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Chapter 12: Problem 9

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 lights, Everglow and Dimlit. They have identical cost functions: $$\begin{array}{c} C_{i}=10 Q_{i}+\frac{1}{2} Q_{i}^{2}(i=E, D) \\ Q=Q_{E}+Q_{D} \end{array}$$ a. Unable to recognize the potential for collusion, the two firms act as short-run perfect competitors. What are the equilibrium values of \(Q_{E^{\prime}} Q_{D^{\prime}}\) and \(P ?\) What are each firm's profits? b. Top management in both firms is replaced. Each new manager independently recognizes the oligopolistic nature of the light bulb industry and plays Cournot. What are the equilibrium values of \(Q_{E^{\prime}} Q_{D^{\prime}}\) and \(P ?\) What are each firm's profits? c. Suppose the Everglow manager guesses correctly that Dimlit is playing Cournot, so Everglow plays Stackelberg. What are the equilibrium values of \(Q_{E^{\prime}}\) \(Q_{D^{\prime}}\) and \(P ?\) What are each firm's profits? d. If the managers of the two companies collude, what are the equilibrium values of \(Q_{E^{\prime}} Q_{D^{\prime}}\) and \(P ?\) What are each firm's profits?

Short Answer

Expert verified
a. Q_E=Q_D=45, P=55, Profit=992.5, b. Q_E=Q_D=30, P=40, Profit=800, c. Q_E=36, Q_D=27, P=37, Profit_Everglow=704, Profit_Dimlit=569, d. Q_E=Q_D=30, P=40, Profit=800

Step by step solution

01

Perfect Competition

Since Everglow and Dimlit are behaving as short-run perfect competitors, they set marginal cost equal to marginal revenue to maximize profits: MC=MR. The marginal cost using the cost function for each firm, \(MC=10+Q_i\). The marginal revenue is given by the derivative of Price times quantity, \(MR=P+Q\cdot(-1)=100-Q\). Setting these equal gives \(Q = 45\) for each Q_E, Q_D. The equilibrium price is \(P = 55\) (As P=100-Q and Q = Q_E+Q_D). Each firm's profit will be \(\pi= PQ_i - C_i = 45*55 - (10*45 + 0.5*45^2)= 992.5 \)
02

Oligopoly - Cournot Competition

In Cournot competition, each firm decides its own output, considering the output of the competitor as given. Based on the reaction function of firm i to the other firm j \( Q_i = (100 - Q_j -10)/2)\), the equilibrium quantity for both firms can be found: equating \( Q_D = (100 - Q_E -10)/2\) and \( Q_E = (100 - Q_D -10)/2\) with each other. Solving the system of equations results in \( Q_E = Q_D = 30 \) and \(P = 40\). Each firm's profit will be \(\pi= PQ - C = 40*30 - (10*30 + 0.5*30^2)= 800 \)
03

Stackelberg Competition

In a Stackelberg competition, one firm (the leader, Everglow here) moves first and the other firm (the follower, Dimlit here) reacts. Everglow will anticipate Dimlit's reaction when choosing its quantity. Applying the reaction function of Dimlit in Everglow's profit function ends up with best response function of Everglow: \(Q_E= (100 - 10) / 2.5 = 36\). Dimlit then reacts to Everglow's quantity: \(Q_D = (100 - 36 -10) / 2 = 27\). The price will be \(P = 37\) and the profits will be \(\pi_E = 704\) and \(\pi_D = 569\)
04

Collusion

In case of collusion, companies maximize their combined profit. The combined quantity will be \(Q = (100 - 10)/1.5 = 60\) and hence each firm produces \(Q_E=Q_D = Q / 2 = 30\). The price will be \(P = 40\) and the profits will be \(\pi= 800 \) for each

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Most popular questions from this chapter

Two firms compete in selling identical widgets. They choose their output levels \(Q_{1}\) and \(Q_{2}\) simultaneously and face the demand curve $$P=30-Q$$ where \(Q=Q_{1}+Q_{2}\). Until recently, both firms had zero marginal costs. Recent environmental regulations have increased Firm \(2^{\prime}\) s marginal cost to \(\$ 15 .\) Firm \(1^{\prime}\) s marginal cost remains constant at zero. True or false: As a result, the market price will rise to the monopoly level Suppose that two identical firms produce widgets and

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