Question

Suppose that the utility function for two commodities is:

U (q₁, q₂) = q₁^α q₂^(1-α)

Let the prices of the two commodities be p₁ and p₂ and let the consumer’s income be M.

(1) Check the properties of marginal utilities. In particular, check whether it satisfies diminishing marginal utilities.

(2) Assuming all income is spent on these two commodities, derive the demand curves for the two commodities.

(3) What happens if U (q₁, q₂) = q₁^α q₂^β?

Answer 1

Solution

$U_1= aq_1^{a-1}q_2^{1-a}=a\frac{U(q_1,q_2}{q_2}$

$U_2=(1-a)q_1q_2^{-a}=(1-a)\frac{U(q_1,q_2)}{q_2}$

$MRS=\frac{dq_2}{dq_1}=\frac{U_1}{U_2}=\frac{aq_2}{1-aq_1}$

The MRS is positive hence satisfies diminishing marginal utilities

The demand curve

$aq_1(\frac{I-P_1q_1}{P_2})^{1-a}+(1-a)q_1(\frac{I-P_1q_1}{P_2})^a(\frac{-P_1}{P_2})=0$

$q_1(P_1,P_2,I)=\frac{aI}{P_1}$

$q_2(P_1,P_2,I)=\frac{(1-a)I}{P_2}$

$3.) \frac{∂q}{∂p} + \frac{q∂q}{∂y} = \frac{βq}{p }+ q(\frac{αq}{y}) = (\frac{q}{p})(β +\frac{αpq}{y})$

Shows the substitution effect

when $U (q1, q2) = q_1^α q_2^β$