Profit Maximization

Profit maximization is the decision rule a firm uses to choose output and price to maximize profit (π). For a single-product firm facing a downward-sloping inverse demand function $p(q)$ and producing at cost $C(q)$, profit is

The core first-order condition (FOC) equates marginal revenue (MR) to marginal cost (MC):

Concrete numeric example: If $p(q)=100-2q$ and $C(q)=20q$ so $MC(q)=20$, then

Setting $MR=MC$ gives $100-4q=20\Rightarrow q^*=20$, and price $p^*=100-2\cdot20=60$. Profit is $\pi=(60-20)\cdot20=800$.

Why MR=MC? In marginal terms, producing one extra unit yields additional revenue $MR$ and additional cost $MC$. Profit increases when $MR>MC$ and decreases when $MR<MC$; at an interior optimum these are equal. This builds directly on the prerequisites: in Demand Functions we learned how $p(q)$ arises from consumer behavior and how elasticity matters; in Cost Functions we learned fixed/variable separation and how $MC$ behaves; and in Lagrange Multipliers we learned how to impose equality constraints.

Several important caveats and refinements:

• •MR differs from price whenever the firm faces a non-flat demand curve: $MR=p+q p'(q)$. For a linear demand $p=q$ slope negative, MR has twice the slope. Concrete numeric: with $p(q)=100-2q$, $p'(q)=-2$, so $MR=100-4q$ (twice the slope).

• •The FOC is necessary but not sufficient. We must check the second-order condition (SOC) for a local maximum:

Numeric check: for $p=100-2q$, $MR'(q)=-4$; if $C(q)=20q$ then $MC'(q)=0$, so $\pi''(q)=-4<0$, confirming a maximum at $q^*=20$.

• •There are corner solutions. If the unconstrained FOC yields an infeasible quantity (e.g., negative $q$ or exceeding capacity), the optimum may be at a boundary. Handling these requires the Kuhn–Tucker conditions (inequality constraints) introduced in the Lagrange Multipliers prerequisite. We will treat those in Section 3.

• •Elasticity matters. The markup over marginal cost depends inversely on the price elasticity of demand $\varepsilon$ (defined below), so more elastic demand forces lower markups and prices, all else equal.

Finally, profit maximization underpins much of industrial organization, regulatory economics, and public policy: price-setting, capacity investment, and welfare analysis are all derived from MR=MC logic (plus constraints and welfare weights). The rest of this lesson turns that logic into operational tools, demonstrates constrained optimization with binding capacity or regulatory caps, and shows how to verify second-order sufficiency.

Derivation of MR and the standard rule. Start with revenue $R(q)=p(q)q$. Differentiate:

Concrete numeric example: with $p(q)=200-5q$, $p'(q)=-5$, so

Set $MR=MC$ to solve for interior solutions. If $MC$ is constant at 40, then $200-10q=40\Rightarrow q^*=16$, $p^*=200-5\cdot16=120$.

Elasticity and the Lerner index. Use the price elasticity of demand $\varepsilon(p)=\frac{dq}{dp}\frac{p}{q}$ (note $\varepsilon<0$ if demand slopes down). Transform MR using elasticity. Start from $MR=p+q p'(q)$ and recall $p'(q)=1/(dq/dp)$; algebra yields the compact form:

Concrete numeric example: Suppose at the candidate price $p=120$ the elasticity is $\varepsilon=-4$. Then $MR=120(1+1/(-4))=120(1-0.25)=90$. If $MC=90$ at that output, MR=MC holds.

From $MR=MC$ we obtain the Lerner index, which expresses the monopoly markup relative to price:

Numeric example: If elasticity $\varepsilon=-5$ and $MC=10$, then

Interpretation: a less elastic demand (|ε| small) permits a larger markup; extremely elastic demand drives the markup down.

Second-order conditions. For a local maximum we require

Since $MR'(q)=p'(q)+p'(q)+p''(q)q=2p'(q)+p''(q)q$, for many standard demand specifications (linear, isoelastic) $MR'$ is negative. If $MC'$ is nonnegative (convex costs), the SOC usually holds.

Concrete SOC checks:

• •Linear demand: $p(q)=a-bq\Rightarrow MR(q)=a-2bq$, so $MR'=-2b$. If $b>0$ and $MC'(q)\ge0$, then $\pi''=-2b-MC'(q)<0$. Example: $a=100,b=2$, $MC'(q)=0$ gives $\pi''=-4<0$.

• •Quadratic cost: $C(q)=cq+\frac{d}{2}q^2\Rightarrow MC=c+dq$, so $MC'=d$. For $d>0$ the SOC becomes more strongly negative.

