Question

Question 15 0/1 pts Consider a consumer with a utility function 11 = 1/23/2. If the prices for goods 1 and 2 are 2 and 4 respectively, and income is 40, what is the consumer's optimal consumption of good 1 and 2? Correct Answer > 0 x = 5; x; = 7.5 x = 2.5; x3 = 7.5 x = 5; x = 5 You Answered x* = 5;x; = 2.5 None of the above

Answer 1

Answer

A consumer maximizes utility at MU per dollar is equal.

$MU_{x_1}=\frac{\partial U}{\partial x_1}=\frac{\partial \left ( x_1^{\frac{1}{2}}*x_2^{\frac{3}{2}} \right )}{\partial x_1}=\frac{x_2^{\frac{3}{2}}}{2x_1^{\frac{1}{2}}}\\ MU_{x_2}=\frac{\partial U}{\partial x_2}=\frac{\partial \left ( x_1^{\frac{1}{2}}*x_2^{\frac{3}{2}} \right )}{\partial x_2}=\frac{3}{2}*x_1^{\frac{1}{2}}x_2^{\frac{1}{2}}\\ MRS=\frac{MU_x_1}{MU_x_2}=\frac{\frac{x_2^{\frac{3}{2}}}{2x_1^{\frac{1}{2}}}}{\frac{3}{2}*x_1^{\frac{1}{2}}x_2^{\frac{1}{2}}}=\frac{x_2}{3x_1}\\ \frac{x_2}{3x_1}=MRS=\frac{P_1}{P_2}=\frac{2}{4}=0.5\\ x_2=0.5*3x_1=x_2=1.5x_1\\ from\ budget\ line\\ 2x_1+4x_2=40\\ 2x_1+4*1.5x_1=40\\ 8x_1=40\\ x_1=5\\ x_2=1.5*5=7.5$

Option 1