Limits of the Kinked-Demand Explanation

What the Kinked-Demand Model Explains

An oligopoly firm’s price decision depends partly on how its rivals respond. The kinked-demand model represents one particular expectation: rivals match a price cut, but do not match a price increase. This asymmetry makes the firm’s demand relatively less responsive to a price cut and relatively more responsive to a price increase, around an established price and quantity.

The model’s central contribution is an explanation of price rigidity: under specified conditions, a change in marginal cost need not change the firm’s profit-maximizing price. Its limitation is just as important. The model takes the price and quantity at the kink as given; it does not derive where that initial point comes from. This treatment is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Definition: In the kinked-demand model, a firm faces a demand curve with a kink at an established price $P_0$ and quantity $Q_0$. Demand is more elastic above $P_0$, where rivals are assumed not to follow a price increase, and less elastic below $P_0$, where rivals are assumed to match a price cut. The corresponding marginal-revenue curve has a discontinuous gap at $Q_0$.

Assumptions and the Marginal-Revenue Gap

To calculate the model’s implications, suppose the firm chooses its price and quantity to maximize profit, and consider a local market situation represented by two linear demand segments meeting at $(Q_0,P_0)$. Write inverse demand as price as a function of the firm’s quantity. For quantities below the kink, price is above $P_0$; for quantities above it, price is below $P_0$:

P(Q)= \begin{cases} a_u-b_uQ, & Q\leq Q_0,\\ a_l-b_lQ, & Q\geq Q_0, \end{cases} \qquad a_u=P_0+b_uQ_0,\quad a_l=P_0+b_lQ_0, \qquad 0<b_u<b_l.

The two intercepts make both segments pass through the kink. The condition $b_l>b_u$ means inverse demand is steeper below the kink. At the kink, the absolute price elasticity of demand on a linear inverse-demand segment is $P/(bQ)$. Thus the lower segment has smaller elasticity there, while the upper segment has larger elasticity. These slopes represent the assumed difference in rivals’ responses; they are not independently derived from the model.

Assume also that marginal cost is constant at $c$ over the relevant range. Fixed cost, if any, does not affect the choice of output. On a segment with inverse demand $P=a-bQ$, total revenue is $TR(Q)=P(Q)Q=aQ-bQ^2$, so marginal revenue is $MR(Q)=a-2bQ$. At the kink, the marginal revenue just to the left and just to the right are therefore:

MR_u(Q_0)=P_0-b_uQ_0, \qquad MR_l(Q_0)=P_0-b_lQ_0.

Because $b_l>b_u$, $MR_l(Q_0)<MR_u(Q_0)$. The difference between these one-sided marginal revenues is the gap.

Why a Range of Costs Can Support One Price

Profit is $\pi(Q)=TR(Q)-cQ$, apart from any fixed cost. Its change from a small increase in output is governed by $MR-c$. At the kink, a profit maximum requires that producing a little less would not raise profit and producing a little more would not raise profit. In derivative terms, the slope of profit just to the left must be nonnegative, and the slope just to the right must be nonpositive.

MR_u(Q_0)-c\geq 0 \quad\text{and}\quad MR_l(Q_0)-c\leq 0 \quad\Longleftrightarrow\quad MR_l(Q_0)\leq c\leq MR_u(Q_0).

This is the key condition: if marginal cost lies in the marginal-revenue gap, the firm maximizes profit at $Q_0$, and the corresponding price remains $P_0$. The result follows because, on each linear segment, marginal revenue decreases as quantity increases. To the left of the kink, profit is increasing as output approaches $Q_0$; to the right, profit is decreasing as output moves away from $Q_0$. Under these assumptions, the kink is therefore a profit maximum.

The result concerns a cost change while the kink and demand segments are held fixed. If marginal cost changes but stays in the interval, the firm’s price and quantity do not move in this model. If cost moves outside the interval, the kink condition fails and the optimum can lie on one of the demand segments instead.

Worked Example: A Cost Change Within the Gap

A firm’s kink is at $Q_0=20$ units and $P_0=\$60$. Its inverse demand is $P=80-Q$ for $Q\leq20$, and $P=120-3Q$ for $Q\geq20$. Marginal cost is initially $\$25$ per unit and then rises to $\$35$. Determine whether either cost supports the kink, and calculate profit at the kink under each cost. Ignore fixed cost.