Edge cases and non-standard shapes. If demand is such that $MR'(q)\ge MC'(q)$ (e.g., increasing marginal revenue, which requires bizarre shapes), the FOC might be a minimum or inflection. Check the bordered Hessian for constrained problems (next section) or verify global concavity of $\pi(q)$.

Summary of actionable steps when solving standard monopoly problems:

1. 1)Compute $MR(q)=p(q)+p'(q)q$. Include a numeric evaluation at trial $q$.
2. 2)Compute $MC(q)$ from $C(q)$ and evaluate.
3. 3)Solve $MR=MC$ for $q^*$, then compute $p^*=p(q^*)$ and $\pi^*=p^*q^*-C(q^*)$.
4. 4)Verify SOC: compute $\pi''(q^*)$ numerically; if negative, you have a local maximum. If positive or zero, reconsider corner solutions.

Concrete worked micro-example: Let $p(q)=100-2q$ and $C(q)=10q+0.5q^2$. Then $MR=100-4q$, $MC=10+q$. Solve $100-4q=10+q\Rightarrow 90=5q\Rightarrow q^*=18$. Price $p^*=100-36=64$. Profit $\pi=64\cdot18-(180+0.5\cdot324)=1152-342=810$. Check SOC: $MR'=-4$, $MC'=1$, so $\pi''=-5<0$ confirming a maximum.

These procedures and checks are immediate applications of the Demand Functions and Cost Functions prerequisites, and they make the MR=MC rule operational and robust to pathological cases.

Real firms often face constraints: a capacity limit, regulatory price caps, or quantity mandates. These create inequality constraints that change both the chosen output and the valuation of relaxing constraints (shadow prices). The correct mathematical framework is Kuhn–Tucker (KKT) conditions: extend Lagrangians for inequality constraints and interpret multipliers.

Capacity constraint: consider $q\le K$ where $K$ is a fixed capacity. The firm's problem is

Set up the Lagrangian with multiplier $\lambda\ge0$:

KKT conditions:

1. 1)Stationarity: $\frac{\partial \mathcal{L}}{\partial q}=MR(q)-MC(q)-\lambda=0$.
2. 2)Complementary slackness: $\lambda(K-q)=0$.
3. 3)Primal feasibility: $q\le K$, $q\ge0$.
4. 4)Dual feasibility: $\lambda\ge0$.

Interpretation: if constraint non-binding ($q<K$), then $\lambda=0$ and the usual $MR=MC$ holds. If binding ($q=K$), then $\lambda=MR(K)-MC(K)\ge0$ — the shadow value equals the amount MR exceeds MC at the constrained quantity. Intuitively, the marginal value of relaxing capacity by one unit equals the extra profit that that unit would generate (MR-MC) at the current binding level.

Concrete numeric example: with $p(q)=100-2q$ and $C(q)=20q$ (so $MC=20$), we found unconstrained $q^*=20$. Suppose capacity $K=15$ binds. Then the firm sets $q=15$. Price is $p(15)=100-30=70$, profit is $(70-20)\cdot15=750$. The multiplier is \(\lambda=MR(15)-MC(15)\). Calculate: $MR(15)=100-4\cdot15=40$, so $\lambda=40-20=20$. That means increasing capacity by one unit increases profit by 20.

Regulatory price cap: suppose regulator imposes $p\le \bar p$. Under a price cap the firm faces a maximum feasible price; demand that is consistent with that price is $q_d(\bar p)$ from the inverse demand. If $\bar p$ is above the unconstrained profit-maximizing price $p^*$, the cap is non-binding. If $\bar p<p^*$, then the firm cannot charge $p^*$; instead, if the cap is binding and quantity is unconstrained (firm can supply any demanded q at $\bar p$), the firm behaves as a price-taker at $\bar p$ and supplies the demanded $q(\bar p)$ such that $p(q(\bar p))=\bar p$. Its profit is $\pi=(\bar p - MC(q))q(\bar p)$.

Concrete numeric example: with $p(q)=100-2q$, $MC=20$, unconstrained $p^*=60$. If the regulator sets $\bar p=50<p^*$, demand is $q(50)=(100-50)/2=25$. Profit then is $(50-20)\cdot25=750$. Notice this is the same numerical profit as the capacity example above — different constraints can produce identical outcomes.

More generally, when both capacity and price cap exist (or other constraints), form a Lagrangian with multipliers for each active inequality and write KKT conditions. The multipliers are interpretable: marginal welfare of relaxing the corresponding constraint or the marginal transfer value embedded in regulation.

Second-order sufficiency with constraints. For inequality-constrained problems, a sufficient condition for a local maximum is concavity of the objective function $\pi(q)$ (or, for multi-dimensional problems, that the Hessian of $\pi$ is negative-definite on the tangent cone of active constraints). Practically, if $\pi(q)$ is concave (e.g., $R$ concave and $C$ convex), any point satisfying KKT is a global maximum. Check numerically by verifying $\pi''(q)\le0$ for single-dimension problems.

