تعادل عمومی با شرکت کنندگان ناهمگن و زمان مصرف گسسته

The paper investigates the term structure of interest rates imposed by equilibrium in a production economy consisting of participants with heterogeneous preferences. Consumption is restricted to an arbitrary number of discrete times. The paper contains an exact solution to market equilibrium and provides an explicit constructive algorithm for determining the state price density process. The convergence of the algorithm is proven. Interest rates and their behavior are given as a function of economic variables.

Interest rates are determined by the equilibrium of supply and demand. Increased demand for credit brings interest rates higher, while an increase in demand for fixed income investment causes rates to go down. To determine the mechanism by which economic forces and investors' preferences cause changes in supply and demand, it is necessary to develop a general equilibrium model of the economy. Such model provides a means of quantitative analysis of how economic conditions and scenarios affect interest rates. Vasicek (2005) investigates an economy in continuous time with production subject to uncertain technological changes described by a state variable. Consumption is assumed to be in continuous time, with each investor maximizing the expected utility from lifetime consumption. The participants have constant relative risk aversion, with different degrees of risk aversion and different time preference functions. After identifying the optimal investment and consumption strategies, the paper derives conditions for equilibrium and provides a description of interest rates. For a meaningful economic analysis, it is essential that a general equilibrium model allows heterogeneous participants. If all participants have identical preferences, then they will all hold the same portfolio. Since 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 all 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 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 can be avoided if the aggregate consumption can be expressed as a function of a Markov process, in which case only this Markov process becomes a state variable. This is often simple in models of pure exchange economies, where the aggregate consumption is exogenously specified. The situation is different in models of production economies. In such economies, the aggregate consumption depends on the social welfare function weights. Because these weights are determined endogenously, it is necessary that the individual consumption levels themselves be functions of a Markov process. This has 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 above approach is exploited here. Vasicek (2005) 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. In Vasicek (2005), the equilibrium conditions are used to derive a nonlinear partial differential equation whose solution determines the term structure of interest rates. While the solution to the equation can be approximated by numerical methods, the nonlinearity of the equation could present some difficulties. The present paper provides the exact solution for the case that consumption takes place at a finite number of discrete times. This solution does not require solving partial differential equations, and explicit computational procedure is provided. If the time points are chosen to be dense enough, the discrete case will approximate the continuous case with the desired precision. Some may in fact argue that, in reality, consumption is discrete rather than continuous, and therefore the discrete case addressed here is the more relevant. The following section of this paper summarizes the relevant results from Vasicek (2005). Section 3 contains the solution for the equilibrium state price density process and the structure of interest rates in the discrete consumption case. Section 4 gives a proof that the proposed algorithm converges to the market equilibrium.

This paper provides explicit procedure to obtain the exact solution of equilibrium pricing in a production economy with heterogeneous investors. Each investor maximizes the expected utility from lifetime consumption, taking place at discrete times. Interest rates are determined by economic variables such as the characteristics of the production process, the individual investors' preferences, and the wealth distribution across the participants. Such model provides a tool for quantitative study of the effect of changes in economic conditions on interest rates. The algorithm is constructive and converges to the equilibrium solution. The convergence is proven for the case of γ k≥1, k =1, 2,…, n , for which the uniqueness of the equilibrium has been established (cf. Karatzas and Shreve, 1998). All other steps of the procedure, however, are valid in general for any positive values of the risk tolerance coefficients. If some of the γ 1, γ 2,…, γ n are smaller than unity and the values View the MathML sourceWk(j)(0) fail to converge to the input values Wk(0), k=1, 2,…, n after a reasonable number of iterations, a search over the space of positive values of ν1, ν2,…, νn need to be made. While this paper concentrates on the case that the participants have isoelastic utility functions (4), it can be extended to more general class of utilities. Suppose the k –th investor maximizes the objective (24), where U k(C ) has a positive, decreasing continuous derivative View the MathML sourceUk′(C) with U′k(0)=∞U′k(0)=∞, View the MathML sourceUk′(∞)=0, k =1, 2,…, n . Denote the inverse of the derivative by View the MathML sourceIk(x)=Uk′−1(x). Then the optimal consumption is given by equation(85) View the MathML sourceCik=Ik(YiΛkpik), Turn MathJax on where Λk is a positive constant satisfying the condition equation(86) View the MathML sourceWk(0)=1Y0E∑i=1mYiIk(YiΛkpik) Turn MathJax on for k=1, 2,…, n (cf. Karatzas and Shreve, 1998, Theorems 3.6.3 and 4.4.5). Put equation(87) View the MathML sourceKi(Y)=∑k=1nIk(YΛkpik)i=1,2,...,m. Turn MathJax on Then Eqs. (30), (31) and (33) through (40) still hold. The algorithm consisting of making an initial choice of the constants Λ1, Λ2,…, Λn, determining Y0, Y1,…, Ym from Eqs. (39) and (31), setting new values of the constants from Eq. (86), and repeating the calculations may still be applicable, although a proof of convergence is not provided here.