What is the deadweight loss of a firm with the demand function p=100000-2q and a constant marginal cost of $2.

Question

I am currently getting $625million as the answer, but I don't think that's right.

Answer 1

When MC=P, 2= 100,000 -2q. 2q=99,998. or q=49,999.

The Reservation Price is 100,000. Any price below 100,000 and consumers get interested. Total Consumer Surplus at this point is given by the formula of a right triangle. CS=(1/2)Base times Height. The base is q=49,999 the height is (the reservation price - Marginal cost ) 100,000 -2 = 99,998.

So total Consumer Surplus is 1/2(49,999*99,998) = 49,999²

Note that with a constant marginal cost their is no deadweight loss in profduction..

With a monopolist with a linear demand curve theMR curve has a price axis intercept at the same reservation price as the demand curve. and is TWICE AS STEEP so MR = 100,000 - 4 Q

Set MR=MC=2 and Get 2 = 100,000-4q or 4 q = 99,998 or q=(99,998/4) which is exactly half. The monopolist will go up to the demand curve to charge P= 100,000 - 2(99,998)/4) = 100,000 - 49,999 = 50,001.

The firm will "earn" the difference between price and Marginal cost (50,001 - 2 = 49,999) on 49,999/2 units or the will gain (49,999²)/2 in Producer Surplus. The New Consumer surplus is the difference between the Reservation Price and the monopolist price (100,000-50,001 = 49,999) times the monopolist quantity all divided by 2. or New CS = (1/2)(49,999*(49,999)/2) = 49,999²/4. The deadweight loss is The Change in CS plus the change in PS. The Change is CS is New CS - old CS which is (49,999²)/4 -(49,999²)= -3/4(49,999)² we add back the change in PS of (1/2)(49,999²) and we are left with a deadweight loss of

DW = 1/4(49,999²) = 624,975,000 the reason you got 625 million is that you did not take into account that MC=2.

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Lenny