Question

Suppose the demand for commodity X is estimated as follows:
Qx=68-1.6Px^2 + 0.6Py + 0.08E

Where:
Qx=quantity of commodity X
Px=N20 is the price of X
Py= N40 is the price of Y
E=N10,000 is the income of the consumer

Calculate:
1. The price elasticity of X
2. The cross-price elasticity of demand for X with respect to the change in the price of Y. Use your result to determine whether X and Y are substitutes or complements
3. The income elasticity of demand for X. Use your result to determine whether X is a normal or inferior commodity

Answer 1

Given

$Qx=68-1.6Px^2+0.6Py+0.08E\\Px=20\\Py=40\\E=10000\\Qx=68-1.6(20×20)+0.6(40)+0.08(10000)\\Qx=68-640+24+800\\Qx=252$

1. The price elasticity of X

$=\frac{dx}{dpx}×\frac{px}{Qx}\\=-1.6×\frac{20}{252}\\=-0.13\\-0.13<1\space inelastic$

2. The cross-price elasticity of demand for X with respect to the change in the price of Y.

$=\frac{dx}{dpy}×\frac{py}{Qx}\\=0.6×\frac{40}{252}\\=0.095\\0.095>0\space substitutes$

3. The income elasticity of demand for X.

$=\frac{dx}{dm}×\frac{m}{Qx}\\=0.08×\frac{10000}{252}\\=3.17\\3.17>0<1\space normal\space good$