Deriving the Marginal-Revenue Gap at a Demand Kink

Why Marginal Revenue Has a Gap

A kinked demand curve has two different slopes at a particular price and quantity. In the standard model, demand is more responsive to a price increase above the kink than to a price decrease below it: customers may leave when this firm raises its price, while competitors are assumed to match a price cut. This tutorial focuses on the consequence for marginal revenue. Because total revenue depends on the demand curve, its slope changes abruptly when output passes the quantity associated with the kink.

The crucial point is that marginal revenue is not generally continuous at the kink. It has one limiting value as quantity approaches the kink from below and another as quantity approaches from above. If marginal cost falls between those two values, producing the kink quantity can maximize profit. The result depends on the two demand slopes, not on treating the kink as a smooth point.

Definition: Let the kink occur at price $P_k$ and quantity $Q_k$. Write demand as $Q_k-b_U(P-P_k)$ for prices $P\ge P_k$, and $Q_k-b_L(P-P_k)$ for prices $P\le P_k$, where $b_U>b_L>0$. The upper-price segment is more responsive: a given price change produces a larger quantity change there. The slopes $b_U$ and $b_L$ measure units of quantity change per dollar of price change.

For calculating marginal revenue, it is convenient to express price as a function of quantity. Quantities below $Q_k$ lie on the upper-price segment; quantities above $Q_k$ lie on the lower-price segment. Solving each demand equation for price gives:

P_U(Q)=P_k+\frac{Q_k-Q}{b_U}\quad (Q\le Q_k), \qquad P_L(Q)=P_k+\frac{Q_k-Q}{b_L}\quad (Q\ge Q_k).

These are inverse-demand equations: each gives the price at which the firm can sell quantity $Q$. Their slopes with respect to quantity are $-1/b_U$ and $-1/b_L$. Since $b_U>b_L$, the upper-price inverse-demand segment is flatter in absolute value than the lower-price segment. The equations apply over the relevant quantity ranges where the demand model describes feasible sales.

Deriving Marginal Revenue on Each Segment

Total revenue is price multiplied by quantity, $TR(Q)=P(Q)Q$. On a differentiable segment, marginal revenue is the derivative of total revenue with respect to quantity. The product rule gives $MR(Q)=P(Q)+Q\,dP/dQ$: the first term is the revenue from selling one additional unit at the current price, and the second accounts for the price change affecting units sold.

MR_U(Q)=P_k+\frac{Q_k-2Q}{b_U} \quad (Q<Q_k), \qquad MR_L(Q)=P_k+\frac{Q_k-2Q}{b_L} \quad (Q>Q_k).

To see the derivation on the upper-price segment, substitute $P_U(Q)=P_k+(Q_k-Q)/b_U$ into the product-rule expression. Since $dP_U/dQ=-1/b_U$, the result is $P_k+(Q_k-Q)/b_U-Q/b_U=P_k+(Q_k-2Q)/b_U$. The lower-price derivation is identical except that its inverse-demand slope is $-1/b_L$.

At the kink, the left-hand marginal revenue means the limit from quantities below $Q_k$, not a separate derivative defined exactly at the kink. The right-hand value is the limit from quantities above $Q_k$. Substituting $Q_k$ into the two segment formulas yields:

MR_{\text{left}}=P_k-\frac{Q_k}{b_U}, \qquad MR_{\text{right}}=P_k-\frac{Q_k}{b_L}.

Because $b_U>b_L>0$, we have $1/b_L>1/b_U$. Therefore $MR_{\text{left}}>MR_{\text{right}}$ when $Q_k>0$. The size of the gap is:

MR_{\text{left}}-MR_{\text{right}} =Q_k\left(\frac{1}{b_L}-\frac{1}{b_U}\right)>0.

This difference is the marginal-revenue gap at the kink. In a graph with quantity on the horizontal axis, marginal revenue falls discontinuously as quantity passes from below $Q_k$ to above it. The size depends on both slopes and the kink quantity. It is not the vertical distance between the two demand segments; those segments meet at the kink, while their marginal-revenue limits do not.

When the Kink Quantity Maximizes Profit

Let the firm's profit be $\pi(Q)=TR(Q)-C(Q)$, where $C(Q)$ is cost and marginal cost is $MC(Q)=C'(Q)$. On either smooth demand segment, the derivative of profit is $MR(Q)-MC(Q)$. A positive value means profit rises as the firm increases output; a negative value means profit falls as it increases output.

