Constructing the Kinked-Demand Model

Why the Demand Curve Has a Kink

In oligopoly, a firm’s demand depends in part on how its rivals react to its price. The kinked-demand model describes one particular expectation: rivals do not follow a price increase, but they do match a price cut. The firm therefore expects a different change in its sales depending on which direction it moves from its prevailing price.

A price increase that rivals do not match makes the firm relatively expensive. It loses some customers to competitors, so its quantity demanded falls comparatively sharply. If the firm cuts price and rivals match, it does not gain as many customers as it would if their prices stayed fixed. The resulting demand curve has two segments meeting at the prevailing price and quantity: a more responsive segment above that price and a less responsive segment below it.

Definition: A kinked-demand curve is a firm’s demand curve with two segments meeting at a prevailing price \(P_0\) and quantity \(Q_0\). In the basic model, the firm expects rivals not to match its price increases, but to match its price cuts. At the kink, the two segments meet at \((P_0,Q_0)\); the firm’s demand response differs on either side.

This is a model of the firm’s perceived demand response around an existing price, not a general demand curve for the entire market. It does not, by itself, calculate the prevailing price. To construct it, identify the baseline price and quantity, then apply the relevant rival-response assumption to price changes above and below that baseline.

Assumptions and a Linear Construction

The standard construction uses a firm selling a differentiated product in an oligopoly. Hold other conditions fixed, including product quality, income, and the number of customers. Let \(p_i\) be the firm’s price, \(p_j\) a representative rival’s price, and \(q_i\) the firm’s quantity demanded. Assume a linear demand relationship:

q_i = \alpha - \beta p_i + \gamma p_j, \qquad \beta > \gamma > 0.

The own-price coefficient \(\beta\) measures the effect on the firm’s quantity of changing its price while the rival’s price is fixed. The cross-price coefficient \(\gamma\) measures the effect of a change in the rival’s price on the firm’s quantity. A rival price increase makes this firm more attractive, so \(\gamma\) is positive. The condition \(\beta>\gamma\) means the firm’s own-price effect is larger than the cross-price effect used in this illustration.

At the prevailing point, suppose both firms charge \(P_0\), and the focal firm sells \(Q_0\). Above \(P_0\), the rival is expected to leave its price unchanged. Below \(P_0\), the rival is expected to reduce its price by the same amount as the focal firm. These expectations produce the two segments as follows.

Find the baseline.
Set \(p_i=p_j=P_0\) in the demand relationship. This gives \(Q_0=\alpha-\beta P_0+\gamma P_0\).

Construct the segment above the kink.
For \(p_i\ge P_0\), keep \(p_j=P_0\). The rival does not match an increase, so only the focal firm’s own-price term changes.

Construct the segment below the kink.
For \(p_i\le P_0\), set \(p_j=p_i\). The rival matches the cut, so both price terms change together.

The resulting piecewise demand relationship is

q_i = \begin{cases} Q_0-\beta(p_i-P_0), & p_i\ge P_0,\\ Q_0-(\beta-\gamma)(p_i-P_0), & p_i\le P_0. \end{cases}

For a price increase, \(p_i-P_0\) is positive and the first expression predicts a fall in quantity. For a price cut, \(p_i-P_0\) is negative; because \(\beta-\gamma\) is positive, the second expression predicts a rise in quantity, but a smaller rise than if the rival had held its price fixed.

Why the Two Segments Differ

The slope of quantity demanded with respect to the focal firm’s price is \(-\beta\) above the kink, where the rival’s price is fixed. Below the kink, both prices move together, so the slope is \(-\beta+\gamma=-(\beta-\gamma)\). Since \(\beta>\gamma>0\), the absolute value of the first slope is greater than that of the second.

\left|\frac{dq_i}{dp_i}\right|_{\text{above}}=\beta \quad > \quad \left|\frac{dq_i}{dp_i}\right|_{\text{below}}=\beta-\gamma.

