Question

A market consists of two individuals. Their demanded equations are Q1 = 16- 4P and Q2 = 20 –2P, respectively.

What is the market demand equation?

At a price of $2, what is the point price elasticity for each person and for the market?

Answer 1

Market demand equation:

The market demand is an aggregation of individual demand functions. Therefore, assuming market demand function is given by Q,

$Q = Q_{1} + Q_{2}$

$= (16-4P)+(20-2P)$

$=16+20-4P-2P$

$\bold {Q=36-6P}$ $(Answer)$

Point price elasticity of demand:

When $P =\$2,$

$Q_{1}=16-4(2)$

$= 16-8$

$= 8 \space units$

$Q_{2} = 20-2(2)$

$= 20-4$

$=16$ $units$

$Q=36-6(2)$

$= 36-12$

$= 24 \space units$

Differential calculus is applied to find the point price elasticity $(\eta_{p})$ using the formula,

$\eta_{p}= \dfrac {dQ}{dP}× \dfrac {P}{Q}$

For individual 1

$\dfrac {dQ}{dP} = \dfrac{d}{dP}(16-4Q)$

$= -4$

$\therefore \space \eta_{p} = -4×\dfrac {2}{8}$

$=\bold {-1(unitary)}$

For individual 2

$\dfrac {dQ}{dP} = \dfrac{d}{dP}(20-2Q)$

$=-2$

$\therefore \space \eta_{p} = -2×\dfrac {2}{16}$

$= \bold {-0.25(inelastic)}$

For the market

$\dfrac {dQ}{dP} = \dfrac{d}{dP}(36-6Q)$

$=-6$

$\therefore \space \eta_{p}=-6×\dfrac{2}{24}$

$= \bold {-0.5(inelastic)}$