Method and assumptions. The firm chooses output on the stated piecewise linear demand curve, marginal cost is constant at each cost level, and the kink remains fixed. First calculate the marginal-revenue values on each side.

For the upper segment, $a_u=80$ and $b_u=1$, so $MR_u=80-2Q$. At $Q=20$, $MR_u=80-40=\$40$. For the lower segment, $a_l=120$ and $b_l=3$, so $MR_l=120-6Q$. At $Q=20$, $MR_l=120-120=\$0$. The gap is therefore from $\$0$ to $\$40$.

Both marginal costs are in that interval: $0\leq25\leq40$ and $0\leq35\leq40$. The kink quantity remains $20$, and the price remains $\$60$, at both cost levels.

At cost $\$25$, profit is $(60-25)\times20=35\times20=\$700$. At cost $\$35$, profit is $(60-35)\times20=25\times20=\$500$. The price and output are unchanged, but profit falls by $\$200$, equal to the cost increase of $\$10$ multiplied by $20$ units. Check: both costs satisfy the gap condition, as required. The common mistake avoided here is to assume that any increase in marginal cost must immediately raise the firm’s price.

What Happens When Cost Leaves the Gap

The gap condition is local to the established kink. If marginal cost exceeds the upper marginal-revenue value, the firm has an incentive to reduce output from the kink; if it is below the lower value, the firm has an incentive to increase output. With linear demand and constant marginal cost, the relevant segment’s usual $MR=MC$ condition then identifies the interior optimum, provided that solution lies on that segment.

Worked Example: Cost Above the Gap

Use the demand curve from the previous example: $P=80-Q$ for $Q\leq20$ and $P=120-3Q$ for $Q\geq20$. Now let constant marginal cost be $\$45$. Find the profit-maximizing output and price, and compare them with the kink.

Method and assumptions. The kink’s upper marginal-revenue value is $\$40$, so $MC=\$45$ is above the gap. The firm’s optimum is expected on the upper segment, where $Q\leq20$. Set that segment’s marginal revenue equal to marginal cost.

On the upper segment, $TR=Q(80-Q)=80Q-Q^2$, so $MR=80-2Q$. Solve $80-2Q=45$: $2Q=35$, so $Q=17.5$. This is on the upper segment because $17.5\leq20$. The price is $P=80-17.5=\$62.50$.

Profit is $(62.50-45)\times17.5=17.50\times17.5=\$306.25$. At the kink, profit would be $(60-45)\times20=15\times20=\$300$, which is lower. Check: the calculated upper-segment quantity is on the segment used to calculate it, and its profit exceeds the kink profit. A common mistake is to solve $MR=MC$ on the lower segment merely because it describes quantities above the kink; here the cost condition instead points to the upper segment.

Why the Initial Price Is Undetermined

The kinked-demand model begins with $P_0$ and $Q_0$ already in place. Its assumptions describe how demand responds to a firm’s price changes around that point, and its marginal-revenue gap shows when a given marginal cost is consistent with keeping the point. They do not supply a rule for choosing $P_0$ or $Q_0$ in the first place.

This distinction can be shown directly. Suppose the same marginal cost is consistent with the gap at two possible kinks. For the first candidate, let $P_0=\$60$, $Q_0=20$, $b_u=1$, and $b_l=3$. Its marginal-revenue values are $\$40$ and $\$0$, so $MC=\$25$ supports that kink. For a second candidate, let $P_0=\$70$, $Q_0=15$, $b_u=2$, and $b_l=4$. Its values are $70-(2)(15)=\$40$ and $70-(4)(15)=\$10$, so the same $MC=\$25$ also supports this kink. Both satisfy $b_l>b_u$. The cost condition alone therefore does not select between a price of $\$60$ and a price of $\$70$.

Worked Example: Two Possible Kinks at the Same Cost

Consider two hypothetical market descriptions for a firm with constant marginal cost $\$25$. In market A, the kink is $(Q_0,P_0)=(20,\$60)$, with slopes $b_u=1$ and $b_l=3$. In market B, the kink is $(15,\$70)$, with slopes $b_u=2$ and $b_l=4$. For each market, find the marginal-revenue gap and determine whether the stated kink is profit-maximizing under the model.

