Perfect Price Discrimination and First-Degree Pricing

Introduction

A monopolist that charges one price must choose a price that balances the number of units sold against the revenue earned per unit. Perfect price discrimination removes that single-price restriction: the seller charges each buyer, or for each unit, the buyer’s willingness to pay. This changes who receives the gains from trade and can change the quantity sold.

The benchmark is useful for analyzing the connection between monopoly pricing and economic efficiency. Under the ideal conditions described below, the seller supplies every unit for which willingness to pay covers marginal cost. The allocation is efficient, but nearly all the gains from trade go to the seller rather than to buyers.

Definition: First-degree price discrimination (or perfect price discrimination) occurs when a seller can charge each buyer the buyer’s maximum willingness to pay for each unit purchased. In the continuous model, the seller charges each successive unit at its marginal willingness to pay. The buyer receives no consumer surplus on units purchased at the stated willingness to pay.

Model and Conditions

Let $P(Q)$ denote inverse demand: the willingness to pay for the marginal unit when total quantity is $Q$. Assume demand is downward sloping, marginal cost is $MC(Q)$, and there are no externalities or other effects on people outside the transaction. For the standard result, the seller can observe or reliably infer buyers’ willingness to pay, can make individualized offers, and can prevent resale between buyers. The seller can also choose quantity freely unless a capacity constraint is specified.

These are strong assumptions. A buyer’s willingness to pay is not usually known precisely, and resale or limits on individualized offers can prevent complete extraction. Perfect discrimination is therefore best treated as a theoretical benchmark, not a description of ordinary pricing.

With a constant marginal cost $c$, the seller serves units as long as their willingness to pay is at least $c$. In a smooth market with a unique crossing, the efficient output satisfies $P(Q)=c$. If marginal cost varies with output, the corresponding condition is $P(Q)=MC(Q)$. A capacity limit can prevent the seller from reaching that output.

Why Output Reaches the Efficient Level

The key difference from a single-price monopolist is the revenue from one additional unit. Under perfect discrimination, the seller can charge that unit at its own willingness to pay without lowering the prices charged for earlier units. Thus, the marginal revenue from the next unit is its willingness to pay, $P(Q)$. The seller expands output while that value exceeds the marginal cost of producing the unit.

P(Q) > MC(Q) \quad \Longrightarrow \quad \text{expanding output increases surplus}

At the efficient output, the value of the marginal unit equals its marginal cost. Producing fewer units would leave out units worth more to buyers than the resources required to make them; producing more would use resources worth more than the additional units are to buyers. This condition identifies the efficient quantity under the stated assumptions. It does not say that buyers receive the surplus.

Theorem: If willingness to pay is observable, resale is prevented, the seller can price each unit individually, and no capacity constraint binds, first-degree price discrimination produces the efficient quantity: the seller supplies units until willingness to pay equals marginal cost. Consumer surplus is zero, producer surplus captures the gains from trade, and deadweight loss is zero. These surplus statements assume there are no externalities and compare the allocation of output, not its fairness.

Calculating Revenue and Surplus

For a continuous demand curve, the seller’s revenue is the sum of the willingness to pay for each unit. The integral below adds the prices of the marginal units from zero to output $Q$. Variable cost is $C(Q)$. Producer surplus is revenue minus variable cost; if there is a fixed cost $F$, economic profit is producer surplus minus $F$. The examples below have no fixed cost.

R(Q)=\int_0^Q P(q)\,dq,\qquad PS=R(Q)-C(Q),\qquad CS=0

The same result follows by measuring total surplus directly. Each unit contributes its willingness to pay minus the marginal cost of producing it. Summing those gains from zero to $Q$ gives total surplus. Since buyers pay their full willingness to pay, they retain none of this surplus as consumer surplus; the seller receives it as producer surplus.

TS=\int_0^Q \bigl(P(q)-MC(q)\bigr)\,dq=PS+CS

The integral also explains why the efficient output has no deadweight loss. At the efficient quantity, every unit with a positive value-minus-cost gap is traded, and no unit with a negative gap is produced. If the seller cannot produce enough to serve all positive-gap units, the omitted gains are deadweight loss.

Worked Example: Unconstrained Linear Demand

A hypothetical seller faces inverse demand $P(Q)=120-Q$ and has constant marginal cost $MC=\$20$. There are no fixed costs, externalities, or capacity limits. Under perfect price discrimination, find output, revenue, producer surplus, consumer surplus, and deadweight loss. Then compare the result with a single-price monopoly.

