Question

Identify the conditions under which a firm operates in a Sweezy, Cournot, Stackelberg, or Bertrand oligopoly and the ramifications of each type of oligopoly for optimal pricing decisions, output decisions, and firm profits.Try these problems: 6, 19

Short answer

$\text{CournotModelofduopoly}=\frac{a+2c}{3}$

$\mathrm{The}Bertrandmodel=\frac{a-c}{2b}$

$\text{Stackelbergmodel}=\frac{a-3c}{4}$

Sweezy kinked demand curve theory describes pricing rigidity in oligopoly market circumstances.

Step-by-step solution

Explanation for Cournot,Bertrand,Stackelberg, and Sweezy

In an oligopoly market, at least two rival businesses offer a homogenous item to maximize their profit with a set marginal cost of manufacturing each good unit, based on other strategy hypotheses. The (inverse) market demand function determines the market price P:

$P=a-bQ$

The overall output is calculated as

$Q=q_1+q_2$they have the marginal cost is equal to$M{C}_1=c_1$ and$M{C}_2=c_2$

Every market player's equilibrium criterion (strategic objectives) are as follows:

a) Cournot: Companies establish amounts at the same time.

b) Bertrand: Costs are established concurrently by firms.

c) Stackelberg: Quantities are established by firms, firm 1 initially, then firm 2.

d) Sweezy: The Sweezy kinked demand curve theory attempts to explain why prices are inflexible in oligopolistic marketplaces.

To find the Cournot model of duopoly 

In a Cournot duopoly market, firms determine their production based on other firms' conjectures$\frac{d{q}_i}{d{q}_j}=0$, which means that if one business changes its output, it does not affect the output choice of the other firm.

Firm 1's profit function is now defined as follows:

$\begin{array}{c}{\pi }_1=T{R}_1-T{C}_1\\ 2b{q}_2+b{q}_1=a-c_2\\ 2b{q}_2=-b{q}_1+a-c\\ q_2=-\frac{q_1}{2}+\frac{a-c}{2b}\end{array}$

Therefore,

${\pi }_1=a{q}_1-b{q}_1^2-b{q}_1(-\frac{q_1}{2}+\frac{a-c}{2b})-c{q}_1$

The first order profit maximization condition is

$\frac{d{\pi }_1}{d{q}_1}=a-2b{q}_1-b{q}_2-c_1=0$

$2b{q}_1+b{q}_2=a-c_1\dots \left(1\right)$

The profit function of firm 2 is:

$\begin{array}{l}{\pi }_2=T{R}_1-T{C}_2\\ {\pi }_2=(a-b(q_1+q_2))q_2-T{C}_2\\ {\pi }_1=a{q}_2-b{q}_2^2-b{q}_2{q}_1-T{C}_2\end{array}$

The first order profit maximization condition is

$\begin{array}{l}\frac{d{\pi }_2}{d{q}_2}=a-2b{q}_2-b{q}_1-c_2=0\\ or,2b{q}_2+b{q}_1=a-c_2\dots \left(2\right)\end{array}$

Equations (1) as well as (2) represent the response functions of firms 1 as well as 2.

If they solve both equations, they get the equilibrium production of firms 1 and 2 as:

$q_1^c=\frac{(a-c)}{3b}and{q}_2^c=\frac{(a-c)}{3b}theyassumeM{C}_1=M{C}_2=c$and market demand is

$Q=q_1+q_2=2\frac{(a-c)}{3b}$

The market equilibrium price is:

$\begin{array}{c}p=a-bQ\\ =a-2b\frac{(a-c)}{3b}\\ =\frac{3a-2a+2c}{3}\\ =\frac{a+2c}{3}\end{array}$

Step 3:Bertrand equilibrium model: 

In the Bertrand model, firms set their prices based on an inverse demand function, with the consumer choosing the lowest-priced business.

Firm 1's profit function is now defined as follows:

${\pi }_1=T{R}_1-T{C}_1=(P_1-c_1)q_1$

They understand the cost:

$P=a-bQorQ=\frac{a-P}{b}$

As a result, to sustain positive production$P_1\pounds a_{andnonnegativeprofit}{P}_1³c_1$

As a result, business 1 selects

$c_1\le P_1\le a$

As a result, business 2 selects

$c_2\le P_2\le a$

As a result, company 1's optimal response function is:

$\begin{array}{c}0P_1>P_2\\ q_1=\frac{a-P_1}{b}{P}_1<P_2\\ \frac{a-P}{2b}{P}_1=P_2=P\end{array}$

As a result, the Bertrand model's equilibrium price was favorable.

$\begin{array}{c}{P}_1=P_2=MC=c\\ q=\frac{a-P}{2b}\\ =\frac{a-c}{2b}\end{array}$

To findthe total demand forthe Stackelberg model

Firm 1 acts as a leader, while Firm 2 acts as a follower and Firm 1 chooses his output based on Firm 2's supposition or the response function of the Cournot model.

Firm 1's profit function is now defined as follows:

${\pi }_1=T{R}_1-T{C}_1$where the conjecture of firm 2 is

$\begin{array}{c}2b{q}_2+b{q}_1=a-c_2\\ or2b{q}_2=-b{q}_1+a-c\\ or,q_2=-\frac{q_1}{2}+\frac{a-c}{2b}\end{array}$

Therefore,

${\pi }_1=a{q}_1-b{q}_1^2-b{q}_1(-\frac{q_1}{2}+\frac{a-c}{2b})-c{q}_1$

The initial order condition of maximization is:

$\frac{d{\pi }_1}{d{q}_1}=a-2b{q}_1+b{q}_1-\frac{a-c}{2}-c=0$

$\begin{array}{c}a-2b{q}_1+b{q}_1-\frac{a-c}{2}-c=0\\ -b{q}_1-\frac{a-c}{2}=0\\ q_1^S=\frac{a-c}{2b}\\ q_2^S=-\frac{a-c}{4b}+\frac{a-c}{2b}=\frac{a-c}{4b}\end{array}$

Therefore,the total demand

$Q=q_1^S+q_2^S=\frac{a-c}{4b}+\frac{a-c}{2b}=\frac{3(a-c)}{4b}$and price is

$\begin{array}{c}P=a-bQ\\ =a-\frac{3(a-c)}{4}\\ =\frac{4a-3a-3c}{4}\\ =\frac{a-3c}{4}\end{array}$

Step 5:To find the Sweezy kinked graph 

Sweezy kinked demand curve theory describes pricing rigidity in oligopoly market circumstances. The image is given below: