Why a Kink Can Make Price Rigid
In the kinked-demand model, a firm believes that rivals will react differently to a price increase than to a price decrease. A price increase is not matched, so the firm expects to lose many customers; a price decrease is matched, so it expects to gain relatively few. The resulting demand curve is relatively flat above the current price and steeper below it. At the kink, the marginal-revenue curve has a gap.
That gap matters for cost changes. If marginal cost changes but remains inside the gap, the firm can keep producing the same quantity at the same kink price. If marginal cost moves outside the gap, the profit-maximizing quantity generally moves to one of the two demand branches, and the price changes with it. The method is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.
Demand Branches and the Marginal-Revenue Gap
Write the upper-branch price as \(P(Q)=P_k+b(Q_k-Q)\), where \(b>0\), and the lower-branch price as \(P(Q)=P_k-d(Q-Q_k)\), where \(d>b\). Both branches pass through the kink. Equivalently, they are linear demand equations with intercepts \(P_k+bQ_k\) and \(P_k+dQ_k\), respectively.
For a linear inverse-demand branch \(P=A-BQ\), total revenue is \(TR(Q)=AQ-BQ^2\), so marginal revenue is \(MR(Q)=A-2BQ\). Applying that result to each branch gives:
At the kink, the marginal revenue just to the left is \(P_k-bQ_k\), while the marginal revenue just to the right is \(P_k-dQ_k\). Since \(d>b\), the right-hand value is lower. The missing marginal-revenue values form a gap:
Why Marginal Cost Inside the Gap Leaves the Kink in Place
Profit is revenue minus variable cost, so on a smooth part of the demand curve its derivative with respect to output is \(MR-c\). At the kink, use the derivative from each side. For output just below \(Q_k\), profit is increasing as output approaches the kink if the upper-branch marginal revenue there is at least \(c\). For output just above \(Q_k\), profit is decreasing if the lower-branch marginal revenue there is at most \(c\).
Substituting the two marginal-revenue values yields \(P_k-bQ_k \geq c \geq P_k-dQ_k\), which is precisely the gap condition. Each branch’s marginal revenue declines as output rises, and it drops at the kink. Therefore, when those one-sided inequalities hold, profit rises up to the kink and falls after it. This establishes why the kink maximizes profit under the model’s assumptions, rather than merely giving a rule to memorize.
Use the upper branch for the left-hand value and the lower branch for the right-hand value.
If \(MR_{\text{lower}}(Q_k)\leq c\leq MR_{\text{upper}}(Q_k)\), keep \(Q=Q_k\) and \(P=P_k\).
For a candidate interior optimum, solve \(MR=c\), verify the solution lies on that branch, and check whether producing zero is better.
Worked Cost Changes
The following examples use the same hypothetical demand curve so the effect of changing marginal cost is easy to compare. Let the kink be \((Q_k,P_k)=(5,30)\), with \(b=1\) and \(d=2\). Then the demand branches are \(P=35-Q\) for \(0\leq Q\leq5\) and \(P=40-2Q\) for \(Q\geq5\). The corresponding marginal revenues are \(MR_{\text{upper}}=35-2Q\) and \(MR_{\text{lower}}=40-4Q\). At the kink, those values are 25 and 20, so the marginal-cost gap is from 20 to 25.
Worked Example: A Cost Increase That Stays Within the Gap
Suppose constant marginal cost rises from 21 to 24. Determine the profit-maximizing output and price before and after the increase, and calculate the change in profit.
Method. Compare each cost with the kink’s marginal-revenue gap, then calculate profit at the kink. Avoid assuming that every cost increase must change price.
Before the increase. The gap is \(20\leq c\leq25\), and \(c=21\) lies in it. Thus \(Q=5\) and \(P=30\). Revenue is \(30(5)=150\), variable cost is \(21(5)=105\), and profit is \(150-105=45\).
After the increase. The new cost \(c=24\) is also in the gap, so output and price remain \(Q=5\) and \(P=30\). Revenue is still \(150\); variable cost is \(24(5)=120\); profit is \(150-120=30\).
Check. At \(c=24\), upper-branch marginal revenue at the kink is 25, which exceeds cost, while lower-branch marginal revenue is 20, which is below cost. Profit is therefore increasing toward the kink from the left and decreasing away from it to the right. Profit falls by \(30-45=-15\), even though price and output do not change.
Worked Example: A Cost Increase That Moves Output to the Upper Branch
Using the same demand curve, let marginal cost rise to 28. Find the optimum and check that the firm should not choose zero output.
Method. Since \(28>25\), cost is above the gap. A lower quantity on the upper branch is a candidate. Solve that branch’s marginal-revenue equation, then verify feasibility and the zero-output boundary.
Set upper-branch marginal revenue equal to marginal cost:
This output is feasible on the upper branch because \(0<3.5<5\). The price is \(P=35-3.5=31.5\). Revenue is \(31.5(3.5)=110.25\), variable cost is \(28(3.5)=98\), and profit is \(110.25-98=12.25\).
