Micro Lab 6

11/22/2024

1. Impure competition

Cournot competition

Question - Varian 16.10 Consider an industry with 2 firms, each having marginal costs equal to zero. The (inverse) demand curve facing this industry is

$$\begin{array}{}\text{(1)}& P(Y)=100-Y\end{array}$$

where $Y=y_1+y_2$$Y=y_1+y_2$ is total output. (a) What is the competitive equilibrium level of industry output? (b) If each firm behaves as a Cournot competitor, what is firm 1's optimal choice given firm 2's output? (c) Calculate the Cournot equilibrium amount of output for each firm. (d) Calculate the cartel amount of output for the industry. (e) If firm 1 behaves as a follower and firm 2 behaves as a leader, calculate the Stackelberg equilibrium output of each firm.

Question - JR 4.13 (Cournot competition in price)

Duopolists producing substitute goods $q_1$$q_1$ and $q_2$$q_2$ face inverse demand schedules:

$$\begin{array}{}\text{(2)}& p_1=20+\frac{1}{2}{p}_2-q_1\text{ and }{p}_2=20+\frac{1}{2}{p}_1-q_2\end{array}$$

respectively. Each firm has constant marginal costs of 20 and no fixed costs. Each firm is a Cournot competitor in price, not quantity. Compute the Cournot equilibrium in this market, giving equilibrium price and output for each good.

Bertrand competition

Question - JR 4.12

In the Bertrand duopoly of section 4.2.2, market demand is $Q=\alpha -\beta p$$Q=\alpha-\beta p$, and firms have fixed costs and identical marginal cost. Find a Bertrand equilibrium pair of prices, ( $p_1,p_2$$p_1, p_2$ ), and quantities, $(q_1,q_2)$$\left(q_1, q_2\right)$, when the following hold.

(a) Firm 1 has fixed costs $F>0$$F>0$. (b) Both firms have fixed costs $F>0$$F>0$. (c) Fixed costs are zero, but firm 1 has lower marginal cost than firm 2, so $c_2>c_1>0$$c_2>c_1>0$. (For this one, assume the low-cost firm captures the entire market demand whenever the firms charge equal prices.)

2. Price discrimination

First-degree P.D.

Question - Varian 14.10. One common way to price discriminate is to charge a lump sum fee to have the right to purchase a good, and then charge a per-unit cost for consumption of the good after that. The standard example is an amusement park where the firm charges an entry fee and a charge for the rides inside the park. Such a pricing policy is known as a two part tariff. Suppose that all consumers have identical utility functions given by $u(x)$$u(x)$ and that the cost of providing the service is $c(x)$$c(x)$. If the monopolist sets a two part tariff, will it produce more or less than the efficient level of output?

Second-degree P.D.

Question - Varian14.18. There are two consumers who have utility functions

$$\begin{array}{}\text{(3)}& \begin{array}{l}{u}_1(x_1,y_1)=a_1{x}_1+y_1\\ u_2(x_2,y_2)=a_2{x}_2+y_2\end{array}\end{array}$$

The price of the $y$$y$-good is 1 , and each consumer has a "large" initial wealth. We are given that $a_2>a_1$$a_2>a_1$. Both goods can only be consumed in nonnegative amounts.

A monopolist supplies the x -good. It has zero marginal costs, but has a capacity constraint: it can supply at most 10 units of the x-good. The monopolist will offer at most two price-quantity packages, $(r_1,x_1)$$\left(r_1, x_1\right)$ and $(r_2,x_2)$$\left(r_2, x_2\right)$. Here $r_i$$r_i$ is the cost of purchasing $x_i$$x_i$ units of the good. (a) Write down the monopolist's profit maximization problem. You should have 4 constraints plus the capacity constraint $x_1+x_2\le 10$$x_1+x_2 \leq 10$. (b) Which constraints will be binding in optimal solution? (c) Substitute these constraints into the objective function. What is the resulting expression? (d) What are the optimal values of $(r_1,x_1)$$\left(r_1, x_1\right)$ and $(r_2,x_2)$$\left(r_2, x_2\right)$ ?

Third-degree P.D.

Question - Varian 14.19. A monopolist sells in two markets. The demand curve for the monopolist's product is $x_1=a_1-b_1{p}_1$$x_1=a_1-b_1 p_1$ in market 1 and $x_2=a_2-b_2{p}_2$$x_2=a_2-b_2 p_2$ in market 2 , where $x_1$$x_1$ and $x_2$$x_2$ are the quantities sold in each market, and $p_1$$p_1$ and $p_2$$p_2$ are the prices charged in each market. The monopolist has zero marginal costs. Note that although the monopolist can charge different prices in the two markets, it must sell all units within a market at the same price. (a) Under what conditions on the parameters $(a_1,b_1,a_2,b_2)$$\left(a_1, b_1, a_2, b_2\right)$ will the monopolist optimally choose not to price discriminate? (Assume interior solutions.) (b) Now suppose that the demand functions take the form $x_i=A_i{p}_i^{-b_i}$$x_i=A_i p_i^{-b_i}$, for $i=1,2$$i=1,2$, and the monopolist has some constant marginal cost of $c>0$$c>0$. Under what conditions will the monopolist choose not to price discriminate? (Assume interior solutions.)