Question

Question 3 Market demand is given byp(Q) 140- Q if Q < 140 There are two firms, each with unit cost of GHC20. Further assume that firms can choose any quantity or , otherwise. Define the reaction functions of the firms Find the Cournot equilibriunm Compare the Cournot equilibrium to the perfectly competitive outcome and to the monopoly outcome . One possible strategy for each firm is to produce half of the monopolist quantity. Would the resulting outcome be better for both firms (Pareto dominant)? Explain why this is not the equilibrium outcome of the Cournot game.

Answer 1

Let us suppose that the two firms are denoted by f1 and f2. Suppose that firm 1 produces q1 and firm 2 produces q2.

Q=q1+q2, p(Q)=p(q1,q2)=140-q1-q2; if Q<140, p(Q)=0, if Q>=140

Each firm aim to maximize the profit generated by manufacturing and selling the product in the market.

The profit function of f1 is  $\pi_1(q_1, q_2)=q_1.p(q_1,q_2)-c(q_1)$

and profit function of f2 is $\pi_2(q_1,q_2)=q_2.p(q_1,q_2)-c(q_2)$

(Note that GHC refers to Ghanaian cedi, the currency of Ghana)

i.e.

​​​​​​$\pi_1(q_1,q_2)=q_1.(140-q_1-q_2)-20q_1$

$\pi_2(q_1,q_2)=q_2.(140-q_1-q_2)-20q_2$

$$If $q_1&plus;q_2\geq140\\ $then$ \ p=0, $ and for any $q_1, q_2 > 0, \pi_1,\pi_2 < 0$ then the optimal option for both the firms would be to produce zero units and have zero profits.$$

Both the firms aim to maximize their profits choosing their own quantities, so using First Order Conditions w.r.t. to their own quantities,

$\frac{\partial\pi_1(q_1,q_2) }{\partial q_1}=0\\\\ \implies 140-2q_1-q_2-20=0\\ and\\\\ \frac{\partial\pi_1(q_1,q_2) }{\partial q_2}=0\\\\ \implies 140-2q_2-q_1-20=0\\$

The reaction function becomes:

$q_1=r_1(q_2)=60-q_2/2;\newline q_2=r_2(q_1)=60-q_1/2$

where r1(q2) is the reaction function of firm 1 taking quantity produced by f2 to be given. and r2(q1) represents the reaction function of firm 2 taking quantity produced by f1 to be given.

b) Cournot Equilibrium can be found by solving the two reaction functions assuming that the other firm is maximizing their output.

$q_1^*=60-q_2*/2;\newline q_2^*=60-q_1^*/2.\\$Putting $q_2^*$ in $r_1(q_2^*)$, we have$\\ q_1^*=60-(60-q_1^*/2)*/2 =30&plus;q_1^*/4 \\\implies q_1^*=40 $ units.$$

Using this q*₁ to find q*2 we get $q_2^*=60-40/2=40$

Therefore, Cournot Equilibrium is $(q_1^*, q_2^*)=(40,40)$

p=140-40-40=GHC 60

The profits earned by both the firms are, $\pi_1=\pi_2=p.40-20.40=60.40-20.40= $GHC $1600$

c) In the monopoly outcome, there is only one firm and it is free to choose the price in order to maximize the profit.

Let the monopoly quantity be Q

$p=140-Q\\ \implies Q=140-p\\ \pi_(p)=(140-p).p-20(140-p)\\$

$\frac{\partial\pi(p) }{\partial p}=0\\\\ \implies 140-2p&plus;20=0\\\implies p^*=$GHC $80\\ $Hence, Q=140-p=60 units.$ \\ $Monopoly profit,$ \pi=60.80-20.60=$GHC $ 3600$

Under perfect competition, both the firms take prices to be given and try to maximize the profit. They behave exactly the same as in Cournot Equilibrium. Hence profits under perfect competition are : $\pi_1=\pi_2= $GHC $1600$

d) If firms produce half the monopoly q1=q2=30, total quantity their total profits would be half the profits they earn in monopoly. profits would be GHC 1800 for both the firms. This case would be still better than the perfect competition case. But this is not the Cournot Equilibrium because if q1 produces 30, f2's best response according to r2(q1) would be to produce 45 units and capture higher market share. and profit for f2 would be GHC 1800> profit of f1 which is GHC 1200. Therefore, f1 would try to seek this opportunity by producing more till the point both the firms end up producing 40 units each as in C.E. Pareto outcome is not a sustainable equilibrium