Question

3. Assume that a monopolist produces a good at constant marginal cost MC(q) = 1. Demand is given by PD (q) = 10 – 2q. There are no other pre-existing distortions in the market. (a) What is the privately optimal quantity and price chosen by the monopolist? For parts (b) and (c), assume that a tax of $t is imposed on every unit of output produced by the monopolist. (b) Derive the optimal quantity and price chosen by the monopolist as functions of T. (c) How much does the price increase when t is raised from $0 to $1? For part (d), assume that a tax t is imposed on each dollar of the monopolist's profits, while the tax on output is removed (r = 0). This means that if the monopolist earns $t in profit, it must pay t * $1 to the government. (d) What is the effect of this tax on the price and quantity chosen by the monopolist? Now suppose the government is deciding whether to raise revenue by taxing the mo- nopolist's output (T>0), or by taxing its profits (t > 0).

Answer 1

a).

Here the demand curve is “P = 10 - 2*q”, => MR = 10 – 4*q. The marginal cost function is “MC=1”. At the equilibrium the MR must be equal to MC.

=> MR = MC, => 10 – 4*q = 1, => q = 9/4 = 2.25.

The equilibrium price is “P = 10 - 2*q = 10 - 2*(9/4) = 10 - (9/2) = 5.5”. So the equilibrium price is “P=5.5”.

b).

Let’s assume a tax of “t per unit” is imposed, the new marginal cost function is “MC = 1+t”. So, at the equilibrium the MR must be equal to MC.

=> MR = MC, => 10 – 4*q = 1+t, => 4*q = 9 - t, => q = 9/4 – t/4, => q = 2.25 – t/4.

The optimum price charge by the monopolist is “P = 10-2*q”, => “P = 10 - 2*(9/4 – t/4)”.

=> P = 10 - 9/2 + t/2, => P = 5.5 + t/2.

So, the optimum price and the quantity as a function of “t” are “P = 5.5 + t/2” and “q = 2.25 – t/4” respectively.

c).

Here the optimum price is “P = 5.5 + t/2”, if the “t=0” then the price is “P = 5.5”. Now, if the tax rate is “t=1” the optimum price is “P = 5.5 + ½ = 6”. So, as the tax per unit increases the optimum price also increases to “6” from “5.5”.

d).

Now, let’s assume that tax is imposed on profit. So, the new profit function is given below.

=> A = [TR – TC]*(1-t), where “t” is a fraction. The FOC for maximization require “dA/dq = 0”.

=> dA/dq = [dTR/dq – dTC/dq]*(1-t) = 0, => [MR – MC]*(1-t) = 0, => MR – MC = 0.

=> MR = MC, get the initial condition. So, the optimum price and the quantity are “P=5.5” and “q=2.25” respectively, independent of tax rate.