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Erzo G.J. Luttmer Department of Economics University of Minnesota Spring 2011 1. Introduction These notes begin with some ground work on a convenient utility function that will often be used in this course. Then we will use this utility function to give an interpretation of what is known as the gravity equation of international trade. A key feature of the utility function discussed below is that it is homothetic: all indi/erence curves have the same slope along any given ray through the origin. In turn, this implies a very strong aggregation result: taking income from one consumer and giving it to another has no e/ect on the aggregate demand for the various goods in the economy. As a result, one might as well assume that there is only one consumer who earns all income in the economy. The actual economy under consideration may have millions of consumers with very di/erent levels of income, but to compute aggregate demand curves it su¢ ces to know aggregate income. The distribution of income is irrelevant. This aggregation property will be used in many places. It is not always essential, but it certainly tends to sharpen conclusions and simplify proofs. 2. A Convenient Utility Function Suppose there are N 2 N goods and consider a consumer with preferences given by the utility function U ( c 1 ;c 2 ;:::;c N ) = N X n =1 1 =" n c 1 1 =" n 1 1 1 =" ! (1) where " and the n are positive parameters that will be interpreted below. Consider a consumer with income x facing prices ( p 1 ;:::;p N ) , both denominated in some arbitrary numeraire. The consumer solves max f c n g N n =1 ( U ( c 1 ;:::;c N ) : N X n =1 p n c n ± x ) : (2) 1

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That is, the consumer maximizes utility subject to a budget constraint. The solution to this problem is determined by the requirement that expenditures satisfy the budget constraint, and that marginal utility of good n 0 marginal utility of good n = p n 0 p n holds for all possible n and n 0 . The marginal rates of substitution must equal the relevant price ratios. Another way to say this is marginal utility of good n = n for all n = 1 ;:::;N , where is some positive multiplier that does not depend on n . This is the Lagrange multiplier on the budget constraint. Taking a derivative of the utility function with respect to consumption of good n shows that the marginal utility of consumption of good n is D n U ( c 1 ;:::;c N ) = ± n c n ± 1 =" : (3) The optimal consumption choices are therefore determined by ± n c n ± 1 =" = n , n = 1 ;:::;N , (4) and the budget constraint N X n =1 p n c n = x (5) The optimality conditions (4) and (5) gives us N + 1 equations in the N + 1 unknowns ( c 1 ;:::;c N ) . To solve these equations, write the marginal conditions (4) as
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