At $Q_k$, a sufficient condition for a profit maximum is that the marginal revenue just to the left is at least marginal cost at the kink, while the marginal revenue just to the right is at most marginal cost there. Equivalently:

Theorem: If profit is concave over the relevant output range, then $Q_k$ maximizes profit when $MR_{\text{left}}\ge MC(Q_k)\ge MR_{\text{right}}$. For example, this condition applies with the piecewise-linear demand above and nondecreasing marginal cost. The condition says that profit is not falling as output approaches the kink from below and is not rising as output moves above it. Without concavity or another check on profit away from the kink, the inequalities alone establish a local maximum; other output levels must also be checked for a global maximum.

The condition follows from the signs of the one-sided profit derivatives. Just below $Q_k$, $MR-MC$ is nonnegative, so profit is increasing toward the kink. Just above it, $MR-MC$ is nonpositive, so profit is decreasing away from the kink. With concave profit, these local signs support a global maximum. Importantly, there is no requirement that one smooth marginal-revenue value equal marginal cost at the kink. The derivative of revenue is not defined there in the ordinary two-sided sense.

If marginal cost is constant, the condition becomes particularly direct: cost must lie within the interval from $MR_{\text{right}}$ to $MR_{\text{left}}$. A cost change that remains in that interval can leave the maximizing quantity at $Q_k$. Since the demand curve still charges $P_k$ at that quantity, the maximizing price also remains $P_k$. This conclusion assumes the kinked demand schedule itself has not shifted.

Worked Example: Calculating the Gap and Testing a Cost Shift

A hypothetical firm faces a kink at price $P_k=\$40$ and quantity $Q_k=120$. Above the kink price, quantity falls by 6 units per dollar of price increase; below the kink price, quantity rises by 2 units per dollar of price decrease. The firm has constant marginal cost, first $\$10$ and then $\$15$, and a fixed cost of $\$300$. Find the marginal-revenue gap, test the maximizing quantity at each cost, and compute profit at that quantity.

Method. Treat the quantity changes per dollar as $b_U=6$ and $b_L=2$. Find the two marginal-revenue limits, compare each cost with both limits, and then calculate profit at the kink.

Step 1: Find the marginal-revenue limits. The left-hand value, approached from quantities below 120, is $40-120/6=40-20=\$20$. The right-hand value, approached from quantities above 120, is $40-120/2=40-60=-\$20$. The gap is $\$20-(-\$20)=\$40$.

Step 2: Test the initial cost. At $MC=\$10$, the condition is $20\ge10\ge-20$, which holds. With linear demand on both segments and constant marginal cost, marginal revenue falls on each segment, so profit rises toward $Q_k$ from below and falls after $Q_k$. The maximizing quantity is 120 and the corresponding price is $\$40$.

Step 3: Test the shifted cost. At $MC=\$15$, $20\ge15\ge-20$ also holds. The maximizing quantity and price remain 120 and $\$40$, even though the marginal cost has increased by $\$5$.

Step 4: Compute profit at each cost. Revenue at the kink is $\$40\times120=\$4{,}800$. At marginal cost $\$10$, total cost is $\$10\times120+\$300=\$1{,}500$, so profit is $\$4{,}800-\$1{,}500=\$3{,}300$. At marginal cost $\$15$, total cost is $\$15\times120+\$300=\$2{,}100$, so profit is $\$4{,}800-\$2{,}100=\$2{,}700$.

Check and common mistake avoided. The two limits are ordered correctly, $20>-20$, and both costs lie between them. The price and quantity stay at the kink, but profit falls by $\$600$, exactly the increase in marginal cost of $\$5$ times 120 units. The example avoids treating a stable choice of price and output as a claim that the firm's profit is unchanged.

Reading the Condition Carefully

The marginal-revenue gap does not guarantee that every cost change leaves the firm's choice unchanged. It identifies a range of marginal-cost values consistent with choosing the kink quantity under the stated assumptions. A cost that moves outside that range does not satisfy the kink condition, so the firm must compare profit at other quantities. Nor does the condition apply unchanged if a cost shift also moves the demand curve or the kink.

With nonconstant marginal cost, evaluate $MC(Q_k)$, not an average cost, at the kink. If marginal cost is increasing and the demand curves are linear, profit is concave on each segment and the marginal-revenue drop supports a maximum when the condition holds. For cost schedules that are not well behaved, checking the one-sided derivatives only establishes a local result; a global comparison may be needed.

