Question

The inverse market demand in a homogeneous-product Cournot duopoly is$\mathbf{\text{P2003}}\mathbf{\left(}{\mathbf{\text{Q}}}_{\mathbf{\text{1}}}{\mathbf{\text{+Q}}}_{\mathbf{\text{2}}}\mathbf{\right)}{\mathbf{\text{andcostsareC}}}_{\mathbf{\text{1}}}\mathbf{\left(}{\mathbf{\text{Q}}}_{\mathbf{\text{1}}}\mathbf{\right)}{\mathbf{\text{26Q}}}_{\mathbf{\text{1}}}{\mathbf{\text{andC}}}_{\mathbf{\text{2}}}\mathbf{\left(}{\mathbf{\text{Q}}}_{\mathbf{\text{2}}}\mathbf{\right)}{\mathbf{\text{32Q}}}_{\mathbf{\text{2}}}$.

a. Determine the reaction function for each firm.

b. Calculate each Firm’s equilibrium output.

c. Calculate the equilibrium market price.

d. Calculate the profit each firm earns in equilibrium.

Short answer

a. Reaction function of firm 1 is$\text{29}-\frac{\text{1}}{\text{2}}{\text{Q}}_{\text{2}}$,and firm 2 is$\text{28}-\frac{\text{1}}{\text{2}}{\text{Q}}_{\text{1}}$

b.Equilibrium output of firm 1 is 20, and firm 2 is 18.

c.Equilibrium market price is$\text{P=86}$

d.The profit firm 1 earns in equilibrium is$\text{\$1200}$,and firm 2 is$\text{\$972}$

Step-by-step solution

Model of imperfect competition 

The Courante oligopoly is a model of imperfect competition in which two firms have the same cost functions and compete with homogeneous goods in a static environment. The duopoly has the following functions:

Inverse demand function:$\text{P=200}-\text{3}\left({\text{Q}}_{\text{1}}{\text{+Q}}_{\text{2}}\right)$

Cost function firm 1:${\text{C}}_{\text{1}}{\text{Q}}_{\text{1}}{\text{=26Q}}_{\text{1}}$

Cost function firm 2:${\text{C}}_{\text{2}}{\text{Q}}_{\text{2}}{\text{=32Q}}_{\text{2}}$

(a) To obtain the reaction functions 

To obtain the reaction functions in the Courante oligopoly, it is important to consider that changes in the marginal income will impact firm 1 will have an impact on the marginal income of firm 2; therefore, if firm 2 increases its production, firm 1 will have lower income.

Therefore, by equating the marginal revenues (MR) firmly with the marginal costs (MC) of firm 1 and we solve for${\text{Q}}_{\text{1}}$as a function of${\text{Q}}_2$, we will obtain the reaction function of firm 1 and vice versa.

Therefore, the following condition must be met:

${\text{MR}}_{\text{Q1}}{\text{=MC}}_{\text{Q1}}{\text{andMR}}_{\text{Q2}}{\text{=MC}}_{\text{Q2}}$

Fulfilling the condition

Fulfilling the condition that

$\text{TotalIncome=Price}\times \text{Quantity}$We multiply the inverse demand function by${\text{Q}}_{\text{1}}$,and then we derive it from obtaining the marginal income for firm 1:

$\text{TotalRevenue}=\text{Price}\times \text{Quantity}$

$\text{TotalRevenue=200}-{\text{3Q}}_{\text{1}}^{\text{2}}-{\text{3Q}}_{\text{1}}{\text{Q}}_{\text{2}}$

$\begin{array}{l}\text{MR=}\frac{{\text{PQ}}_{\text{1}}}{{\text{dQ}}_{\text{1}}}{\text{=200Q}}_{\text{1}}-{\text{3Q}}_{\text{1}}^{\text{2}}-{\text{3Q}}_{\text{1}}{\text{Q}}_{\text{2}}\\ \text{MR=}\frac{{\text{PQ}}_{\text{1}}}{{\text{dQ}}_{\text{1}}}\text{=200}-{\text{6Q}}_{\text{1}}-{\text{3Q}}_{\text{2}}\end{array}$

And deriving from the marginal cost of the firm$1$:

$\text{MC=}\frac{{\text{CQ}}_{\text{1}}}{{\text{dQ}}_{\text{1}}}\text{=26}$

Equating marginal revenue and marginal cost 

Equating marginal revenue and marginal cost and solving for,${\text{Q}}_{\text{1}}$we will obtain the reaction function of firm 1:

$\begin{array}{l}\text{200}-{\text{6Q}}_{\text{1}}-{\text{3Q}}_{\text{2}}\\ \text{=26200}-{\text{6Q}}_{\text{1}}-{\text{3Q}}_{\text{2}}\\ {\text{=26Q}}_{\text{1}}\\ \text{=}\frac{\text{200}-\text{26}-{\text{3Q}}_{\text{2}}}{\text{6}}{\text{Q}}_{\text{1}}\\ \text{=29}-\frac{\text{1}}{\text{2}}{\text{Q}}_{\text{2}}\end{array}$

Equating the income and marginal costs

By obtaining the income and marginal costs and equating them, we can obtain the reaction function for firm 2:

