Assumptions: 1) Large number of buyers and sellers with none large enough to effect market prices. 2) Product is homogeneous. 3) No long run barriers to entry or exit. 4) Market demand given by P = A - Q = 100 - Q.

Results: 1) Long run economic profits = 0.

2) Market P = MC = Min AC.

3) Long run supply curve is flat, short run supply curve is upward sloping.

The profits tax is not the most interesting (or lucrative) way to tax this type of firm, although this issue is not irrelevant here. Instead, let's consider a unit tax. For simplicity, it will be levied on demanders. This shifts the effective demand curve faced by suppliers downward by the amount of the tax (for any given quantity demanded, demanders are now willing to pay less to suppliers by the amount of the tax. Had the tax been put on suppliers, the effective supply curve faced by demanders would have been shifted upward. These two cases give the same result in terms on new quantity and effective net-of-tax prices to suppliers and demanders.

Figure 1 examines a market served initially by n₀ perfectly competitive firms. Initially (before the tax is put on demanders) the market is in equilibrium at point A with a price of P₀ and a quantity of Q_A. Figure 2 depicts one of the many perfectly competitive firms. Each firm is producing q_i0 = Q_A/n₀ which is the minimum efficient scale (MES) of each firm. Here P₀ = MC = Min AC. Each firm is breaking even by charging a price equal to its minimum average short and long run costs which is P₀.

Putting a unit tax of T on demanders (it's just as easy to put it on suppliers) will shift the demand curve downward by T. This will move the equilibrium down the SR supply curve from point A to point B. At point B where the market price has fallen to P_Net, the n₀ firms are each producing at less than MES, q_i1 = Q_B/n₀. In the long run, the supply curve will shift leftward as some firms flee the industry due to negative profits. The new equilibrium will be at point C where n₁ (n₁ < n₀) firms will each produce q_i0 = Q_C/n₁ at MES with P₀ = MC = Min AC.

Q_A = (Sum over i = 1 to n₀ firms) of each firm's output q_io

Q_B = (Sum over i = 1 to n₀ firms) of each firm's output q_i1

Q_C = (Sum over i = 1 to n₁ firms) of each firm's output q_io, where n₁ < n₀.

Assumptions: 1) One seller in market for good with no close substitutes. 2) Barriers to entry protect the seller from competition.

Results: Monopolist is free to choose any combination of P and Q_D which is feasible, given the demand curve, to maximize it's profits. This happens where P is set so that the marginal revenue from the last unit sold equals the marginal cost of producing that unit.

Basic Monopolist Problem without Taxation:

Demand Curve: P = A - Q = 100 - Q

Cost Function: C = c*Q = 20*Q

Max Profit = Max (Revenue - Costs)

Profit = P*Q - C = (100-Q)*Q - 20*Q = 80Q - Q²

dProfit/dQ = 80 - 2*Q = 0. Q = 40.

So to maximize profits, the monopolist would choose the following price quantity combination.

Q₀ = 40, P₀ = 60, Profit₀ = 2400 - 800 = 1600

Figure 3 shows the basic monopolist problem in an increasing marginal cost industry. For the simple case, the marginal revenue (MR) curve is half the horizontal distance from the vertical axis as the demand curve (D). The monopolist chooses a quantity where MR = marginal cost (MC) and then lets the demand curve set the price for this restricted quantity. Socially, this is bad because MU (as indicated by the demand curve) is above MC so society values the output more than it would cost to make it. The position of average cost (AC) and MC are fairly arbitrary in this picture as long as they lie in a region where the monopolist chooses to produce a non-zero quantity.

Unit Tax:

Let the cost function include the unit tax, t.

Cost Function: C = 20*Q + t*Q

Now the profit maximizing quantity for the monopolist is no longer 40 but is now

Q₁ = 40 - t/2 < Q₀

P_Gross = 100 - Q = 60 + t/2 > P_Gross

P_Net = P_Gross - t = 60 - t/2 < P_Net

Profit₁ = P₁*Q₁ - C₁ = 1600 - 40*t + t²/4 < 1600 for any positive level of output.

