Marginal Revenue, Residual Demand Curves, Quotas and Monopolists

February 27, 2013 12:24 PM

How do I calculate the differences in welfare for a state when an import quota is applied to an industry that is controlled by a monopolist at home, vs an industry that is perfectly competitive at home?

Hi all,
I have a problem set for grad school that is asking me to calculate something that wasn't explained in class and doesn't appear to be in my textbooks as well. I am being asked to compare the differences in welfare losses to a home market when an import quota is applied to a perfectly competitive market, but I don't know how to calculate this for a market dominated by a monopolist. I'm given the home market's demand function and the marginal cost function for producers, which in the problem is the same for perfectly competitive firms and a monopolist, as well as the world price for the good. I think I need to calculate the marginal revenue function as well as a residual demand curve that would exist for the monopolist if a quota was applied, but I am not sure how to do this.

I'm not asking anyone to solve the problem for me, but could someone please point me to a book, internet resource, etc, where I can find an explanation for how to go about calculating this? I have searched around for a while now, but haven't found anything so I've turned to askmefi.

Thanks!

posted by tokaidanshi to Education (3 answers total)

I'll try, but I might not be super helpful. I'm in an advanced micro undergraduate class, and we are using Nicholson's Microeconomic Theory. I found a section on welfare losses and monopolies, but we haven't covered monopolies yet, so this is just me typing it out. I think you can skip to the bottom and just calculate out the ratio of consumer surpluses, with monopoly/perfect competition. You were given the average cost function, so you can multiple that by q to get total costs. Because total costs = P^e (see below), you probably should multiply that through instead of q. Use demand to calculate elasticity.

This is what my book pretty much says:

"Assume that the monopoly has constant marginal and average costs "c," and that the demand curve has a constant elasticity of the form "Q=P^e", then the competitive price will be Pc=c and the monopoly price is Pm=c/(1+1/e). Consumer surplus is the integral from P0 to infinity of Q(P)dP. (P0 = "p-naught")

This integral is actually -P0^(e+1)/(e+1).

Under perfect competition, consumer surplus, CSc, is -c^(e+1)/(e+1), and under monopoly, consumer surplus, CSm, is -[(c/(1+1/e))^(e+1)]/(e+1) (that's "c divided by 1 plus 1/e, all to the e+1, all divided by e+1, all negative).

Taking the ratio of the two surpluses gives you:

CSm/CSc = (1/(1+1/e))^(e+1)"

Again, I think you can just skip to the CSm/CSc step. I think everything else was the proof.

Did that help? Please please please if anyone stumbles across this and knows I'm wrong, correct it, because I just typed it out of the book.
posted by obviousresistance at 10:34 PM on February 27, 2013

!!!

I found a PDF of Nicholson's book here.

Look at page 499, starting on the bottom.
posted by obviousresistance at 10:39 PM on February 27, 2013

Funnily enough, we ended up going over this exact topic/example today in class. Memail me with any questions.
posted by obviousresistance at 1:22 PM on February 28, 2013