### Find the indirect utility function

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1. Suppose U(x, y) = x + y

a. Find the condition for tangency of the budget constraint to an indifference curve.

b. Suppose Px = 6, Py = 6 and income is I. What maximizes the consumer's utility?

c. Suppose Px = 6, Py = 7 and income is I. What is the utility maximizing bundle?

d. What expenditure is required to achieve a utility of 10? How would your answer change if Px= 8?

e. Find the indirect utility function(s).

f. Find the expenditure function(s).

2. Faiyaz will live and earn income for two periods. He consumes two goods, y which is not addictive and c which is addictive. In the first period, his utility will be given by ciyi, but in the second period, first period consumption of the addictive good will raise the utility of consuming c in the second period, so that his utility in the second period is given by ycic2 + c2y2 where y >= 0. For simplicity, assume no discounting of future utility, an interest rate of zero, and that prices of the goods, Pc and Py, do not change between periods.

a. Set up one Lagrangian for utility maximization over the two periods and find the first order conditions for utility maximization.

b. What's the relationship between ci and c2?

c. What's the relationship between yi andy2?

d. Assuming an interior solution, does the size of the addiction parameter, y, affect the MRS at the utility maximizing bundle? If so, how?

e. Does the magnitude of the addiction parameter, y, affect the composition of the utility maximizing bundle? If so, how? (To check your expression, note that if y = 0, your demand equations should collapse to yi = y2 = I/4Py and ci = c2 = I/4Pc).

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