Incidence is shared by firm (since P_Net < P₀) and by the consumers (since P_Gross > P₀), and since the quantity exchanged is reduced. This is depicted in Figure 4.

Profits Tax:

Let the profit tax rate be (1-T). {I'm making it "1-T" to simplify the math a bit.}

Monopolist tries to maximize T*(Revenues - Costs).

Profit = T*[P*Q - C] = T*[80*Q - Q²]

dProfit/dQ = T*[80 - 2*Q] = 0

So monopolists decision is nearly the same as in the basic problem.

Q = 40, P = 60, Profit = T*1600.

Here the profits tax looks very efficient. There are some subtle problems however. It is surprisingly difficult to define profit (as has been discovered by people who've sued Motown Records or various movie studios for having their ideas stolen). Should the government give money to firms which lose money (have negative profits)? If not, what should be done about firms with highly variable profits (perhaps alternating positive and negative from year to year)? They would be more heavily taxed than a firm with steady year-by-year profits. This would seem like a penalty on higher risk investments. There's a fairness issue for stockholders as well which will be discussed later. Perhaps you would like to see a picture of this but there's really nothing to see beyond just Figure 3 with some (1-T) of the profit spirited away.

[Consider a profits tax of 30% per year. A firm that made a steady profit of $100 per year would pay $30 per year. A firm that would lose $200 every even year but then make $400 every other year would earn the same average yearly profit but would end up paying twice as much in taxes over a two year cycle. One way around this problem is allowing firms to carry forward (or back) losses to reduce their taxes in other years. However, should firms that have no current profits be allowed to sell these "losses" to other firms to allow those firms to cut their taxes? Sometimes an otherwise valueless and bankrupt firm will get purchased by another firm for the value of the tax breaks due for the first firm's past losses. Whew, this stuff gets complicated!]

In a 1838 blockbuster book, Recherches sur les Principes Mathematiques de la Théorie des Richesses, Augustin Cournot created an influential model of oligopoly. Recall that for perfect competitors, each produces a quantity which puts it at the minimum of its average cost curve, while the monopolist finds it optimal to restrict the quantity produced to drive up the price and therefore the profit, and the oligopolists are somewhere in between.

Let the demand curve be P = A - Q where Q is total market quantity. Let A = 10, and let the cost of producing a unit of the good be constant at c which is 4.

Perfect competition: There are lots of firms producing at price c. The quantity is given by: c = A - Q => Q = A - c = 10 - 3 = 7. Each of the "n" firms would produce Q/n = 7/n, would charge a price of $4 and would earn profit of zero.

Monopoly: One firm. Profit = P*Q - c*Q = (A - Q)*Q - c*Q

Maximize profit by taking the derivative with respect to Q.

A - 2*Q - c = 0. Q = (A-c)/2 = (10-4)/2 = 3.

P = A - Q = 10 - 3 = 7.

Profit = (A - Q)*Q - c*Q = 7*3 - 4*3 = 9.

(Try differing Q's to see that the profit can't be greater than 9.)

Oligopolistic Duopoly: Two firms.

In an oligopoly, each firm has incentive to restrict its output to drive up the market price and incentive to increase its output to try to steal market share from its competitors. Cournot makes the simplifying assumption that when choosing the profit maximizing quantity to produce, each firm in the oligopoly takes the other firms' output as given in determining the market price. For this example, let there be only two firms. Further, it is assumed that the output of each firm is homogeneous.

P = A - q₁ - q₂, C₁ = c* q₁

Profit = P* q₁ - C₁ = (A - q₁ - q₂)* q₁ - c* q₁

dProfit/dQ = (A - q₁ - q₂) - q₁ -c = 0

q₁ = (A - q₂ -c)/2

This formula for q₁ is what is known as the reaction function for firm one. This formula gives firm one's profit maximizing level of output as functions of various parameters and the other firm's level of output. If the other firm's output were zero then firm one would be a monopolist and would produce quantity (A-c)/2. This is as high as firm one's profits can ever get. If the other firm chose to produce q₂ = A-c (the perfect competition total market quantity), firm one would not find it profitable to produce anything and so would produce zero.

