Question

1. Amy's utility function is U(x₁ , x₂) = x₁x₂, where x₁ and x₂ are Amy's consumption of banana and apple, respectively. The price of apples is $1, the price of bananas is $2, and his income is $40.

(a) Find out the Amy's optimal consumption bundle. (Note that Amy's utility function is Cobb-Douglas.)
(b) If the price of apples now increases to $6 and the price of bananas stays constant, what would Amy's income have to be in order to just be able to afford his old bundle?
(c) Based on (b), measure the changes in Amy;s consumption of apple due to the substitution effect and the income effect.

Answer 1

Ans. Utility function, U = x1 * x2

Marginal Utility of x1, MU1 = dU/dx1 = x2

Marginal Utility of x2, MU2 = dU/dx2 = x1

=> Marginal Rate of Substitution, MRS = dx2/dx1 = MU1/MU2 = x2/x1

At utility maximizing level,

MRS = p1/p2

=> x2/x1 = p1/p2

=> x2 = x1*p1/p2 ---> Eq1

Substituting Eq1 in budget constraint, x1*p1 + x2*p2 = M, we get,

x1*p1 + x1*p1*p2/p2 = M

=> x1 = M/2p1 and thus, x2 = M/2p2 --> Eq2

a) At p1 = $2 and p2 = $1 and M = $40

Fron Eq 2

x1 = 40/2*2 = 10 units of bananas

and x2 = 40/2*1 = 20 units of apples

b) If new p2 = $6, p1 = $2 and M = $40

From Eq2, we get,

x2 = 40/6*2 = 3.33 units and x1 = 40/2*2 = 10 units

So, compensating income required to be paid to Amy so that she is able to afford the old bundle of (10,20) is,

M' = 10*2 + 20*6 = $140

c) For substitution effect,

p1 = $2, p2 = $6 and M' = $140

From Eq2,

x2 = 140/2*6 = 11.667 units

So, decrease in demand for apples due to substitution effect = Original demand for apples - 11.667 = 20 - 11.667 = 8.333 units

For Income effect,

p1 = $2, p2 = $6 and M = $40

Fron Eq2,

x2 = 40/2*6 = 3.333 units

Decrease in demand for apples due to income effect = Demand from substitution effect - 3.333 = 11.667 - 3.333 = 8.333 units

Thus, total decrease in demand for apples = Income Effect + Substitution effect = 8.333 + 8.333 = 16.666 units

Thus, new demand for apples = 20 - 16.666 = 3.333 units

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