Oldrich Alfons Vasicek
General equilibrium with heterogeneous participants and discrete consumption times
Journal of Financial Economics | Volume 108, Issue 3 (Jun 2013), 608–614
Vasicek (JFE 2013) presents a computable general equilibrium model that relates interest rates to the underlying economic variables such as the risks and returns of real production, the risk tolerances and time preferences of the investors, and the distribution of wealth in the economy. The model provides a means of quantitative analysis of how economic conditions and scenarios affect interest rates.

The continuous-time economy includes risky production subject to uncertain technological changes. Consumption takes place at a finite number of discrete times. Each investor maximizes the expected utility from lifetime consumption. The participants have constant relative risk aversion, with different degrees of risk aversion and different time preference functions.

An equilibrium with heterogeneity

For a meaningful economic analysis, it is essential that a general equilibrium model allows heterogeneous participants. If all participants have identical preferences, then they all hold the same portfolio. Because there is no borrowing and lending in the aggregate, there is no net holding of debt securities by any participant, and no investor is exposed to interest rate risk. Moreover, if the utility functions are the same, it does not allow for study of how interest rates depend on differences in investors' preferences.

The main difficulty in developing a general equilibrium model of production economies with heterogeneous participants had been the need to carry the individual wealth levels as state variables, because the equilibrium depends on the distribution of wealth across the participants. This had precluded an analysis of equilibrium in a production economy with any meaningful number of participants; most explicit results for production economies had previously been limited to models with one or two participants.

The paper shows that the individual wealth levels can be represented as functions of a single process, which is jointly Markov with the technology state variable. This allows construction of equilibrium models with just two state variables, regardless of the number of participants in the economy.

Consider a continuous time economy with n participants endowed with initial wealth Wk(0), k = 1, 2,…, n. It is assumed that investors can issue and buy any derivatives of any of the assets and securities in the economy. The investors can lend and borrow among themselves, either at a floating short rate or by issuing and buying term bonds. The resultant market is complete. It is further assumed that there are no transaction costs and no taxes or other forms of redistribution of social wealth. The investment wealth and asset values are measured in terms of a medium of exchange that cannot be stored unless invested in the production process.

The economy contains a production process whose rate of return dA/A on investment is   frac{dA}{A} =  mu sdot text{dt} +  sigma sdot text{dy} ,;/var/tmp/iawltxhtml/mathcache//udisplaymathf1c1e69a74083e4ec92d17d492d904f1.svg where y(t) is a Wiener process. The process A(t) represents a constant return-to-scale production opportunity. The amount of investment in production is determined endogenously.

The parameters of the production process can themselves be stochastic, reflecting the fact that production technology evolves in an unpredictable manner. It is assumed that their behavior is driven by a Markov state variable X(t), μ = μ(X(t),t), σ = σ(X(t),t). The state variable can be interpreted as representing the state of the production technology. The process X(t), which can be a vector, may be correlated with the production process A(t).

The consumption is restricted to a set of specific discrete dates t1 < t2 < ... < tm = T. The economy exists in continuous time, and between the consumption dates the participants are continuously trading and production is continuous. Each investor maximizes the expected utility from lifetime consumption,   max E  sum_{i=1}^m p_{i,k}  sdot U_{k}(C_{i,k}) , ;/var/tmp/iawltxhtml/mathcache//udisplaymath33782a99b1f2dc1126568c52355b2a1d.svg where Ci,k is the consumption at time ti and Uk is a utility function given by  U_k(C) =  frac{C^{( gamma_k-1)/ gamma_k}}{ gamma_k-1} ;/var/tmp/iawltxhtml/mathcache//udisplaymathbd4919078b28a8189a4db3df17e4fb29.svg if 0<γk, γk≠1 and log(C) when γk = 1.

An economy cannot be in equilibrium if arbitrage opportunities exist. A necessary and sufficient condition for absence of arbitrage is that there exist a process Y(t), called the state price density process, such that the price P of any asset in the economy satisfies the equation  P(t) = E_t  left[P(s) sdot frac{Y(s)}{Y(t)}  right] .;/var/tmp/iawltxhtml/mathcache//udisplaymathcd2649a797dd5261841e4a15ad57789d.svg

Equilibrium is fully described by specification of the process Y(t), which determines the pricing of all assets in the economy, such as bonds and derivative contracts. Bond prices in turn determine the term structure of interest rates. The state price density process also determines each participant's optimum investment strategy by means of the formula for Wk(t) below. Solving for the equilibrium means solving for the process Y(t).

A simple solution and algorithm

The paper shows that the optimal consumption of the k-th investor is a function of his own preference parameters and the state price density process only, given by  c_{i,k} = v_{k}  sdot p_{i,k}^{ gamma_k}  sdot Y^{- gamma_k}(t_i)  ;/var/tmp/iawltxhtml/mathcache//udisplaymath2cf13dbae9a44756da6f11ad010cbe4e.svg for i = 1, 2, ..., m, k = 1, 2, ..., n, where vk is a constant determined by the initial wealth Wk(0). The investor's wealth Wk(t) at time t under an optimal investment and consumption strategy is  W_k(t) = v_k  sdot  frac{1}{Y(t)}  sdot E_t  left[ sum_{i:t_i t} p_{i,k}^{ gamma_k} Y^{- gamma_k+1} (t_i)  right]. ;/var/tmp/iawltxhtml/mathcache//udisplaymathe977b53cb5d6c3849c21b6a71a31d497.svg

In equilibrium, the total wealth W(t) =  sum W_k(t);/var/tmp/iawltxhtml/mathcache//math8120bed296accb202150644e695d78a8.svg must be invested in the production process (the market portfolio). Any lending and borrowing, including lending and borrowing implicit in issuing and buying contingent claims, is among the participants in the economy, and its sum must be zero. This requirement produces equations for the state price density process Y(t).

These equations have a unique solution for Y(t1), Y(t2), …, Y(tm) in the form Y(ti) = Fi[ Y(ti – 1), A(ti – 1), X(ti – 1), A(ti), X(ti) ] , where Fi for i = 1, 2, …, m are functions determined by the algorithm. These variables in turn specify the state price density process Y(t) in continuous time. The algorithm requires no more complicated mathematical tools than finding the root of a monotone function. This represents the exact solution to the equilibrium economy, provided the initial wealth distribution is as specified.

To determine the values of the constants v1, v2,…, vn, our paper utilizes the fact that any choice of the constants is consistent with a unique equilibrium described by the process Y(t), except that the initial wealth levels Wk'(0) obtained from the equation for the optimal strategy for t = 0 do not agree with the given initial values Wk(0). Repeatedly replacing vk by vk ⋅ Wk(0)/Wk'(0) and recalculating Y(t) converges to the required equilibrium. The proof of convergence is given in the paper.

While our paper concentrates on the case that the participants have iso-elastic utility functions, the approach can be extended to more general class of utility functions.

Hieronymus Bosch (1450–1516): The Conjurer. Flemish, 1502.. “No one is so much a fool as a willful fool.” This is the proverb depicted in this Bosch painting. It shows how people are fooled by their lack of alertness and insight. If it was today, could Bosch have painted the subprime crisis from the bankers' point of view?





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