Question

Let a consumer’s utility function be 𝑈 = 𝑞1

6𝑞2

4 + 1.5𝐼𝑛𝑞1 + 𝐼𝑛𝑞2

and his budget constraint

3𝑞1 + 4𝑞2 = 100.

a. Find his optimum commodity bundle of 𝑞1

and 𝑞2

. 

b. Verify whether second order is fulfilled or not. 

c. Determine the marginal utility of money

d. Estimate the new optimal utility if the consumer’s income rises by ₦1

Answer 1

$$\frac{\delta U}{\delta q_1}=6q_1^5q_2^4+\frac{1.5}{q_1}$$

$$\frac {\delta U}{\delta q_2}=4q_1^6q_2^3+\frac{1}{q_2}$$

$\frac{6q_1^5q_2^4+\frac{1.5}{q_1}}{3}=\frac {4q_1^6q_2^3+\frac{1}{q_2}}{4}$

$$q_1=20$$

$$q_2=10$$

$$3\times10+4\times20=100$$

$$\lambda=9.6\times10^{10}$$

If I=101

Since the growth in consumer income is relatively small compared to previous income, the Lagrange multiplier will not change significantly.