At \(Q=0\), profit is zero. Since the interior candidate earns \(12.25\), it is better than producing nothing. Also, marginal revenue at zero is 35, greater than the cost of 28, so profit initially rises from zero output. At the kink, both one-sided marginal revenues, 25 and 20, are below cost; output should not be extended to the kink.
Check. The candidate lies on the branch used to calculate it, earns positive profit, and has marginal revenue equal to marginal cost. The optimum is \(Q=3.5\), \(P=31.5\).
Worked Example: Checking the Zero-Output Boundary
Now let marginal cost be 36. Find the firm’s best output. Explain why solving marginal revenue equal to marginal cost does not produce a feasible upper-branch optimum.
Method. Compare cost with marginal revenue at zero output before accepting an interior solution. The upper-branch equation would give \(35-2Q=36\), so \(Q=-0.5\), which is infeasible.
At \(Q=0\), upper-branch marginal revenue is 35, already less than the marginal cost of 36. As output increases along the upper branch, marginal revenue falls further. On the lower branch, marginal revenue is also below cost: it equals 20 at the kink and declines as output rises. Thus \(MR-c<0\) for every positive output.
Profit is therefore decreasing from its value of zero at \(Q=0\). The firm maximizes profit by choosing \(Q=0\), earning zero. This boundary check is essential; a negative solution to \(MR=MC\) is not a feasible quantity and must not be reported as the optimum.
Check. Every positive quantity has negative marginal profit, so no positive output improves on the zero-output choice.
Worked Example: A Cost Decrease That Moves Output to the Lower Branch
Return to the same kinked demand curve and suppose marginal cost falls to 15. Find the profit-maximizing output, price, and profit.
Method. Since \(15<20\), cost is below the gap. The firm has an incentive to increase output beyond the kink, so solve marginal revenue on the lower branch and verify the resulting quantity.
Set lower-branch marginal revenue equal to cost:
The solution is feasible because \(6.25>5\), so it is on the lower branch. Price is \(P=40-2(6.25)=27.5\). Revenue is \(27.5(6.25)=171.875\), variable cost is \(15(6.25)=93.75\), and profit is \(171.875-93.75=78.125\).
At the kink, lower-branch marginal revenue is 20, greater than cost 15, so profit rises as output increases past five. At \(Q=6.25\), marginal revenue equals cost, and beyond that quantity lower-branch marginal revenue falls below cost. The optimum is therefore \(Q=6.25\), \(P=27.5\), with profit \(78.125\).
Common Mistakes and Practical Interpretation
Price rigidity in this model does not mean that costs have no effect on profit. In the first example, cost rises while price and output stay fixed, and profit falls. The model’s specific prediction is that the chosen price and output can remain unchanged for cost movements within the marginal-revenue gap. Once cost crosses an endpoint, the profit-maximizing output moves to a different segment and price changes along that segment.
- Using one marginal-revenue equation on both branches. The upper and lower demand segments have different slopes, so each has its own marginal-revenue equation. Use the equation for the branch containing the candidate quantity.
- Accepting an infeasible solution. A solution to \(MR=MC\) must be checked against the branch’s quantity range. If it falls outside that range, it is not the answer for that branch.
- Ignoring zero output. If marginal cost is at least marginal revenue at \(Q=0\), the upper-branch interior solution may be negative or absent. Check the boundary \(Q=0\); if marginal revenue is below cost throughout positive output, zero production is optimal.
- Assuming every cost change changes price. When marginal cost stays in the gap, output and price remain at the kink even though profit changes.
These conclusions rely on the stated kinked-demand assumptions and constant marginal cost. The analysis identifies how the model responds to a cost change; it does not establish that every firm’s demand has such a kink or that rival responses always take the assumed form.
Check Your Understanding
For each problem, identify the relevant branch and check feasibility. Unless a question says otherwise, assume zero fixed cost and that the firm can choose zero output.
- A firm has kink \((Q_k,P_k)=(4,24)\), with \(b=1\) and \(d=2\). Constant marginal cost is 18. Find its profit-maximizing output and price. (Answer: \(Q=4,\ P=24\).)
- A firm has kink \((6,18)\), with \(b=0.5\) and \(d=1\), and marginal cost \(c=17\). The upper demand branch is \(P=21-0.5Q\). Find the feasible optimum and price. (Answer: \(Q=4,\ P=19\).)
- A firm has kink \((8,40)\), with \(b=1.5\) and \(d=2.5\), and constant marginal cost \(c=16\). Find the optimum on the appropriate branch and its price. (Answer: \(Q=8.8,\ P=38\).)
- A firm has kink \((5,25)\), with \(b=1\) and \(d=2\). If \(c=32\), explain why an interior solution is not feasible and give the optimum. (Answer: \(Q=0\), because upper-branch marginal revenue at zero is 30, below marginal cost 32.)
- A firm has kink \((10,50)\), with \(b=2\) and \(d=3\). Compare outcomes when marginal cost is first 23 and then 32. For each cost, give output, price, and profit. (Answer: At \(c=23\), \(Q=10,\ P=50,\ \pi=270\). At \(c=32\), \(Q=9.5,\ P=51,\ \pi=180.5\).)