Question

Cleopatra spends all her money on bags (B) and shoes (S). Her utility function is U(B, S) = 2B0.5S
0.5
. Her income is
$150. Price of shoes is pS = 2. Price of bags depends on the number of bags purchased. Bags cost $5 each if she purchases
between 1 and 8 bags. If she purchases more than 8 bags, the price falls to $2 for all subsequent books.
(a) Draw Kleopatra's budget constraint where bag is on the horizontal axis. (10 Points)
(b) Solve for Cleopatra's optimal bundle. (15 Points)
(c) Suppose Cleopatra's utility function is U(B, S) = 0.5ln(B) + 0.5ln(S). What is her optimal bundle? (10 Points)
(Hint: Compare the MRS for both utility functions.)

Answer 1

(a) The budget constraint represents all the possible combinations of the two goods which do not exceed the income budget.

Now, if Cleopatra spends rs 150 total only on bags, she can buy 150/2 = 75 bags, this is her vertical intercept,

if cleaopatra spends rs 150 total on shoes, she can buy 150/2 = 75 shoes, this is her horizontal intercept.

Taka a look at fig 1 to see the budget constraint of cleopatra, Y = PS * QS + PB * QB , where PS = price of shoes, QS = quantty of shoes, PB = price of bags, QB = quantity of bags, Y = income

b) Utiity function , U(B,S) = 2B*0.5S

MRS of utility function = MU_B/MU_S = P_B/P_S,

MU_B = 2*0.5 * S = S, MU_S = B,

So, P_B/P_S = MU_B /MU_S = S/B = 2/2 = 2s = 2B, S = B, So substituting it in budget constraint ,

2 * S + 2 * B = 150, 2 * S + 2 * S = 150, 4S = 150, S = 37.5, so

the optimal bundle is 37.5 units of shoes and 37.5 units of bags,

c) Utiity function , U(B,S) = 0.5 ln(B) + 0.5 ln(S)

MU_B = 0.5 *1/B, MU_S = 0.5 * 1/S,

P_B/P_S = 2/2 = MU_B /MU_S = S/B , 2/2 = S/B , S =B,

Substituting it into the budget constraint,

2 * S + 2 * B = 150, 2 * S + 2 * S = 150, 4S = 150, S = 37.5, so

the optimal bundle is 37.5 units of shoes and 37.5 units of bags,