This is the core justification for the model’s shape: an unaccompanied price increase has a larger quantity effect than a price cut that rivals match. At the kink, the two expressions give the same quantity \(Q_0\) when \(p_i=P_0\), so the curve is continuous there.

A larger absolute slope means a greater quantity change for a given price change. It does not automatically mean that the upper segment has price elasticity greater than one or that the lower segment has elasticity below one. For a linear demand segment, the absolute point elasticity depends on both its slope and the price-to-quantity ratio at the point. The model’s essential conclusion is the relative difference in responsiveness.

Worked Example: Deriving Both Segments from Rival Responses

A firm’s demand is \(q_i=120-4p_i+2p_j\), where prices are in dollars per unit and quantity is in units per week. At the prevailing point, both firms charge \(10\) dollars. Construct the firm’s demand curve above and below that price, then find its quantity at prices of \(11\) dollars and \(9\) dollars.

Method and baseline. Substitute \(p_i=p_j=10\) into the demand equation:

$$ Q_0=120-4(10)+2(10)=120-40+20=100. $$

The kink is therefore at \((P_0,Q_0)=(10,100)\).

Above the kink. For \(p_i\ge10\), the rival does not match an increase, so \(p_j=10\). Substitute the fixed rival price:

$$ q_i=120-4p_i+2(10)=140-4p_i. $$

At \(p_i=11\), \(q_i=140-4(11)=140-44=96\) units per week.

Below the kink. For \(p_i\le10\), the rival matches the cut, so \(p_j=p_i\). Therefore,

$$ q_i=120-4p_i+2p_i=120-2p_i. $$

At \(p_i=9\), \(q_i=120-2(9)=120-18=102\) units per week.

Check. At \(p_i=10\), the upper expression gives \(140-4(10)=100\), and the lower expression gives \(120-2(10)=100\). Both meet at the stated kink. The method avoids treating the rival’s price as fixed after a price cut.

Worked Example: Reading Responsiveness and Elasticity at the Kink

A hypothetical supplier currently charges \(12\) dollars per unit and sells \(80\) units per day. A study of its perceived demand indicates that, for each \(1\)-dollar price increase above the current price, its quantity falls by \(8\) units, because rivals hold their prices fixed. For each \(1\)-dollar price cut below the current price, its quantity rises by \(2\) units, because rivals match the cut. Write the two demand segments, calculate quantity at \(13\) dollars and \(10\) dollars, and compare the one-sided absolute price elasticities at the kink.

Method and equations. The prevailing point is \((12,80)\). The upper segment has slope \(-8\); the lower segment has slope \(-2\). Thus:

q = \begin{cases} 80-8(p-12), & p\ge12,\\ 80-2(p-12), & p\le12. \end{cases}

At \(p=13\), use the upper segment: \(q=80-8(13-12)=80-8=72\) units. At \(p=10\), use the lower segment: \(q=80-2(10-12)=80-2(-2)=84\) units.

For a differentiable demand segment, the absolute point elasticity is \(\left|\frac{dq}{dp}\right|\frac{p}{q}\). Evaluate each one-sided value at \(p=12\), \(q=80\):

|\varepsilon_{\text{above}}|=8\left(\frac{12}{80}\right)=\frac{96}{80}=1.2, \qquad |\varepsilon_{\text{below}}|=2\left(\frac{12}{80}\right)=\frac{24}{80}=0.3.

The upper segment is more elastic at the kink in this example, and its elasticity exceeds one; the lower segment’s elasticity is below one. Those numerical classifications follow from the given slopes and baseline values, not from the kinked-demand assumptions alone.

Common mistake. Using the lower segment for a price increase would give \(80-2(13-12)=78\) units, which is incorrect: that segment assumes rivals match a price cut. The correct quantity at \(13\) dollars is \(72\) units from the upper segment.