Worked Example: Inferring Slopes From Demand Observations

A hypothetical seller's demand kink is at $P_k=\$30$, $Q_k=80$. On the higher-price segment, a price of $\$35$ corresponds to quantity 40. On the lower-price segment, a price of $\$26$ corresponds to quantity 88. Marginal cost shifts from $\$12$ to $\$18$, and fixed cost is $\$100$. Determine whether the maximizing price and quantity change, and calculate profit at each cost if the kink remains the maximizing choice.

Method. Recover each segment's slope from its change in quantity divided by its change in price. Then use those slopes to find the marginal-revenue limits and compare both costs with the resulting interval.

Step 1: Recover the upper-price slope. The price rises from $\$30$ to $\$35$, a change of $\$5$, while quantity falls from 80 to 40, a change of 40 units. Thus $b_U=40/5=8$ units per dollar.

Step 2: Recover the lower-price slope. The price falls from $\$30$ to $\$26$, a change of $\$4$, while quantity rises from 80 to 88, a change of 8 units. Thus $b_L=8/4=2$ units per dollar. These estimates satisfy $b_U>b_L$, as required for the stated kink pattern.

Step 3: Derive the marginal-revenue interval. The left-hand limit is $30-80/8=30-10=\$20$. The right-hand limit is $30-80/2=30-40=-\$10$. The gap is $\$20-(-\$10)=\$30$.

Step 4: Check both marginal costs. Both $20\ge12\ge-10$ and $20\ge18\ge-10$ hold. Under the linear-demand and constant-marginal-cost assumptions, the maximizing quantity remains 80 and its price remains $\$30$ at both costs.

Step 5: Compute profit and check. Revenue is $\$30\times80=\$2{,}400$. At marginal cost $\$12$, total cost is $\$12\times80+\$100=\$1{,}060$, giving profit $\$2{,}400-\$1{,}060=\$1{,}340$. At marginal cost $\$18$, total cost is $\$18\times80+\$100=\$1{,}540$, giving profit $\$2{,}400-\$1{,}540=\$860$. The profit reduction is $\$480$, equal to the $\$6$ cost increase times 80 units. The slope estimates and both marginal-cost comparisons are consistent with the unchanged kink choice.

A Nonconstant Marginal Cost

The same comparison works when marginal cost varies with output, but the relevant value is marginal cost at $Q_k$. The following example also checks the signs of the profit derivative on both entire demand segments, rather than relying only on the values at the kink.

Worked Example: Checking Both Segments With Increasing Marginal Cost

A hypothetical firm's kink is at $P_k=\$50$ and $Q_k=100$. The upper-price demand slope is $b_U=5$, and the lower-price slope is $b_L=1$, each measured in units per dollar. Marginal cost is $MC(Q)=\$10+\$0.10Q$, then shifts upward by $\$5$ at every output level. Fixed cost is $\$200$. Determine whether the maximizing output and price remain at the kink, and calculate profit before and after the shift.

Method. Derive each segment's marginal revenue, compare it with marginal cost at the kink, and verify the derivative signs over each side. Then integrate marginal cost to obtain total cost at 100 units.

Step 1: Find the marginal-revenue limits. On the upper-price segment, $MR_U(Q)=50+(100-2Q)/5=70-0.4Q$, so the left-hand limit at 100 is $70-0.4(100)=\$30$. On the lower-price segment, $MR_L(Q)=50+(100-2Q)/1=150-2Q$, so the right-hand limit is $150-2(100)=-\$50$. The gap is $\$30-(-\$50)=\$80$.

Step 2: Compare marginal cost at the kink. Before the shift, $MC(100)=10+0.10(100)=\$20$. After the shift, it is $\$25$. Both values lie between $-\$50$ and $\$30$.

Step 3: Verify the direction of profit change on both segments. Before the shift, for $Q\le100$, $MR_U-MC=(70-0.4Q)-(10+0.1Q)=60-0.5Q$, which is positive throughout that range. For $Q\ge100$, $MR_L-MC=(150-2Q)-(10+0.1Q)=140-2.1Q$, which is negative throughout that range. After the shift, the corresponding expressions are $55-0.5Q$ on the upper-price segment and $135-2.1Q$ on the lower-price segment. The first is positive for $Q\le100$, and the second is negative for $Q\ge100$. Thus profit rises up to 100 units and falls beyond 100 units both before and after the shift.

