«MATHEMATICS OF OPERATIONS RESEARCH Vol. 27, No. 4, November 2002, pp. 637–646 Printed in U.S.A. COMPUTING EQUILIBRIA IN FINANCE ECONOMIES P. ...»
MATHEMATICS OF OPERATIONS RESEARCH
Vol. 27, No. 4, November 2002, pp. 637–646
Printed in U.S.A.
COMPUTING EQUILIBRIA IN FINANCE ECONOMIES
P. JEAN-JACQUES HERINGS and FELIX KUBLER
The general equilibrium model with incomplete asset markets is ideally suited for the study of
problems in cross-sectional asset pricing and portfolio theory. In this paper, we develop a homotopy
algorithm to approximate equilibria in these models. Since the algorithm is tailor made for so-called ﬁnance economies, the number of nonlinear equations that has to be solved for, and therefore the computing time, is an order of magnitude smaller than that of existing general-purpose algorithms.
The algorithm is shown to be generically convergent. We implement the algorithm using HOMPACK. To illustrate its performance, we present various numerical examples and report running times.
1. Introduction. During the last two decades, there has been substantial interest in the general equilibrium model with incomplete asset markets, the GEI model. One of the important features of this model is its integrated approach to the real and ﬁnancial sectors of an economy. For almost all GEI economies, a competitive equilibrium exists. The tools required to show existence of an equilibrium are demanding, and involve many results from differential topology, including the concept of the Grassmann manifold.
The complications of the GEI model imply that it is no longer possible to compute equilibria by the same methods that are used for the standard general equilibrium model. For instance, convergence of the algorithm of Scarf (1967) or the homotopy algorithm of Eaves (1972) is not guaranteed. By using algorithms that operate on the Grassmann manifold, Brown et al. (1996b) and DeMarzo and Eaves (1996) have produced computational methods that converge for a generic GEI economy. Brown et al. (1996a) develop a generically convergent algorithm by means of switching homotopies. This algorithm does not directly involve the Grassmann manifold, although its convergence proof does. For numerical purposes, one may also want to use the homotopy algorithm of Schmedders (1998) that does not involve homotopy switching. A drawback of that algorithm is that it is an open question whether as to it displays generic convergence to an equilibrium.
Our interest in computing solutions in the GEI model is derived from our desire to study the pricing of ﬁnancial assets. For instance, to study whether the lessons of the capital asset pricing model remain valid in a setting with market incompleteness, heterogeneous investors, assets whose distributions have fat tails, and so on. For all these applications, one needs to approximate a continuous multivariate probability distribution by a ﬁnite probability distribution. In order to achieve a reasonable approximation, a big-state space is required.
In a companion paper (Herings and Kubler 2000), we focus on the application to the capital asset pricing model and we need to compute equilibria for models with, for instance, H = 3 agents and J = 8 ﬁnancial assets, and we need S = 32 768 states of nature to approximate a multivariate lognormal distribution by a ﬁnite probability distribution.
All existing algorithms transform the equilibrium problem into a problem that involves so-called state prices. This is essential to show convergence but has the drawback that the Received July 11, 2000; Revised March 21, 2001 and May 22, 2002.
MSC 2000 subject classiﬁcation. Primary 91B28, 91B50, 50C30.
OR/MS subject classiﬁcation. Primary: Finance/asset pricing.
Key words. Computational methods, asset pricing, general equilibrium, incomplete markets.
0364-765X/02/2704/0637/$05.00 1526-5471 electronic ISSN, © 2002, INFORMS 638 P. J.-J. HERINGS AND F. KUBLER number of equations increases rapidly in the number of states. For instance, the homotopy proposed by Brown et al. (1996b) involves 2S nonlinear equations. Another complication in applying that algorithm is that it involves closed-form solutions for the demand functions for assets of the agents, but such closed-form solutions are usually impossible to obtain when asset markets are incomplete. The solution to that problem is to state the problem not in terms of demand functions themselves but in terms of the ﬁrst-order conditions of agents that yield the demand function. This approach is suggested in Garcia and Zangwill (1981) and is followed by Schmedders (1998). It increases the number of equations further, to 2HS + H − 1 J + H so the use of Schmedders’ (1998) algorithm to the models of the size H = 3 J = 8 and S = 32 768 involves solving 196,627 nonlinear equations.
