Residual Demand and Strategic Variables

Why Residual Demand Matters

In an oligopoly, one firm’s sales and revenue depend not only on its own decision but also on what it expects its rivals to do. A firm choosing output asks what market price will result from its output together with rival output. A firm choosing price asks how much customers will buy from it at that price, given rival prices and the way buyers allocate purchases. In either case, the firm’s relevant demand is conditional on its view of rival behavior.

The strategic variable is the decision variable a firm directly chooses, such as its quantity or its price. Residual demand is the demand left for that firm after rival actions, or anticipated rival responses, are taken into account. These ideas are related but distinct: the strategic variable describes the firm’s choice, while residual demand describes how market conditions map that choice into the firm’s sales or the price it receives.

Definition: A firm’s residual demand is the relationship between its own strategic choice and the sales it can make, or the price it can obtain, conditional on specified rival actions or expected rival responses. It is not generally a fixed curve: it can change when the firm’s beliefs about rivals change.

Residual Demand When Firms Choose Quantities

Suppose firms sell a homogeneous product, total market output is $Q=q_i+q_{-i}$, and inverse market demand gives the price $P(Q)$. Here $q_i$ is firm $i$’s output and $q_{-i}$ is combined rival output. If firm $i$ expects rival output to depend on its own choice according to $q_{-i}^{e}(q_i)$, its anticipated residual inverse demand is

p_i^R(q_i)=P\bigl(q_i+q_{-i}^{e}(q_i)\bigr).

This expression is conditional: it describes the price the firm expects at each own output, given the rival response it anticipates. If rivals’ output is expected to stay fixed, then $q_{-i}^{e}$ is constant. If rivals are expected to change output in response, the firm’s residual inverse demand has a different slope.

Result: If inverse market demand is differentiable and $q_{-i}^{e}$ is differentiable, the local change in anticipated market price as firm $i$ changes its output is

\frac{dp_i^R}{dq_i} =P'(Q)\left(1+\frac{dq_{-i}^{e}}{dq_i}\right).

This is a local relationship, evaluated at the output under consideration. It assumes the firm’s expected rival-output function and market inverse demand are differentiable there.

The formula follows from the chain rule. Total output is $Q(q_i)=q_i+q_{-i}^{e}(q_i)$, so its derivative is $dQ/dq_i=1+dq_{-i}^{e}/dq_i$. Since $p_i^R=P(Q(q_i))$, differentiating gives $dp_i^R/dq_i=P'(Q)(1+dq_{-i}^{e}/dq_i)$. With downward-sloping inverse demand, $P'(Q)<0$. The sign and size of the response term still matter: it is not enough to know that rivals’ expected response is negative.

For example, write $r=dq_{-i}^{e}/dq_i$. If $-1<r<0$, rivals reduce their output when firm $i$ expands, but by less than the firm’s increase. Total output still rises, so market price falls, but less sharply than it would with fixed rival output. If $r=-1$, the rival reduction exactly offsets the firm’s increase, leaving total output locally unchanged and price locally unchanged. If $r<-1$, total output falls and price rises locally. If $r>0$, rivals expand along with firm $i$, so total output rises more sharply and price falls more sharply than with fixed rival output.

Worked Example: A Quantity Choice with an Expected Rival Response

Market inverse demand is $P=200-2Q$, where $Q=q_1+q_2$. Firm 1 expects firm 2 to choose $q_2^{e}=40-0.25q_1$ over the range being considered. Find firm 1’s anticipated residual inverse demand, the expected price and revenue at $q_1=20$, and the local price slope there. Compare the slope with the case in which firm 2’s output stays fixed.

Method. Substitute the expected rival-output function into total output and then into inverse market demand. This avoids treating total output as firm 1’s output alone.

First, total output as a function of $q_1$ is

$$ Q=q_1+(40-0.25q_1)=40+0.75q_1. $$

Therefore, firm 1’s anticipated residual inverse demand is

$$ p_1^R(q_1)=200-2(40+0.75q_1)=120-1.5q_1. $$

At $q_1=20$, expected rival output is $40-0.25(20)=35$, so total output is $20+35=55$. Market price is $200-2(55)=90$, which also equals $120-1.5(20)=90$. Firm 1’s revenue is $20(90)=\$1{,}800$.

The local price slope from the residual inverse demand is $-1.5$. The chain-rule calculation gives the same result: $P'(Q)=-2$ and $dq_2^{e}/dq_1=-0.25$, so

\frac{dp_1^R}{dq_1}=-2(1-0.25)=-1.5.

