Leonard J. Mirman

Optimal economic growth and uncertainty: The discounted case

The cornerstone of one-sector optimal economic growth models is the existence and stability of a steady-state solution for optimal consumption policies. The optimal consumption policy is the stable branch of the saddle point solution of the system of differential equations governing the dynamics of the economy. Examples of this type of behavior can be found in Cass [2] and Koopmans [4]. However, the stable branch solution is a knife-edge policy in the sense that any perturbation, no matter how small, results in instability and eventual annihilation. (This phenomenon is true when the Euler conditions are adhered to after the perturbation). Small pertubations might occur due to observation errors or the lack of knowledge of the exact production functions. It seems reasonable to expect that all sorts of human errors influence decision variables. Hence unless perfect knowledge of all variables, present and future, were known with complete certainty, and unless all decisions were made with exactness, economies of the one-sector deterministic type would lead at best to suboptimal consumption and investment policies. One would expect analogous instability results if production due to aggregation error, say, were not known with certainty. Moreover, it should be expected that planners know the future with something less than certainty. Since it seems impossible, in the face of uncertainty of the future,

Uncertainty and Optimal Consumption Decisions

linear production function, that for some utility functions the optimal initial consumption in the random case decreases for all values of initial wealth as compared with the initial consumption in the deterministic case. For other utility functions the optimal consumption always increases. Hence it seems, from these examples, that two divergent forces are at work. The first is the desire to consume more initially as a hedge against the uncertain future. The second force is the desire to consume less initially so as to increase the future consumption prospects. (It is assumed, of course, that increased inputs increase outputs for all possible random events, or states of the world). The relative strength of each of these forces, as implied by the utility function, is the key to the relationship between random consumption and deterministic consumption in this model. The major conclusion of this paper is that the qualitative difference between optimal consumption decisions in the two different models is very strongly influenced by the shape of the utility function. In particular the third derivative of the utility function plays a rather large role. It is this derivative that determines the attitude toward the skewness of a distribution in the theory of portfolio choices, as may be seen from the analysis of Pratt [7] and Tobin [10]. Even in these models, however, the third derivative cannot be ignored, since ignoring skewness distorts the results. Moreover, there does not seem to be any intuitive economic reason to make any assumptions concerning the third derivative of the utility function. The extent to which the utility function influences savings and consumption decisions is exhibited in a precise manner. It may be shown that the qualitative relationship between random and deterministic consumption depends in general on the initial wealth. It is not true, as one would infer from the papers cited above, that random consumption is always either greater than or less than deterministic consumption independently of the initial wealth. In other words, for many utility functions the initial wealth turns out to be a decisive factor in the qualitative relationship between the random and deterministic case. Naturally this relationship will also normally depend on -the probabilistic structure of the model. The key result of this paper is a theorem which gives a necessary and sufficient condition for determining the qualitative relationship between random consumption and deterministic consumption. This condition, which is both necessary and sufficient, is in a particularly simple form in that it depends only on the known parameters of the model (i.e., the production function, the utility function, and the distribution of the random variable) and also on the optimal deterministic policy which, in general, is much simpler to exhibit than its counterpart in the random case.

On optimal growth under uncertainty

Abstract In a recent paper Brock and Mirman showed that in a one-sector model of economic growth under uncertainty the long-run behavior of the optimal capital stock is governed by the basic properties of an acyclic ergodic Markov process. This paper considers a similar model and has two purposes. First, necessary and sufficient conditions for optimal policy functions are derived in a regime in which future utilities are discounted. These conditions lead, in an example, to an explicit optimal policy function, which is used to display the steady-state solution for the capital stock under an optimal policy. Secondly, in the Brock and Mirman paper it was assumed that the production functions are ordered. We show that all the properties proved by Brock and Mirman are satisfied even when the production functions are not ordered.

Demand Compatible Equitable Cost Sharing Prices

We propose here a new approach for equitable cost sharing pricing based upon the Shapley value for nonatomic games. However, it is shown that the proposed price mechanism can be justified on economic terms since it is uniquely determined by a set of axioms involving only cost functions and quantities consumed and not any notion of game theory. Moreover, taking into account the utilities of the consumers, one can prove the existence of an equilibrium under this price mechanism for a general class of cost functions. This approach has the advantage of not involving any interpersonal comparisons of utilities.

