Question

Demand for light bulbs can be characterized by Q=100−p, where Q is in millions of boxes of lights sold and p is the price per box. There are two producers of light bulbs, firm 1 and firm 2. They have identical cost functions: Ci(Qi)=10Qi+21Qi2,i=1,2. a) (2pt) Unable to recognize the potential for collusion, the two firms act as perfectly competitive (price-taking) firms. What are the equilibrium values of the firms' outputs and market price? What are each firm's profits? b) (2pt) Top management in both firms is replaced. Each new manager independently recognizes the oligopolistic nature of the light bulb industry and engage in Cournot competition. What are the equilibrium values of the firms' outputs and market price? What are each firm's profits? c) (2pt) Suppose the manager of firm 1 guesses correctly that firm 2 is best responding to firm 1's output, so firm 1 behaves as the Stackelberg leader. What are the equilibrium values of the firms' outputs and market price? What are each firm's profits? 1 d) (1pt) If the managers of the two firms collude by forming a cartel to maximize joint profits. How many light bulbs will be produced? What are each firm's profits? e) (1pt) Suppose firm 2 abides by the cartel agreement, but Firm 1 cheats by increasing production. How many light bulbs will firm 1 produce? What will be each firm's profits?

Answer 1

a) In a perfectly competitive scenario, the equilibrium values of the firms' outputs and market price can be determined by setting the market demand equal to the sum of the individual firms' outputs.

b) In Cournot competition, the equilibrium values can be found by solving for the Nash equilibrium, where each firm's output is a best response to the other firm's output.

c) In the Stackelberg leader scenario, Firm 1 chooses its output level first, and then Firm 2 chooses its output level as a best response to Firm 1's output.

d) The profits for each firm can be calculated by substituting their equilibrium output levels into their respective profit functions.

Step-by-step explanation:

a) Equilibrium values

When the two firms act as perfectly competitive (price-taking) firms, the equilibrium values can be determined by setting the market demand equal to the sum of the firms' individual outputs.

The market demand is represented by Q = 100 - p, where Q is the quantity and p is the price per box.

The equilibrium values can be found by substituting the market demand into the individual firms' cost functions and solving for the firms' outputs.

Firm 1:

Cost function: C1(Q1)=10Q1+21Q1^2

Substituting the market demand: Q = 100 - p

Profit function: π1 = p * Q1 - C1(Q1)

By maximizing the profit function, we can find the equilibrium value of Firm 1's output.

Firm 2:

Cost function: C2(Q2)=10Q2+21Q2^2

Profit function: π2 = p * Q2 - C2(Q2)

By maximizing the profit function, we can find the equilibrium value of Firm 2's output.

The market price can be found by substituting the equilibrium values of the firms' outputs into the market demand equation.

b) Equilibrium values

When the firms engage in Cournot competition, they will independently choose their output levels to maximize their profits. The equilibrium values can be found by solving for the Nash equilibrium, where each firm's output is a best response to the other firm's output.

The market price can be found by substituting the equilibrium values of the firms' outputs into the market demand equation.

The profits for each firm can be calculated by substituting their equilibrium output levels into their respective profit functions.

c) Equilibrium values

When Firm 1 behaves as the Stackelberg leader, it will choose its output level first, and then Firm 2 will choose its output level as a best response to Firm 1's output. The equilibrium values can be found by solving these two steps.

The market price can be found by substituting the equilibrium values of the firms' outputs into the market demand equation.

d) The profits for each firm can be calculated by substituting their equilibrium output levels into their respective profit functions.