1)(4pts) Optimal choice: example #1

Consider the utility function U(X,Y)=XY, which describes the enjoyment Emi gets from consuming tacos (x) and sandwiches (y) over a period of 1 week.

a)Does Emi like both tacos and sandwiches? Does she like variety?

b)Let Emi have budget I=$24, and let prices be Px=$2, Py=$4. Find Emi's optimal basket of goods x and y. Is this an interior or a corner solution?

c)What will happen if tacos go on sale for $1? Find the new optimal bundle.

d)Continuing from these prices, suppose that, in order to boost sales, the sandwich vendor introduces a discount card. Each week, every sandwich a consumer buys after the first 3 will be on a 50% off sale. (Purchase records are not transferred from week to week.) Draw the new budget constraint and express it algebraically. Will this change have an effect on Emi's purchase? Check how much of each good Emi will want to buy now.

e)Solve Emi's utility maximization problem for general parameters Px, Py and I, to find the demand functions x?(Px,Py,I) and y?(Px,Py,I), and indirect utility V(Px,Py,I).

Plug in prices and income from parts (b) and (c) to check your earlier numerical solutions.(Note that you can't use these functions to answer the question at part (d), since they implicitly assume that prices don't vary with the quantity bought)

f)Calculate the income and own-price elasticity of demand for good x, and well as the cross-price elasticity of demand for good x with respect to the price of good y.

Starting from I=$24, Px=$2, Py=$4:

g)Draw the income consumption curve. (You have to figure out what axes you need and which parameters can change.)

h)Draw the corresponding Engel curves for goods x and y (Once again, pay attention to the axes.)

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