4. [6 + 87 21 pts] Consider a good whose market demand is givern by P = 120-Q. T

Question

4. [6 + 87 21 pts] Consider a good whose market demand is givern by P = 120-Q. There are n firms producing this good: all the firms are identical and have costs of production c(1) 1. (i) Consider the game where the n firms compete in quantity; they set their outputs simultaneously, and the market price is given by P 120-Q, where Q is the total output of the n firms. Each firm is interested in maximizing its own profit, and takes into the account the effect that its own production has on the market price. Derive how much each firm produces in the Nash equilibrium of this Cournot oligopoly game. (ii) Take the case of n 5 firms. The CEOs of the firms get together and realize that since they are going to play this game repeatedly over time, it would be good to form a cartel in which each firm commits to producing q- 10 (which is the quantity that maximizes joint profits). If any of them deviate from this arrangement, they will all go back to playing the one-shot Cournot Nash equilibrium of part (i) forever. All firms discount future payoffs at the rate of . How large must this discount rate be in order to support the cartel's resolution as a Nash equilibrium of this infinitely repeated game?

Explanation / Answer

Consider the given problem here the market demand curve be, “P=120-Q”, where “Q=Q1+Q2+ … +Qn”.

There are “n” firms and the cost function are identical for all firms.

i).

let’s assume that the demand curve be, “P = a - Q”, where a=120. So, the profit function of the 1st firm is given by.

P1 = P*Q1 – C1 = (a-Q1-Q2…-Qn)*Q1 – C1 = a*Q1 – Q1^2 – Q1*Q2 - … - Q1*Qn – C1.

So, FOC for profit maximization require, dP1/dQ1 = 0, => (a – 2*Q1 – Q2 - … - *Qn) – MC1 = 0.

=> (a– Q2 – Q3 - … - *Qn) – MC1 = 2*Q1, where “MCi = 1” for all firms.

=> (1/2)*(a – 1) – (1/2)*Q2 - (1/2)*Q3 - … - (1/2)*Qn = Q1, this is the “reaction function” of the 1st firms.

So, similarly we can derive the reaction function for “ith” firms.

=> (1/2)*(a – 1) – (1/2)*Q1 - (1/2)*Q2 - … - (1/2)*Qn = Qi ………………….(2)

Now, here all firms are identical with respect to “cost” function as well as with respect to “Reaction function”, => Qi = q = same for all firms.

Now, by substituting the above condition into (2) we get the optimum solution.

=> Qi = (1/2)*(a – 1) – (1/2)*Q1 - (1/2)*Q2 - … - (1/2)*Qn = (1/2)*(a – 1) – (n-1)*(1/2)*Qi.

=> Qi + (n-1)/2 * Qi = [1+ (n-1)/2]*Qi = [2 + n – 1]*Qi/2 = (n + 1)*Qi/2 = (1/2)*(a-1).

=> (n + 1)*Qi/2 = (1/2)*(a-1), => Q*i = (a-1)/(n+1)

So, under “cournot oligopoly” each firm will produce at Q*i=(a-1)/(n+1), where “a=120”, this is NE of this game.

ii).

Now, suppose that “n=5”, if they cartel where each will produce “q=10”, total output is “Q=5*10=50.

=> The profit of the each firm is given by, P*q – C = (120 – 50)*10 – 1*10 = 700 -10 = 690 > 0.

Now, here under cournot model qi=(120-1)/6 =19.8, so here each producer have 2 strategy “qi=10” or “qi=19.8”.

So, if one firm will charge “qi=19.8” and other 4producer will charge, “10” then total output, “10*4+19.8=59.8”.

So, the profit of the ith firm is “P*qi – Ci, => 60.2*19.8 – 1*19.8 = 1191.96 – 19.8 = 1172.16 > 690.

The profit of the other firms, “P*q – C, => 60.2*10 – 1*10 = 602 – 10 = 592 < 690.

Now, profit if all firm all produce under cournot model, “21*19.8 – 19.8 = 415.8 – 19.8 = 396, where P=21, qi=19.8.

So, a firm will support the cartel if the following condition will hold.

=> (690 + 690*d + 690*d^2 + …..) > (1172.16 + 396*d + 396*d^2 + 396*d^3 + …), “d” be the discount factor.

=> 690 + 690*d/(1-d) > 1172.16 + 396*d/(1-d), => 690*d/(1-d) > (1172.16 – 690) + 396*d/(1-d),

=> 690*d/(1-d) - 396*d/(1-d) > (1172.16 – 690) = 482.16.

=> d/(1-d)*(690 – 396) = 294*d/(1-d) > 482.16, => d/(1-d) > 482.16 / 294 = 1.64.

=> d/(1-d) > 1.64, => d > 1.64*(1-d) = 1.64 – 1.64*d, => 1.64*d + d > 1.64, => d > 1.64/2.64 = 0.62.

=> to support the cartel the “discount factor” should be “more than 0.62”.

So, if the discount factor is more than "0.62", then the NE of this infinite game is the corresponding cartel outcome.

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