Question

a, A consumer has $150 to spend on two types of cheese: French cheese (C_f) at $10 per lb and a less expensive California cheese (C_c) priced at $2. How much of each cheese should she purchase if her utility is:

U(C_F, C_C) = C_F^2/3 C_C^1/3

b. When she arrived at the cheese store, she discovered that the price of the French cheese had fallen to $5 per lb. If the price of the California cheese remains at $2 per lb, how much of each cheese should she purchase to maximize utility under these changed conditions?

c. Explain why this consumer is better off in part (b) than in part (a). How would you put a monetary value on this utility increase?

Answer 1

a. We are given the utility function as: U(C_F, C_C) = C_F^2/3 C_C^1/3

Firstly we find the marginal utilities by partially differentiating the utility function with C_F &C_c

MU(C_F) = 2/3 C_F^2/3-1 C_C^1/3 = 2/3 C_F^-1/3 C_C^1/3

MU(C_c) = 1/3 C_F^2/3 C_C^1/3-1 = 1/3 C_F^2/3 C_C^-2/3

Then we solve for the slope of indifference curve which is given by the marginal rate of substitution:

MRS = MU(C_F)/ MU(C_c)

MRS = 2/3 C_F^-1/3 C_C^1/3 /1/3 C_F^2/3 C_C^-2/3

MRS = 2 C_C^1/3 C_C^2/3 / C_F^2/3 C_F^1/3

MRS = 2 C_C^{1/3 +2/3} / C_F^{2/3 + 1/3}

MRS = 2 C_C/ C_F

Let the given price of French Cheese be P_F = $10 and price of California Cheese be Pc = $2; the income of the consumer be M= 150. The budget constraint can be written as:

M = P_F C_F + Pc C_C

150 = 10C_F + 2C_C

Slope of Budget line = P_F/Pc = 10/2 = 5

At equilibrium: Slope of Indifference curve = Slope of Budget Line

MRS = P_F/Pc

2 C_C/ C_F = 5

C_C/ C_F = 5/2

C_C/ C_F = 2.5

C_C = 2.5 C_F

Using the value of C_C in the budget constraint we get the value of C_F

150 = 10C_F + 2C_C

150 = 10C_F + 2(2.5 C_F )

150 = 10C_F + 5C_F

150 = 15C_F

C_F = 150/15

C_F = 10

C_C = 2.5 C_F

C_C = 2.5(10) = 25

Thus, the initial consumption bundle be A(10,25)

At the Initial consumption bundle A(10,25) the utility can be calculated as:

U(C_F, C_C) = C_F^2/3 C_C^1/3

U(10, 25) = 10^2/3 25^1/3

U(10, 25) = 10^0.66 25^0.33

U(10, 25) = 4.57 * 2.89 = 13.22

b. If the price of French cheese falls to P_F’ = $5, then keep all other things constant the new budget constraint be:

M = P_F’ C_F + Pc C_C

150 = 5C_F + 2C_C

Slope of Budget Constraint = P_F’/ Pc = 5/2 = 2.5

At equilibrium:

MRS = P_F ‘/PC

2 C_C/ C_F = 2.5

C_C/ C_F = 2.5/2

C_C/ C_F = 1.25

C_C = 1.25 C_F

Using the value of C_C in the budget constraint we get the new value of C_F

150 = 5C_F + 2C_C

150 = 5C_F + 2(1.25 C_F)

150 = 5C_F + 2.5 C_F

150 = 7.5 C_F

C_F = 150/7.5

C_F = 20

C_C = 1.25 C_F

C_C = 1.25(20) = 25

Hence the new consumption bundle be B(20,25)

At the new consumption bundle B(10,25) the utility can be calculated as:

U(C_F, C_C) = C_F^2/3 C_C^1/3

U(20, 25) = 20^2/3 25^1/3

U(20, 25) = 20^0.66 25^0.33

U(20, 25) = 7.22 * 2.89 = 20.89

Increase in Utility = U(20,25) - U(10, 25) = 20.89 - 13.22 = 7.67

c. As the price of French cheese fall from $10 to $5 the consumer’s purchasing power increase and he/she able to purchase more amount of French cheese as compared to earlier. The consumption of French cheese rises from 10 lb to 20 lb.

The utility which the consumer gets by consuming the new bundle is greater than the utility that he/she gets by consuming the initial bundle i.e. U_B(20,25) > U_A(10, 25). Higher utility means the consumer can move to a higher indifference curve and gets a greater satisfaction in consuming the two goods at the new price of French cheese. Hence, the consumer is better off in part (b).