Question

Sally consumes two goods, pizza (P) and hamburgers (H). Her utility function is given by the expression U = 3PH². The current markiet price of pizza is $10, while the market price of hamburgers is $5. Sally's current income is $500.

a. Roughly sketch a set of two indifference curves for Sally.

b. Write an expression for Sally's budget constraint. Graph the budget constraint and determine its slope.

c. Determine the combination of pizza and hamburgers that maximizes Sally's utility, given her budget constraint.

d. Calculate the impact on Sally's optimum consumption bundle of an increase in the price of pizza to $10. What would happen to her utility as a result of the price increase?

Answer 1

U = 3PH²

P_P = 10

P_H = 5

Income (M) = 500

(a)

(b) Budget constraint:

P_P P + P_H H = M

(10)P + 5 (H) = 500

When P = 0, then H = 100.

When H=0, then P= 50.

And slope of the Budget constraint = P_P /P_H = 10/5 = 2.

By putting these values , we get the budget constraint as shgown in the figure above.

(c) Now, to find the optimal bundle , calculate marginal utilities. By taking the partial derivative of U with respect to P or H , we get MU_P or MU_H respectively.

MU_P = 3H²

MU_H = 6PH

MRS = MU_P/ MU_H

At optimal , MRS = P_P /P_H

3H² / 6PH = 10/5

H /2P = 2

H = 4P

Now, by putting H=4P in budget costraint , we get the optimal values:

10 P + 5(4P) = 500

10 P + 20 P = 500

30 P = 500

P = 16.67

H = 4(16.67) =66.68

Therefore, sally consumes 16.67 pizza and 66.68 hamburgers to maximise her utility.

(d) The price of pizza is already $10. Therefore, there is no change in optimal bundle.

If there is change in the price of pzza other than $10 or there is change in the price of hamburgers then, you can infer that value in budget constraint and see the change in optimal bundle.