The algorithms of Brown et al. (1996b) and Schmedders (1998) are designed to deal with the general version of the GEI model. In many applications, one is interested in what is known as the ﬁnance version of the GEI model, or ﬁnance economy, for short. In the ﬁnance version of the GEI model, the modeling of the ﬁnancial sector is the same as in the general version of the GEI model. The consumption sector, however, is drastically simpliﬁed, in that in each time period, at each state of the world, there is only one commodity, called income. In this paper, we develop an algorithm that is tailor made for ﬁnance economies.
The restriction to ﬁnance economies leads to a great reduction in the number of equations to be solved for and, thereby, to strong improvements in computing times. If closed-form solutions for demand functions are available, then the number of nonlinear equations to be solved for by our algorithm equals J − 1 Otherwise, we specify the ﬁrst-order conditions and the number of nonlinear equations amounts to H + 1 J + 1 − 2 which is 34 in the application reported on earlier.
Our algorithm is a homotopy algorithm. The use of deformations to solve nonlinear systems of equations dates back at least to Lahaye (1934). A constructive proof of Brouwer’s ﬁxed-point theorem for differentiable functions, based on a homotopy, was given in Kellogg et al. (1976). They use the nonretraction principle of Hirsch (1963) in their proof. We also exploit the differentiability that is present in the problem. We follow the system of differential equations as presented in Davidenko (1953) to follow the homotopy path in our implementation and show that this is possible to do for almost all ﬁnance economies. For recent surveys on homotopies, the reader is referred to Judd (1998) or Eaves and Schmedders (1999).
Compared to traditional general equilibrium theory, ﬁnance economies pose a number of additional difﬁculties. The prices of assets are not necessarily positive but may well be zero or negative, which rules out some of the algorithms that are used in traditional general equilibrium theory, for instance, the simplicial variable dimension algorithm of Doup et al. (1987) or its differentiable counterpart described in Herings (1997). Crucial to the convergence proof of homotopy methods applied to traditional general equilibrium models is the boundary behavior of the excess demand function. When prices of commodities converge to zero, demand for commodities explodes. As a price of zero has no special meaning in the case of ﬁnancial assets, that boundary behavior cannot be used. The convergence proof of our homotopy algorithm builds on the approach to show existence of an equilibrium in ﬁnance economies as outlined in Hens (1991).
The paper is organized as follows. Section 2 introduces the notation and the model of a ﬁnance economy. In §3, we present an algorithm that is tailored to compute equilibria in ﬁnance economies. Special attention is given to the problem that closed-form solutions for demand functions of assets rarely exist in ﬁnance economies. A second, related algorithm is introduced that does not require closed-form solutions. In §4, we show generic convergence of the algorithm, that is, for an open set of ﬁnance economies with full Lebesgue measure, the algorithm converges to an equilibrium. Section 5 discusses the implementation of the algorithm, and in §6, we describe numerical examples. Section 7 concludes.
COMPUTING EQUILIBRIA IN FINANCE ECONOMIES
Assumption 3. rank A = J and A·J 0 (The notation x 0 means that all components of the vector x are nonnegative and at least one component is positive.) Usually Assumption 3 is replaced by the weaker assumption that there is ∈ J such that A 0 Assumption 3 is without loss of generality. Indeed, if there is ∈ J such that A 0 we take A as asset J and we delete an asset j for which j = 0 Equilibria of the original economy are obtained by a simple transformation of the equilibria of the resulting economy. Under Assumption 3, it holds that qJ 0 for all arbitrage-free prices for assets q It follows immediately that the solution to the optimization problem of an agent, as described in (1) of Deﬁnition 2.1, remains unchanged when all prices are multiplied by 0 We can therefore restrict ourselves to arbitrage-free asset prices for which qJ = 1 and we deﬁne Q = q ∈ J −1 × 1 ∃ ∈ S q = A to be the set of arbitrage-free ++ prices for assets with this property.
Given arbitrage-free prices for assets q ∈ Q the demand for assets by agent h denoted g h q is the asset portfolio that solves the following maximization problem
If prices for assets are arbitrage free, then the maximization problem is well deﬁned.
Assumption 1 guarantees that the solution to the optimization problem is unique.
