Cournot Oligopoly

An alternative model of oligopoly uses quantity setting rather than price as the competitive weapon. Cournot assumes that if firm 1 has already determined its output/sales, then firm 2 will make its choice of output on the assumption that firm 1 will not change its output in any given period. The total output of the two firms will then determine market price.

We initially explain the model by the use of Figure 9.6. The market demand curve is AD, the marginal revenue curve is AM and the marginal cost curve, assuming constant costs, is CE. In part (a) firm 1 initially acts as a monopolist and sets the monopoly price OP1 and quantity OQ1. Firm 2 conjectures that firm 1 will continue to sell OQ1, leaving it with the residual demand curve FED and the marginal revenue curve FN. Firm 2 maximizes profit and sells QiQ2. Total sales are now OQ2.

In the next round, firm 1 assumes that firm 2 will continue to sell QiQ2. Firm 1's residual demand curve now has an intercept on the horizontal axis which is derived by deducting QiQ2 from AD, with the same slope as the original demand curve; this is shown as the demand curve ST in part (b) of Figure 9.6. Firm 1 chooses its profit-maximizing output OQ3. The residual demand curve for firm 2 then becomes UT and it selects output Q3Q4. This process continues until each firm faces identical demand curves; these are shown in part (c), where firm 1 sells OQ5 and firm 2 produces OQ6.

N

S

. S

U

\

\ N

V

\

E

\

\ S \ \

\

\ X \ s \ s

\

Q5 M

N R

S

T W

C

(c) Final outcome Figure 9.6 Cournot duopoly

(c) Final outcome Figure 9.6 Cournot duopoly

Both firms charge price OP3, and total supply to the market is two-thirds the competitive level of sales.

Assuming a market demand curve of P = 100 — Q and a constant marginal cost of 10, the competitive output will be 90 and the monopoly output 45. In a Cournot oligopoly, convergence of the sales of both firms is shown in Table 9.1, for five rounds. Firm 1 initially sets the monopoly price and makes profits of £2,025. The sales of firm 1 reduce from the monopoly output of 45 toward 30, and the sales of firm 2 increase from 22.5 toward 30. Eventually, both firms have a residual demand curve of P = 70 — Q, both selling 30 with a market price of £40.

Cournot equilibrium and reaction functions

An alternative approach to explaining equilibrium in Cournot oligopoly is to construct reaction curves for both firms in a duopoly. A reaction curve, or best response

Table 9.1 Cournot duopoly, collusion and cheating (£)

Period

Firm 1's sales

Firm 2's sales

Market sales

Market price

Firm 1

Profits Firm 2

Industry

Cournot duopoly

1

45.00

0.00

45.0

55.0

2,025.0

0.0

2,025.00

2

45.00

22.5

67.5

32.5

1,012.5

506.25

1,518.00

3

33.75

28.125

61.875

38.125

949.21

791.02

1,740.23

4

30.94

29.53

60.47

39.53

913.65

872.02

1,785.67

5

30.24

29.88

60.12

39.88

903.57

892.81

1,796.38

Final

30.00

30.00

60.00

40.00

900.00

900.00

1,800,00

Collusion

22.5

22.5

45.0

55.0

1,012.5

1,012.5

2,025,0

Cheating

Firm 1

23.5

22.5

46.0

54.0

1,034.0

990.0

2,024.0

Firm 2

22.5

23.5

46.0

54.0

990.0

1,034.0

2,024.0

Note Assumes market demand is given by P — 100 — Q, marginal revenue by 100 — 2Q and marginal cost is equal to 10.

Source Author function, for firm 1 defines the profit-maximizing output for firm 1, given the output of firm 2. Given that firm 2 sells Q2 units of output, firm 1's output can be expressed as Q1 — R1(Q2). For firm 2 the reaction function is given by Q2 — R2(Q1).

Reaction functions for a duopoly are shown in Figure 9.7, where firm 1's sales are measured on the horizontal axis and firm 2's on the vertical axis. The horizontal intercept of firm 1's reaction curve Qf1 assumes that firm 2 sells nothing and that firm

1 behaves as a profit-maximizing monopolist. The vertical intercept of firm 1's reaction curve QC assumes that firm 1 sells nothing and firm 2 sells the competitive output where price is equal to marginal cost. Using the demand equation P — 100 — Q, the horizontal intercept would be at sales of 45 and the vertical intercept at sales of 90. Firm 2's reaction curve is derived in a similar way, with the vertical intercept QM having a value of 45 and the horizontal intercept QC a value of 90.

If firm 1 initially behaves as a monopolist, then it will sell output OQf on reaction curve R1. Firm 2 will respond by choosing point A on its reaction curve (R2), selling output OQ2. Firm 1 reacts by moving to point B on its reaction curve, producing OQ2. Firm 2 will respond by moving to point C on its reaction curve, producing output OQ2. The process continues until Cournot equilibrium is reached at point E, where firm 1 sells OQ1 and firm 2 sells OQ|; this is a position from which neither firm would want to move, given the other firm's output.

Reaction functions can also be derived algebraically. To maximize profits, firm 1 must set marginal revenue equal to marginal cost for any given level of firm 2's output. Therefore, when P — 100 — Q:

Total revenue (IR) — QiP — Qi(100 — Qi — Q2) or R — 100Qi — Qj — QiQ2 Marginal revenue (MR) — STR1/SQ1 — 100 — 2Q1 — Q2

Since marginal cost is equal to 10, marginal revenue equal to marginal cost can be expressed as:

Qi = f (Q2) = [(90 - Q2)/2] = 45 - 2 Q2 Q2 = f (Qi) = [(90 - Qi)/2] = 45 - 2Qi

If firm 2 sells 40 units, then firm i would choose to produce [(90 - 40)/2], or 25. If firm 2 produces 30, then firm i would produce [(90 - 30)/2], or 30. The equilibrium output is to be found at point E in Figure 9.6. At this point, both firms make profits of £900, which is derived by deducting marginal cost from price multiplied by output or (40 - i0) * 30.

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