitive equilibria for an incomplete markets model taken from Krusell and Smith (1998). Again, Miao’s characterization of equilibria is not readily computable. After this relatively technical introduction, we start in section 2 with three illustrative examples. In section 3 we present our general framework of analysis. Section 4 studies the numerical implementation of our algorithm and its convergence properties. Then, we apply these numerical procedures to various types of models. Sections 5 and 6 are devoted to simulation of equilibria for economies with borrowing constraints, and section 7 replicates these methods for an OLG economy. We conclude in section 8.

3

2

Three Examples

These three examples highlight some of the pitfalls encountered in the computation of non-optimal economies by many familiar algorithms based on social planning problems or approximations of the Euler equations.

2.1

An Asset Pricing Model with Borrowing Constraints

We begin with an exchange economy populated by 2 types of agents that simply differ in the stochastic process driving their endowments. Time is discrete t = 0, 1, · · · . Uncertainty is modeled by a Markov chain (zt )t≥0 . At each date-event z t = (z0 , ..., zt ), agents can trade quantities of the unique aggregate good and shares of a real asset or “Lucas” tree. For each agent i = 1, 2, preferences are represented by the intertemporal objective "∞ # X  E β t ui cit . (1) t=0

The discount factor is β ∈ (0, 1), and E is the expectations operator. The one-period utility function ui depends on the quantity consumed cit . This function is assumed to satisfy standard properties. At every z t = (z0 , ..., zt ), an agent i is endowed with eit (z t ) units of the aggregate good. Let qt (z t ) be the price of a unit share of the real asset, and dt (z t ) > 0 the dividends attached to such unit. Then, every agent i faces the following sequence of budget constraints: i cit (z t ) + θt+1 (z t )qt (z t ) = eit (z t ) + θti (z t−1 )(qt (z t ) + dt (z t )),

(2)

θti ≥ −Ω, for all z t ,

(3)

where Ω is an exogenous borrowing limit. We will also consider another scenario in which agents may renege on their debts at the cost of exclusion from financial markets. In the event of default, individual consumption is just limited to the endowment allocation. For every affordable consumption plan, default will then cease to occur if the following sequence

4

of individually rational debt constraints is satisfied: Ez t

∞ X

 β τ ui ciτ ≥ V i,aut (z t ), for all i and z t ,

(4)

τ =t

where V i,aut (z t ) denotes the expected discounted value of consuming the endowment allocation from the time t of default into the infinite future. Note that this limited enforcement constraint depends on endogenous variables over future equilibrium paths. We have just described a simple asset pricing model with two agents and with exogenous and endogenous borrowing constraints [e.g. Alvarez and Jermann (2001) and Kehoe and Perri (2002)]. The model can be given a Markovian structure by letting endowments e and dividends d depend on the current realization zt . Still, the equilibrium solution cannot be generated by an associated social planning problem because of incomplete financial markets as well as additional borrowing restrictions. Indeed, it is not clear that a Markov equilibrium exists [cf. Kubler and Schmedders (2002)]. Moreover, because of endogenous constraint (4) each consumption variable enters various first-order conditions, which becomes problematic for computational methods approximating the Euler equations [see, however, Kehoe and Perri (2002) and Marcet and Marimon (2010)].

2.2

A Growth Model with Taxes

We consider a deterministic growth economy with a continuum of identical infinitely livedhouseholds [Peralta-Alva and Santos (2010)]. Each agent solves the optimization problem max

∞ X

β t log(ct )

t=0

s.t. ct + kt+1 ≤ πt + (1 − τt ) rt kt + Tt k0 given, 0 < β < 1, ct ≥ 0, kt+1 ≥ 0 for all t ≥ 0.

5

Here, ct denotes individual consumption, and kt is the individual capital stock. The discount factor, β = 0.95. Taxes on capital income {τt } are assumed to depend on the average capital stock Kt ; more specifically, let  if K ≤ 0.160002  0.10 0.05 − 10(K − 0.165002) if 0.160002 ≤ K ≤ 0.170002 τ (K) =  0 if K ≥ 0.170002. Tax revenues are rebated back to the the representative agent as lump-sum transfers Tt . The production sector is represented by an individual firm. At each date, the firm hires capital to maximize one-period profits πt = max K 1/3 − rt Kt . Kt

In a (Kt , Kt+1 )-plane Figure 1 depicts the flow diagram characterizing the set of competitive equilibria; that is, those solutions in which every agent takes as given the sequences of capital rental rates rt , taxes τt , transfers Tt , and profits πt . The model has three steadystate equilibria, K ∗ = 0.152148, K ∗ = 0.165002 and K ∗ = 0.178198. The multiplicity of steady states comes from the non-linear tax function. The middle steady state is unstable and has two complex eigenvalues while the other two steady states are saddle-path stable. All remaining equilibria converge to one of the two saddle-path steady states. Note that the flow diagram may cross the 45-degree line at non-stationary solutions because an initial value K0 will not qualify as a steady-state solution for an Euler equation H(K0 , K1 , K2 ) = 0 with K0 = K1 but K1 6= K2 , where H is a functional describing the Euler equation. The aggregate capital stock K would be the natural state variable. But as it may be clear from Figure 1, there is no continuous equilibrium function for the law of motion Kt+1 = G(Kt ). Thus, the model does not admit a continuous M arkov equibrium. This is because of the multiplicity of cycling equilibria around the unstable steady state. To select over these various paths, we can work with an expanded state space of variables (K0 , c0 ) or (K0 , K1 ). None of these expanded state spaces can restore continuity of an equilibrium function. For reasons of computational convenience, we will work with the state space of 6

variables (K, m) where m is the shadow value of investment, m = u0 (c)[f 0 (K)]. An initial capital value K0 may be associated with various shadow values of investment m, and so this correspondence of values is not necessarily continuous. Therefore, we have presented a simple decentralized economy with a representative agent where standard computation methods based on the existence of a continuous equilibrium solution G(K) will not be adequate. To illustrate how these standard methods may actually behave, we now apply a projection algorithm approximating the Euler equation. This is a collocation method with piecewise linear interpolation. Figure 1 also displays the b computed continuous equilibrium function K 0 = G(K), which contains four extra steady states. There is then a middle region that is not attracted to one of the two saddlepath stable steady states. The second part of Figure 1 just depicts the flow diagram that circles around the unstable steady state, K ∗ = 0.165002, and the computed solution b K 0 = G(K) under the projection method. As discussed in Peralta-Alva and Santos (2010) our program converged up to computer precision in only 3 iterations. Convergence in this case is deceptive because no continuous policy function does exist. Inspection of the Euler equation over the numerical approximation revealed an average residual of 0.0073. The maximum Euler equation residual is slightly more pronounced in a small area near the unstable steady state. But even in that area, the error is not extremely large: In three tiny intervals the Euler equation residuals are just around 0.06. To conclude, from these computational tests a researcher may be led to believe that the model admits a continuous equilibrium function. These patterns are also replicated in some stochastic versions of the model. As we introduce some small noise into the production function, the continuous approximation contains some additional middle ergodic sets whereas the exact solution only contains two ergodic sets.

7

Figure 1a: True model dynamics vs. approximate continuous solution.

Figure 1b: True model dynamics vs. approximate continuous solution near the unstable steady state K ∗ = 0.165002.

2.3

An OLG Economy

There are no reliable algorithms for the computation of competitive equilibria in OLG economies. Again, dynamic programming techniques are of limited application for these open-ended economies where it is known that the simple assumption of perfect foresight 8

may generate multiple equilibria and sun-spots. We present an example below from Kubler and Polemarchakis (2004) that does not admit a Markov equilibrium over the natural state space. Citanna and Siconolfi (2010) establish generic existence of this Markovian property of equilibrium under the additional assumption that the number of agents is sufficiently large. Of course, for computational reasons many economies of practical interest contain a limited number of agents which are given as primitives of the model. As in our previous example the recursive representation of equilibrium in Citanna and Siconolfi (2010) is not necessarily continuous. Time is discrete t = 0, 1, 2 · · · . The exogenous shock zt follows a Markov process with support Z = {z1 , z2 } . At each date, a new generation made up of 2 agents appears in the  economy. A generation lives for 2 periods. Let i, z t denote an agent of type i = 1, 2 born at date-event z t = (z0 , z1 , · · · , zt ). At each date, there are 2 perishable commodities available for consumption. Let good 1 be the numeraire commodity, and p the relative price of good 2. There are two assets in this economy. The first asset is a one-period risk-free bond trading at price q b (z t ). A Lucas tree is also available trading at price q s (z t ). The t

t

tree generates a random stream of dividends d(zt ). Let (θb,i,z , θs,i,z ) be a pair of bond and  t share holdings of agent i, z t . Shares cannot be sold short: θs,i,z ≥ 0. Each individual faces the following budget constraint: t

t

t

t

p(z t ) · ci,z (z t ) + θb,i,z (z t )q b (z t ) + θs,i,z (z t )q s (z t ) ≤ p(z t ) · ei,z (zt ) t

t

t

(5)

t

p(z t+1 )·ci,z (z t+1 ) ≤ θb,i,z (z t )+θs,i,z (z t )[d(zt+1 )+q s (z t+1 )]+p(z t+1 )·ei,z (z t+1 ) all z t+1 |z t . (6) Let us now study the model specification of Example 2 in Kubler and Polemarchakis (2004) in which the real asset is not available.1 The intertemporal objective of agent of 1

Because of an indeterminacy problem of the Euler equation pointed out below, we can approximate the equilibrium of this more limited economy by letting the stock of trees go to zero.

