Answer to Question #198029 in Microeconomics for robel legese

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 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

Solution:

a.). To determine the optimal level of output and price in the short run for monopolistic competition, we set the MR = MC. At this point, the profits are maximized.

First derive the inverse demand function:

Q = 20 – 0.5P

20 – 0.5P = Q

-0.5P = Q – 20

P = -2Q + 40

P = 40 – 2Q

Derive TR:

TR = P x Q

TR = (40 – 2Q) Q = 40Q – 2Q²

TR = 40Q – 2Q²

Now derive the marginal revenue:

MR = $\frac{\partial TR} {\partial Q} = 40 - 4Q$

Derive MC from the TC:

TC= 4Q² -8Q+15

MC = $\frac{\partial TC} {\partial Q} = 80 - 8$

Now set MR = MC and find the optimal output:

MR = MC

40 – 4Q = 8Q – 8

40 + 8 = 8Q +4Q

48 = 12Q

Q = 4

The optimal level of output (Q) = 4

Now substitute in the inverse demand function to get the optimal price level:

P = 40 – 2Q

P = 40 – 2(4)

P = 40 – 8 = 32

P = 32

The optimal level of price (P) = 32

 

b.). Economic profit (loss) = TR – TC

Economic profit (loss) = (40Q – 2Q²) – (4Q² -8Q+15)

= (40(4) – 2(4²)) – (4(4²) – 8(4) + 15))

 = (160 – 32) – (64 – 32 + 15)

 = 128 – 47

= 81

Economic profit = 81

 

c.). The economic profit of the firm has been shown graphically as follows: