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STRATEGIC SITUATION

Individuals face a strategic situation if the payoff to each person depends directly on the actions chosen by other persons, and each person is aware of this fact when he chooses his action.

Game theory is a formal way of modeling strategic situations. The most widely used solution concept in game theory is the Nash equilibrium.

A set of strategies constitute a Nash equilibrium if, holding the strategies of all other players unchanged, no single player can attain a higher payoff by choosing a different strategy.

Let S_i be the set of all possible strategies for player i. Let s_i denote an element of S_i, and let s_-i denote the strategies of all players other than i. The payoff to individual i is represented by y_i(s₁, ..., s_N). The group of strategies (s₁^*, ..., s_N^*) constitute a Nash equilibrium if, for each player i,

y_i(s_i^*, s_-i^*) >= y_i(s_i,s_-i^*) for all s_i in S_i.

A Nash equilibrium

• need not be Pareto efficient
• is "self-enforcing"
• need not be unique
• need not exist in pure strategies

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COURNOT MODEL

Assumptions:

• quantity is the strategic variable (Nash equilibrium in quantities)
• price clears the market
• homogeneous goods (not essential)

Let P = P(q₁,q₂) be the inverse demand function. Let C₁(q₁) and C₂(q₂) be the cost functions of firms 1 and 2. If (q₁^*, q₂^*) is a Nash equilibrium, then

q₁^* maximizes P(q₁,q₂^*)q₁ - C₁(q₁) q₂^* maximizes P(q₁^*,q₂)q₂ - C₂(q₂)

These two conditions can be written in terms of two first order conditions:

P + q₁^*P' - C₁' = 0 P + q₂^*P' - C₂' = 0

• price exceeds marginal costs
• marginal costs need not be equalized across firms, i.e., production is inefficient
• compare the monopoly pricing equation: P + Q^MP' - C' = 0 Oligopolists only take into partial account the effect of their output expansion on output price. Therefore equilibrium price is lower and equilibrium quantity is higher than the monopoly solution.
• we can re-arrange the first order conditions to get (P - MC_i)/P = a_i/e where a_i is the market share of firm i and e is the industry's elasticity of demand.

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DIAGRAMATIC REPRESENTATION

The best response function (also known as reaction function) shows the optimal (i.e., profit-maximizing) output q₁ as a function of q₂. It is implicitly defined by the first order condition for firm 1:

P'(q₁ + q₂)q₁ + P(q₁ + q₂) - C₁'(q₁) = 0. We can write q₁ = BR₁(q₂). Apply a similar reasoning to firm 2, we can write q₂ = BR₂(q₁).

The Nash equilibrium can alternatively be defined as the solution to the two equations

q₁ = BR₁(q₂). q₂ = BR₂(q₁). Diagramatically, the Nash equilibrium can be represented by the intersection of the best response curves.

In the Cournot model, the best response function is often assumed to be downward sloping because an increase in other firms' output will tend to reduce the incentive for each firm to increase output. (That is, d/dq_j(dy_i/dq_i) tends to be negative.)

When best response functions are downward sloping, we call the strategic variables strategic substitutes.

We can also draw iso-profit curves for firm 1 that shows combinations of q₁ and q₂ that give firm 1 equal profits. These curves have the following characteristics:

• the preference direction is downwards
• the isoprofit curve has an inverted U shape and it reaches a maximum on the BR₁ curve

Putting the best response functions and the iso-profit curves together, we can see

• there is potential for profitable collusion
• there are enforcement problems associated with collusion

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EXAMPLES

1. Let

• D = 1 - P,
• C₁ = c₁q₁,
• C₂ = c₂q₂,

Firm 1 maximizes (1 - q₁ - q₂)q₁ - c₁q₁. The first order conditions is (1 - q₁ - q₂) - q₁ - c₁ = 0. Rearrange this to q₁ = (1 - q₂ - c₁)/2 =BR₁(q₂). Note

• the BR function is downward sloping
• the BR function will shift out when marginal cost falls

Write a similar first order condition for firm 2: (1 - q₁ - q₂) - q₂ - c₂ = 0. Solve the two first order equations simultaneously to get the Nash equilibrium output: q₁^* = (1 - 2c₁ + c₂)/3. q₂^* = (1 + c₁ - 2c₂)/3. Substitute these equilibrium output back into the original profit function to get the equilibrium profits y₁^* = [(1 - 2c₁ + c₂)/3]². y₂^* = [(1 + c₁ - 2c₂)/3]². Notice that firm 1's output and profits depends (positively) on firm 2's cost.