9

type 1 is given by 

 −

1024 1 1024   4 −  t 4  4 + Ezt+1 |z t −  t t c1,z c11,z (zt+1 ) c21,z (zt+1 ) 1

while that of agent of type 2 is given by  −

1 c2,z 1

  + Ezt+1 |z t −  t 4

 1 t c12,z (zt+1 )

4 − 

1024

 4  .

t c22,z (zt+1 )

Each individual receives a random endowment of good 1 in their first period of life. Specifit

t

t

t

1,z 2,z cally, e11,z (zt ) = 10.4, e2,z 1 (zt ) = 2.6 if zt = z1 , and e1 (zt ) = 8.6313, e1 (zt ) = 4.3687 if

zt = z2 . Endowments during the second period of life are deterministic and include positive   t t amounts of both goods. Namely, e1,z z t+1 = (12, 1) and e2,z z t+1 = (1, 12) . Kubler and Polemarchakis show that this model does not have a recursive equilibrium over the natural state space of current shocks and portfolio holdings. In particular, bond holdings turn out to be equal to zero in the two states. Hence, to determine consumption when old we must know the realization of the endowment when young. At any state history z t−1 with zt−1 = z1 , and for any possible value of the shock today  t−1  t−1     t−1 t−1 2,z t , c2,z t t , c1,z t = (2.6, 10.4), and p = 1. = (10.4, 2.6), c z z c1,z z z 1 2 1 2 Likewise, for any state history z t−1 with zt−1 = z2 , ad for any possible value of the shock  t−1  t−1     t−1 t−1 2,z t , c2,z t t , c1,z t = (4.6, 11.6) , and = (8.4, 1.4), c z z today c1,z z z 1 2 1 2 p = 7.9. Finally, we computed this model using a projection method with collocation and piecewise linear interpolation. This collocation method approximates the Euler equation to search for a continuous equilibrium function over the natural state space – albeit the model does not admit a continuous Markov equilibrium. The computed equilibrium function delivers reasonable Euler equation residuals (i.e., of the order of 10−5 ). A researcher may again be led to believe that this function is a good approximate solution; however, the computed prices and allocations are quite different from those of the exact equilibrium. 10

Statistics (µtrue , µprojection ) 2 , σ2 (σtrue projection )

q (1.0,0.6) (0.0,0.05)

t−1

c11,z (9.7,9.7) (1.0,0.2)

t−1

c1,z 2 (2.0,1.7) (0.36,0.81)

t−1

c2,z 1 (3.6,3.8) (1.0,0.09)

t−1

c22,z (11.0,11.3) (0.36,0.08)

Table 1: Statistical properties of the true equilibrium vs. an equilibrium generated by the projection method. Statistics: Mean µ and variance σ 2 . In summary, in equilibrium the relative price of good 2 is a function of the endowment in the previous period. The price is not signaled by the natural state space – current shocks and portfolio holdings – as there is no trade among generations. The equilibrium relative price of good 2 can take on two values and asset holdings take on one single value. This observation may explain the large differences in Table 1 between the simulated moments generated by the true and computed solutions. Indeed, the computed function by the projection method takes on a single value for the relative price of good 2 midway between the two possible equilibrium values.

3

General Theory

In this section, we first set out a general analytical framework that encompasses various economic models. We then present our numerical approach and main results on existence and global convergence to the Markovian equilibrium correspondence.

3.1

General Framework

Time is discrete t = 0, 1, 2, · · · . The state space includes a vector of exogenous shocks z and a state vector of endogenous variables x. Vector x contains all predetermined variables, such as agents’ holdings of physical capital, human capital, and financial assets. The exogenous state vector follows a Markov chain (zt )t≥0 over a finite set Z = {1, 2, ..., Z}. This Markovian process is described by positive transition probabilities π (z 0 |z) for all z, z 0 ∈ Z. The initial state, z0 ∈ Z, is known to all agents in the economy. Then z t = (z0 , z1 , z2 , ..., zt ) ∈ Zt+1 is a history of shocks, often called a date-event or node. As already discussed, we let m 11

denote the vector of shadow values of investment for all assets and all agents, and y the vector of all other current endogenous variables. This latter vector may involve equilibrium prices and choice variables such as consumption and investment. Our analysis focuses on computation of sequential competitive equilibria (SCE) {x(z t ), y(z t )}t≥0 . As shown in the example of the OLG economy, even if the exogenous shock zt is driven by a Markov process, the set of SCE may not admit a recursive representation over the natural state space. For a large class of models we show existence of the Markovian property over the state space of variables (x, z, m). That is, to the natural state space of vectors (x, z) we append the shadow values of investment m. The addition of variable m is convenient for computation because the Euler equation can be written in a simple form. Before getting into the workings of our algorithm, let us first outline a simple framework for the characterization of SCE. We consider that the law of motion of the state vector x is conformed by a system of non-linear equations: ϕ (xt+1 , xt , yt , zt ) = 0.

(7)

Function ϕ may embed technological constraints as well as individual budget constraints. For some models (e.g., the growth model in section 2) we can explicitly solve for xt+1 as a function of (xt , yt , zt ) . But in some other applications such as in models with adjustment costs, xt+1 may not admit an analytical solution. Second, we assume that the vector of shadow values of investment is a continuous function of current variables: mt = h (xt , yt , zt ) .

(8)

This is usually a vacuous assumption under continuously differentiable production and utility functions. Third, a SCE {x(z t ), y(z t ), m(z t )}t≥0 must satisfy a system of equations Φ (λt , xt , yt , zt , Et [mt+1 ]) = 0,

(9)

together with additional constraints Ψ({x(z t ), y(z t ), m(z t )}t≥0 ) ≥ 0, 12

(10)

involving all current and future equilibrium variables {x(z t ), y(z t ), m(z t )}t≥0 . Function Φ may describe individual optimality conditions (such as Euler equations), market-clearing conditions, various types of budget restrictions, and resource constraints. Inequality constraints may be written as equalities by appending a vector of Lagrange multipliers λ. We assume that all feasible vectors belong to a compact set so that the Euler equations are necessary and sufficient conditions to characterize individual optimal solutions for concave programs. Function Ψ may represent endogenous constraints such as the participation constraint (4) in the above asset pricing model. As illustrated in section 5 below, in the case of endogenous constraints vector λ may actually correspond to ratios of Lagrange multipliers. Finally, a SCE {x(z t ), y(z t ), m(z t )}t≥0 is assumed to exist, and the set of all SCE {x(z t ), y(z t ), m(z t )}t≥0 is uniformly bounded. In our applications below, under standard regularity conditions we show existence of a SCE, as well as uniform upper and lower bounds for equilibrium allocations and prices.

3.2

Recursive Equilibrium Theory

In order to compute the set of dynamic equilibria for the model economy we define the equilibrium correspondence V ∗ : (x, z) 7−→ V ∗ (x, z) , as the set of equilibrium vectors of shadow values of investment m for any given state (x, z) . To insure that participation constraint (10) always holds,2 we also define correspondence A∗ (x0 , z0 ) as the set of values y0 such that there is a sequence {x(z t ), y(z t ), m(z t )}t≥0 with y0 = y(z0 ) that satisfies (7)-(10) at all times. We prove existence of a fixed point for an operator B : (V, A) 7−→ B(V, A) that links state variables to future equilibrium states. Operator B embodies all temporary equilibrium conditions such as agents’ Euler equations, endogenous and exogenous constraints, and market-clearing conditions from any initial value z to all immediate successor nodes z+ . This operator is analogous to the expectations correspondence of Duffie et al. (1994), albeit 2 We could have imposed the limited enforcement constraint (10) directly, but our set notation is not specific to this constraint, and allows us to include other endogenous constraints. See section 5 below for a practical application of correspondence A.

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it may contain a smaller set of endogenous variables. Using operator B, we can derive the set of all SEC. More precisely, let (V 0 , A0 ) = B (V, A) (x, z) be defined as follows: m ∈ V 0 (x, z) and y ∈ A0 (x, z) if there are m+ (z+ ) ∈ V (x+ , z+ ) and y+ (z+ ) ∈ A(x+ , z+ ) for all z+ ∈ Z and ϕ (x+ , x, y, z) = 0 such that 

 X

Φ λ, x, y, z,

π (z+ |z) m+ (z+ ) = 0,

z+ ∈Z

for some vector of multipliers λ, and (10) is satisfied.3 It follows that operator B(V, A) is well defined since we have assumed that a SCE exists. One can readily see that operator B is monotone: If V ⊂ V 0 and A ⊂ A0 then B(V, A) ⊂ B(V 0 , A0 ).4 Moreover, if (V, A) has a closed graph then B(V, A) will also have a closed graph as the above functions ϕ, h, Φ, Ψ are continuous. Assumption 3.1 Operator B preserves compactness: If V and A are compact valued then B(V, A) is also compact valued. We now show existence of a fixed-point solution (V ∗ , A∗ ) and a general form of global convergence. Let A0 (x, z) denote the set of y that satisfy both (7) and (10) for period 0. (For the first model of section 2 we can simply check if participation constraint (4) is satisfied at period 0 by appending the highest possible levels of consumption for future periods even if those are not feasible.) Theorem 3.1 (convergence) Let V0 be a compact-valued correspondence such that V0 ⊃ V ∗ , and let A0 be defined as above. Let (Vn , An ) = B (Vn−1 , An−1 ) for all n ≥ 1. Then, operator B has a fixed-point solution, i.e., (V ∗ , A∗ ) = B(V ∗ , A∗ ), where V ∗ = limn→∞ Vn , 3 From vector (x, y, z) we can then generate a sequence {x(z t ), y(z t ), m(z t )}t≥0 with y+ (z+ ) ∈ A(x+ , z+ ). As shown in some applications below, using recursive arguments these latter infinite sequences need not be stored. 4 For correspondences V, V 0 we say that V ⊂ V 0 if V (x, z) ⊂ V 0 (x, z) for all (x, z). We shall use the usual notion of distance over sets given by the Hausdorff metric.