If rival output instead stayed fixed at 35, total output would rise one-for-one with $q_1$, giving a price slope of $-2$. The anticipated rival reduction partly offsets firm 1’s added output, so price falls less sharply. The response slope is between $-1$ and 0, which is the condition for that conclusion. A negative response slope alone would not suffice.

Common mistake avoided: Using $P=200-2q_1$ ignores the rival’s output. The market price depends on $Q=q_1+q_2$, not on firm 1’s output by itself.

Price Choice and Residual Demand

A firm may choose price rather than quantity. With differentiated products, customers may regard one firm’s product as an imperfect substitute for another’s. The firm’s demand can then depend on its own price and rival prices. If the firm anticipates a rival-price response $p_j^{e}(p_i)$, it substitutes that response into its own demand function to obtain residual demand as a function of its chosen price.

For example, suppose firm $i$’s demand is $q_i=D_i(p_i,p_j)$. Its anticipated residual demand is $q_i^R(p_i)=D_i(p_i,p_j^{e}(p_i))$. This does not mean rival behavior is already an equilibrium: it is the firm’s demand conditional on its specified expectation about the rival. To describe an equilibrium, expectations and actual rival choices must be mutually consistent, a question developed in the analysis of particular oligopoly models.

Worked Example: Pricing with a Differentiated Rival Product

A firm faces demand $q_1=130-4p_1+2p_2$, where prices are in dollars and quantities are units. It expects the rival’s price to follow $p_2^{e}=8+0.5p_1$. The firm’s variable cost is \$8 per unit, with no fixed cost. Derive residual demand and find the profit-maximizing price under this expected response.

Method. Substitute the expected rival price into demand, then maximize the resulting profit function. This is a conditional price decision, not a claim that the expected response is an equilibrium.

Substituting $p_2^{e}=8+0.5p_1$ gives

$$ q_1^R(p_1)=130-4p_1+2(8+0.5p_1)=146-3p_1. $$

At any price for which this quantity is positive, revenue is $R(p_1)=p_1(146-3p_1)$. With variable cost \$8 per unit and no fixed cost, profit is

\pi_1(p_1)=(p_1-8)(146-3p_1) =-3p_1^2+170p_1-1168.

Differentiating gives $d\pi_1/dp_1=170-6p_1$. Setting this equal to zero gives $p_1=170/6=85/3$, or approximately \$28.33. The second derivative is $-6<0$, so this stationary point is a maximum. At this price, residual demand is $146-3(85/3)=61$ units. The expected rival price is $8+0.5(85/3)=131/6$, or approximately \$21.83. Profit is $(85/3-8)(61)=(61/3)(61)=3721/3$, or approximately \$1,240.33.

As a check, the firm’s price exceeds its unit variable cost and residual demand is positive. The calculated profit is the product of the \$20.33 margin and 61 units, rounded to cents only at the final step.

Common mistake avoided: Holding $p_2$ fixed while differentiating would describe a different expectation. Here the rival-price response changes the slope of firm 1’s residual demand from $-4$ to $-3$, so it also changes the profit-maximizing price.

Homogeneous Products: A Price Choice Can Create a Kink

With a homogeneous good, buyers generally choose the lower-priced seller, subject to such details as production capacity and how sales are divided when prices tie. Consequently, residual demand under price choice need not be a smooth curve. It may change abruptly when a firm’s price crosses a rival’s price. A derivative-based marginal-revenue calculation is not appropriate at a jump or kink unless the relevant segment is differentiable and the chosen point lies within it.

The allocation rule matters. In the next example, there are two firms selling the same product. Market demand at price $p$ is $D(p)=100-2p$. The rival’s price is expected to stay at \$22. Firm 1 has capacity of 45 units, sells all market demand if it is the cheaper firm up to that capacity, receives half of market demand at a tie, and sells nothing when it charges more than the rival. These assumptions define the residual demand in this example; other allocation rules can produce different residual demand.

Worked Example: Residual Demand Around a Rival’s Price

Using the assumptions above, calculate firm 1’s sales and revenue at prices of \$21, \$22, and \$23. Explain why the resulting residual demand is not a smooth price-demand curve.

Method. Apply the stated allocation rule separately in each price case. Do not assume that a small price change always causes a small change in sales.

At $p_1=\$21$, firm 1 is cheaper. Market demand is $100-2(21)=58$ units, but capacity limits sales to $\min(58,45)=45$. Revenue is $21(45)=\$945$.

