Question

Petra consumes only two goods, pizza (P) and hamburgers (H) and considers them to be perfect substitutes, as shown by his utility function: U(P, H) = P + 4H. The price of pizza is $3 and the price of hamburgers is $6, and Paul’s monthly income is $300.

a. Write the equation for Petra’s budget line.

b. If hamburgers is on the vertical axis, what is the slope of the budget line?

c. Graph Petra’s budget line. Place the hamburgers servings on the vertical axis and pizza on the horizontal axis. Make sure to indicate the values of where the budget line hits each axis.

d. On the same graph, draw several of Petra’s indifference curves, including one that shows where Petra will maximize his utility. Make sure to clearly indicate which indifference curve that maximizes utility.

e. Petra is a utility maximizer. Write down the full optimization problem with the objective function and the constraint.

f. Solve for the values of P and H that maximizes Petra’s utility

PLEASE SOLVE F TOO!!!!!

Answer 1

Petra consumes only two goods, pizza (P) and hamburgers (H) with a utility function U(P, H) = P + 4H. Here MRS = -MUP/MUH = -1/4.

The price of pizza P1 is $3 and the price of hamburgers P2 is $6, and Paul’s monthly income M is $300.

a. Write the equation for Petra’s budget line.

This is given by M = P*P1 + H*P2

300 = 3P + 6H

b. If hamburgers is on the vertical axis, the slope of the budget line is -1/2. This is because slope is -P1/P2 and we have P1 = 3 and P2 = 6.

c. Graph of Petra’s budget line is shown below where hamburgers servings are on the vertical axis and pizza on the horizontal axis. If all the income is spent on pizza, the horizontal intercept is 300/3 = 100. Similarly the vertical intercept is 300/6 = 50.

d. There are several of Petra’s indifference curves, including one that shows where Petra will maximize his utility by U1.

e. Petra is a utility maximizer. Maximize the utility as

MAX U = P + 4H - λ(3000 - 3P - 6H)

f. From e we find partial derivatives and see that

1 - 3λ = 0, 4 - 6λ = 0 and 3P + 6H = 3000

From the first two equation we don't get anything concrete so we see that there is no interior solution. We now look at MRS and price ratio. |MRS| = 1/4 and P1/P2 = 1/2. Since MRS < P1/P2, Petra is consuming only Hamburgers so the optimal bundle is at the corner and it is 300/6 = 50  Hamburgers