14

and A∗ = limn→∞ An . Moreover, (V ∗ , A∗ ) is the largest fixed point of operator B, i.e., (V, A) = B(V, A) implies (V, A) ⊂ (V ∗ , A∗ ). Theorem 3.1 provides the theoretical foundations of our algorithm. The basic idea is to iterate on correspondences of shadow values V (x, z) and optimal controls and auxiliary variables A(x, z) satisfying feasibility and endogenous constraints starting from some well chosen pair (V0 , A0 ). Then, operator B generates sequences of non-empty compact sets (Vn , An ) that shrink to the equilibrium pair (V ∗ , A∗ ). In practice, this iterative process may stop until a desirable level of convergence is attained. We would like to remark that operator B iterates over sets rather than functions. Hence, if there are multiple equilibria we can find all of them. By construction, for any (x, z, m) ∈ graph(V ∗ ), under the action of operator B we can generate a new vector (x+ , z+ , y, m+ ) that can be extended into a SCE {x(z t ), y(z t ), m(z t )}t≥0 . Since the fixed point of operator B is an upper semicontinuous correspondence, it is possible to select a measurable policy function y = g y (x, z, m), a transition function m+ (z+ ) = g m (x, z, m; z+ ), as well as a law of motion for capital ϕ (x+ , x, y, z) = 0. Let us summarize these future equilibrium values over the extended state space as g(x, z, m; z+ ) = (x+ , z+ , m+ ). Then, g is an equilibrium selection, and provides a Markovian characterization of a subset of dynamic equilibria.5

4

Numerical Implementation

Numerical implementation of our theoretical framework requires the construction of a computable method that approximates the fixed point of operator B. In this section, we develop and study properties of such an algorithm. Our method proceeds as follows. First, we partition the state space into a finite set of J simplices with mesh size h. Compatible with this partition we consider a sequence of step 5

It should be clear that g(·; z+ ) denotes a coordinate function of g(·) corresponding to the successor node z+ |z.

15

correspondences, which take constant set-values on each simplex.6 Step correspondences are shown to have good approximation properties. We also introduce a finite-dimensional outer approximation of the image of this correspondence; this outer approximation is made up of N cubes or finite-dimensional elements. Then, using these approximations we obtain a computable operator B h,N with accuracy parameters (h, N ). We show that the sequence of correspondences defined by iterations of operator B h,N converges to a fixed point that contains equilibrium correspondence (V ∗ , A∗ ). Moreover, as (h, N ) approach the limit, we prove that the associated sequence of fixed points converges to (V ∗ , A∗ ). At a later stage, we address the issue of convergence of the moments obtained from simulations of our numerical approximations. This problem has been hardly addressed in the literature, and it has to cope with the fact that (V ∗ , A∗ ) may not have a continuous selection.

4.1

The Numerical Algorithm

Assume that all equilibrium state vectors (x, z, m) belong to some set S, which is a subset  J of the product space S = X × Z × M . Let X j j=1 be a finite family of simplices with 0

non-empty interior such that ∪j X j = X and int(X j ) ∩ int(X j ) is empty for every pair 0

X j , X j . Define the mesh size h of this discretization as  h = max diam X j . j

Consider any given correspondence V : X × Z = → 2M , where 2M denotes the subsets of vectors for space M containing the shadow values of investment m. An approximation V h  compatible with the partition X j takes on constant set-values V h (x, z) on each simplex X j . We then define the step correspondence: V h (x, z) = ∪x∈X j V (x, z), for each given z and all x ∈ X j .

(11)

Now, we construct an operator B h (V, A) between step correspondences as B h (V, A)(x, z) = ∪x∈X j B(V, A)(x, z), for each given z. By similar arguments as above, we can prove that B h 6

Step correspondences are the analog of step functions. That is, a step correspondence is a piecewise constant correspondence having only finitely many pieces or set-values.

16

has a fixed-point solution. To obtain a computable representation of these correspondences we also discretize the image space. For a given set V (x, z) we say that C N (V (x, z)) ⊇ V (x, z) is an N -element outer approximation of V (x, z) if C N (V (x, z)) can be generated by N elements. We require this numerical representation to preserve monotonicity: V (x, z) ⊂ V 0 (x, z) implies C N (V (x, z)) ⊂ C N (V 0 (x, z)). This is essential to guarantee monotonicity of a computable version of operator B. We also require limN →∞ C N (V (x, z)) = V (x, z). Using these approximations, we can define a new operator B h,N that starts by computing a step correspondence B h (V, A) of B(V, A). Each set-value is then adjusted by the N  element outer approximation to get C N B h (V, A) . Sections 5 to 7 illustrate examples of such operators, and their application to different dynamic models. Here we show that our discretized operator B h,N has good convergence properties: The fixed point of this operator (V ∗,h,N , A∗,h,N ) contains (V ∗ , A∗ ) and it approaches this limit point as we refine the approximations. The proof of this result extends the convergence arguments of Beer (1980) to a dynamic setting.

Theorem 4.1 For given h, N, let V0 ⊇ V ∗ and A0 as defined above. Let (Vnh,N , Ah,N n )= h,N h,N ⊇ A∗ for all n; (ii) ⊇ V ∗ and Ah,N B h,N (Vn−1 , Ah,N n n−1 ) for all n ≥ 1. Then, (i) Vn

Vnh,N → V ∗,h,N , Ah,N → A∗,h,N uniformly as n → ∞; and (iii) V ∗,h,N → V ∗ , A∗,h,N → A∗ , n as h → 0 and N → ∞. Of course, in applications these computations must stop in finite time. Hence, the output of our numerical algorithm would be summarized by some correspondences (Vnh,N , Ah,N n ) under the action of an operator B h,N . By Theorem 4.1, we have that graph(C N (Vnh,N , Ah,N n )) can be made arbitrarily close to graph(V ∗ , A∗ ) for appropriate choices of n, h, and N . From the application of operator B h,N on (Vnh,N , Ah,N n ), we can choose an approximate policy function y = gny,h,N (x, z, m), and a transition function m+ (z+ ) = gnm,h,N (x, z, m; z+ ). From these approximate equilibrium functions we can generate SCE paths {xt (z t ), yt (z t )}∞ t=0 .

17

4.2

Convergence of the Simulated Moments

To assess model’s predictions, analysts usually calculate moments of the simulated paths ∞ xt (z t ), yt (z t ) t=0 from a numerical approximation. The idea is that the simulated moments should approach steady-state moments of the original model. Under continuity of the policy function, Santos and Peralta-Alva (2005) establish various convergence properties of the simulated moments. They also provide examples of non-existence of stochastic steady-state solutions for non-continuous functions, and lack of convergence of empirical distributions to some invariant distribution of the model. Hence, it is not clear how economies with distortions should be simulated, since for these economies the continuity of the policy function does not usually follow from standard economic assumptions. We now outline a reliable simulation procedure that circumvents the lack of continuity of the equilibrium law of motion. We append two further steps to the usual model simulation. First, we discretize the image space of the approximate equilibrium selection so that this function can take on a finite number of points. Then, the simulated moments are generated by a finite Markov chain that has an invariant distribution, and every empirical distribution from the simulated paths converges almost surely to some ergodic invariant distribution of the Markov chain. Second, following Blume (1982) and Duffie et al. (1994) we randomize over continuation values of operator B. We construct a new operator B cv that is a convex-valued correspondence in the space of probability measures. This correspondence has an invariant distribution µ∗ ∈ B cv (µ∗ ). Moreover, as we refine the approximations the simulated moments from our numerical approximations are shown to converge to the moments of some invariant distribution µ∗ . (i) Discretization of the equilibrium law of motion: As before, let the set of all equilibrium state vectors s = (x, z, m) lie in the expanded state space S. Then, recall that at the end of the preceding section we defined the equilibrium selection g : S → S that governs the law of motion of the selected vector of state variables s. Let us now assume that gnh,N : S → S is an approximate equilibrium function for accuracy parameters 18

(h, N, n) as defined above. Hence, gnh,N gives rise to a time-homogeneous Markov process on S. Next, let Hγ be a set with a finite number of points in S such that d(Hγ , S) < γ. h,N,Hγ

Therefore, each point in S is within a γ-ball of some point in H. Now, let gn

(s; z+ ) =

arg mins+ ∈Hγ d(s+ , gnh,N (s; z+ )) for every z+ . If there are various solution points s+ we h,N,Hγ

arbitrarily pick one solution s+ . Hence, the new discretized function gn

takes values

over the finite set Hγ , and gives rise to a Markov chain that has an invariant distribution ∗,h,N,Hγ

νn

. Further, the moments of a simulated path {st }∞ t=0 converge almost surely to ∗,h,N,Hγ

those of some ergodic invariant distribution νn

[e.g., see Stokey, Lucas and Prescott

(1989), Ch. 11]. (ii) Randomization over continuation equilibrium sequences: Let B(·, A∗ ) : V ∗ → V ∗ be denoted as BA∗ : V ∗ → V ∗ . Note that this operator is defined in the space of equilibrium states (x, z, m). We can also view operator BA∗ : V ∗ → V ∗ as a correspondence in the space of probability measures µ on S. Thus, abusing notation we shall say that ν ∈ BA∗ (µ) if there is a selection g as chosen above such that ν = g · µ, where g · µ denotes the action of function g on probability measure µ [e.g., see Stokey, Lucas and Prescott (1989)]. Following Blume (1982) and Duffie et al. (1994), we convexify the image of BA∗ . Thus, if ν and ν 0 are two probability measures that belong to the range of BA∗ we assume that every convex cv denote the combination λν + (1 − λ)ν 0 also belongs to the range of BA∗ . We let BA ∗

randomization7 of operator BA∗ over the space of probability measures µ on S. The new cv is a convex-valued, upper semicontinuous correspondence. Since S is assumed operator BA ∗

to be compact, the set of probability measures µ on S is also compact in the weak topology cv has a fixed-point solution; that is, there exists an of measures. Therefore, operator BA ∗ cv (µ∗ ). invariant probability, µ∗ ∈ BA ∗

(iii) Convergence of the simulated moments to population moments of the model: h,N,Hγ

Using standard simulation procedures, for a given function gn

we generate an ap-

7 Duffie et al. (1994) argue that this convex operator generates some weak form of sunspot equilibria since the randomization proceeds over equilibrium distributions rather than over an external parameter or extraneous sunspot variable.