These reaction functions give how much a firm would produce in reaction to what its competitor produces. Plotting the two curves against each other (Figure 5) quickly indicates what the equilibrium output for each firm will be.

For identical firms,

q₁ = (A - q₂ - c)/2, q₂ = (A - q₁ - c)/2

q₁ = [A - (A - q₁ - c)/2 - c]/2 = (A + q₁ - c)/4

q₁ = (A - c)/3 = q₂

Note that the total equilibrium output [q₁ + q₂ = 2*(A-c)/3] is greater than the monopoly level of output [(A-c)/2] yet less than the perfect competitor [zero profit, (A-c)] level of output.

The Cournot model is important in many fields of economics and has the delightful feature that once this equilibrium is reached, neither firm can increase its profits by unilaterally changing its level of output. Thus the equilibrium would be stable.

Profit for firm 1 = P*q₁ - c*q₁ = (A - q₁ - q₂)*q₁ - c*q₁

Maximize profit by taking the derivative with respect to q₁.

A - 2*q₁ - q₂ - c = 0 => q₁ = (A-q₂-c)/2.

Since firm two solves the analogous problem, q₂ = (A-q₁-c)/2.

Solving for q₁ using the solution to q₂ gives:

q₁ = (A-q₂-c)/2 = (A - (A-q₁-c)/2 - c)/2 = (A-c)/2 - (A-c)/4 + q₁/4

(3/4)*q₁ = A/4 - c/4

q₁ = (A-c)/3 = 2 = q₂.

Profit for firm one = (A - q₁ - q₂)*q₁ - c*q₁

= [A - 2*(A-c)/3]*(A-c)/3 - c*(A-c)/3

= (A - c)²/9 = 36/9 = 4.

Try this with differing q₁'s see that this is as well as each firm can do in isolation (without explicit collusion).

What would be the incidence of a unit tax in the Cournot model? What is the effect on total quantity produced, Profit_i, net and gross market price? For these issues, how does a unit tax compare with a profits tax?

I will denote pretax variables as just the letter, unit tax variables with a ', and profit tax variables with a ", i.e. q, q', q".

Maximize Profit' over q₁ = (A - q₁' - q₂')* q₁' - (c + t)* q₁'

q₁' = (A - c - t)/3 = q₂' < (A - c)/3 = q_i

P'_Gross = A - q₁' - q₂' = (A + 2*c + 2*t)/3 > (A+2*c)/3 = P

P'_Net = P'_Gross - t = (A + 2*c + 2*t)/3 > (A + 2*c)/3 = P

q₁' + q₂' = 2*(A - c - t)/3 < 2*(A - c)/3 = q₁ + q₂

Profit' = P'_Gross * q₁' - c* q₁' - t* q₁' = [(A - c - t)/3]² < [(A - c)/3]² = Profit

So a unit tax on producers will

i) lower the amount produced by each firm and thus lower total quantity produced,

ii) raise the gross price faced by consumers but lower the net or after tax price received by firms (so the incidence of this tax is spread to both consumers and producers)

iii) lowers the profits to both firms.

Remarkably, and this makes a nice practice exercise, the result of a unit tax on demanders comes out to be just the same as one on suppliers.

What would be the incidence of a profits tax in the Cournot model?

A profits tax of (1-T) would leave net price, gross price, quantities produced by individual firms, and total quantity unchanged. New profits for each firm will be just the old profits multiplied by T. So the incidence would be upon current share holders with the same caveats as for the monopoly model.

Do you find this exercise too trivial? If you desire a purely academic challenge (which won't help with any of the tests), try working the problem with n₀ competing firms and a more interesting cost curve.

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