Chapter 12, Problem 20P

(a)

To determine

The stream of profits generated by the firms.

Explanation of Solution

When both produce 2,000 otters, each one would receive only the profit of $400, whereas when both produce 1,000 otters, each would receive a profit of $500. When the collusion is introduced, the objective of the collusion would be to restrict the output of otters and to maximize the profits of the members. Thus, the collusion would lead to the situation where each would produce 1,000 otters and maximize their profit t $500 each.

When the firms adopt the grim strategy where the two players will cooperate until when one cheats, the other will immediately stop the cooperation and react. When the profits of each future year are calculated by the help of the discounting, the profit of $500 will be multiplied with each year discounting. Thus, the stream of discounts will be as follows:

$\text{Stream of Profits}=500+\left(\text{500}\times \text{d}\right)+\left(\text{500}\times {\text{d}}^2\right)+\mathrm{...}+\left(\text{500}\times {\text{d}}^{\text{n}}\right)$

Economics Concept Introduction

Game theory: Game theory is the study of how people who make decisions in situations in which attaining their goals depend on their interactions with others.

(b)

To determine

The stream of profits generated by the firms.

Explanation of Solution

When the firm decides to reduce the output by decreasing the output from the year 2 onward, it means that the firm would produce $700 profit in the first year by expanding the output and from year 2 onward and the output would be limited in such a way that the profit earned would become $400. Thus, the stream of profits that the country could make can be depicted as follows:

$\text{Stream of Profits}=700+\left(\text{400}\times \text{d}\right)+\left(\text{400}\times {\text{d}}^2\right)+\mathrm{...}+\left(\text{400}\times {\text{d}}^{\text{n}}\right)$

(c)

To determine

The profitability of the firm when the value of d is 0.5, 0.99 and 0.01.

Explanation of Solution

When the firm abides by the agreement, the firm can get the profit explained by the part a, and when it cheats, the profit changes to the one explained in the part b. This can be calculated by substituting the value of d into the equations as follows:

When the firm abides by the agreement,

$\begin{array}{c}\text{Stream of Profits}=500+\left(\text{500}\times \text{d}\right)+\left(\text{500}\times {\text{d}}^2\right)+\mathrm{...}+\left(\text{500}\times {\text{d}}^{\text{n}}\right)\\ =500+\frac{500\text{d}}{1-\text{d}}\\ =\frac{500}{1-0.5}\\ =1,000\end{array}$

When the firm cheats, the profit sum would be as follows:

$\begin{array}{c}\text{Stream of Profits}=700+\left(\text{400}\times \text{d}\right)+\left(\text{400}\times {\text{d}}^2\right)+\mathrm{...}+\left(\text{400}\times {\text{d}}^{\text{n}}\right)\\ =700+\frac{400\text{d}}{1-\text{d}}\\ =700+400\\ =1,100\end{array}$

Thus, when the d is 0.5, the firm would cheat in order to maximize their profit. Similarly, the value of d with 0.99 and 0.01 can be calculated as follows:

When the firm abides by the agreement:

$\begin{array}{c}\text{Stream of Profits}=500+\left(\text{500}\times \text{d}\right)+\left(\text{500}\times {\text{d}}^2\right)+\mathrm{...}+\left(\text{500}\times {\text{d}}^{\text{n}}\right)\\ =500+\frac{500\text{d}}{1-\text{d}}\\ =\frac{500}{1-0.99}\\ =50,000\end{array}$

When the firm cheats, the profit sum would be as follows:

$\begin{array}{c}\text{Stream of Profits}=700+\left(\text{400}\times \text{d}\right)+\left(\text{400}\times {\text{d}}^2\right)+\mathrm{...}+\left(\text{400}\times {\text{d}}^{\text{n}}\right)\\ =700+1\left(\frac{400\text{d}}{1-\text{d}}\right)\\ =300+40,000\\ =40,300\end{array}$

Therefore when the value of d is 0.99, the firm would abide by the agreement as the profit received from cheating is very low. When the value of d is 0.01, the profit can be calculated as follows:

When the firm abides by the agreement:

$\begin{array}{c}\text{Stream of Profits}=500+\left(\text{500}\times \text{d}\right)+\left(\text{500}\times {\text{d}}^2\right)+\mathrm{...}+\left(\text{500}\times {\text{d}}^{\text{n}}\right)\\ =500+\frac{500\text{d}}{1-\text{d}}\\ =\frac{500}{1-0.01}\\ =505.05\end{array}$

When the firm cheats, the profit sum would be as follows:

$\begin{array}{c}\text{Stream of Profits}=700+\left(\text{400}\times \text{d}\right)+\left(\text{400}\times {\text{d}}^2\right)+\mathrm{...}+\left(\text{400}\times {\text{d}}^{\text{n}}\right)\\ =700+1\left(\frac{400\text{d}}{1-\text{d}}\right)\\ =700+4.04\\ =704.04\end{array}$

Thus, when the value of d is 0.01, the country would cheat as it provides the higher profit to the firm.

(d)

To determine

The level of d where the firm is indifferent between cheating and abiding.

Explanation of Solution

The point at which the value of d where the country becomes indifferent between cheating and abiding with the law can be calculated by setting the sums from part a that indicates the abiding with agreement outcome equal to the sum from the part b that indicates the cheating outcome. This can be calculated as follows:

$\begin{array}{c}500+\left(\text{500}\times \text{d}\right)+\left(\text{500}\times {\text{d}}^2\right)+\mathrm{...}+\left(\text{500}\times {\text{d}}^{\text{n}}\right)=700+\left(\text{400}\times \text{d}\right)+\left(\text{400}\times {\text{d}}^2\right)+\mathrm{...}+\left(\text{400}\times {\text{d}}^{\text{n}}\right)\\ \frac{500}{1-\text{d}}=700+\frac{400}{1-{\text{d}}^1}\\ 500=700\left(1-\text{d}\right)+400\text{d}\\ 700-500=700\text{d}-400\text{d}\\ \text{200}=300\text{d}\\ \text{d}=\frac{200}{300}\\ =\frac{2}{3}\end{array}$

Thus, the value of d equal to 2/3 which is almost equal to 0.67, the firm would be indifferent between the cheating and abiding to the agreement.