Question

Section 3: Utility Maximization

Dave receives utility from buying clothes and grocery items (At the ABC shop). His budget is $1000. The ABC shop is offering a $200 coupon if the amount spent on groceries is equal or greaterthan $520. Let Y be the quantity of units of clothes consumed with a price of $1 per unit and X thequantity of grocery items consumed with a price of $2 per unit/item. His utility function is:

Y

U (X, Y) = 0.5lnY +0.5lnX

A

a. Find the optimal allocation of groceries (X) and clothes (Y) for Dave if his budgetis $1000.

b. Find the optimal allocation of groceries (X) and clothes (Y) if his budget is $1200.

c. Based on your answers on part a and b, Could you tell what would Daveoptimal allocation be if he receives the offer from ABC shop?

5. (Mark 10) Assume that instead of receiving an offer the ABC shop. Dave receives $200 on clothes from his grandmother (So he is forced to consume this amount of clothes) How would thisalter Dave’s budget? Draw the budget line an find the optimal allocation of groceries (X) and clothes (Y) in this case.

6. (Mark 5) Out of the three possible allocations, which one gives Dave the highest utility? Is it feasible?

Answer 1

a. The consumer's problem is:

At equilibrium, marginal rate of substitution is equal to the price ratio:

Substituting this value into the budget constraint:

b. The consumer's problem is:

At equilibrium, marginal rate of substitution is equal to the price ratio:

Substituting this value into the budget constraint:

c. Expenditure on grocery in part a is . Expenditure on grocery in part b is

According to the scheme mentioned in the question, the $200 coupon will only be provided if the consumer spends more than $520 on groceries with a budget of $1000. Since this is not the case in part a, the optimal allocation when the consumer receives the offer will be the answer in part a, which is (250,500).

The budget line and optimal choice of the consumer when the scheme is offered is shown in the figure below:

Since the consumer does not spend $520 on groceries with a budget of $1000, he does not get the $200 coupon.

d. When the extra $200 received can only be spent on clothes, the consumer's budget line is as follows:

Plotting the budget line:

The consumer's problem is:

At equilibrium, marginal rate of substitution is equal to the price ratio:

Substituting this value into the consumer's budget constraint: