Question

Consider a version of the Cournot duopoly game, where firms 1 and 2 simultaneously and independently select quantities to produce in a market. The quantity selected by firm $i$ is denoted $q_i$ and must be greater than or equal to zero, for $i=1,2$. The market price is given by $p=100-2q_1-2q_2$. Suppose that each firm produces at a cost of 20 per unit. Further, assume that each firm's payoff is defined as its profit. (If you completed Exercise 5 of Chapter 3, then you have already dealt with this type of game.) Suppose that player 1 has the belief that player 2 is equally likely to select each of the quantities 6,11 , and 13 . What is player l's expected payoff of choosing a quantity of 14 ?

Short answer

The expected payoff for player 1 when choosing a quantity of 14 is approximately 354.67.

Step-by-step solution

Calculating Firm 1's Payoff with Player 2's Quantity 6

First, calculate the payoff of player 1 when player 2 selects a quantity of 6. The market price is given by the equation $p=100-2q_1-2q_2$. By substituting $q_1=14$ and $q_2=6$ into the equation, we find the market price: $p=100-2(14)-2(6)=60$. The cost for player 1 producing 14 units is $20\times 14=280$. Thus, the profit (payoff) is $60\times 14-280=560-280=280$.

Calculating Firm 1's Payoff with Player 2's Quantity 11

Next, calculate the payoff of player 1 when player 2 selects a quantity of 11. Substitute $q_1=14$ and $q_2=11$ into the price equation: $p=100-2(14)-2(11)=50$. With a cost of $280$ (calculated as in the previous step), player 1's profit is $50\times 14-280=700-280=420$.

Calculating Firm 1's Payoff with Player 2's Quantity 13

Then, calculate the payoff for when player 2 selects a quantity of 13. Use the market price formula with $q_1=14$ and $q_2=13$: $p=100-2(14)-2(13)=46$. The profit is $46\times 14-280=644-280=364$.

Calculate Player 1's Expected Payoff for Quantity 14

To find the expected payoff, average the profits obtained for each of player 2's potential quantities, considering that each is equally probable (i.e., probability of 1/3 each). Calculate the expected payoff as: $$\text{Expected Payoff}=\frac{1}{3}(280)+\frac{1}{3}(420)+\frac{1}{3}(364)=\frac{1064}{3}\approx 354.67$$

Key concepts

Cournot Duopoly

The Cournot Duopoly is an essential concept in game theory, where two firms compete in the same market by deciding on the quantity of output they will produce. Each firm makes its decision simultaneously and independently without knowing what the competitor will choose. This strategic interaction leads to an equilibrium where neither firm has an incentive to unilaterally change its output decision, known as the Cournot-Nash Equilibrium.

In this setup, each firm's goal is to maximize its profit, considering the quantity choice of its competitor. The firms affect the market price through their cumulative production, which in turn feeds back into their decisions. The Cournot model applies to industries where firms have some degree of market power, but not enough to influence the market alone, making strategic quantity setting crucial.

In our exercise, firms are tasked with selecting their production quantity. The firm's output decisions directly influence the market price, which depends on the sum of the quantities produced by both firms.

Market Price Calculation

Calculating the market price in a Cournot Duopoly involves determining how the total quantity supplied by both firms affects the price. The market price equation in this scenario is given by $p=100-2q_1-2q_2$, where $q_1$ and $q_2$ are the quantities chosen by each firm.

This equation highlights the inverse relationship between the total quantity supplied $q_1+q_2$ and the market price $p$. As the total production increases, the price decreases. It's a linear demand curve where doubling the quantity produced by one firm decreases the market price correspondingly.

For example, when Firm 1 produces 14 units and Firm 2 produces 6, the market price is calculated by plugging these quantities into the formula: $p=100-2(14)-2(6)=60$. This shows how even small adjustments in either firm's production have direct effects on the pricing, and therefore, on profits.

Profit Maximization

Profit maximization is the primary objective for firms in a Cournot duopoly. Each firm aims to choose a production level that yields the highest profit, given the cost of production and the market price. To compute profit, you subtract the total cost from total revenue.

Total revenue is determined by multiplying the market price $p$ by the quantity $q_1$ that the firm produces. The cost is straightforward, given by the production cost per unit times the number of units produced, here $20\times q_1$.

For example, if the market price is 50 and Firm 1 produces 14 units, the firm's revenue is $50\times 14=700$. With production costs of $20\times 14=280$, the profit is the difference: $700-280=420$. So, profit maximization involves choosing $q_1$ such that profit $(pq-tq)$ is as large as possible, where $t$ is the cost per unit.

Expected Payoff Calculation

Expected payoff calculation is critical in situations of uncertainty where a player is unsure about the other player's quantity selection. Each potential outcome is weighed by its probability of occurrence, providing a "weighted average" outcome for decision-making.

In the exercise, player 1 considers three possible quantities that player 2 might produce: 6, 11, and 13. Assuming all are equally likely, each has a probability of $\frac{1}{3}$. Firm 1 calculates the payoff for each scenario and then computes the expected payoff as a sum of these weighted outcomes.

The expected payoff is calculated: $$\text{Expected Payoff}=\frac{1}{3}(280)+\frac{1}{3}(420)+\frac{1}{3}(364)=\frac{1064}{3}\approx 354.67$$ This value helps the firm make informed decisions considering all potential competitor actions, balancing the risks and rewards associated with different outputs.