Question

ABC Sports, a store that sells various types of sports clothing and other sports items, is planning to introduce a new design of Arizona Diamondbacks’ baseball caps. A consultant has estimated the demand curve to be

Q = 2,000 - 100P

where Q is cap sales and P is price.

a. How many caps could ABC sell at $6 each?

b. How much would the price have to be to sell 1,800 caps?

c. Suppose ABC were to use the caps as a promotion. How many caps could ABC give away free?

d. At what price would no caps be sold?

e. Calculate the point price elasticity of demand at a price of $6.

Answer 1

a) $\bold {Answer}$

$Q = 1,400 \space Caps$

$\bold {Solution}$

When P = $6, $Q = 2,000 - 100(6)$

$= 2,000 - 600$

$= 1,400 \space caps$

b) $\bold {Answer}$

$P = \$2 \space per \space Cap$

$\bold {Solution}$

When $Q = 1,800 \space Caps$

$=> 1,800 = 2,000 - 100P$

$=> 100P = 2,000 - 1,800$

$=> 100P = 200$

$P = \dfrac {200}{100}$

$P = \$2$

c) $\bold {Answer}$

$Q = 2,000 \space Caps$

$\bold {Solution}$

We need Q when P = $0

Thus, $Q = 2,000 - 100(\$0)$

$= 2,000 - 0$

$= 2,000 \space Caps$

d) $\bold {Answer}$

$P = \$20 \space per \space Cap$

$\bold {Solution}$

We need P when Q = 0 caps,

$Thus, \space 0 = 2,000 - 100P$

$=> 100P = 2,000$

$=> P = \dfrac {2,000}{100}$

$\therefore \space P = \$20$ per cap

e) $\bold {Answer}$

$\eta = -0.429$

$\bold {Solution}$

$\eta = \dfrac {dQ}{dP} × \dfrac {P_{0}}{Q_{0}}$

When $P = \$6, \space Q = 1,400 \space Caps$

$\dfrac {dQ}{dP} = \dfrac {d}{dP} (2,000-100P)$

$= -100$

$=> \eta = \dfrac {-100}{1} × \dfrac {6}{1,400}$

$= \dfrac {-1}{1}×\dfrac {6}{14}$

$= -0.4285714285$

$= -0.429$