Worked symbolic relation for shadow price under binding capacity: If capacity binds at $q=K$, then

Concrete numeric: earlier, with $K=15$, $\lambda=40-20=20$, so marginal value of capacity is 20.

Edge cases worth noting:

• •If the capacity is very tight so $q=K$ but $MR(K)<MC(K)$, then $\lambda$ would be negative which violates dual feasibility. That case would mean the capacity constraint cannot be binding at an optimum — rather, the true optimum would be interior or another constraint.

• •If regulatory caps are expressed in quantities or revenues rather than prices, the Lagrangian must incorporate those transformed constraints; always map constraints through the inverse demand or direct demand to the chosen decision variable.

This constrained optimization machinery is directly built on Lagrange Multipliers (the prerequisite). It produces not only optimal choices (output, price) but also shadow prices that tell regulators how binding constraints affect firms' incentives and how much welfare could be gained from marginally relaxing a constraint.

Profit maximization and MR=MC are used in many applied settings. Below are several concrete applications, each with short analytical descriptions and numerical notes that show how the MR=MC logic is applied and extended.

1) Welfare and deadweight loss. Monopoly pricing generates deadweight loss relative to competitive pricing ($p=MC$). Compute consumer and producer surplus changes using numeric examples. For $p(q)=100-2q$ and $MC=20$, monopoly sets $q^*=20$, $p^*=60$. Competitive output solves $p=MC\Rightarrow100-2q^{c}=20\Rightarrow q^{c}=40$, so competitive price $p^{c}=20$. Deadweight loss is the triangular area between demand and marginal cost from $q^*=20$ to $q^{c}=40$,

Concrete interpretation: the firm’s markup produces a welfare loss of 400 in the same monetary units used for prices and quantities.

2) Ramsey pricing (regulatory second-best). When a regulator must allow a firm to cover fixed costs but wants to minimize welfare loss, the regulator solves a constrained optimization that uses Lagrange multipliers across multiple products. The conditions resemble a generalized Lerner formula where the weighted markup equals a scaled inverse elasticity. For two goods, numbers matter: suppose goods have elasticities $\varepsilon_1=-2$, $\varepsilon_2=-4$ and the regulator uses weights proportional to demand quantities; the optimal markups will allocate more markups to the less elastic market.

3) Multi-product monopoly and cross-price effects. For two products with prices $(p_1,p_2)$ and demands $q_i(p_1,p_2)$, the FOC generalizes to a vector equality:

where $MR_i=\partial R/\partial q_i= p_i+\sum_j q_j \frac{\partial p_j}{\partial q_i}$ (or written in inverse demand form). Numeric examples typically require specifying a demand matrix; solve for both prices jointly using linear algebra.

4) Price discrimination. Under first-degree (perfect) discrimination, a firm can capture all consumer surplus by setting $p$ equal to each consumer’s willingness to pay; output equates to competitive output (no DWL). Under third-degree discrimination across segments with different elasticities, each segment satisfies the Lerner rule separately with its own elasticity. Numeric example: segment A with $\varepsilon_A=-2$ and MC=10 implies markup 0.5 of price: $(p_A-10)/p_A=0.5\Rightarrow p_A=20$. Segment B with $\varepsilon_B=-4$ gives $(p_B-10)/p_B=0.25\Rightarrow p_B\approx13.33$.

5) Empirical application: structural estimation of demand and cost. Econometricians estimate demand (Demand Functions prerequisite) and cost parameters to compute MR and MC and simulate policy changes. Concrete numbers are used to compute markups and counterfactual prices under regulation.

6) Industrial organization and strategic interaction. The MR=MC condition is the firm-level rule for price-setting in monopoly; in oligopoly (Cournot), each firm sets $MR_i=MC_i$ treating rivals’ outputs as fixed (best-response condition). Numeric Cournot examples use linear demands to produce closed-form equilibria.

7) Investment and capacity choice. When capacity is costly and dynamic, the shadow price of capacity (the Lagrange multiplier) derived above enters the investment decision: expand capacity until the marginal cost of capacity equals its discounted shadow benefit. Numeric dynamic models calibrate those multipliers to determine optimal investment paths.

Each of these applications relies on the primitives covered in the prerequisites: Demand Functions for mapping price ↔ quantity and elasticity, Cost Functions for MC and convexity, and Lagrange Multipliers for incorporating constraints and interpreting shadow values. Looking forward, mastering constrained profit maximization enables work on regulatory design, dynamic pricing, auction design, and empirical IO counterfactuals (e.g., mergers).