Kinked Demand Curve Model

A duopoly is a market situation where there are two sellers, for each set of actions and reaction of the duopolist, there corresponds a solution. Thus there is no unique solution for duopoly equilibrium. The solution depends on the actions of one duopolist and the counter of another duopolist.^duopoly-definition

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In this model the duopolist start with a predetermined set of prices and quantity. They examine whether a deviation from this¹ by either increase in price or decrease in price can be advantageous.

If they find that a change is not advantageous, they will maintain the initial position itself which is the equilibrium otherwise they will try for change.

Suppose one of the duopolists (say $I$) increased his price, than the other duopolist (say $II$) will find that it is advantageous for him and hence he will keep his price unchanged.

But if one of the duopolists ($I$) decreased his price, then the other ($II$) will also decrease its price, to the extent of maintaining equal share in output.²

Thus, price decrease will follow but not price increase.

Finally, if they find a change is not advantageous, they don’t move from their initial position, which itself will become Static Equilibrium. In this process, there cannot be a uniform demand curve, but an effective demand curve, a part of which is guided for price increase and the other part for price decrease.

Thus, there is a kink³ in the effective demand curve, hence this model is called a Kinked Demand Curve Model.

Example

$$\begin{array}{rlrl}{P}_1& =100-2q_1-q_2& c_1& =2.5q_1^2\\ P_2& =95-q_1-3q_2& c_2& =25q_2\end{array}$$

Suppose the predetermined price and quantity are $P_1=70$, $q_1=10$, $P_2=55$ and $q_2=10$.

CASE A: Increase in price

Suppose $I$ increases his price from $70$; then $II$ will keep his price unchanged at $P_2=55$.

$$q_2=\frac{40-q_1}{3}$$

$\therefore$ $I$’s demand function is given by

$$\begin{array}{rl}{P}_1& =100-2q_1-\frac{40-q_1}{3}\\ P_1& =\frac{260-5q_1}{3},\text{which is valid for price, }{P}_1>70\end{array}$$

Now, looking at the revenue,

$$\begin{array}{rl}{R}_1& =\frac{260q_1-5q_1^2}{3}\\ M{R}_1& =\frac{260-10q_1}{3}\end{array}$$

Since, $P_1>70$, $q_1<10$

$$\begin{array}{rl}\therefore 10q_1& <100\\ -10q_1& >-100\\ 260-10q_1& >160\\ \frac{260-10q_1}{3}& >53\frac{1}{3}\\ M{R}_1& >53\frac{1}{3}\\ M{C}_1& =5q_1<50\end{array}\text{\hspace{1em}(change sign)(add 260 OBS)}$$

$\therefore$ $MR$ and $MC$ cannot be equated $\therefore$ one cannot try for increasing his price.

CASE B: Decrease in price

Suppose $I$ decreases his price $P_1<70$. Then $II$ will follow by decreasing his price till equal quantity is followed i.e. $\boxed{q_2=q_1}$.

$\therefore$ $I$’s demand function will be

$$\begin{array}{rl}{P}_1& =100-2q_1-q_1\\ ⟹P_1& =100-3q_1\end{array}$$

$R_1=100q_1-3q_1^2$ and so, $M{R}_1=100-6q_1$.

But, $q_1>10$,⁴

$$\begin{array}{rl}6q_1& >60\\ -6q_1& <-60\\ 100-6q_1& <100-60\\ M{R}_1& <40\\ \text{but, }M{C}_1& =5q_1>50\end{array}$$

$\therefore$ $MR$ cannot be equated to $MC$. Thus initial position itself is the Static Equilibrium position.

Transclude of Kinked-Demand-Curve-Model-2024-09-24-10.50.46.excalidraw

Footnotes

1. can be in the form of prices ↩

2. If I increase the price, he will not decrease the price. But if I decrease the price, he also has to decrease the price to stay in competition ↩

3. doubt how does this kink arise? ↩

4. because of the decrease in the price ↩