微观经济学讲义(黄有光)4.docVIP

Advanced Microeconomics Topic 3: Consumer Demand Primary Readings: DL – Chapter 5; JR - Chapter 3; Varian, Chapters 7-9. 3.1 Marshallian Demand Functions Let X be the consumers consumption set and assume that the X = Rm+. For a given price vector p of commodities and the level of income y, the consumer tries to solve the following problem: max u(x) subject to p(x = y x ( X The function x(p, y) that solves the above problem is called the consumers demand function. It is also referred as the Marshallian demand function. Other commonly known names include Walrasian demand correspondence/function, ordinary demand functions, market demand functions, and money income demands. The binding property of the budget constraint at the optimal solution, i.e., p(x = y, is the Walras’ Law. It is easy to see that x(p, y) is homogeneous of degree 0 in p and y. Examples: (1) Cobb-Douglas Utility Function: From the example in the last lecture, the Marshallian demand functions are: where . (2) CES Utility Functions: Then the Marshallian demands are: where r = (/(( -1). And the corresponding indirect utility function is given by Let us derive these results. Note that the indirect utility function is the result of the utility maximization problem: Define the Lagrangian function: The FOCs are: Eliminating (, we get So the Marshallian demand functions are: with r = (/((-1). So the corresponding indirect utility function is given by: 3.2 Optimality Conditions for Consumer’s Problem First-Order Conditions The Lagrangian for the utility maximization problem can be written as L = u(x) - (( p(x - y). Then the first-order conditions for an interior solution are: (1) Rewriting the first set of conditions in (1) leads to which is a direct generalization of the tangency condition for two-commodity case. Sufficiency of First-Order Conditions Proposition: Suppose that u(x) is continuous and quasiconcave on Rm+, and that (p, y) 0. If u if differentiable at x*, and (x*,