From the demand function for assets of agent h g h Q → J the total demand function for assets G Q → J follows as G = H g h Prices for assets q ∗ induce a competitive h=1 equilibrium for an economy if and only if G q ∗ = 0 The following properties are useful when showing convergence of the algorithm.
If the economy satisﬁes Assumptions 1–3, then the following properties
1. The function G Q → J is twice continuously differentiable.
2. For all q ∈ Q q · G q = 0
3. If q n n∈ is a sequence in Q converging to q ∈ Q or diverging, that is, q n 2 → then for all q ∈ Q q · G q n →. ( Q represents the relative boundary of Q.) ˜ ˜ Proof. See Hens (1991).
Our algorithm starts from some initial price system q 0 ∈ Q We will address the issue of choosing a sensible q 0 in §5. A price system q with the last component deleted is denoted by q in particular, q 0 speciﬁes the ﬁrst J − 1 components of the initial price system q 0 ˆ ˆ The function G with the last component deleted is denoted by G We propose to compute 0 1 × Q → J −1 deﬁned by equilibria in a ﬁnance economy by means of the homotopy
We are looking for solutions to t q = 0 If 1 q = 0 then; since qJ = 1 Lemma 3.1.2 implies that GJ q = 0 It follows that q is a competitive equilibrium price system.
4. Generic convergence. A homotopy is in general constructed in such a way that there is a unique solution to 0 q = 0 solutions to 1 q = 0 are solutions to the problem of interest; and the unique solution to 0 q = 0 is linked by a path of solutions to t q = 0 for varying t to one solution to 1 q = 0 By following this path, a solution to the problem of interest is found. When the unique solution to 0 q = 0 is indeed linked by a path to a solution to 1 q = 0 then the homotopy is said to converge.
For an excellent discussion on the numerical techniques available to follow the path we refer to Allgower and Georg (1990).
It cannot always be guaranteed that our homotopy converges. There may exist economies such that the set of solutions −1 0 does not link the unique solution to 0 q =0
COMPUTING EQUILIBRIA IN FINANCE ECONOMIES
It is also clear that q 0 − q n · q 0 − q n 0 for n sufﬁciently large. As a consequence, the ˆ ˆ ˆ ˆ right-hand side of Equation (1) is strictly positive for n sufﬁciently large, a contradiction.
Solutions to the homotopy equations stay away from 0 1 × Q and are not diverging, which shows 4.1.4.
To show 4.1.
1, notice that the continuity of yields that −1 0 is closed in 0 1 ×Q Since, moreover, solutions to the homotopy equations stay away from 0 1 × Q and are not diverging, it follows that −1 0 is compact.
The proof of 4.1.1 is completed by showing that qˆ 0 q and, generic in initial endowments, qˆ 1 q and t qˆ t q have full rank for points in −1 0 A simple calculation reveals that
5. Implementation. The speed of homotopy algorithms depends mainly on two factors, the number of equations and the arc length of the homotopy path. A quick comparison shows the great beneﬁts of developing a special purpose homotopy tailored to the ﬁnance GEI model. The homotopy algorithms as reported in Brown et al. (1996a) and Schmedders (1998) are designed to deal with the general GEI model with multiple commodities per state but can be applied to ﬁnance economies.
The homotopy proposed by Brown et al. (1996a) needs closed-form solutions for excess demand functions and should therefore be compared with our homotopy Applied to twoperiod ﬁnance economies, their algorithm has 2S equations, whereas ours only has J − 1 The algorithm of Schmedders (1998) does not require closed-form solutions for excess demand functions and also uses the ﬁrst-order conditions. The number of equations of his algorithm amounts to 2HS + H − 1 J + 1 + 1 whereas the number of equations in our algorithm ∗ equals H + 1 J + 1 − 2 In both cases, we roughly need a fraction J /2S only of the equations of alternative algorithms. This is especially favorable when S is high, as is, for instance, the case in any application where a continuous multivariate distribution of endowments and asset returns in Period 1 has to be approximated accurately by a discrete one. A speciﬁc example concerns our companion paper (Herings and Kubler 2000), where we need S = 32 768 states of nature to reasonably approximate a continuous multivariate distribution of the endowments of three agents and eight ﬁnancial assets.