Question

In: Economics

Suppose two firms, "A" and "B," form a duopoly in the market for a special type...

Suppose two firms, "A" and "B," form a duopoly in the market for a special type of computer chip. Each firm has a constant marginal cost of $2. The daily market demand for this chip is given by the following equation:

P = 8 – 0.005 Q = 8 – 0.005 (qA+qB).

a. Find an expression for firm #A's revenue, as a function of its own quantity and firm B’s quantity: RevA(qA,qB). [Hint: By definition, RevA = P qA. Here, replace P by the equation for the demand curve.]

b. Find an expression for firm A's marginal revenue, as a function of its own quantity and firm B’s quantity: MRA(qA,qB). [Hint: MRA(qA,qB) = d RevA(qA,qB) / dqA., where the derivative is taken holding qB constant.]

c. Find an expression for firm #A's reaction function, showing how much firm A will produce for any given level of quantity set by the other firm: qA* = f(qB). [Hint: Set MRA = MC and solve for qA as a function of qB .]

d. Since both firms have the same cost, you may assume the equilibrium is symmetric (that is, assume qA* = qB*). Compute firm A's equilibrium quantity qA*.

e. Compute total market quantity Q* and the Cournot equilibrium price P*.

f. Assume marginal cost equals average cost and compute the total profit of both firms together.

g. Compute the social deadweight loss. [Hint: Sketch a graph first.]

Solutions

Expert Solution

Consider the given problem here there are two firms “A” and “B” each have same “MC = 2” and the market demand curve is given by, “P = 8 – 0.005*Q”, where “Q = Qa+ Qb”.

a).

So, the revenue function of “A” is given by, “Ra= P*Qa = (8 – 0.005*Q)*Qa = 8*Qa – 0.005*(Qa+Qb)*Qa.

=> Ra = 8*Qa – 0.005*Qa^2 – 0.005*Qb*Qa.

So, the above equation show the “Revenue function” of “A” as function of “Qa” and “Qb”.

b).

So, the “marginal revenue” function of “A” is given below.

=> ?Ra/?Qa = 8 – 2*0.005*Qa – 0.005*Qb => MRa = 8 – 0.01*Qa – 0.005*Qb.

So, the above equation shows the “MR” of “firm a”.

c).

Here we will get the “reaction function” of “A” by equating “MRa” with “MC”. So, the reaction function of “A” is given by.

=> MRa = MC, => 8 – 0.01*Qa – 0.005*Qb = 2, => 6 – 0.005*Qb = 0.01*Qa.

=> Qa = 600 – 0.5*Qb, be the reaction function of “firm A”.

d).

Now, both the firms have same cost function ,=> “Qa=Qb. SO, by plugging this condition into the above “Reaction function” we will get the optimum quantity production.

=> Qa = 600 – 0.5*Qb, => => Qa = 600 – 0.5*Qa, => 1.5*Qa = 600, => Qa = 600/1.5 = 400.

So, the quantity production by each firm is “Qa=Qb=400”.

e).

So, here the total production is given by, “Q= QA+QB = 2*400 = 800. So, at this quantity the corresponding “P” is given by, “P = 8 – 0.005*Q = 8 – 0.005*800 = 4. So, the quantity and the price combination is given by, (Q, P)=(800, 4).

f).

So, the profit function of “A” is given below.

=> ?A = P*QA - ACa*QA = (P – AC)*QA = (4-2)*400 = 2*400 =800.

Similarly, the profit function of “B” is given by, “=> ?B = P*QB - ACb*QB = (P – AC)*QB = (4-2)*400 = 2*400 =800”.

g).

Now, to find out DWL, we 1st have to find out the “Q” corresponding to “P=MC=2”. So, at MC=2, the corresponding “Q” is “1200”.

So, the DWL is given by the area ABC in the fig. SO, the DWL = (1/2)*(4-2)*(1200-800) = 400.


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