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

marginal utility(MU) is the addition to the total utility when an additional unit of a good is consumed. The law of diminishing marginal utility states that when more and more units of one good are consumed, the marginal utility from each successive unit declines. 

The marginal utility of good q₁ is the derivative of total utility with respect to q₁. The marginal utility of good q₂ is the derivative of total utility with respect to q₂.

$1)\\ U(q_1,q_2) =q_1^αq_2^{(1-α)}\\MUq_1=\frac{∂U(q_1,q_2)}{∂q_1}\\=α\frac{q_2^{(1-α)}}{q_1^{(1-α)}}\\MUq_2=\frac{∂U(q_1,q_2)}{∂q_2}\\=(1-α)\frac{q_1^α}{q_2^α}$

The marginal utility is diminishing because the quantity is in the denominator. So, the marginal utility of q₁ decreases with the increase in q_1. The marginal utility of q₂ decreases with the increase in q₂. 

2)

A consumer consumes where its utility is maximized. The consumer maximizes its utility where the marginal rate of substitution(MRS) is equal to the price ratio. The marginal rate of substitution is the ratio of marginal utility of good 1 to the marginal utility of good 2. 

$MRS=\frac{\frac{αq_2^{(1−α)}}{q_1^{(1−α)}}}{\frac{(1−α)q_1^α}{q_2^α}}$

$=\frac{αq_2}{(1−α)q_1}$

$MRS=\frac{p_1}{P_2}\\\frac{αq_2}{(1−α)q_1}=\frac{p_1}{P_2}$

$q_2=\frac{(1−α)q_1p_1}{αp_2}$

Budget constraint

$p_1q_1 +p_2q_2=M$

$p_1q_1 +p_2(\frac{(1−α)q_1p_1}{αp_2})=M$

$p_1q_1 +\frac{(1−α)q_1p_1}{α}=M$

$p_1q_1 \frac{(1+1−α)}{α}=M$

$q_1=\frac{α×M}{p_1}$ ...demand function for q₁

$q_2=\frac{(1−α)p_1(\frac{α×M}{p_1})}{αp_2}$

$=\frac{(1−α)×M}{p_2}$ ...demand function for q₂

3)

As the power of q₂ changes from $1-α\space to\space β$ , the demand function of q₂. The demand function of q₁ remains the same.

$q_2 =\frac{β×I}{p_2}$