Question

if philip's utility function is... Question: If philip's utility function is U=2q1^0.5+q2, what are his demand functions for two goods? if philip's utility function is U=2q1^0.5+q2, what are his demand functions for two goods

Answer 1

given utility function $U(x,y)=2q_1^{0.5}+q_2$

$MRS=\frac{MU_{q_1}}{MU_{q_2}}$

$=\frac{\frac{\delta U(q_1,q_2)}{\delta q_1}}{\frac{\delta U(q_1,q_2)}{\delta q_2}}$

$=\frac{0.5q_1^{-0.5}}{1}$

$=\frac{1}{2q_1^{0.5}}$

budget line $p_1q_1+p_2q_2=M$

slope of budget line is $\frac{p_1}{p_2}$

The optimal consumption is where MRS equal slope of budget line. it is shown below: $\frac{1}{2q_1^{0.5}}=\frac{p_1}{p_2}$

$\frac{p_2}{2p_1}=q_1^{0.5}$

squaring both sides

$(\frac{p_2}{2p_1})^2=q_1$

substitute this value of q₁ into budget line

$p_1q_1+p_2q_2=M\\p_1(\frac{p_2}{2p_2})^2+p_2q_2=M\\\frac{p_2^2}{4p_1}+p_2q_2=M\\P_2Q_2=M-\frac{p_2^2}{4p_1}\\q_2=\frac{M}{p_2}-\frac{p_2}{4p_1}$

Thus , the demand function of q₁ $(\frac{p_2}{2p_1})^2$ is and q₂ is $\frac{M}{p_2}-\frac{p_2}{4p_1}$