Question

Assume a firm engaging in selling its product and promotional activities in monopolistic

competition face short run demand and cost functions as Q = 20-0.5P and TC= 4Q2

-8Q+15,

respectively. Having this information (5 marks)

a) Determine the optimal level of output and price in the short run.

b) Calculate the economic profit (loss) the firm will obtain (incur).

c) Show the economic profit (loss) of the firm in a graphic representation

Answer 1

The demand function is given by$Q = 20 − 0.5P\space or\space P = 40 − 2Q$

and the cost function is given by$TC = 4Q^2 − 8Q + 15$

Since, in monopolistic competition, equilibrium takes place where Marginal Revenue (MR) = Marginal Cost (MC),

we find MR and MC and equate them. to find our equilibrium P and Q.

By definition, 

$MR=\frac{d}{dQ}(Total\space Revenue)$

$=\frac{d}{dQ}(P\times Q)$

$=\frac{d}{dQ}(Q\times(40-2Q)$

$=\frac{d}{dQ}(40Q-2Q^2)$

$=40-4Q$

and,

$MC=\frac{d}{dQ}(Total\space Cost)$

$=\frac{d}{dQ}(TC)$

$=\frac{d}{dQ}(4Q^2-8Q+15)$

$=8Q-8$

Equating MC and MR we get, 

$MC= MR\\8Q-8 =40 − 4Q \\12Q=48\\Q=4$

(a) Hence, number of quantities produced (Q) = 4.

Therefore, the price (P)  = 40 − 2Q = 40 − (2 × 4) = 40 − 8 = 32

per unit. 

(b)

Now, the profit function of the firm, can be written as, 

$\pi(Q)=Total\space Revenue-Total\space Cost\\=(P\times Q)-(TC\times Q)$

If the above expression is calculated for a numerical value, the following is obtained. 

$\pi(4)=(32\times4)-[(4\times4^2)-8(4)+15]\\=128-[64-32+15]\\=128-47\\=81$

  The economic profit is calculated to be 81 units.

(c)

The economic profit can be shown diagrammatically as the following,

P_c is the competitive price and P_mc is the price under monopolistic competition. Q_mc is the equilibrium output.

The grey shaded area is the total amount of profit.