Question

A firm operates in a perfectly competitive market. The market price of its product is 4
birr and the total cost function is given by: TC= 1/3Q3-5Q2+50,
a. What level of output should the firm produce to maximize its profit?
b. Determine the level of profit at equilibrium.
c. What minimum price is required by the firm to stay in the market?

Answer 1

a) $\bold{Answer}$

Output, $Q = 11 \space units$

$\bold {Solution}$

Perfectly competitive firms are price takers. Therefore,

AR = MR = 4 birr

Profit maximising level of output is found when MR = MC.

Now,

$MC = \dfrac {d}{dQ} (TC)$

$= \dfrac{d}{dQ}\big(\dfrac {1}{3}Q^3 - 5Q^2 + 50 \big)$

$= Q^2 -10Q$

$\therefore \space Q^2 - 10Q = 4$

$=> Q^2 -10Q -4 = 0$

$Q = \dfrac { -(-10) \pm \sqrt {(-10)^2-4(1)(-4)}}{2}$

$= \dfrac {10 \pm \sqrt {116}}{2}$

$= \dfrac {20.770329614}{2}$

$= 10.385164807$

$\approx {11 \space units}$

b) $\bold {Answer}$

$π = 155.33 \space birr$

$\bold {Solution}$

Q = 11

$TR = Q × P$

$= 11 \space units × 4 \space birr$

$= 44 \space birr$

$TC = \dfrac {1}{3}(11)^3 -5(11)^2 +50$

$= \dfrac {1331}{3} - 605 +50$

$= -111.3333333 \space birr$

(The firm might have saved costs)

$π = TR - TC$

$= 44 \space birr - (-111.33333 \space birr)$

$= 155.333333 \space birr$

$\approx 155.33 \space birr$

c) $\bold {Answer}$

$P = -18.75 \space birr$

$\bold {Solution}$

Shutdown condition is:

$AR \geq Min( AVC)$

$TC = \dfrac {1}{3} Q^3 - 5Q^2 +50$

$FC = 50 \space birr$

$TVC = \dfrac {1}{3}Q^3 -5Q^2$

$AVC = \dfrac {\dfrac {1}{3}Q^3 -5Q^2}{Q}$

$= \dfrac {1}{3}Q^2 -5Q$

For minimum AVC:

$\dfrac {d}{dQ}(AVC) = \dfrac {d}{dQ} \big(\dfrac {1}{3}Q^2-5Q \big)$

$= \dfrac {2}{3}Q -5$

Now, $\dfrac {2}{3} Q -5 = 0$

$=> 2Q - 15 = 0$

$=> 2Q = 15$

$=> Q = \dfrac {15}{2}$

$Thus, \space Q = 7.5 \space units$

$\therefore \space AVC = \dfrac {1}{3}(7.5)^2 -5(7.5)$

$= \dfrac {56.25}{3} - 37.5$

$= 18.75 - 37.5$

$= -18.75 \space birr$

Thus, the minimum acceptable price is -18.75 birr (assuming negative prices exist)