$\begin{array}{l}\text{MR=}\frac{{\text{PQ}}_{\text{2}}}{{\text{dQ}}_{\text{2}}}{\text{=200Q}}_{\text{2}}-{\text{3Q}}_{\text{1}}{\text{Q}}_{\text{2}}-{\text{3Q}}_{\text{2}}^{\text{2}}\\ \text{MR=}\frac{{\text{PQ}}_{\text{2}}}{{\text{dQ}}_{\text{2}}}\text{=200}-{\text{3Q}}_{\text{1}}-{\text{6Q}}_{\text{2}}\end{array}$

And deriving from the marginal cost of the firm:

$\text{MC=}\frac{{\text{CQ}}_{\text{2}}}{{\text{dQ}}_{\text{2}}}\text{=32}$

Equating marginal revenues and costs

Equating marginal revenues and costs and solving for${\text{Q}}_{\text{2}}$to obtain the reaction function of firm 2:

$\begin{array}{l}\text{200}-{\text{3Q}}_{\text{1}}-{\text{6Q}}_{\text{2}}\\ {\text{=326Q}}_{\text{2}}\\ \text{=200}-\text{32}-{\text{3Q}}_{\text{1}}{\text{Q}}_{\text{2}}\\ \text{=}\frac{\text{200}-\text{32}-{\text{3Q}}_{\text{1}}}{\text{6}}{\text{Q}}_{\text{2}}\\ \text{=28}-\frac{\text{1}}{\text{2}}{\text{Q}}_{\text{1}}\end{array}$

Therefore, the reaction function of firm 1 is $\text{29}-\frac{\text{1}}{\text{2}}{\text{Q}}_{\text{2}}$and firm 2$\text{28}-\frac{\text{1}}{\text{2}}{\text{Q}}_{\text{1}}$

(b) To obtain the equilibrium output

To obtain the equilibrium output of each firm, we can substitute the reaction equation of firm 2 in firm 1 and solve for${\text{Q}}_{\text{1}}$:

$\begin{array}{c}{\text{Q}}_{\text{1}}\text{=29}-\frac{\text{1}}{\text{2}}(\text{28}-\frac{\text{1}}{\text{2}}{\text{Q}}_{\text{1}}){\text{Q}}_{\text{1}}\\ \text{=29}-{\text{14+1/4Q}}_{\text{1}}\\ {\text{Q}}_{\text{1}}-{\text{1/4Q}}_{\text{1}}{\text{=15Q}}_{\text{1}}-{\text{1/4Q}}_{\text{1}}\\ {\text{=1/4Q}}_{\text{1}}\text{+15}-{\text{1/4Q}}_{\text{1}}{\text{3/4Q}}_{\text{1}}\text{=15}\end{array}$

Multiplying both sides by 4

$\begin{array}{l}\text{4}\times {\text{3/4Q}}_{\text{1}}\\ \text{=15}\times {\text{43Q}}_{\text{1}}\\ {\text{=60Q}}_{\text{1}}\\ \text{=20}\end{array}$

Substituting the value 

Substituting the value obtained in${\text{Q}}_{\text{1}}$we can substitute it in the reaction function of firm to obtain the equilibrium output of firm 2 :

$\begin{array}{l}{\text{Q}}_{\text{2}}\text{=28}-\frac{\text{1}}{\text{2}}\text{(20)}\\ {\text{Q}}_{\text{2}}\text{=28}-\text{10}\\ {\text{Q}}_{\text{2}}\text{=18}\end{array}$

Therefore, output by the equilibrium of firm$1$ is$20$_,and Firm$2$ is.$18$

(c) To obtain the equilibrium market price

To obtain the equilibrium market price, we can substitute the equilibrium values of the firm’s outpu$\text{1and2}({\text{Q}}_{\text{1}}\text{and}{\text{Q}}_{\text{2}})$tin the inverse demand function of the duopoly.

$\begin{array}{l}\text{P=200}-\text{3}\left({\text{Q}}_{\text{1}}{\text{+Q}}_{\text{2}}\right)\\ \text{P=200}-\text{3(20+18)}\\ \text{P=200}-\text{114}\\ \text{P=86}\end{array}$

Therefore, the market price of equilibrium is$\text{P=86}$

(d) Obtain the benefits 

The benefits of the duopoly under the Cournot scheme, as in any market, the following relationship is fulfilled:

$\text{Benefit(II)=TotalRevenue}-\text{TotalCost}$

For Firm 1 we can obtain the following benefit:

$\begin{array}{l}\text{Benefit(Π)=P}\times {\text{Q}}_{\text{1}}-{\text{26Q}}_{\text{1}}\\ \text{Benefit(Π)=(86}\times \text{20)}-\text{26(20)}\\ \text{Benefit(Π)=\$1,200}\end{array}$

For the firm, we can obtain the following benefit:

$\begin{array}{l}\text{Benefit(Π)=P}\times {\text{Q}}_{\text{2}}-{\text{32Q}}_{\text{2}}\\ \text{Benefit(Π)=(86}\times \text{18)}-\text{32(18)}\\ \text{Benefit(Π)=\$972}\end{array}$

Therefore, profit earned in equilibrium by firm 1 is $\text{\$1,200}$, and firm 2 is$\text{\$972}$