Trout Rader

PREFERENCES WHICH HAVE OPEN GRAPHS

Gale and Mas-Collel and Shafer and Sonnenschein introduced powerful new theorems on the existence of competitive equilibrium which used the assumption that the graph of the strict preference relation P is an open set. We show when this condition is satisfied and when it is not.

Utility over time: The homothetic case

The purpose of this paper is to show how the economics of choice over time can be strengthened by introduction of restrictive assumptions, namely, additivity and homotheticity of utility. We consider applications to labor SUPPlY 2 savings behavior, and long-run distribution and welfare measurement. Labor supply is mainly a function of the ratio of non-labor income to the wage rate. Savings generally increase with increases in the rate of interest. In the long run, distribution is not much dependent upon initial wealth, and instead is mainly determined by attitudes towards time. In the long run, the “as-if” social utility function is often well measured by consumer surplus. The applications appear to be strong. Therefore, it is appropriate to begin with a general argument for the main assumptions to be imposed. Two early contributions to the economics of time are by Becker [ 1 ] and Koopmans [ 111. The former presents a general framework without any specific assumptions about utility over time. At most, there is a homogeneous technology to produce consumption from goods and time (Pollak and Wachter [ 131). Koopmans [ 1 l] adds highly intuitive assumptions to show utility additive and discounted over time. Koopmans’ analysis could be extended to cases where utility is additive merely between remote time periods (see Rader [ 15, pp. 161-1851). This would correspond to the realistic case where the influence of distant consumption on preferences is negligible. Additivity must be reinterpreted in terms of a shifting environment of current consumption plus consumption sufficiently contiguous in time. The analysis following could be adapted to this most compelling case. Merely adding utility over time does have implications for demand, for example, if complementaries occur between two goods they must occur between many others (e.g., [ 15, pp. 228-2331). However, the results are meager and insufficient for the kind of analysis economists generally want to apply. Therefore, we are led to add the assumption of homotheticity, namely, 219 0022.0531/81/050219-18

Nice demand functions - II

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Factor Price Equalization with More Industries Than Factors

For utility n times differentiable, demand is shown to be n −1 times differentiable almost everywhere. If utility is analytic, then demand is analytic almost everywhere and also satisfies condition ( N ).

6 – Classical Dynamics

Publisher Summary This chapter discusses factor price equalization with more industries than factors. The local version of factor price equalization is that small changes in factor endowments leave factor prices unchanged, given that goods prices are fixed. Factor prices are locally constant as functions of factor endowments. A global version is that two countries whose factor prices are locally constant and that face the same international prices and have the same technology also have exactly the same factor prices. Factor price equalization, both local and global, depends critically on the comparison of the number of industries and the number of factors. For example, on the negative side, if the number of factors exceeds the number of goods, small changes in factor endowments will change factor prices, and local factor price equalization does not hold (Samuelson, Diewert and Woodland, and Jones and Scheinkman). On the positive side, both local and global factor price equalization results have been shown for the cases where there are exactly as many factors as industries. A big difference between the cases of equal and unequal numbers of industries and factors is that in the former case, the cone of factor price equalization is unique, whereas it need not be in the latter, more realistic, case.

7 – Adjustment Dynamics

Publisher Summary This chapter focuses on classical dynamics. Classical dynamics is usually viewed as the analysis of the economy over time, assuming that the economy is always in equilibrium. The chapter discusses conditions under which the economy tends to the kinds of limiting states postulated in long-run Ricardian analysis. Related to the Malthusian theory is the result that over time in an economy with Koopmans-type consumers, the families with the lowest rate of discount will eventually dominate. Some powerful assumptions must be made to get theorems about dynamics in large countries. However, for small countries, some general points can be made without any special assumption except that international prices have a time path. The chapter shows that over time as capital accumulates, factor prices will pass through regions of factor price nonequalization. The tariffs may arrest the decline in capital returns and encourage growth. Varying reproduction along Malthusian lines leads to the numerical dominance by the most myopic- type and least self-sacrificing type of consumer, who may nonetheless be at a level near serfdom, depending on how severe the myopia is.

4 – General Equilibrium

Publisher Summary This chapter focuses on adjustment dynamics. A powerful theory of disequilibrium economics is of great importance in two ways. First, if it shows the stability of equilibrium, then there is a method of finding equilibrium—simply set up the disequilibrium system and trace its path to equilibrium. Second, most economic variables are likely to be in at least slight disequilibrium; therefore, a more realistic theory would be forthcoming relative to the macrostatics ordinarily used by applied economists. Several different price adjustment systems are reviewed in the chapter. First, the systems independent of asset accumulation, both Walrasian and Keynesian, are discussed. The analysis is appropriate to a very short run, for example, a month. The chapter also discusses asset accumulation but without growth in productive capital. This applies to intermediate-run adjustments, perhaps, over six months. The chapter also shows how Marshallian- type and Keynesian-type systems incorporate constraints on the Walrasian-type system.

The Existence of a Utility Function to Represent Preferences

Publisher Summary The existence of equilibrium is of importance to both normative and positive economics. From a positive viewpoint, theories of the economy are often equilibrium theories, and therefore, it is absolutely necessary to know that the equilibrium exists. In proving the existence of equilibrium, various convexity assumptions are used, just as in welfare economics. Therefore, the generality of positive economics theory is not great as it stands, especially insofar as it assumes competition. The chapter presents conditions for the continuity and convexity of excess demand. These properties are used to show competitive equilibrium. The chapter also discusses the integration of monopolies, oligopolies, and other noncompetitive individuals into the economy. Also, the local uniqueness of equilibrium is shown for almost all wealths. The nature of prices over infinite time remains a problem. Minkowskis theorem is proved by a highly abstract construction involving an uncountable algebraic base and the Axiom of Choice.