Answer to Question #239950 in Microeconomics for Joel

Question

If a consumer's objective function is given by U=X1X2, subject P1X1+P2X2=M established the demand function for X1 and X2 that maximize the consumer satisfaction

Answer 1

Consumer get maximum satisfaction when his/her utility maximize

Given

$U=X_1X_2, \\subject \\P_1X_1+P_2X_2=M$

Using langrarian we can maximize utility

$L=X_1X_2+\lambda[M-P_1X_1-P_2X_2] \\\frac{\delta L}{\delta X_1}=X_2-\lambda P_1=0.....(1)\\ \frac{\delta L}{\delta X_2}=X_1-\lambda P_2=0.....(2)\\ \frac{\delta L}{\delta \lambda}=M-P_1X_1-P_2X_2=0.....(3)$

Divide equition 1 by 2

$\frac{X_2}{X_1}=\frac{\lambda P_1}{\lambda P_2}=\frac{X_2}{X_1}=\frac{P_1}{P_2}\\P_2X_2=P_1X_1$ Put in equition 3

$M=P_1X_1+P_2X_2\\M=P_1X_1+P_1X_1$

Therefore the demand of X₁ and X₂ that will maximize consumer satisfaction will be

$X_1=\frac{M}{2P_1}\\X_2=\frac{M}{2P_2}$