19

proximate equilibrium path {st }∞ t=0 . Let f : S → R+ be a function of interest. Then, P h,N,Hγ T 1 defines t=0 f (st ) represents a simulated moment or some other statistic. Since gn T a Markov chain, it follows that {st }∞ t=0 must enter an ergodic set in finite time. It follows ´ P ∗,h,N,Hγ that T1 Tt=0 f (st ) must converge almost surely to f (s)dνn as T → ∞ for some ∗,h,N,Hγ

ergodic invariant distribution νn ∗,h,N,Hγ

νn

. We now link convergence of ergodic distributions

cv (µ∗ ) so that the simulated statistics of numerical approximations to some µ∗ ∈ BA ∗

converge almost surely to those of some invariant distribution µ∗ . ∗,h,N,Hγ

Theorem 4.2 Let {(νn

} be a sequence of invariant distributions corresponding to h,N,Hγ

the sequence of approximate functions {gn

∗,h,N,Hγ

}. Then, every limit point of {νn

cv (µ∗ ), converges weakly to some invariant distribution µ∗ ∈ BA ∗

}

as h → 0, N → ∞,

n → ∞, and Hγ → S. To summarize, Theorems 4.1 and 4.2 present new results on numerical recursive solutions over correspondences; these results are essential to insure convergence of both computed solutions and simulated moments. Convergence is established over discrete approximations along the following margins: 1. Discretization of the domain : h mesh size of the family of simplices {X j }. 2. Discretization of the image : N number of elements of the outer approximation. 3. F inite iterations: n number of iterations of operator B h,N . 4. F inite M arkov chain : γ maximum distance of every point in S to some point in set Hγ . 5. F inite simulation : T length of a simulated path {st }t≥0 . cv convex-valued operator in the 6. Randomization of equilibrium distributions: BA ∗

space of distributions.

20

Thus, for every  > 0 we can make the aforementioned parameters sufficiently close to h,N,H

γ their respective limits so that for a given path {st }∞ , t=0 generated under function gn ´ P cv such that there are invariant distributions µ∗ , µ0∗ of BA f (s)dµ∗ −  ≤ T1 Tt=0 f (st ) ≤ ∗ ´ f (s)dµ0∗ + almost surely. Consequently, for a sufficiently fine approximation the moments

from simulated paths are close to some set of moments of the invariant distributions of the cv has a unique invariant distribution µ∗ then µ0∗ = µ∗ and the model. Of course, if BA ∗ ´ ´ P above expression reads as f (s)dµ∗ −  ≤ T1 Tt=0 f (st ) ≤ f (s)dµ∗ + .

5

Asset Pricing Models with Market Frictions

An important family of macroeconomic models incorporates financial frictions in the form of sequentially incomplete markets, borrowing constraints, transactions costs, cash-in-advance constraints, and margin and collateral requirements.8 Fairly general conditions rule out the existence of financial bubbles in these economies, and hence equilibrium asset prices are determined by the expected value of future dividends [Santos and Woodford (1997)]. There is, however, no reliable algorithm for the numerical approximation and simulation of these economies. Here, we illustrate the workings of our algorithm in the economy of Kehoe and Levine (2001). These authors provide a characterization of steady-state equilibria for an economy with idiosyncratic risk under exogenous and endogenous borrowing constraints. We complement their qualitative analysis with numerical simulation to appraise quantitatively the effects of both borrowing constraints on consumption and asset prices.

5.1

Economic Environment

We now recount the model of Kehoe and Levine (2001) which goes along the lines of our first example in section 2. There are two states of uncertainty: Each household may be getting a current high endowment eh or a low endowment el . There is no aggregate risk: One household gets the high endowment whilst the other one gets the low endowment at 8 For instance, see Campbell (1999), Heaton and Lucas (1996), Huggett (1993), Krebs and Wilson (2004), Mankiw (1986), and Telmer (1993).

21

every date. There is only one unit of the Lucas tree with a constant dividend. The bond market is closed, and hence savings can only be channeled through the real asset. Kehoe and Levine (2001) contemplate two scenarios for the modelling of financial markets. In the first scenario, each household has to satisfy the participation constraint (4), and is allowed to trade contingent shares of the tree that are honored depending upon the state of uncertainty. In essence, in this first scenario, households are subject to an overall budget constraint and to the sequence of individually rational debt constraints (4). In the second scenario, each household can trade uncontingent shares of the Lucas tree subject to the exogenous borrowing constraint (3) with Ω = 0. Of course, each household must satisfy budget balance (2) at all times. Kehoe and Levine refer to the fist scenario as the debt constrained economy, and to the second scenario as the liquidity constrained economy. Following these authors, for each scenario a sequential competitive equilibrium (SCE) is a collection of stochastic paths of individual consumptions, asset holdings, and asset prices such that (i) Individual consumption and asset holding allocations solve the constrainedutility maximization problem of each household i = 1, 2, and (ii) Goods and financial markets clear.

5.2

Quantitative Experiments

For convenience of the presentation, we consider the baseline case of Kehoe and Levine (2001). The equilibrium for the debt constrained economy takes on a simple form. Roughly, consumption is high for the household that gets the high endowment, and consumption is low for the other household for whom the limited enforcement constraint is binding. For the liquidity constrained economy, the solution does not take on such a simple form, and needs to be computed. Basically, the ergodic set comprises the whole domain of capital holdings, and allocations depend on the shock and the distribution of asset holdings. Again, let π be a transition probability of switching endowments. Both households share the same Bernoulli utility function u(c) = log(c), and the same discount factor. Dividends,

22

d, are constant. Our computations center upon the following baseline values β = 1/2; el = 9; eh = 24; d = 1; π = 1/2. The Equilibrium Correspondence Note that in equilibrium θ1 = 1 − θ2 . Hence, in the sequel we let θ be the share holdings of household 1, and es be the endowment of household 1, for s = l, h. Then, the equilibrium correspondence V ∗ (θ, es ) is a map from the space of possible values for share holdings and endowments for agent 1 into the set of possible equilibrium shadow values of investment for each agent (m1 , m2 ). For the economy with endogenous debt constraints, asset holdings and prices are state contingent and thus both θ, q are vectors in R2 . For the economy with exogenous constraints, both θ, q are scalars. For this latter economy the shadow values of investment are defined as follows: 1 [d + q] , e1 + θ(d + q) − θ+ q 1 [d + q] , m2 (θ, e2 ) = 2 e + (1 − θ)(d + q) − (1 − θ+ )q m1 (θ, e1 ) =

(12) (13)

where (e1 , e2 ) = (el , eh ), or (e1 , e2 ) = (eh , el ). For any pair of equilibrium shadow values of investment (m1 , m2 ) ∈ V ∗ (θ, es ), there must be share prices q, multipliers λ, tomorrow’s share holdings θ+ , and shadow values of investment (m1+ , m1+ ) ∈ V ∗ (θ+ , e+ ) such that for the economy with exogenous constraints we must have i qDui (ei + θi (d + q) − θ+ q) = λi + β i Emi+ .

Here λi ≥ 0, with strict inequality if today’s borrowing constraint binds. As before, E is the expectations operator. Analogously, for the economy with endogenous constraints we must have qDui (ei + θi (d + q) − θi · q) = λi β i π[ei+ |ei ]mi+ . Observe that in this latter equation we presume that shares are state contingent, and λi ≥ 1 is a ratio of multipliers corresponding to the participation constraints. That is, 23

λi =

1+µi +µi+ , 1+µi

where µi ≥ 0 is a multiplier associated with today’s participation constraint,

and µi+ ≥ 0 is a multiplier associated with tomorrow’s participation constraint at state ei+ |ei . Therefore, λi > 1 only if tomorrow’s participation constraint is binding.9 Our Algorithm Our method proceeds as follows. We start with a correspondence V0 such that V0 (θ, es ) ⊇ V ∗ (θ, es ) for all (θ, es ) with s = l, h, and A0 as defined in Theorem 3.1. It is easy to come up with the initial candidate V0 , since the low endowment el is a lower bound for consumption, and the marginal utility of consumption can be used to bound asset prices as discounted values of dividends. Then, operator B generates a sequence of correspondences defined by the recursion (Vn+1 , An+1 ) = B(Vn , An ) for initial condition (V0 , A0 ). This sequence converges to equilibrium correspondence (V ∗ , A∗ ). For the purposes of presentation, let us first consider the scenario of the exogenous borrowing constraint (3) where correspondence A is not really operative. Our mapping B dictates that for (m1 , m2 ) ∈ BVn (θ, es ) it must be possible to find continuation shadow values of investment (m1+ , m2+ ) ∈ Vn (θ+ , es+ ), a bond price q, and multipliers (λ1 , λ2 ), such that the individual’s intertemporal optimality conditions are satisfied q = λ1 + βEm1+ , + θ(d + q) − θ+ q q = λ2 + βEm2+ . 2 e + (1 − θ)(d + q) − (1 − θ+ )q e1

If we cannot find values that satisfy the previous conditions, then (m1 , m2 ) ∈ / BVn (θ, es ). A new correspondence Vn+1 = B(Vn ) is defined after proceeding with these computations over every possible value (θ, es ) for s = l, h. For the scenario with the limited enforcement constraint (4), our algorithm requires iterating over candidate values that preclude default. In the present model the reservation value of default is autarky. Let V i,aut (es ) be the expected utility value of consuming the endowment allocation starting from es for s = l, h. Computing the value of participation 9

Note that this is the Euler equation for t = 0 to build operator B. Using the envelope theorem for the value function J i defined below, we can derive the Euler equation for any future date t ≥ 1.

24

requires an initial guess for a value function J0i (m1 , m2 , θ, es ) ≥ V i,aut (es ) for i = 1, 2 for some q ∈ A0 (θ, es ) and all (λ1 , λ2 ).10 As already argued, the initial guess function J0i may be computed from the expected utility function by assuming maximum consumption over all future date-events starting from es for s = l, h. A new candidate value for participation is computed as follows. First, let Θ(m1 , m2 , θ, es ) = {(m1+ , m2+ , θ+ )}, be the set of continuation vectors (m1+ (el ), m2+ (el ), θ+ (el ), m1+ (eh ), m2+ (eh ), θ+ (eh )) ∈ graph(Vn ) such that there are q and (λ1 , λ2 ) so that the Euler equations, and the individually rational debt constrains u(ci ) + βEJni (m1+ , m2+ , θ+ , es ) ≥ V i,aut (es ), are satisfied, for i = 1, 2. Now, let i Jn+1 (m1 , m2 , θ, es ) =

m1+ ,m2+ ,θ+

(

max u(ci ) + βEJni (m1+ , m2+ , θ+ , es+ ). 1 2 )∈Θ(m ,m ,θ,es )

(14)

Note that this maximization goes over future continuation vectors (m1+ , m2+ , θ+ ) ∈ graph(Vn ) for given initial state variables (m1 , m2 , θ, es ). Therefore, (m1 , m2 ) ∈ Vn+1 (θ, es ) if (m1 , m2 , θ, es ) i belongs to the intersection of the domains of Jn+1 for each i = 1, 2, and (q, λ1 , λ2 ) ∈

An+1 (θ, es ). Our algorithm can then be used to generate a sequence of approximations to the equilibrium correspondence via the recursion (Vn+1 , An+1 ) = B(Vn , An ). For the numerical implementation of the algorithm, let us just consider the debt constrained economy. We assume a pre-specified interval of share holdings [θl , θh ], which in this case is [0, 1]. We then partition the state space by selecting a set of vertex points with grid size h. The step correspondence approximating V0 at θ over a simplex [θi , θi+1 ] can be defined as V0h (θ, eh ) = ∪θ1 ∈[θi ,θi+1 ] V0 (θ, eh ) V0h (θ, el ) = ∪θ1 ∈[θi ,θi+1 ] V0 (θ, el ). 10

1 Observe that for a given q and continuation shares θ+ we can compute the consumption vector (c1 , c2 ) from budget constraint (2).

25

The image of this correspondence are the shadow values of investment (m1 , m2 ). Hence,  a simple outer approximation C N B h (V ) would be a finite collection of squares for vectors (m1 , m2 ). This completes the numerical implementation of operator B h,N , defined over computable step correspondences. The various tasks involved in this process can adequately be performed by parallel computing. Quantitative Results We now compare the quantitative implications for consumption volatility and asset prices for the two different scenarios. The debt constrained economy inherits a simple dynamic structure with two steady-state values for consumption. The liquidity constrained economy, however, presents richer dynamics. The ergodic set is made up of the whole distribution of shares θ ∈ [0, 1] as agent 1 buys shares of the asset in the presence of the good shock, and sells shares of the asset in the presence of the bad shock. Table 2 below reports sample statistics for equilibrium time series from both economies. In this table, q refers to the price of a state uncontingent share. This is the price of the asset for the liquidity constrained economy and the sum of the two current prices of the asset for the debt constrained economy. Model Exogenous constraint Endogenous constraint

mean(q) 2.11 1.07

std(q) 1.23 0.00

mean(c1 ) 16.91 17.00

stdev(c1 ) 7.36 4.52

Table 2: Simulated moments for the debt and liquidity constrained economies – mean and standard deviation (stdev). Perfect risk sharing would require constant consumption across states equal to 17. The endogenous participation constraint prevents perfect risk sharing and consumption displays some volatility in the debt constrained economy. Since the unique equilibrium is a symmetric stochastic steady state and the agent with the good shock (who is unconstrained) 26

determines the price of the asset, the price of a state uncontingent share is constant. As is well understood, however, the volatility of the pricing kernel of this economy is higher than that of a complete markets economy but we do not report state contingent prices. The economy with exogenous borrowing constraints and uncontingent trading displays more volatile equilibrium consumption allocations and asset prices. The basic reason for this higher volatility is market incompleteness coupled with a higher variability of asset holdings. Indeed, the borrowing constraint binds less than 4 percent of the time. This is an interesting result for asset pricing that may be extended to more general finance scenarios. In conclusion, the liquidity constrained economy generates more asset price volatility. Differences in the absolute price of the asset comes from the behavior of the interest rate [cf., Kehoe and Levine (2001)]. These are economies with purely idiosyncratic risk. The introduction of aggregate risk may have considerable effects on the volatility of asset prices. The numerical solution of models with aggregate risk has proven quite challenging but can be readily integrated into our computational method.

6

International Risk Sharing

A growing literature has developed to explore the performance of business cycle models under limited risk sharing because of market imperfections. As documented in various papers [e.g., Backus, Kehoe and Kydland (1992)] standard versions of the neoclassical growth model cannot account for certain co-movements of macroeconomic aggregates. We now show that our reliable algorithm can naturally be applied to the computation of twocountry models with real and financial frictions.

6.1

Economic Environment

We just outline an extended version of the economy of Kehoe and Perri (2002) in which we include shocks on preferences and taxes. Consider a two-country model with a representa-

27

tive household in each country. There is a unique aggregate good. Total factor productivity (TFP) of each country is affected by a vector of shocks z that follow a Markov chain. There is a constant returns to scale technology. Labor and capital stocks cannot be moved across countries, but limited international borrowing is possible. Assets include capital and bonds. The representative household of country i = 1, 2 has preferences over stochastic sequences of consumption and labor given by the utility function "∞ # X  E β t ui cit , lti , zt .

(15)

t=0

Function ui (·, ·, zt ) : R2 → R is increasing in ci ≥ 0 and decreasing in li ∈ [0, 1], strictly  concave, and twice continuously differentiable. Stochastic consumption plans cit t≥0 are financed by commodity endowments, after-tax capital returns, labor income, and lumpsum transfers. These values are expressed in terms of the single good, which is taken as the numeraire commodity of the system at each date-event. For a given rental rate rti and wage wti in country i, the representative household offers kti (z t−1 ) ≥ 0 units of capital accumulated from the previous period, and supplies lti (z t ) units of labor. l (z t )) denote bond holdings One-period bonds can be traded at all times. Let bi (z t , ξt+1 l (z t ) is a representative element of a given partition of the possible of agent i, where ξt+1 0

l (z t ) equals all z t+1 |z t , and ξ l (z t ) ∩ ξ l (z t ) = 0 whenever successors z t+1 |z t . Hence, ∪l ξt+1 t+1 t+1

l0 6= l. A bond is a promise to deliver 1 unit of the consumption good whenever z t+1 ∈ l (z t ), and 0 otherwise. This specification allows for a full set of state contingent bonds ξt+1 l (z t ) is a unique element for each l. An uncontingent bond pays one unit of the good if ξt+1 l (z t )) be the price of a bond issued at z t . for any possible future state. Let q(z t , ξt+1

The representative household of country i is subject to the following sequence of budget constraints:

28

    i l (z t ))q(z t , ξ l (z t )) = w i z t li z t + cit z t + kt+1 z t + bi (z t , ξt+1 (16) t t t+1      (1 − τki (K i ))rti z t kti z t−1 + (1 − δ) kti z t−1 + eit z t + bi (z t−1 , ξtl (z t−1 )) + Tti z t , for all z t , t ≥ 0, given k0i . Endowments eit (z t ) are strictly positive and depend on the current realization zt . As in the growth model of section 2, capital income is taxed according to function τk , which may depend on the aggregate capital stock, Kti , or some other state variables. This tax function is assumed to be positive, continuous, and bounded away from 1. Tax revenues are rebated    back to the representative consumer as lump-sum transfers Tti z t = τki (K i )rti z t Kti z t . As in the preceding section we consider two scenarios. In the debt constrained economy consumers have a complete menu of contingent bonds. There are therefore complete financial markets, but debt repudiation entails permanent exclusion from financial markets. Then, to prevent default as an equilibrium outcome the following individually rational debt constraint must always be satisfied. Ez t

∞ X

βi



 i ui ciτ , lτi , zτ ≥ V i,aut (Kt−1 (z t ), z t ), for all t ≥ 0.

(17)

τ =t

Here, V i,aut is the expected discounted utility value for autarky as a result of zero bond i (z t ) is the average level of physical trading for country i at all dates after z t . Hence, Kt−1

capital of country i starting at node z t . In the liquidity constrained economy, households can trade quantities bi (z t ) of a single uncontingent bond that yields one unit of the commodity for all states, subject to the following exogenous constraint: bi (z t ) ≥ −Ωi ,

(18)

where Ωi is some large positive number. In each country i = 1, 2, the production sector is made up of a continuum of identical units that have access to a constant returns to scale technology in individual factors. Thus, 29

without loss of generality we shall focus on the problem of a representative firm. After observing the current shock z the firm rents K i units of capital and hires Li units of labor. The total quantity produced of the single aggregate good is given by a production function   Ait F Kti , Lit , where Ait is a TFP index and F Kti , Lit is the direct contribution of the firm’s inputs to the production of the aggregate good. At every date-event z t , factors of production are hired by the firm at the point in which the marginal productivity of capital equals the rental rate rti and the marginal productivity of labor equals the wage wti . We shall maintain the following standard conditions on production function F . Let D1 F (K, L) be the derivative of F with respect to K. Assumption 6.1 F : R+ × R+ → R+ is increasing, concave, continuous, and linearly homogeneous. This function is continuously differentiable at each interior point (K, L); moreover, limK→∞ D1 F (K, L) = 0 for all L > 0.

6.2

Competitive Equilibrium

Definition 6.1 : A SCE is a tax function τki (K), and a collection of vectors  i (z t ), bi (z t , ξ l (z t )), K i (z t ), Li (z t ), r i (z t ), w i (z t )} t l t {cit (z t ), lti (z t ), kt+1 i=1,2 , q(z , ξt+1 (z )) t≥0 t t t t+1 t+1 that satisfy the following conditions: i , bi } (i) Constrained-utility maximization: For i = 1, 2 the sequence {cit , lti , kt+1 t t≥0 solves

the maximization problem for the objective (15) subject to the sequence of budget constraints (16), as well as constraint (17) for the debt constrained economy, and constraint (18) for the liquidity constrained economy. (ii) Market clearing in the goods, capital, labor, and bond markets. We are just extending the definition of SCE of Kehoe and Perri (2002) for an international economy with taxes. Note that in this economy international borrowing allows for imports of the aggregate good produced abroad – available for consumption and investment – but the representative firm can only hire local inputs – capital and labor. There does

30

not seem to be a general proof of existence of competitive equilibria for infinite-horizon economies with distortions. We are aware of a related contribution by Jones and Manuelli (1999), but their analysis is not directly applicable to models with incomplete markets or externalities. Hence, the appendix outlines a proof of the following result. Proposition 6.2 A SCE exists.