Method and output. With perfect discrimination, the seller compares willingness to pay for the marginal unit with marginal cost. Set $120-Q=20$, giving $Q=100$. Each unit is sold at its willingness to pay, so price varies across units from $\$120$ for the first marginal unit to $\$20$ for the last.

Revenue. Integrate the inverse demand curve from 0 to 100:

R(100)=\int_0^{100}(120-q)\,dq =120(100)-\frac{100^2}{2} =12{,}000-5{,}000 =\$7{,}000.

Cost and surplus. Variable cost is $20(100)=\$2{,}000$. Therefore $PS=7{,}000-2{,}000=\$5{,}000$. Buyers pay their willingness to pay, so $CS=\$0$. The efficient quantity is 100, so $DWL=\$0$.

Single-price comparison. For this linear demand curve, marginal revenue is $MR=120-2Q$. A single-price monopolist sets $MR=MC$: $120-2Q=20$, so $Q=50$, and the market price is $P(50)=\$70$. Revenue is $70(50)=\$3{,}500$, variable cost is $20(50)=\$1{,}000$, and producer surplus is $\$2{,}500$. Consumer surplus is the triangle $\frac12(50)(120-70)=\$1{,}250$. The lost gains from units 50 through 100 are a triangle with base 50 and height $70-20=\$50$, so deadweight loss is $\frac12(50)(50)=\$1{,}250$.

Check. The discriminatory outcome’s total surplus is $\$5{,}000$. At the single-price outcome, $PS+CS=\$2{,}500+\$1{,}250=\$3{,}750$; adding the $\$1{,}250$ deadweight loss gives the same $\$5{,}000$. The comparison shows that perfect discrimination increases output and removes the allocation loss, while transferring surplus away from buyers.

Method and common mistake avoided. The method is to integrate willingness to pay for revenue, then subtract variable cost. Do not treat the highest willingness to pay as the price for every unit, and do not confuse zero consumer surplus with zero total surplus.

Discrete Buyers and Capacity Limits

For a small number of buyers or indivisible units, use the same logic without integration. List each buyer’s willingness to pay and compare it with the marginal cost of serving that buyer. A buyer is served if willingness to pay exceeds the cost of the associated unit. With perfect discrimination, the seller charges each served buyer their willingness to pay. When capacity is limited, serve the buyers with the highest positive gains from trade, provided buyers and units are otherwise comparable.

Worked Example: Individual Prices for Discrete Buyers

A hypothetical seller can serve four buyers, each wanting one indivisible unit. Their willingness to pay is $\$60$, $\$45$, $\$35$, and $\$20$. Each unit costs $\$25$ to produce. There is no capacity limit and no fixed cost. Find the efficient sales, revenue, producer surplus, consumer surplus, and deadweight loss.

Method and sales. Compare each willingness to pay with the $\$25$ marginal cost. The first three buyers have positive gains from trade: $\$60-\$25=\$35$, $\$45-\$25=\$20$, and $\$35-\$25=\$10$. The fourth buyer’s willingness to pay is below cost: $\$20-\$25=-\$5$. Serve the first three buyers and do not produce the fourth unit.

Revenue and surplus. Charge each served buyer their own willingness to pay. Revenue is $60+45+35=\$140$, and variable cost is $3(25)=\$75$. Producer surplus is $140-75=\$65$. Each served buyer pays their full willingness to pay, so consumer surplus is zero. The three positive gains from trade sum to $35+20+10=\$65$, all received by the seller. No positive-gain transaction is omitted, so deadweight loss is zero.

Check and common mistake avoided. The fourth unit is excluded because its value is less than its cost; serving it would reduce total surplus by $\$5$. Do not interpret perfect discrimination as a requirement to serve every buyer. It permits individualized prices, but efficient production still depends on whether willingness to pay covers cost.

A capacity constraint can prevent the seller from reaching the efficient quantity even when it can charge individualized prices. In that case, consumer surplus can remain zero while deadweight loss is positive. The loss comes from valuable units that cannot be supplied, not from the price charged on units that are supplied.

Worked Example: Capacity-Constrained Perfect Discrimination

A hypothetical seller faces inverse demand $P(Q)=100-Q$, has constant marginal cost $MC=\$20$, and can produce at most 30 units. There are no fixed costs or externalities. Under perfect price discrimination, find actual output, revenue, producer surplus, consumer surplus, and deadweight loss.

Method and actual output. Without the capacity limit, efficient output would satisfy $100-Q=20$, so $Q=80$. Since capacity is only 30, the seller produces $Q=30$. The marginal buyer’s willingness to pay is $100-30=\$70$, but under perfect discrimination the earlier units are sold at their own willingness to pay.

