Question

A mathematical demand function for new Toyotas sold per year for a dealer is as follows:

Qr-200-0.01P+0.005PM-10PG+0.017 +0.0034.

where

Q=quantity purchased,

P= the average price of Toyotas,

PM= the average price of Mazdas,

P= the price of gasoline,

I= per capita income, and

A = dollars spent annually on advertising.

a. Find the point price elasticity of demand for the Toyotas if Pr= $25,000, PM= $20,000, P=$1.00, 1-$15,000, and A = $10,000.

b. Is the price elasticity of demand elastic, unitary elastic, or inelastic? Why?

c. Find the arc cross elasticity of demand for Toyotas and Mazdas between PT=$20,000 and PM $22,000. (All other figures except P, remain the same as in part (a).) d. Are Toyotas and Mazdas substitutes or complements? Why?

Answer 1

Solution:

a.). The point price elasticity of demand = $\frac{\triangle Q_{T} }{\triangle P_{T}} \times \frac{P_{T} }{Q_{T}}$

Determine Q_T = 200 – 0.01(25,000) + 0.0005(20,000) – 10(1) + 0.01(15,000) + 0.003(10,000)

Q_T = 200 – 250 + 10 – 10 + 150 + 30 = 130

Q_T = 130

$\frac{\triangle Q_{T} }{\triangle P_{T}}$ = -0.01

P_T = 25,000

Q_T = 130

=$-0.01 \times \frac{25,000 }{130} = -1.92$

(Ignore the negative sign)

PEd = 1.92

 

b.). PEd = 1.92, which is greater than 1, means that the demand is price elastic (Demand is sensitive to price changes)

 

c.). The arc cross elasticity of demand = $\frac{\triangle Q_{T} }{\triangle P_{M}} \times \frac{P_{M} }{Q_{T}}$

$\frac{\triangle Q_{T} }{\triangle P_{M}} = 0.0005$

P_M = 22,000

= $0.0005 \times \frac{22,000 }{130} = 0.08$

The arc cross elasticity of demand = 0.08

d.). Mazda’s and Toyotas are substitutes since the cross elasticity of demand is greater than zero. This means that as the price of Mazda increases, the demand for Toyota increases.