Optimal Economic Growth and Uncertainty: The No Discounting Case

The cornerstone of one-sector optimal economic growth models is the existence and stability of a steady-state solution for optimal consumption policies. The optimal consumption policy is the stable branch of the saddle point solution of the system of differential equations governing the dynamics of the economy. Examples of this type of behavior can be found in Cass [2] and Koopmans [4]. However, the stable branch solution is a knife-edge policy in the sense that any perturbation, no matter how small, results in instability and eventual annihilation. (This phenomenon is true when the Euler conditions are adhered to after the perturbation). Small pertubations might occur due to observation errors or the lack of knowledge of the exact production functions. It seems reasonable to expect that all sorts of human errors influence decision variables. Hence unless perfect knowledge of all variables, present and future, were known with complete certainty, and unless all decisions were made with exactness, economies of the one-sector deterministic type would lead at best to suboptimal consumption and investment policies. One would expect analogous instability results if production due to aggregation error, say, were not known with certainty. Moreover, it should be expected that planners know the future with something less than certainty. Since it seems impossible, in the face of uncertainty of the future,

Strategic dynamic interaction: Fish wars

Abstract The motivation for this paper is fishing legislation that divides a fishery into regions, each of which is exploited by a single firm. As the regions have common borders, an analysis based on models with one firm and one region would lead to erroneous results. An alternative motivation is the problem of a fishery with several interacting species, each exploited by a single firm. In this paper we assume that fish move between regions. Therefore, the stock in one region depends not only on previous stock and catch in the region, but also on the stock and catch in neighboring regions. In the multiple species interpretation, these movements correspond to the interactions between species. We use a multisectoral differential duopoly model in order to capture the strategic interactions of the firms over time. The model is solved for an example in which varying levels of dependence between regions can be compared.

Savings and Consumption with an Uncertain Horizon

This paper studies the effect of lifetime uncertainty on optimal consumption decisions. It is shown that for risk averters changing the distribution of lifetime uncertainty decreases consumption due to the higher probability of having a longer life and increases consumption due to the desire for sure consumption in the present. The stronger of these effects determines the effect of lifetime uncertainty on optimal consumption decisions. The major result is that if the utility function is Cobb-Douglas and the rate of return is not too large relative to the amount of future discounting then lifetime uncertainty will always increase consumption.

One-Sector Nonclassical Optimal Growth: Optimality Conditions and Comparative Dynamics

The authors consider a one-sector nonclassical model of optimal economic growth, characterized by a convex-concave production function. They provide, in a dynamic-programming context, a characterization of all local (interior) maximum of the miximand of the Bellman equation. These conditions are the Euler equation and a second order condition, namely, that the marginal propensity to consume is less than one. An example is used to illustrate these conditions. Also, several comparative dynamic results are derived. In particular, it is shown that the maximum and minimum selections out of the optimal consumption correspondence shift down as the discount factor increases. Copyright 1991 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.

Information and Market Equilibrium

Under the assumption of complete markets, a fundamental result of competitive market analysis (whether for speculative or commodity markets) is that prices contain all information necessary for optimal decisionmaking by individual economic units. The role that prices play in disseminating information is analyzed in the context of two different models. First an economy under uncertainty without complete markets is analyzed. Conditions are specified under which equilibrium prices reflect (or transmit) all available information to market observers. It is shown that uninformed market observers can deduce inside information about the environment from the change in the equilibrium price when there is a one-to-one correspondence between the market price and the useful part of the information received. For a special case, sufficient conditions for invertibility are derived. Then a Bayesian hypothesis is used to study the price expectations formed on the basis of information obtained about the economy from observations of past market prices. The Bayesian price expectations are shown to converge, as price observations accumulate, to the expectations of an observer who knows the true structure of the economy.

Existence and Uniqueness of Equilibrium in Distorted Dynamic Economies with Capital and Labor

In this paper, we provide a set of sufficient conditions under which recursive competitive equilibrium exist and are unique for a large class of distorted dynamic equilibrium models with capital and elastic labor supply. We develop a monotone map approach to the problem. The class of economies for which we are able to obtain our existence result is apparently considerably larger than those considered in previous work. Additionally unlike previous work, we are able to also prove that this equilibrium is unique. We conclude by applying the new results to some important examples of monetary economies often used in applied work.