Revenue and surplus. Revenue is the area under inverse demand from 0 to 30:

R(30)=\int_0^{30}(100-q)\,dq =100(30)-\frac{30^2}{2} =3{,}000-450 =\$2{,}550.

Variable cost is $20(30)=\$600$, so producer surplus is $2{,}550-600=\$1{,}950$. Consumer surplus is zero because each served unit is charged at its willingness to pay.

Deadweight loss. The units from 30 to 80 are not produced even though their willingness to pay exceeds marginal cost. At $Q=30$, the value-minus-cost gap is $100-30-20=\$50$; at $Q=80$, it is zero. The omitted gains form a triangle with base $80-30=50$ and height $\$50$:

DWL=\frac12(80-30)(50)=\$1{,}250.

Check and common mistake avoided. Total surplus with capacity is $PS+CS=\$1{,}950$. Adding the $\$1{,}250$ deadweight loss gives $\$3{,}200$, the surplus available at efficient output: $\int_0^{80}(80-q)\,dq=\$3{,}200$. Do not report the height of the loss triangle, $\$50$, as the loss itself; deadweight loss is its area, $\$1{,}250$.

Interpreting the Result and Avoiding Traps

Perfect price discrimination separates efficiency from distribution. With no binding capacity limit, it can produce the efficient output while leaving buyers with zero consumer surplus. That does not mean buyers receive no value from the transactions: it means the price they pay captures that value. Producer surplus is larger, but the efficient allocation alone does not establish that the outcome is equitable.

A common exam trap is to calculate revenue correctly and then call it producer surplus. Producer surplus must account for the cost of production. Another is to conclude that deadweight loss must be zero whenever consumer surplus is zero. That conclusion is valid only when output is efficient under the model’s assumptions. A capacity limit, or another restriction on transactions, can leave positive gains from trade unrealized.

Revenue is not producer surplus. Subtract variable cost from revenue; subtract fixed costs as well if the question asks for profit.
Zero consumer surplus is not zero total surplus. The seller may capture the gains from trade as producer surplus.
Check the output constraint. Compare capacity with the quantity at which willingness to pay equals marginal cost.
Calculate the area of omitted gains. When the value-minus-cost gap changes with quantity, deadweight loss is an area, not merely the gap at one endpoint.
Keep the benchmark assumptions in view. If willingness to pay is not observable or resale is possible, the seller may not achieve first-degree pricing.

Key takeaway: Under first-degree price discrimination, a seller charges each buyer their willingness to pay. Without a binding constraint, output expands until willingness to pay equals marginal cost, consumer surplus is zero, producer surplus captures the gains from trade, and deadweight loss is zero. A capacity limit can leave deadweight loss even when all served buyers pay their full willingness to pay.

Check Your Understanding

For each problem, assume the seller can identify willingness to pay, prevent resale, and charge each buyer or unit individually. Unless stated otherwise, assume no fixed costs or externalities.

A seller faces inverse demand $P(Q)=90-2Q$ and constant marginal cost of $\$10$, with no capacity limit. Find efficient output, revenue, producer surplus, consumer surplus, and deadweight loss. (Answer: $Q=40$; revenue $=\$2{,}000$; producer surplus $=\$1{,}600$; consumer surplus $=\$0$; deadweight loss $=\$0$.)
Four buyers each want one unit, with willingness to pay $\$72$, $\$54$, $\$38$, and $\$16$. Marginal cost is $\$24$ per unit and there is no capacity limit. How many units are sold, and what are revenue, producer surplus, consumer surplus, and deadweight loss? (Answer: 3 units; revenue $=\$164$; producer surplus $=\$92$; consumer surplus $=\$0$; deadweight loss $=\$0$.)
Inverse demand is $P(Q)=84-Q$, marginal cost is $\$24$, and capacity is 20 units. Find actual output, revenue, producer surplus, and deadweight loss. (Answer: output $=20$; revenue $=\$1{,}480$; producer surplus $=\$1{,}000$; deadweight loss $=\$800$.)
Four buyers have willingness to pay $\$100$, $\$70$, $\$50$, and $\$40$, one unit each. Marginal cost is $\$30$, with no capacity limit. Find output, revenue, and producer surplus. (Answer: output $=4$; revenue $=\$260$; producer surplus $=\$140$.)
Inverse demand is $P(Q)=150-3Q$, marginal cost is $\$30$, and capacity is 25 units. Find actual output, revenue, producer surplus, and deadweight loss. (Answer: output $=25$; revenue $=\$2{,}812.50$; producer surplus $=\$2{,}062.50$; deadweight loss $=\$337.50$.)

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