2. Let

• D = 1 - P
• n firms, each with the same marginal cost = c

Consider a typical firm i. Firm i maximizes

(1 - q₁ - ... - q_n)q_i - cq_i. The first order condition is (1 - q₁ - ... - q_n) - q_i - c = 0. In a symmetric equilibrium, q₁ = ... = q_n = q. We can then rewrite the first order condition in the form: 1 - nq - q - c = 0. (Note it is important to impose symmetry only after deriving the first order condition.) Solve the above to get q = (1 - c)/(n + 1). Q = nq = n(1 - c)/(n + 1) P = 1 - Q = (1 + nc)/(n + 1) Note

• dP/dn < 0
• P approaches c when n approaches infinity

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STACKELBERG MODEL

Consider a two-stage game where firm 1 (the leader) chooses its output level q₁ in the first stage. In stage 2, firm 2 (the follower) chooses its output level q₂, knowing what q₁ is.

The Stackelberg leader is implicitly assumed to have some ability to commit to its chosen output level q₁. It will not change q₁ after knowing what q₂ is, even though such a change might be profitable ex post.

To solve the Stackelberg game, we proceed backwards. In stage 2, firm 2 maximizes profits taking q₁ as given. The optimal q₂ is given by

q₂ = BR₂(q₁). This BR function is just like the one in the Cournot model.

In stage 1, firm 1 chooses q₁ to maximize its profits, taking into account firm 2's optimal response:

maximize P(q₁ + BR₂(q₁))q₁ - C₁(q₁). The first order condition is (1 + BR'₂)P'q₁ + P = MC₁.

Compare with the Cournot outcome, where P'q₁ + P = MC₁, we can see that

q₁^S > q₁^C. Moreover, since BR'₂ < 0, q₂^S = BR₂(q₁^S) < BR₂(q₁^C) = q₂^C.

EXAMPLE

• D = 1 - P
• C₁ = c₁q₁
• C₂ = c₂q₂

As in Example 1 in the section on Cournot model, Firm 2's best response function is

BR₂(q₁ = (1 - q₁ - c₂)/2.

Firm 1 maximizes

(1 - q₁ - (1 - q₁ - c₂)/2)q₁ - c₁q₁. The solution gives q₁^S = (1 + c₂ - 2c₁)/2. Substitute this back into the best response function for firm 2, we get q₂^S = (1 - 3c₂ + 2c₁)/4.

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BERTRAND MODEL

The Bertrand model can be interpreted as a game in which prices are the strategic variables.

Example 1.

Assume

• homogeneous products
• constant marginal costs with no capacity constraint

Let D(p) be the market demand function. Then the demand function for firm 1 is D₁(p₁,p₂)=

D(p₁) if p₁ < p₂;
D(p₁)/2 if p₁ = p₂;
0 if p₁ > p₂.
Notice that this demand function is discontinuous.

• price = c₁ (minus a very small number) if c₁ > c₂
• with more than 2 firms, price is equal to the second lowest marginal cost
• if firms have the same marginal cost, price = marginal cost.
• pure strategies equilibrium may not exist if marginal costs are not horizontal

Example 2.

Assume

• differentiated products; e.g., D₁(p₁,p₂) is the demand curve for firm 1. D₁ is decreasing in the first argument and increasing in the second
• cost functions are C₁(q₁) and C₂(q₂)

Suppose the demand function takes the form D₁ = a - bp₁ + dp₂. Then profits for firm 1 is

y₁ = (a - bp₁ + dp₂)p₁ - C₁(a - bp₁ + dp₂) We have

• dy₁/dp₁ = (a - 2bp₁ + dp₂) + bC'₁(a - bp₁ + dp₂)
• d/dp₂(dy₁/dp₁) = d + bdC''₁(a - bp₁ + dp₂)

The incentive the raise p₁ is higher when p₂ is higher. This suggests that the best response function is upward sloping.

When the best response function is upward sloping, we call the strategic variables strategic complements.

In a Bertrand model with differentiated goods

• we usually assume that the best response functions are upward sloping
• the isoprofit curves for firm 1 has the following characteristics
  • the preference direction is upwards
  • it has a U shape and it reaches its lowest point on the BR₁ function

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CONJECTURAL VARIATIONS MODEL

The conjectural variations model cannot be interpreted as the Nash equilibrium of a game. Yet the model is useful for describing the degree of monopoly power in oligopolistic markets.

Consider a duopoly with homogeneous goods. Let P = P(q₁ + q₂). Define a parameter L_ij to be i's conjecture of j's output response when q_i increases. I.e., L_ij = dq_j/dq_i.

Firm i maximizes P(q_i + q_j)q_i - C_i(q_i). The first order condition is:

P'q_i(1 + L_ij) + P - C'_i = 0.

• if L = -1, then P = C' (perfect competition)
• if L = 0, then P'q_i + P = C' (Cournot model)
• if L = 1/s - 1 (s is market share), then P'Q + P = C' (monopoly)

Another possibility is to impose the condition that L_ij has to be equal to the equilibrium dq_j/dq_i. This is known as the consistent conjectural variations model.

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Lecture Notes

Klemperer

Fat cats