6.3

Bounds on Equilibrium Allocations and Prices

The appendix shows existence of positive constants K max and K min such that for every P2 i 0 i (z t ))} max ≥ equilibrium sequence of physical capital vectors {kt+1 t≥0 if K i=1 k0 (z ) ≥ P i (z t ) ≥ K min for all z t . Hence, in what follows the domain of K min then K max ≥ 2i=1 kt+1 aggregate capital will be restricted to the interval [K min , K max ]. We also show that every equilibrium sequence of factor prices {rti (z t ), wti (z t )}t≥0 is bounded. To implement operator B, we need to bound the equilibrium shadow values of investment. For this purpose, it is convenient to use the following dynamic programming argument. We define an auxiliary value function of an individual sequential optimization problem.

For a given sequence of factor and bond prices and aggregate capital   (r0 (z0 ), w0 (z0 ), q(z0 ), K(z0 )) = {rt (z t ), wt (z t ), qt z t , Kt+1 z t }t≥0 , let J i (k0i , bi0 , z0 , r0 (z0 ), w0 (z0 ), q(z0 ), K(z0 )) = max E

∞ X

β t ui (ct (z t ), lt (z t ), zt )

t=0

subject to the sequence of budget constraints (16), as well as constraint (17) for the debt constrained economy, and constraint (18) for the liquidity constrained economy, for given k0i , bi0 . That is, J i (k0i , bi0 , z0 , r0 (z0 ), w0 (z0 ), q(z0 ), K(z0 )) is the maximum utility attained for initial k0i , bi0 , over an expected future sequence of equilibrium prices and tax rebates. For every bounded sequence (r0 (z0 ), w0 (z0 ), q(z0 ), K(z0 )), the value function J i (k0i , z0 , bi0 , r0 (z0 ), w0 (z0 ), q(z0 ), K(z0 )) is well defined, and continuous. Moreover, mapping J i (·, ·, z0 , r0 (z0 ), w0 (z0 ), q(z0 ), K(z0 )) is increasing, concave, and differentiable with 31

respect to the initial conditions k0i and bi0 [cf. Rincon-Zapatero and Santos (2009)]. Let Dk,b J i (·, ·, z0 , r0 (z0 ), w0 (z0 ), q(z0 ), K(z0 )) be the partial derivative of function J i (·, ·, z0 , r0 (z0 ), w0 (z0 ), q(z0 ), K(z0 )) with respect to (k0 , b0 ). Then, Dk,b J i (·, ·, z0 , r0 (z0 ), w0 (z0 ), q(z0 ), K(z0 )) varies continuously with (k0i , bi0 , r0 (z0 ), w0 (z0 ), q(z0 ), K(z0 )). The next result readily follows from these regularity properties of the value function. Proposition 6.3 For all SCE  i (z t ), bi (z t , ξ l (z t )), K i (z t ), Li (z t ), r i (z t ), w i (z t )} t l t {cit (z t ), lti (z t ), kt+1 i=1,2 , q(z , ξt+1 (z )) t≥0 t t t t+1 t+1 P with K max ≥ 2i=1 k0i (z 0 ) ≥ K min , there is a constant vector γ b = (γ, γ) for γ > 0 such that 0 ≤ Dk,b J i (·, ·, z0 , r0 (z0 ), w0 (z0 ), q(z0 ), K(z0 )) ≤ γ b for all z t . Observe that these bounds apply to the following types of utility functions: (i) Both function u(·, ·, z) and its derivative are bounded; (ii) function u(·, ·, z) is bounded, and its derivative function is unbounded; and (iii) both function u(·, ·, z) and its derivative are unbounded. Phelan and Stacchetti (2001) deal with case (i) and Krebs (2004) and Kubler and Schmedders (2003) consider utility functions of type (iii). We provide a uniform method of proof that covers all three cases, as well as production functions with bounded and unbounded derivatives, and exogenous and endogenous constraints. As a matter of fact, Proposition 6.3 fills an important gap in the literature for production economies with heterogeneous consumers and market frictions, since no general results are available on upper and lower bounds for equilibrium allocations and prices. For any initial distribution of capital k0 = (k01 , k02 ), bonds b0 = (b10 , b20 ) and a given shock z0 , we define the Markov equilibrium correspondence as   {Dk,b J i (k0i , bi0 , z0 , r0 (z0 ), w0 (z0 ), q(z0 ), K(z0 ))}i=1,2 : ∗ V (k0 , b0 , z0 ) = . There is a SCE

(19)

Hence, the set V ∗ (k0 , b0 , z0 ) contains all current equilibrium shadow values of investment returns mi0 , for every household i. Corollary 6.4 Correspondence V ∗ is non-empty, compact-valued, and upper semicontinuous. 32

This corollary is a straightforward consequence of Propositions 6.2 and 6.3. These bounds insure that our operator B maps compact sets into compact sets [cf., Assumption 3.1]. The construction of B follows the same steps of the preceding section.

6.4

Quantitative Experiments

We now explore the quantitative implications of the above two financial scenarios. For comparison purposes we will also report results for the model with complete markets which can be solved using standard dynamic programming techniques. We assume a one-period utility with stochastic shock ν i (z) given by  η 1−σ c (1 − l)1−η u (c, l, z) = ν (z) , 1−σ i

i

(20)

and a Cobb-Douglas production function AF (K, L) = AK α (L)1−α .

(21)

We also consider the following standard parameter values: α = 0.36, η = 0.36, and σ = 2. From quarterly data, we let β = 0.99 and δ = 0.025. We consider a discrete VAR process for the technology shocks with four possible states: (A1 = 0.95613, A2 = 0.95613), (A1 = 0.95613, A2 = 1.04480), (A1 = 1.04480, A2 = 0.95613), (A1 = 1.04480, A2 = 1.04480). These states evolve according to the  0.83022  0.10821 π=  0.10971 0.01354

transition matrix 0.07849 0.77567 0.00793 0.07934

0.07803 0.00865 0.77629 0.07960

 0.01326 0.10747  . 0.10607  0.82752

We use our method to compute SCE of the this two-country model with endogenous and exogenous borrowing constraints. In both scenarios we find that the equilibrium correspondence converges to a function (up to numerical accuracy of 10−6 ), which indicates that the SCE is unique for given initial conditions. This is the only model of the paper where computational time is a substantial issue. The basic form of our algorithm is fairly easy to 33

implement: It only requires to search for (x, z, m) so that the conditions of operator B are to be satisfied. As this process of search is independent across states, it is straightforward to use parallel computing. In terms of running times, as in most algorithms the choice of initial guess matters greatly. The initial guess we employed was the solution of a similar economy but with complete markets and no distortions, which can easily be secured with a standard dynamic programming algorithm. Our grid considers 51 equally spaced points for K and 501 points for m for each country i = 1, 2. We ran our C++ MPI code using an IBM Server 1350 Cluster, with 50 Xeon 2.3GHZ processors. The average time per iteration of operator B was 24 minutes. The program took 94 iterations to converge to a desired level of accuracy. Table 3 reports the simulated moments for the complete markets economy, the liquidity constrained economy, and the debt constrained economy. The resulting simulated sample moments are in line with those reported in Kehoe and Perri (2002) who use a slightly different calibration and a different computational method. Only the debt constrained economy offers a chance of generating reasonable correlations. In the first three scenarios, preferences are non stochastic (ν(z) = 1), and there are no taxes (τ = 0). The last column of Table 3 reports a slightly different experiment for the liquidity constrained economy with stochastic preferences and taxes. We assume that ν i = 1.05 if Ai > 1, and ν i = 0.95 if Ai ≤ 1. Hence, the representative household is more optimistic (or more willing to consume) in the event of a good productivity shock. Also, τ i = .30 if Ai > 1, and τ i = .25 if Ai ≤ 1. Hence, taxes are procyclical. With respect to the debt constrained economy, this last calibration improves the bilateral correlations of investment and labor, but does not do as well for the correlations of consumption c and GDP .

34

Bilateral correlations c GDP Investment labor

Data

complete markets

liquidity constrained

debt constrained

pref erences/tax shocks

0.32 0.51 0.29 0.43

0.8003 −0.5947 −0.9117 −0.9341

-0.8767 -0.7568 −0.9953 -0.8714

0.2264 0.0170 0.6037 −0.1062

-0.36 -0.28 0.41 0.19

Table 3: Statistical properties of the economies with complete markets, and with exogenous or endogenous constraints. In summary, in this section we apply our reliable algorithm to a two-country general equilibrium model with several real and financial frictions: Incomplete markets, exogenous and endogenous constraints, preference shocks, and taxes. We establish bounds for equilibrium allocations and prices as a key condition for the numerical implementation of our algorithm. Our model simulations broadly confirm the findings of Kehoe and Perri (2002): Endogenous debt constraints seem instrumental to fix some international business cycles anomalies. We here obtain a related result with pro-cyclical preference shocks and taxation. Our computational method can accommodate some other extensions (e.g., time-to-build, adjustment costs), or can be applied to related models of international investment [Bai and Zhang (2010)].

7

A Stochastic OLG Economy

OLG models have become quite relevant in the analysis of several macro issues, such as the funding of social security, the optimal profile of savings and investment over the life cycle, the effects of various fiscal and monetary policies, and the evolution of future interest

35

rates and asset prices under current demographic trends.11 As already stressed, there are no known convergent procedures for the computation of sequential competitive equilibria in OLG models even for frictionless economies with complete financial markets. We now illustrate that our approach delivers a reliable, computable algorithm for the solution of competitive equilibria in a general class of OLG models.