Method and assumptions. In each case, use $MR_u=P_0-b_uQ_0$ and $MR_l=P_0-b_lQ_0$, then check whether marginal cost lies between them.

In market A, $MR_u=60-(1)(20)=\$40$, and $MR_l=60-(3)(20)=\$0$. Since $0\leq25\leq40$, the kink is consistent with profit maximization.

In market B, $MR_u=70-(2)(15)=\$40$, and $MR_l=70-(4)(15)=\$10$. Since $10\leq25\leq40$, this kink is also consistent with profit maximization.

Both candidates meet the condition even though their prices and quantities differ. Check: the slopes satisfy $b_l>b_u$ in both cases, and the marginal-revenue values are in the correct order. The common mistake avoided is treating the cost-gap test as a method for deriving the initial price; it tests whether a proposed kink is supported, not how that kink was established.

Scope, Interpretation, and Exam Traps

The model is best read as a conditional explanation of why an existing oligopoly price may be slow to change. Its conclusions depend on an assumed asymmetry in rivals’ responses, a fixed kink, and the cost and demand conditions used in the calculation. It does not explain why rivals behave asymmetrically, establish that they actually do so, or show that an observed price must be the outcome of this mechanism.

It is also not a general claim that oligopoly prices never change. A cost change outside the marginal-revenue gap can move the optimum off the kink. A change in the market conditions that determine demand can also alter the kink itself, so the fixed-kink cost result no longer applies without further analysis.

Confusing rigidity with price determination. The gap can support a given price; it does not predict the initial price or quantity.
Reversing the marginal-revenue values. With $b_l>b_u$, the lower-side value is $MR_l=P_0-b_lQ_0$, which is below $MR_u=P_0-b_uQ_0$.
Using the condition outside its assumptions. The interval result assumes the kink and demand segments are held fixed and marginal cost is constant over the relevant range.
Assuming every cost increase raises price. A cost increase within the gap leaves the model’s profit-maximizing price and quantity unchanged.

Key takeaway: The kinked-demand model explains price rigidity conditionally: if marginal cost lies between the marginal-revenue values on either side of an established kink, that price and quantity maximize profit. The model does not determine the initial kink, and its rigidity prediction need not hold when cost or demand conditions move beyond its assumptions.

Check Your Understanding

For each problem, identify the relevant kink condition before calculating. Treat marginal cost as constant and ignore fixed cost unless stated otherwise.

A firm has a kink at $P_0=\$50$, $Q_0=10$, with $b_u=0.5$ and $b_l=1.5$. Calculate both marginal-revenue values and state the range of marginal costs that supports the kink. (Answer: $MR_u=\$45$, $MR_l=\$35$; the kink is supported for $35\leq MC\leq45$.)
A kink is at $P_0=\$70$, $Q_0=20$. The upper inverse-demand segment is $P=90-Q$, the lower segment is $P=110-2Q$, and $MC=\$55$. Find the profit-maximizing output and price. (Answer: $Q=17.5$, $P=\$72.50$.)
A firm has a kink at $P_0=\$80$, $Q_0=12$, with $b_u=1$ and $b_l=2.5$. Marginal cost rises from $\$54$ to $\$64$, while demand and the kink remain fixed. State whether output and price change, and give both marginal-revenue bounds. (Answer: $MR_l=\$50$, $MR_u=\$68$; both costs are within the gap, so output remains $12$ and price remains $\$80$.)
One possible kink is $P_0=\$45$, $Q_0=8$, $b_u=1$, $b_l=2$. Another is $P_0=\$60$, $Q_0=10$, $b_u=2$, $b_l=4$. If $MC=\$33$, test each kink. What does this establish about the price the model determines? (Answer: Both kinks satisfy the condition: their gaps are $[\$29,\$37]$ and $[\$20,\$40]$, respectively. The model does not determine a unique initial price.)
A firm has a kink at $P_0=\$100$, $Q_0=25$, with $b_u=1$, $b_l=2$. Its constant marginal cost is $\$40$. Using the appropriate segment, find its profit-maximizing output and price. (Answer: The gap is $[\$50,\$75]$, so cost is below it and the optimum is on the lower segment $P=150-2Q$: $MR=150-4Q=40$, $Q=27.5$, and $P=\$95$.)

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