EconS 527 Homework 3 Answer Key. Consider a consumer with Cobb-Douglas utility function

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1 EconS 527 Homework 3 Answer Key 1. Consider a consumer with Cobb-Douglas utility function over two goods a) Find the Walrasian demand function for goods and. The Lagrangian of this UMP is then The first order conditions are Solving for on both first order conditions, we obtain which solving for yields Using now the budget constraint (which is binding), we have and plugging this expression of into the above equality, yields the Walrasian demand for good 2 and similarly solving for, we obtain the Walrasian demand for good 1, Hence, the Walrasian demand function is which, as usual for the Cobb-Douglas utility function, represents that the consumer spends a fraction of his wealth on each good, i.e. for good 1 and for good 2. b) What restrictions on would ensure that the consumer demands positive amounts of the goods (interior solution)?

2 In order to have a positive demand for the first good, we just need, since by definition. On the other hand, a positive demand for the second good, is satisfied if and only if, that is, if. Combining both conditions, we can guarantee that the consumer demands non-negative amounts of both goods when 2. Let denote the number of phone calls, and denote spending on other goods. The expression of the budge line under Plan A, is, or, as depicted in the solid line of the following figure that originates at and which crosses the horizontal axis at. Under Plan B, Tyler s budget line,, is, or, as illustrated in the figure by the dashed line that originates at and crosses the horizontal axis at. These two budge lines intersect each other at, i.e.,. Hence, Therefore, and intersect at bundle (66.67, 26.67)

3 3. a) Increasing prices and wealth by a common factor, we obtain That is, increasing both prices and wealth by the same factor does not change this consumer s demand. Intuitively, if we double the price of all change goods but also double his income, the individual s demand is unaffected. b) Recall that Walras Law states that for a strictly positive price vector ( and a positive wealth level ( ),, or alternatively,. Hence, in this context, and further rearranging, we obtain Therefore, Walras Law is satisfied, confirming that the individual spends all his income on goods 1, 2 and 3. c) Let us use a counterexample, which yields a demand of which yields a demand of We know that WARP is satisfied if for any pair of prices and wealth ( ) and ( and then In our example, the bundle that the consumer selects at the final price wealth pair is affordable under initial prices and wealth, (since )

4 However, the consumption bundle at initial prices and wealth, and wealth. In particular,, is affordable under final prices Hence, since, bundle is exactly affordable at final prices and wealth, implying that the conclusion of WARP, is not satisfied. Therefore, WARP is violated. d) Let us first recall the Slutsky matrix: where every component is defined as (Slutsky equation) Hence, Slutsky equation informs about what is the change in the demand for good after the price of good varies, once the consumer s wealth is appropriately compensated. Let us now find each of the components of the Slutsky matrix for this particular exercise. Therefore, the Slutsky matrix is

5 4. a) The consumer solves a UMP given by max subject to Using the shortcut, we obtain interior solutions, or, which together with budget constraint yields a Walrasian demand of for every good (1) In addition, we can obtain the Lagrange multiplier,, from the first-order condition, or which, combined with (1) yields and solving for we obtain Hence, the marginal value of relaxing the constraint (i.e., the shadow price of wealth) is. b) The indirect utility function is Hence, the marginal utility of wealth is

6 Interestingly, this result is generalizable to settings in which, given the separable nature of the utility function, the consumer focuses on a subset of goods where, { and solves a separated UMP for each of these subsets of goods, i.e., one UMP for goods, another UMP for goods, etc.. The consumer s solution to these separated UMPs must coincide with that in part (a), where the consumer simultaneously considers all goods.