Step 4: Calculate profit. At the kink, revenue is $\$50\times100=\$5{,}000$. Before the shift, total variable cost is the integral of $10+0.10Q$ from 0 to 100: $10(100)+0.05(100)^2=\$1{,}500$. Including fixed cost, total cost is $\$1{,}700$, so profit is $\$5{,}000-\$1{,}700=\$3{,}300$. After the shift, marginal cost is $15+0.10Q$, so variable cost is $15(100)+0.05(100)^2=\$2{,}000$. Including fixed cost, total cost is $\$2{,}200$, and profit is $\$5{,}000-\$2{,}200=\$2{,}800$.

Check and common mistake avoided. The derivative signs on both sides confirm the kink is the profit maximum for each cost schedule; the cost at the kink alone would not establish that result for an arbitrary, nonconcave profit function. Profit falls by $\$500$, the added fixed-per-unit component of marginal cost, $\$5$, multiplied by 100 units. The example avoids comparing the kink's marginal revenue with average cost or overlooking the change in cost across all units.

Common Exam Trap

A frequent error is to impose the smooth condition $MR=MC$ at the kink and continue using one segment's marginal-revenue formula after quantity has crossed into the other segment. The formula derived from one inverse-demand slope is valid only on that segment.

For the first worked example, suppose marginal cost is $\$15$. Using only the upper-segment formula and setting it equal to cost gives:

40+\frac{120-2Q}{6}=15 \quad\Longrightarrow\quad 120-2Q=-150 \quad\Longrightarrow\quad Q=135.

This is not a valid upper-segment solution: that formula applies only for $Q\le120$, whereas 135 is above the kink. At $Q=135$, the applicable lower-segment marginal revenue is $40+(120-270)/2=-\$35$, not $\$15$. The correct comparison is between marginal cost $\$15$ and the one-sided limits $\$20$ and $-\$20$; since it lies between them, the maximizing choice is the kink quantity, 120, at price $\$40$.

Key takeaway: Calculate marginal revenue separately on the two inverse-demand segments. The kink's left-hand marginal revenue is $P_k-Q_k/b_U$, and its right-hand marginal revenue is $P_k-Q_k/b_L$. When marginal cost at the kink lies between these values, and profit is concave, the kink quantity maximizes profit. A cost shift that stays within this interval can change profit without changing the price or quantity chosen.

Check Your Understanding

For each problem, use the segment slopes and the one-sided marginal-revenue limits. Answers are given at the end of each question.

A firm's kink is at $P_k=\$24$, $Q_k=60$, with $b_U=5$ and $b_L=2$. Find the left- and right-hand marginal-revenue limits and the size of the gap. (Answer: $\$12$, $-\$6$, and $\$18$.)
A kink occurs at $P_k=\$18$, $Q_k=40$. At price $\$20$, quantity is 28; at price $\$15$, quantity is 46. Find both segment slopes and marginal-revenue limits. If constant marginal cost changes from $\$8$ to $\$13$, state which cost supports the kink quantity. (Answer: $b_U=6$, $b_L=2$; limits $\$11.33$ and $-\$2$; only $\$8$ supports the kink quantity.)
A firm has $P_k=\$70$, $Q_k=90$, $b_U=9$, and $b_L=3$. Constant marginal cost rises from $\$45$ to $\$55$. Find the gap and determine whether the maximizing price and quantity change, assuming concave profit. (Answer: limits $\$60$ and $\$40$, gap $\$20$; both costs lie inside the interval, so the choice remains $Q=90$ at $P=\$70$.)
A kink has $P_k=\$32$, $Q_k=80$, $b_U=4$, and $b_L=1.6$. Constant marginal cost is $\$8$, then rises to $\$13$. Under the standard linear-demand model, which cost supports the kink quantity? (Answer: limits $\$12$ and $-\$18$; $\$8$ supports the kink quantity, but $\$13$ is above the interval.)
A firm's kink is at $P_k=\$40$, $Q_k=50$, with $b_U=5$ and $b_L=1$. Marginal cost is $MC(Q)=\$8+\$0.20Q$, then shifts upward by $\$5$ at every output. Assuming linear demand and concave profit, determine whether the kink quantity remains optimal. (Answer: the marginal-revenue limits are $\$30$ and $-\$10$; marginal cost at the kink rises from $\$18$ to $\$23$, both within the interval, so the kink quantity and price remain $Q=50$ and $P=\$40$.)

This treatment of the marginal-revenue discontinuity is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

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