1. The color copy machine market is dominated by a large firm with significant p

Question

1.

The color copy machine market is dominated by a large firm with significant production capacity. The market demand for color copy machines is:
Q = 4000 - 5 P
The dominant firm has projected the supply by the small firms in the market to be:
Qs = - 600 + 3 P

Questions:
I. Write the demand curve faced by the dominant firm.
II. What price should the leader charge to drive all the small firms out of the market?
III. Write the marginal revenue function of the dominant firm.
IV. Assuming the leader's cost function is:
TC = 6000 + 175 Q + 1/32 Q2,
determine the following:

Questions:

a. The price

b. The total quantity demanded (purchased)

c. The leader's share of the market

d. The small firms' share of the market

e. The leader's profit

Explanation / Answer

The total cost function given as TC = 6000 + 175 Q + 1/32 Q2 is to be used only in the 4th part of the question. So for the first three questions we will try to come to rational decisions.

Now the total Demand is Q = 4000 - 5 P

and this is what the smaller firms will supply : Qs = - 600 + 3 P

the firm can satisfy the remaining demand(after what is left after smaller companies sell their produce) or it can edge out all the smaller firms or take a place somewhere in between.

Let us assume that it only supplies the shortfall by smaller firms.

So it will sell (4000 - 5P) - (-600 + 3P) = 4600 - 8P <<<<demand curve for dominant firm

To drive the smaller firms out of the market the dominant firm needs to find a price at the quantity sold by smaller firms is 0.

Or Qs = 0;

Or 600 = 3P

Or P = $200

Revenue for the dominant firm = Q*P = (4600 - 8P)*P

Or revenue = 4600P - 8P2

Or maginal revenue = 4600 - 16P (differentiating wrt P)

Now we have, TC = 6000 + 175 Q + 1/32 Q2

MC = 175 + 1/16 Q

At max profit MC = MR

Or  175 + 1/16 Q = 4600 - 16P

Or 175 + 4000/16 - 5/16 P = 4600 -16P

Or 425 - 4600 = -15.6875

Or P = $266.153

and Q = 2471 (pluggin P = 266.173 in eqn for demand for dominant firm)

and total quantity for industry, 2669 (substitute P = 266.173 in eqn for enitire demand of industry Q = 4000 - 5 P)

So market share = 2471/2669 = 92.57% <<<this is consistent with our initial inference that a price closer to 200, the dominant firm will have a high market share.

and profit for firm is revenue - total cost = 4600P - 8P2 - 6000 + 175 Q + 1/32 Q2 @P = $266.13 and @Q = 2471

Profit = 657794.39 - 623232.54 = $34561.9 <<<lower price and higher market share.

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