Question

Suppose that your tastes over coke(X₁) and burgers(X₂) can be summarized by the utility function: u(X₁, X₂) = (X₁²X₂)^1/3.

(a) Calculate the optimal quantity of coke and burgers' consumption as a function of P₁, P₂ and I.

(b) Illustrate the optimal bundle A when P₁ = 2, P₂ = 10 and weekly income I = 180. What numerical label does this utility function assign to the indifference curve that contains bundle A?

(c) Using your answer, show that both coke and burgers are normal goods when your tastes can be summarized by this utility function.

(d) Suppose the price of coke goes up to $4. Illustrate your new optimal bundle and label it, C.

(e) How much coke and burgers would you buy if you had received just enough of a raise to keep you just as happy after the increase in the price of coke as you were before(at your optimal income of $180)? Illustrate this as bundle B.

(f) How large was your salary increase in the previous part?

(g) Now suppose the price of burgers (P₂) falls to $5 (and suppose the price of coke and your income are $2 and $180 as they were originally at bundle A). Illustrate your original budget, your new budget, the original optimum, A, and the new optimum, C, in a graph.

(h) Calculate the income effect and the substitution effect (Hicksian) for both burgers and coke consumption from this change in the price of burgers. Illustrate this in your graph.

(i) True or False: Since income and substitution effects point in opposite directions for coke, coke must be an inferior good. Explain.

Answer 1

Utility function is given by:

(a)

Lets Prices be P₁ and P₂ for x₁ and x₂ respectively and Income be I

Budget line will be given by the following equation:

Lets set up the utility maximization problem:

Lets write first order constiditons to arrive at the optimal x₁ and x₂

FONCs

From 1st and 2nd FONC:

we get,

Using the Budget constraint:

we get

b)

Utility from the bundle will be 27.84

(c)

Consumption of both x₁ and x₂ increases as income increases. Hence, we can say that Both x₁ and x₂ are normal goods.

(d)

Now,

Graph for part B and D