7.1

Economic Environment

For simplicity of the presentation we just consider an extended version of the OLG economy of section 2. The economy is conformed by a sequence of overlapping generations that live for two periods. The primitive characteristics of the economy are defined by a stationary Markov chain. Each generation is made up of I agents, who are present in the economy for two periods. The economy starts with an initial generation who is present only in the initial period t = 0. This generation is endowed with the aggregate supply of assets θ0 . At each node z t , there exist spot markets for the consumption good and J securities. These securities are specified by the current vector of prices, qt (z t ) = (· · · , qtj (z t ), · · · ), and the vectors of future dividends dr (z r ) = (· · · , djr (z r ), · · · ) promised to deliver at future information sets z r |z t for r > t. We assume that the vector of security prices qt (z t ) is nonnegative – a condition implied by free disposal of securities – and the vector of dividends dt (z t ) is non-negative and depends on the current realization zt . The utility function U i is separable over consumption of different dates. Then the intertemporal objective U i is defined as  t   t  t  X     U i ci,z ; z t , z t+1 = ui ci,z z t , zt + β v i ci,z z t+1 , zt+1 π z t+1 |z t .

(22)

z t+1 ∈Z

Assumption 7.1 For each z ∈ Z the one-period utility functions ui (·, z), v i (·, z) : R+ → 11

For instance, see Conesa and Krueger (1999), Geanakoplos, Magill and Quinzii (2003), Gourinchas and Parker (2002), Imrohoroglu, Imrohoroglu, and Joines (1995), Storelesletten, Telmer and Yaron (2004), and Ventura (1999).

36

R ∪ {−∞} are increasing, strictly concave, and continuous. These functions are also continuously differentiable at every interior point c > 0. n t o  t t Given prices {qt z t }t≥0 , a consumption-savings plan ci,z (z t ), ci,z (z t+1 ), θi,z (z t ) must obey the following two-period budget constraints:

t

t

t

t

ci,z (z t ) + θi,z (z t ) · qt (z t ) ≤ ei,z (zt ), f or θi,z (z t ) ≥ 0, t

(23)

t

t

ci,z (z t+1 ) ≤ θi,z (z t ) · (qt+1 (z t+1 ) + dt (zt+1 )) + ei,z (zt+1 ), all z t+1 |z t .

(24)

0

For an initial stock of securities θi,z each agent i at time t = 0 seeks to maximize the 0

0

total quantity of consumption ci,z (z0 ) for given endowments of the aggregate good ei,z (z0 ) 0

and the vector of securities θi,z . More precisely, 0

0

0

ci,z (z0 ) = θi,z · (q0 (z0 ) + d0 (z0 )) + ei,z (z0 ).

7.2

(25)

Competitive Equilibrium

    I  t t z t , ci,z z t+1 , θi,z z t , qt z t i=1 t≥0 n t o t t i,z t i,z t+1 i,z t such that each consumption-savings plan c (z ), c (z ), θ (z ) solves the constrained-

As before, a SCE is a collection of vectors



ci,z

t

utility maximization of the agent, and goods and assets markets clear. To circumvent technical issues concerning existence of a SCE, we still maintain the short-sale constraint θt ≥ 0 for all t. Note that in this economy the aggregate commodity endowment is bounded by a portfolio-trading plan [Santos and Woodford (1997)], and hence asset pricing bubbles cannot exist in a SCE. Therefore, equilibrium asset prices must be bounded at each date. It follows that the existence of a SCE can be established by standard methods [e.g., Balasko and Shell (1980), and Schmachtenberg (1988)]. JI as follows: Then, we define the Markov equilibrium correspondence V ∗ : Θ × Z → R++

37

V ∗ (θ0 , z0 ) =

    

 

ci,z

t

0

· · · , (q0j (z0 ) + dj0 (z0 ))Dc v i (ci,z (z0 ) , z0 ), · · ·   i,z t t+1 t  i,z t t+1 I  t t z ,c z |z , θ z , qt z i=1



t≥0

:

  

is a SCE  

.

(26) From the above results on the existence of SCE for OLG economies, we obtain Proposition 7.1 Correspondence V ∗ is nonempty, compact-valued, and upper semicontinuous.

7.3

Numerical Example: A Monetary Model

We consider a simplified version of the OLG model with money taken from Benhabib and Day (1982) and Grandmont (1985). This simple model is useful for computation because it can be solved with arbitrary accuracy. Hence, it is possible to compare the true solution of the model with other numerical approximations. Extensions to a stochastic environment are easy to handle with our algorithm but may become problematic for other algorithms. Each individual receives an endowment e1 of the perishable good when young and e2 when old. There is a single asset, money, that pays zero dividends at each given period. The initial old agent is endowed with the existing money supply M. Let Pt be the price level at time t. An agent born in period t chooses consumption c1t when young, c2t+1 when old, and money holdings Mt to solve the constrained optimization problem max u (c1t ) + βv (c2t+1 ) subject to c1t +

Mt = e1 , Pt

c2t+1 = e2 +

Mt . Pt+1

A SEC for this economy is a sequence of prices (Pt )t≥0 , and sequences of consumption and money holdings {c1t , c2t+1 , Mt }t≥0 such that an individual solves the budget-constrained 38

utility maximization problem and markets clear. A SEC can be characterized by the following first-order condition:     M 1 M 1 0 0 u e1 − = βv e2 + . Pt Pt Pt+1 Pt+1 Let bt = M/Pt be real money balances at t. Then, bt u0 (e1 − bt ) = bt+1 βv 0 (e2 + bt+1 ) . It follows that all competitive equilibria can be generated by an offer curve in the (bt , bt+1 ) space. A simple recursive equilibrium would be described by a function bt+1 = g (bt ) . Let us now consider the following parameterization: 1 u (c) = c0.45 , v (c) = − c−7 , β = 0.8, M = 1, e1 = 2, e2 = 26/7 − 21/7 . 7 For this simple example, the offer curve is backward bending (see Figure 2). Hence, the equilibrium correspondence is multi-valued. Then, standard methods – based on the computation of a continuous equilibrium function bt+1 = g (bt ) – may portray a partial view of the equilibrium dynamics. There is a unique stationary solution at about b∗ = 0.4181, which is the point of crossing of the offer curve with the 45-degree line.

Figure 2: Offer curve.

39

Implementation of our Algorithm. Following section 7.2 the implementation our numerical algorithm is fairly straightforward. In fact, since the shadow values of the marginal returns to investment lie in a compact set, we can follow the same computational steps of previous sections. For this example, we find that the policy correspondence and time series from our method generate an Euler equation residual of order 10−6 . Actually, the solution obtained with our algorithm is indistinguishable from the “exact” solution. Comparison with other Computational Algorithms A common practice in OLG models is to start the search with an equilibrium guess function b0 = gb(b), and then iterate over the temporary equilibrium conditions. We applied this procedure to our model. Depending on the initial guess, we find that either the upper or the lower arm of the offer curve would emerge as a fixed point. This strong dependence on initial conditions is a rather undesirable feature of this computational method. In particular, if we only consider the lower arm of the actual equilibrium correspondence then all competitive equilibria converge to autarchy. Indeed, the unique absorbing steady state associated with the lower arm of the equilibrium correspondence involves zero monetary holdings. Hence, even in the deterministic version, we need a global approximation of the equilibrium correspondence to analyze the various predictions of the model. As shown in Figure 3, the approximate equilibrium correspondence has a cyclical equilibrium in which real money holdings oscillate between 0.8529 and 0.0953. It is also known that the model has a three-period cycle. But if we iterate over the upper arm of the offer curve, we find that money holdings converge monotonically to

¯ M p

= 0.4181 (as illustrated by the dashed lines

of Figure 3). As a matter of fact, the upper arm is monotonic, and can at most have cycles of period two, whereas the model generates equilibrium cycles of various periodicities.

40

Figure 3: Equilibrium cycles.

In conclusion, for OLG economies, standard computational methods based on iteration of continuous functions do not guarantee convergence to an equilibrium solution, and may miss some important properties of the equilibrium dynamics. In these economies it is pertinent to compute the set of all SCE.

8

Concluding Remarks

This paper provides a theoretical framework for the computation and simulation of dynamic competitive-markets economies in which the welfare theorems may fail to hold because of market frictions or the existence of an infinite number of generations. We have applied these methods to various macroeconomic models with heterogeneous agents, incomplete financial markets, endogenous and exogenous borrowing constraints, taxes, and money. Our numerical algorithm was especially helpful for the simulation of an international business cycle model and for an OLG economy. These models are not amenable to computation by social planning problems because of the existence of real and financial frictions. They are not amenable to computation by projection methods with continuous equilibrium functions because a continuous recursive representation of equilibrium may not exist. And they are 41

not amenable to computation by perturbation methods because the ergodic region may be quite large: Agents accumulate assets to accommodate idiosyncratic and aggregate risk. The application of both projection and perturbation methods may be rather cumbersome in problems with market frictions, and endogenous and exogenous constraints. For optimal economies, sequential competitive equilibria are generated by a continuous policy function that is the fixed-point solution of a contractive operator. Continuity of the policy function allows for various methods of approximation and functional interpolation, and is essential to validate laws of large numbers for the simulated paths. Differentiability and contractive properties are useful for the derivation of error bounds that can guide the computation process. But as discussed in section 2, for economies with distortions or with an infinite number of generations a continuous Markov equilibrium may not exist. We establish a general result on the existence of a Markovian equilibrium solution in a suitably expanded space of state variables, and provide upper and lower bounds for equilibrium allocations and prices. We construct a numerical algorithm that has desirable approximation properties and guarantees convergence of the moments computed from simulated paths. There are three main properties of our algorithm that should be of interest for quantitative work in this area. First, the existence of a Markovian competitive equilibrium is obtained in an enlarged space of state variables. Our choice of the marginal utility values of assets returns is dictated by computational considerations. This is a minimal addition to the state space to restore existence of a Markovian equilibrium and with the property that the added variables enter linearly into the Euler equation. Second, the algorithm iterates in a space of candidate equilibrium sets – rather than in a space of functions. Iteration over candidate equilibrium sets captures the whole set of competitive equilibria, and insures convergence to the fixed-point solution – even if Markov equilibria are not continuous. For the OLG economy of section 7 it was pertinent to have a global understanding of the equilibrium dynamics. And third, the algorithm provides a reliable method for model simulation. There are many papers following the lead of Kydland and Prescott (1980) and

42

Abreu, Pierce and Stacchetti (1990) on recursive characterizations of equilibria, but none of these contributions is concerned with the numerical implementation. Hence, our results should provide a useful benchmark for the construction of other algorithms. We establish some desirable approximation properties of the computed solutions. We then resort to a further discretization of the equilibrium law of motion so that it becomes a Markov chain. For a Markov chain the convergence of the simulated moments follows from the ergodic theorem. Using an approximation argument on fixed-points of upper semicontinuous correspondences, we establish convergence of the simulated moments to those of the invariant distributions of the model. Other ways to restore laws of large numbers for the simulated paths of these economies would be by imposing monotonicity assumptions on the equilibrium dynamics [Santos (2010)] or by expanding artificially the noise process [e.g., Blume (1979)]. These latter approaches seem to be of more limited economic interest. Of course, our methods must face some computational challenges. Iteration over sets is computationally much more costly than iteration over functions. Therefore, the expansion of the state space along with iteration over sets should certainly be manifested into an additional computational burden. Since the many computational tasks in our algorithm can be decentralized, the development of high-performance, parallel computing will certainly make our methods more attractive. Our general convergence results also lack error bounds. This lack of accuracy should be expected because our models cannot be restated as optimization programs, and miss some common concavity, differentiability, and contractive properties. In terms of numerical implementation, the innovative techniques for error estimation proposed by Judd, Yeltekin, and Conklin (2003) require convexity of the approximations, but convexity cannot be imposed on our algorithm because it may arbitrarily expand the set of equilibria. It is therefore of great interest to extend these innovative techniques on estimation of error bounds to the present context. Finally, the numerical implementation of our algorithm starts with an initial correspondence of potential equilibrium values. In most numerical work it is imperative to bound the ergodic region in order to minimize computer

43

costs. This task, however, may become much more delicate for non-optimal economies. In our applications above we have developed various procedures to bound equilibrium allocations and prices by ruling asset pricing bubbles and by defining a value function for each household over future equilibrium paths. This value function is convenient because it can embed exogenous and endogenous budget constraints, and real and financial frictions. Hence, all these market imperfections do not have to be dealt with explicitly in establishing bounds for equilibrium allocations and prices. Our techniques should certainly be valuable for similar equilibrium bounds in related models with heterogeneous agents and market distortions.

44

9

Appendix

In this Appendix we prove some key results formally stated in sections 3 and 4. For convenience, we also offer a proof of existence for the model of section 6, and establish equilibrium bounds. All other claims in the paper may rely on similar arguments. b0 as the set of all feasible y. Let (Vbi , A bi ) = Proof of Theorem 3.1: Let Vb0 ⊃ V0∗ , and A bi−1 ) for all i ≥ 1. To insurance monotone convergence, let us now redefine these B(Vbi−1 , A ∞ b b sets as Vn = ∪∞ i=n Vi and An = ∪i=n Ai for all n ≥ 0. Then (Vn , An ) = B(Vn−1 , An−1 ) and

(Vn , An ) ⊂ (Vn−1 , An−1 ) for all n ≥ 1. It follows that the sequence {(Vn , An )} must converge U U U U to a set (V U , AU ). Further, (V U , AU ) = ∩∞ n=1 (Vn , An ). Therefore, (V , A ) = B(V , A ).

We next prove that (V U , AU ) = (V ∗ , A∗ ). Indeed, by the monotonicity of operator B we get that (V ∗ , A∗ ) ⊂ (V U , AU ); also, (V U , AU ) ⊂ (V ∗ , V ∗ ) since every fixed point conforms an equilibrium – given that the transversality conditions are trivially satisfied in this model. To complete the proof of the theorem, just note that (V U , V U ) ⊂ (V ∗ , A∗ ) ⊂ (Vn , An ) for all n ≥ 1. Since we have already established that (Vn , An ) → (V U , AU ), we get that Vn → V ∗ and An → A∗ . It is clear from these arguments that (V ∗ , A∗ ) is the largest fixed-point of operator B. Proof of Theorem 4.1: (i) Obvious. Operator B h,N is monotone, (V0 , A0 ) ⊇ (V ∗ , A∗ ) and B h,N (V ∗ , A∗ ) ⊃ (V ∗ , A∗ ). (ii) Proof follows similar arguments as in proof of Theorem 3.1. Actually, (Vnh,N , Ah,N n )⊃ (V ∗,h,N , V ∗,h,N ), and our discretized procedure allows for a finite number of set-values. Hence, pointwise convergence implies uniform convergence. (iii) Note that operator B h,N converges to B as h → 0 and N → ∞. Since (V ∗ , A∗ ) ⊂ (V ∗,h,N , A∗,h,N ), we get that (V ∗,h,N , A∗,h,N ) → (V ∗ , A∗ ) as h → 0 and N → ∞.

45

Proof of Theorem 4.2: The proof follows directly from Blume (1982), Theorems 2.1 and 3.1. The sequence of operators {B h,N,Aγ } converges to B, and the set of fixed points is an upper semicontinuous correspondence. Moreover, the convexified operator B cv has a fixed point µ∗ ∈ B cv (µ∗ ). Proof of Proposition 6.2: The existence of a SCE can be established by approximating the infinite-horizon economy by a sequence of finite economies. This is the strategy followed by Jones and Manuelli (1999), but their proof does not apply to sequential competitive economies. Of course, the hardest part is to provide upper bounds for equilibrium quantities over all the finite-horizon economies. These bounds follow from Proposition 6.3 below. Hence, following Jones and Manuelli (1999), we consider the following steps for the proof of a SCE: (i) Existence of an equilibrium for a finite horizon economy. This result is covered by the general proofs of existence of competitive equilibria for economies with taxes, externalities, and incomplete markets [Arrow and Hahn (1971), Levine and Zame (1996), Mantel (1975), and Shafer and Sonneschein (1976)]. (ii) Uniform bounds for equilibrium allocations and prices of finite-horizon economies. As already pointed out, these bounds are established in Proposition 6.3 below. (iii) Existence of SEC as a limit point of finite equilibria. The preceding steps (i) and (ii) guarantee that there is a collection of vectors  i (z t ), bi (z t , ξ l (z t )), K i (z t ), Li (z t ), r i (z t ), w i (z t )} t l t {cit (z t ), lti (z t ), kt+1 i=1,2 , q(z , ξt+1 (z )) t≥0 t t t t+1 t+1 that can be obtained as limits of equilibria of finite economies. It is obvious that for such limiting solution the market clearing conditions must be satisfied at each z t , and that one period-profits are maximized. Moreover, for each agent i the limiting allocation i (z t ), bi (z t , ξ l (z t )) must satisfy the sequence of budget constraints (16), (cit (z t ), lti (z t ), kt+1 t+1

as well as the endogenous or exogenous constraints. This allocation is optimal since the discounted utility function is continuous in the product topology over the set of feasible   consumption/leisure plans cit z t , 1 − lti z t t≥0 which are preferred to the endowment al  location eit (zt ) , 1 t≥0 . This is because feasible consumption plans cit z t t≥0 are bounded

46

 above, and the endowment process eit (zt ) t≥0 is bounded below by a positive quantity and the endowment of leisure is always equal to one. Proof of Proposition 6.3: We first show that there are positive constants K max and K min  i (z t )) such that for every equilibrium sequence of physical capital vectors kt+1 if K max ≥ t≥0 P2 P2 i t min for all z t . The existence of K max i 0 min then K max ≥ i=1 kt+1 (z ) ≥ K i=1 k0 (z ) ≥ K follows directly from Assumption 6.1, since the marginal productivity of capital converges to zero as K goes to ∞ for every fixed 0 ≤ L ≤ 1. Also, it obvious that K min ≥ 0. We now claim that there are constants rmax and wmax such that for every equi  librium sequence of factor prices rti z t , wti z t t≥0 we have 0 ≤ rti (z t ) ≤ rmax and 0 ≤ wti (z t ) ≤ wmax for all z t . The existence of wmax follows from continuity properties of the utility function. The household is endowed with one unit of labor. Hence, if the wage is arbitrarily high it would be optimal to consume a large amount of consumption by giving up a small quantity of leisure. If along an equilibrium path we have that rti is arbitrarily large, then kti must go to zero. From the Euler equation, consumption cit must also go to zero. But this is not possible under either exogenous or endogenous constraints, as eit > 0 is bounded below by a positive quantity, and in the debt constrained economy the household can switch to autarky. Moreover, using a simple arbitrage argument, we have that qt is also bounded. Hence, the value function J i (k0i , bi0 , z0 , r0 (z0 ), w0 (z0 ), q(z0 ), K(z0 )) is well defined. As already pointed out the derivative Dk,b J i (·, ·, z0 , r0 (z0 ), w0 (z0 ), q(z0 ), K(z0 )) is continuous in (k0i , bi0 , z0 , r0 (z0 ), w0 (z0 ), q(z0 ), K(z0 )).12 Moreover, by a simple notational change it follows from (16) that function J i can be rewritten as J i (ai0 , bi0 , z0 , r0 (z0 ), w0 (z0 ), q(z0 ), K(z0 )) w0 (z0 ), K(z0 )), where ai0 = ei0 (z0 ) + (1 − τ ) r0 k0i . Then we can conclude that 0 ≤ Dk,b J i (k0i , bi0 , z0 , r0 (z0 ), w0 (z0 ), K(z0 )) ≤ γ b, since ei0 (z0 ) is bounded below by a positive number, and all feasible vectors (k0i , bi0 , z0 , r0 (z0 ), w0 (z0 ), K(z0 )) lie in a compact set. 12 Note that if bi0 is a large negative number then the value function is well defined, but the agent will switch to autarky. In the autarky region the derivative of J i with respect to bi0 is zero. Hence, at the point of switching to autarky, the derivative of J i will not be continuous but the differential is a compact correspondence.

47

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