Question

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Practice Problem 2: Crash Course in Optimization: Part II 1. Consider a monopolist who pays an output tax $t for each unit of output produced. Thus, total tax payments are simply tQ, where is the quantity produced. Suppose the government is considering increasing the tax. Some people have argued that, because the firm is a monopolist, it will simply increase its price by the amount of the tax. In mathematical terms, they are claiming that -=1, where P* is the price chosen by the monopolist. Your job is to see if their claim is correct. Assume throughout that the monopolist seeks to maximize after-tax profit, i.e., profit minus tax payments. OP* at Consider first the case where the monopolist faces a linear demand curve given by P=a-bQ and has a linear cost function given by C=cQ. where c is the constant) marginal and average cost. (a) What are the endogenous variables in this problem and what are the exogenous variables/parameters? (b) Find the monopolist's optimal choice of output (*) and the corresponding optimal price (P*) as functions of the parameters. (c) Is the claim that the monopolist will increase price by the amount of the tax true in this case? If not, will it increase its price by more or less than the increase in the tax? How can you tell? Now consider a slightly more general specification where the demand curve is still linear (same as above), but now the cost function is a general convex cost function, i.e., C=C(O), where CQ) > 0 and CO20. Find the first order condition for the optimal choice of Q. (d) Find an expression for 2. What if anything, can you say about its sign, i.e., can you say whether output will increase, decrease, or remain unchanged when the tax rate increases? ot (e) Find an expression for F. (Hint: Remember how P* is related to Q*) Is the claim correct in this more general case? If not, will P* increase by more or less than the tax? How can you tell? dt (f) Will the price increase by more when the cost function is linear or when it is convex? How can you tell?

Answer 1

D).

Consider the given problem here the market demand curve is “P=a-b*q”, => the TR = P*q = a*q-b*q^2.

=> TR = a*q-b*q^2, => MR = d(TR)/dq = a-2b*q, = > MR = a-2b*q.

Now, the cost function is “C=c*q^2”, => C’() = 2c*q > 0” for all “q > 0” and “C’’() = c > 0”.

Now, the profit function is given below.

=> A = TR - TC - t*q, => the FOC for maximization is “dA/dq = 0”.

=> dA/dq = d(TR)/dq – d(TC)/dq - t = 0, => MR – MC - t = 0, => MR=MC+t.

=> a-2b*q=2c*q+t, => a-t = 2(c+b)*q, => q = (a-t)/2(b+c), the equilibrium quantity.

The above quantity expression can also be written as follows.

=> q = a/2(b+c) - t/2(b+c), => dq/dt = (-1)/(b+c) < 0, => as tax increases implied the equilibrium quantity decreases and vice versa.

E).

The market equilibrium price is given by, => P = a-b*q, => P = a-b*[(a-t)/2(b+c)]

=> P = [2a*(b+c) - b*(a-t)]/2(b+c), => P = [2ab + 2ac - a*b + b*t]/2(b+c).

=> P = [ab + 2ac + bt]/2(b+c), => dP/dt = b/2(b+c) > 0, => as the tax increases implied the equilibrium price increases and vice versa.

Now, as we know that “b < b+c”, => b/2(b+c) < 1, => as tax increases by “1 rupees” the equilibrium price increases by less than 1 rupees.

F).

Now, let’s assume that the cost function is linear, => the equilibrium rice is “P = (a+c+t)2”, => dP/dt = 1/2. So, as the tax increases by “1 rupees”, => the equilibrium price increases by “1/2 rupees”.

Now, if the cost function is convex, => the equilibrium rice is “P = [ab + 2ac + bt]/2(b+c), => dP/dt = b/2(b+c) = (1/2)*(b/b+c) < 1/2. So, as the tax increases by “1 rupees”, => the equilibrium price increases by less than “1/2 rupees”.

So, the price increases by more under the linear cost function compare to the convex cost function.