At $p_1=\$22$, the firms tie. Market demand is $100-2(22)=56$ units, so firm 1 receives half, or 28 units. Revenue is $22(28)=\$616$.

At $p_1=\$23$, firm 1 charges more than the rival and sells zero under the stated rule. Revenue is $\$0$.

Thus the three price-sales pairs are $(21,45)$, $(22,28)$, and $(23,0)$. In particular, the move from \$21 to \$22 changes sales because the allocation switches from the capacity-limited lower-price seller to an equal share at a tie. The price-sales relationship is not one smooth line to which a single derivative can be applied across all three cases.

Common mistake avoided: Applying the equal-split rule at \$21 would give the wrong sales. The firms split demand only at the specified tie price; below it, firm 1 is the cheaper seller.

How to Analyze a Residual-Demand Problem

A reliable analysis keeps three elements separate: the firm’s own choice, the rival behavior being assumed, and the demand or market-clearing relation connecting them. The following process helps prevent a common strategic-variable mix-up.

Identify the strategic variable.
State whether the firm chooses quantity, price, or another action. Do not substitute a price response into a quantity-choice model, or vice versa.

Specify rival behavior.
Write down whether rival actions are fixed or are expected to respond to the firm’s choice. State the relevant response function or allocation rule.

Construct residual demand.
For quantity choice, combine own and rival quantities before applying market demand. For price choice, substitute expected rival prices into the firm’s demand and respect any capacity or tie-breaking rules.

Check the domain and interpretation.
Confirm that quantities are nonnegative, any capacity constraint is respected, and the result is conditional on the stated rival behavior. A residual-demand calculation alone does not establish equilibrium.

Why the Distinction Matters

Residual demand is the link between a firm’s strategic choice and its economic outcome. In a quantity setting, anticipated rival output determines how much total output changes when the firm expands; that in turn determines the price effect. In a price setting, anticipated rival prices shape the firm’s sales at each price. The same market conditions can therefore imply different residual-demand relationships under different strategic variables or different expectations about rivals.

A useful exam check is to ask what is held fixed and what is allowed to respond. “Rivals’ output is fixed” and “rivals reduce output when this firm expands” are different assumptions, even if the current rival output is the same. Likewise, a rival’s current price is not automatically fixed when the question specifies an expected price response. For homogeneous goods, also check whether the problem specifies capacity and how buyers are allocated at a tie; without those details, sales at a given price may not be uniquely determined.

Key takeaway: Residual demand is conditional on rival behavior. Derive it using the firm’s actual strategic variable and the stated rival response. In quantity models, a negative rival response makes price fall less sharply only when its slope lies strictly between $-1$ and $0$; in price models, the residual demand may be smooth, kinked, or discontinuous depending on product differentiation and the allocation rules.

Check Your Understanding

For each problem, state the relevant strategic variable and apply the specified rival behavior before calculating.

Inverse market demand is $P=160-Q$, where $Q=q_1+q_2$. Firm 1 expects $q_2^{e}=30-0.2q_1$. Find its residual inverse demand and expected market price at $q_1=25$. (Answer: $p_1^R=130-0.8q_1$; at $q_1=25$, price is $110$.)
Inverse market demand is $P=90-3Q$. At the output considered, a firm’s expected rival-output response has slope $-1$. What is the local price slope with respect to the firm’s own output, and why? (Answer: $0$; the rival’s local output reduction exactly offsets the firm’s increase, so total output and price are locally unchanged.)
A firm faces differentiated-product demand $q_1=80-2p_1+p_2$ and expects $p_2^{e}=5+0.25p_1$. Find residual demand and sales at $p_1=20$. (Answer: $q_1^R=85-1.75p_1$; sales at $p_1=20$ are $50$ units.)
Market demand is $D(p)=120-2p$. A homogeneous-good firm expects a rival price of \$30, has capacity 40, and sells all demand up to capacity when it is cheaper, half of demand at a tie, and zero when dearer. Find its sales and revenue at prices of \$29 and \$30. (Answer: At \$29, demand is 62, sales are 40 and revenue is \$1,160; at \$30, sales are 30 and revenue is \$900.)
With inverse market demand $P=100-2Q$, an expected rival-output response has local slope $-1.4$. Find the local price slope with respect to own output and interpret its sign. (Answer: $+0.8$; total output falls locally as own output rises, so market price rises locally.)

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