Answer to Question #249504 in Macroeconomics for Aman

Question

Suppose Mr. Alemu consumes two commodities, X and Y. The income of Mr.Alemu is $200, and price of X is 5 and the price of Y is 15. The demand function for the comnmodity is given as:

Qx=100-0.75)y +0.251px1/3 + 2p3/2

where Qx is quantity demand of commodity X. 7Px is the price of commodity X, p, is

the price of commodity Y and I is income.

Then:

A. Find the price elasticity of demand. Decide whether it is elastic, unitary clastic or

Answer 1

Price elasticity of demand $= \frac{dQ_x}{dP} \times \frac{P }{ Q}$

$Q_x=100 - 0.75y +0.251 px^{\frac{1}{3}} + 2py^{\frac{3}{2}}$

$Qx=100 - 0.75\times200 +0.251 px^{\frac{1}{3}} + 2 \times 15^{\frac{3}{2}}$

$Qx=100 - 150 + 0.251 px^{\frac{1}{3}} + 2 \times 58.09475$

$Qx = - 50 + 0.251 px^{\frac{1}{3}} + 116.1895$

$Qx = - 50 + 0.251 px^{\frac{1}{3}} + 116.1895$

$Qx = 66.1895 + 0.251 px^{\frac{1}{3}}$

$Qx = 66.1895 + 0.251 \times 5^{\frac{1}{3}}$

$Qx = 66.1895 + 0.251 \times 1.709976$

$Qx = 66.1895 + 0.4292$

$Qx = 66.6187$

$Price\space elasticity\space of\space demand = \frac{dQx}{dP} \times \frac{P }{ Q}$

Differentiating equation 1, with respect to P, we get $\frac{dQx}{dP}$

$\frac{dQx}{dP} = \frac{1}{3} \times \frac{0.251 }{px^{\frac{2}{3}}}$

$\frac{dQx}{dP} = \frac{0.0836 }{ px^{\frac{2}{3}}}$

$Price \space elasticity\space of\space demand = \frac{dQx}{dP }\times\frac{ P }{ Q}$

$Price \space elasticity\space of\space demand = [\frac{0.0836}{ px^{\frac{2}{3}}}]\times* [\frac{5 }{ 66.6187}]$

$Price\space elasticity\space of \space demand = [\frac{0.0836 }{ 5^{\frac{2}{3}}}]* [\frac{5 }{ 66.6187}]$

$Price \space elasticity\space of\space demand = [\frac{0.0836 }{ 2.924018}] * [\frac{5}{66.6187}]$

$Price\space elasticity\space of\space demand = 0.0285 \times 0.0750$

Price elasticity of demand = 0.0021

Price elasticity of demand is very near to 0, thus we can conclude that the price elasticity of demand is inelastic in nature.