Question

The equation of an estimated demand function is as follows: -

Qd_A (Quantity demand for A) = 200.5 - 2.5 P_a - 1.5P_b + 3.5 I

where, P_a = Price of A

P_b = Price of B [It is a related product]

I = Income

(i) Determine the demand when P_a = $ 500, P_b = $ 100 and I = $ 3000

(ii) Estimate the price elasticity, cross-price elasticity and income elasticity of the demand according to point method.

(iii) Estimate the elasticity of the demand according to proportion method if P_a2 = $ 515, P_b2 = $ 105 and I = $ 3500.

Answer 1

Given

$Q^d_A= 200.5−2.5P_A −1.5P_B+3.5 I ...... (1)$

Where Price of A = Pa and Pb is the price of B and I is income 

 When P_a = $ 500, P_b = $ 100 and I = $ 3000

Then 

$Q^d_A=200.5−2.5(500)−1.5(100)+3.5(3000)\\⇒ Q^d_A=9300.5$

(I)

Price elasticity of the demand 

Differentiate equation 1 w r t PA

$\frac{dQ^d_A}{dP_A}= −2.5$

Price elasticity(E)$=\frac{dQ^d_A}{dP_A}×\frac{P_A}{Q^d }$

$⇒E=−2.5×\frac{500}{9300.5}\\⇒E=−0.1344\\⇒|E|=0.1344$

(ii)

Cross elasticity

Differentiate equation 1 w r t PB

$\frac{dQ^d_A}{dP_B}= −1.5$

Cross-Price elasticity$=\frac{dQ^d_A}{dP_B}×\frac{P_B}{Q^d }$

$EP_B=−1.5×\frac{100}{9300.5}\\⇒EP_B=−0.0161$

(iii)

Income elasticity:

Differentiate equation 1 w r t I

$\frac{dQ^d_A}{dI}= 3.5$

$Income \space elasticity = \frac{dQ^d_A}{dI}×\frac{IQ}{A^d}$

$= 3.5×\frac{3000}{9300.5}\\=1.129$