Answer to Question #95931 in Calculus for Bidemi Adekunle

Question

Qdx = 250 + 0.65I – 0.2Px2 + 0.5PxPy + 0.21Ax

Where
I = $250
Px= 7
Py = 4
Ax = 20
Determine
a) Price elasticity of demand for commodity x and interprete your result
b) Cross elasticity of demand for commodity x and y and interprete your result
c) Income elasticity of demand for commodity x and interprete your result
d) Advertisement elasticity of commodity x and interprete your result

Answer 1

$$Q_X^D=250+0.65I-0.2(P_X)^2+0.5P_XP_Y+0.21A_X$$

$I=250\$$

$P_X=7$

$P_Y=4$

$A_X=20$

$$Q_X^D=250+0.65(250)-0.2(7)^2+0.5(7)(4)+0.21(20)=$$ $$=420.90$$

a) Determine price elasticity of demand for commodity x 

$$Price \ elasticity\ of\ demand={\partial Q_X^D \over \partial P_X}\cdot{P_X \over Q_X^D}$$ $${\partial Q_X^D \over \partial P_X}=-0.4P_X+0.5P_Y=-0.4(7)+0.5(4)=-0.8$$ $$Price \ elasticity\ of\ demand=-0.8\cdot{7 \over 420.90}=-0.0133$$

$$|Price \ elasticity\ of\ demand|=0.0133<1$$

Therefore, the demand is inelastic.

b) Determine Cross elasticity of demand for commodity x and y

$$Cross \ elasticity\ of\ demand={\partial Q_X^D \over \partial P_Y}\cdot{P_Y \over Q_X^D}$$ $${\partial Q_X^D \over \partial P_Y}=0.5P_X=0.5(7)=3.5$$ $$Cross \ elasticity\ of\ demand=3.5\cdot{4 \over 420.90}=0.0333$$

$$Cross \ elasticity\ of\ demand=0.0333>0$$

Therefore, the goods are substitute: as the price of one good increases, the demand for the other good increases.

c) Determine income elasticity of demand for commodity x

$$Income \ elasticity\ of\ demand={\partial Q_X^D \over \partial I}\cdot{I \over Q_X^D}$$ $${\partial Q_X^D \over \partial I}=0.65$$ $$Income \ elasticity\ of\ demand=0.65\cdot{250 \over 420.90}=0.3861$$

$$0<Income \ elasticity\ of\ demand<1$$

Therefore, the goods are normal and are typically referred to as necessity goods, which are products and services that consumers will buy regardless of changes in their income levels.

d) Determine advertisement elasticity of commodity x

$$Advertisement \ elasticity\ of\ demand={\partial Q_X^D \over \partial A_X}\cdot{A_X \over Q_X^D}$$ $${\partial Q_X^D \over \partial A_X}=0.21$$

$$Advertisement \ elasticity\ of\ demand=0.21\cdot{20 \over 420.90}=0.0100$$

The advertisement elasticity should be positive because there is the possibility of extension of demand and market for the good with advertising expenditure.

$$Advertisement \ elasticity\ of\ demand=0.01<1$$

Therefore, the demand is relatively inelastic.