Knowledge that Transforms

[We introduce and analyze a class of variable elasticity of substitution (VES) production functions for which the substitution parameter varies linearly with the capital-labor ratio around the intercept term of unity. The VES function contains as special cases the more important special cases of the well known CES function. In terms of some familiar economic relationships, the VES posits a linear view of the world in contrast to the log-linear view posited by the CES function.]

A model is considered in which a covariance-stationary exogenous process is related to an endogenous process by an unrestricted, infinite, linear distributed lag. It is shown that when an underlying continuous time model is sampled at unit intervals to yield endogenous and exogenous discrete time processes, the discrete time processes are related by a discrete time equivalent of the underlying continuous model. The relationship between the underlying continuous lag distribution and its discrete time equivalent is close when the exogenous process is smooth. Even then, however, it is interesting to note that (i) a monotone continuous time distribution does not in general have a monotone discrete time equivalent and (ii) a one-sided continuous time distribution does not in general have a one-sided discrete time equivalent. The implications of the results for statistical practice are considered in the latter part of the paper.

A typical proof of the existence of a perfectly competitive market equilibrium employs an appropriately continuous price-to-price mapping that depends on excess demands. If trade tax-subsidy distortions are introduced into the model, the excess demand mappings may have disconnected image sets and destroy the continuity of the price-to-price mapping. This difficulty is overcome by developing a technique which explicitly takes account of the dependence of demand on both income and prices and simultaneously solves for equilibrium prices and income levels for all agents. This technique is then applied to establish two existence theorems for models of international trade with trade tax-subsidy distortions.

[Consider a Bayesian decision problem in which F is the prior distribution over some parameter space T. If —ψ(d, t) is the product of the loss function and the likelihood function, then the Bayesian solution, d_F, maximizes extlesstex-math extgreater$E_\F\(d)= extbackslashint _\T\ extbackslashpsi (d,t)dF(t)$ extless/tex-math extgreater. Suppose \F^n\ is a sequence of distribution functions that approach F^0 in the sup-metric topology. Our main theorem gives conditions under which extlesstex-math extgreater$d_\F extasciicircum \\ extbackslashrightarrow d_\F extasciicircum\0$ extless/tex-math extgreater and extlesstex-math extgreater$E_\F extasciicircum\0\\(d_\F extasciicircum \\) extbackslashrightarrow E_\F extasciicircum\0\\(d_\F extasciicircum\0\\)$ extless/tex-math extgreater.]

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.