Question

You are the manager of a firm that manufactures front and rear windshields for the automobile industry. Due to economies of scale in the industry, entry by new firms is not profitable. Toyota has asked your company and your only rival to simultaneously submit a price quote for supplying$\mathbf{10}\mathbf{,}\mathbf{000}$front and rear windshields for its new Highlander. If both you and your rival submit a low price, each firm supplies$\mathbf{50},\mathbf{000}$front and rear windshields and earns a zero profit. If one firm quotes a low price and the other a high price, the low-price firm supplies$\text{100,000}$front and rear windshields and earns a profit of $\mathbf{11}$\(million and the high-price firm supplies no windshields and loses$\mathbf{2}$\)million. If both firms quote a high price, each firm supplies$\mathbf{50},\mathbf{000}$front and rear windshields and earns a $\mathbf{6}$$million profit. Determine your optimal pricing strategy if you and your rival believe that the new Highlander is a “special edition” that will be sold only for one year. Would your answer differ if you and your rival were required to resubmit price quotes year after year and if, in any given year, there was a$\mathbf{60}$percent chance that Toyota would discontinue the Highlander? Explain.

Short answer

The decision to choose (High price) and get$\$11$ will be different from the initial Nash equilibrium (Low price) of$\$0$ if there is a chance that Toyota will discontinue the Highlander.

Step-by-step solution

Finding the matrix for the one-shot game

In case of simultaneous game, our company competes with a single rival company for the price of the front and rear windshields of the Toyota vehicle. In this case, the firms might adopt the following strategies to earn higher profits.

For simplicity we assume that our company is (Player 1) and the rival firm is (Player 2).

The matrix for the one-shot game for both players can be represented as follows:

Finding the dominant one-shot strategy

Our dominant one-shot strategy will be a low price while the dominant strategy of the rival firm will be the low price option too. Therefore, the Nash equilibrium will be generated in the strategy (low, low) in which both players get $(\$0,\$0)$of profit.

Finding the probability

The above decision may change if we know that it will be required to establish a pricing strategy year after year with the rival firm and there may be a chance that Toyota would discontinue the Highlander. Therefore, the probability that the production of the vehicle will be terminated will be$\text{θ=60\%}$, where $\text{0<θ<1}$,

Thus, our firm will adopt the pricing strategy as if it were a trigger strategy scenario, choosing the option that will generate the highest payoff. Thus, the strategy will be (High, High) or a profit of$(\$6,\$6)$ billion.

Determining the optimal pricing strategy

If we decide not to cheat and maintain our cooperation strategy to obtain the highest payoff, we will obtain$\$6$billions profit. Here, the probability is important to keep in mind that where$\text{θ}$ is closer to 1. The more likely we are to cheat to obtain a greater profit the game will end in the next period.

The firm that decides to opt for a strategy of cheat it will obtain the following profit for either of the two firms:

${\text{Π}}_{\text{cheat}}\text{=11}$

Therefore,

${\text{Π}}_{\text{cheat}}{\text{>Π}}_{\text{coop}}$

Thus, there is benefit of cheating the rival firm today and obtaining a greater benefit instead to cooperate. This is because the probability of the game ending in the next period is high enough.

This is why our decision to choose (High price) and get$\$11$ will be different from the initial Nash equilibrium (Low price) of$\$0$ if there is a$60\%$ chance that Toyota will discontinue the Highlander.