Worked Example: Finding the Kink from Two Demand Equations

For a hypothetical firm, analysts estimate the demand segment above the prevailing price as \(q=150-5p\), and the segment below it as \(q=114-2p\). Find the price and quantity at which the segments meet. Then find quantity at a price of \(14\) dollars above the kink and \(9\) dollars below it.

Method. The kink is where both segment equations predict the same quantity. Set them equal and solve for price:

150-5p=114-2p, \qquad 36=3p, \qquad p=12.

Substitute \(p=12\) into either segment:

q=150-5(12)=150-60=90, \qquad q=114-2(12)=114-24=90.

The segments meet at \((12,90)\). At \(p=14\), use the segment above the kink: \(q=150-5(14)=150-70=80\). At \(p=9\), use the segment below the kink: \(q=114-2(9)=114-18=96\).

Check. Both equations give \(90\) units at \(12\) dollars, confirming continuity. The upper segment has the larger absolute slope, \(5\) rather than \(2\), as expected under the stated response pattern. The method avoids selecting a kink by inspection when the two equations must actually intersect there.

Common Interpretation Traps

Reversing the segments. Above the prevailing price, rivals do not match the increase; below it, rivals match the cut. Choose the segment based on the direction of the price change from \(P_0\).
Using market demand instead of the firm’s demand. The curve describes quantity demanded from one firm as its own price changes and rivals respond. It is not automatically the total quantity demanded in the market.
Confusing slope with elasticity. The upper segment has a greater absolute slope in the basic construction, but whether its elasticity is greater than one depends on price and quantity as well.
Assuming the curve determines the kink’s location. The model takes the prevailing price and quantity as the point where the segments meet. The rival-response assumptions describe the shape around that point; they do not independently calculate it.

Key takeaway: Identify the prevailing point first. Use the no-match assumption to construct the more responsive segment above it, and the match-the-cut assumption to construct the less responsive segment below it. Both segments must pass through the same price–quantity point.

Check Your Understanding

For each problem, identify the prevailing point before selecting a segment.

A firm’s demand is \(q=90-3p_i+p_j\), and both firms currently charge \(10\) dollars. Rivals do not match increases and match cuts one-for-one. Write both segments and find quantity at \(p_i=12\) and \(p_i=8\). (Answer: kink \((10,70)\); above \(q=100-3p_i\), so \(q(12)=64\); below \(q=90-2p_i\), so \(q(8)=74\).)
A firm sells \(50\) units at its prevailing price of \(20\) dollars. Above that price, quantity falls by \(4\) units for each \(1\)-dollar increase; below it, quantity rises by \(1\) unit for each \(1\)-dollar cut. Find the two demand equations and the quantity at \(22\) dollars and \(17\) dollars. (Answer: above \(q=50-4(p-20)\), giving \(42\); below \(q=50-(p-20)\), giving \(53\).)
The upper and lower demand segments are \(q=130-4p\) and \(q=94- p\), respectively. Find their intersection and quantity at \(p=11\) on the appropriate segment. (Answer: kink at \((12,82)\); at \(p=11\), use the lower segment, giving \(q=83\).)
At a kink, price is \(15\) dollars and quantity is \(60\). The upper-segment slope is \(-6\), and the lower-segment slope is \(-2\). Calculate the absolute one-sided elasticities at the kink and state which is larger. (Answer: upper \(6(15/60)=1.5\); lower \(2(15/60)=0.5\); the upper elasticity is larger.)
A firm’s baseline price and quantity are \(8\) dollars and \(120\) units. A model gives a quantity response of \(5\) fewer units per dollar above the kink and \(2\) additional units per dollar below it. A candidate uses the lower segment to estimate quantity at \(10\) dollars. Identify the error and calculate the correct quantity at \(10\) dollars and at \(6\) dollars. (Answer: \(10\) dollars is above the kink, so use the upper segment and obtain \(120-5(2)=110\); at \(6\) dollars use the lower segment and obtain \(120+2(2)=124\).)

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