PS13+exercise+solutions

# PS13+exercise+solutions - p 14 PC 12 PF 10 8 6 4 2 0 0 1 2...

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Intermediate Microeconomics, 2010 Problem Set 13: Exercise Solutions This topic explored more about the quasilinear economies. We saw two cases, taxes and monopoly, in which the equilibrium outcome turns out to be ine±cient. A tax of t dollars per unit that is collected from ²rms is equivalent to an increase in the ²rm’s marginal cost by the amount t . Therefore, instead of the competitive allocation being characterized by v 0 ( q ± ) = c 0 ( q ± ), it is now v 0 ( q ± ) = c 0 ( q ± ) + t With a monopoly, we are deviating from the standard perfect competition model. (Recall from Econ 101 that the monopolist does not have a supply curve. Instead it has a supply point.) To solve this problem, we ²rst must ²nd the inverse demand curve for the consumer. Then, we will plug this into the ²rm’s pro²t function to get pro²t only as a function of quantity. Next, we will di³erentiate the pro²t function with respect to quantity, set the resulting expression equal to zero, and ²nd the quantity that solves that equation. Then, we will plug that quantity into the inverse demand curve to get the price the monopolist sets. 1. Taxes. Consider the quasilinear economy from the last problem set: U ( m;q ) = m + ± 15 q ± 1 2 q 2 ² c ( q ) = q 2 and Y = 100 . Recall that we found the competitive allocation to be [ q ± ;q ± ± ;m ± ] = [5 ; 5 ; 25 ; 50] and p ± = 10 . Now we will consider the consequences of a tax: What happens with a speci±c tax t = 1 : 5 which has to paid by the ±rm? What is the competitive allocation with a tax, x t = [ q t ;q t t ;m t ] . What is the tax revenue T ? What is the deadweight loss? I am going to drop the superscript t in what follows for convenience. First, the easy way. We know that v 0 ( q ) = c 0 ( q ) + t: Why? Recall how you shift up the supply curve when a tax is implemented by the amount of the tax. What is the supply curve? It’s basically the marginal cost

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