A Worked Workshop on Monopoly Pricing and Welfare

Putting Monopoly Pricing and Welfare Together

A complete monopoly problem often asks for more than the firm’s profit-maximizing price. It may also ask how that outcome compares with competition, how much surplus buyers and sellers receive, how much surplus is lost, or whether the firm can charge different prices to different customers. The calculations are connected: the demand curve determines both willingness to pay and revenue, while the cost curve determines the firm’s private incentive and the efficient production benchmark.

The workshop below uses a single-price monopolist facing a downward-sloping inverse demand curve. Unless stated otherwise, assume the firm is the only seller, chooses output to maximize profit, faces no price regulation, and cannot make customers resell the product to one another. For comparisons with perfect competition, assume a homogeneous product, price-taking buyers and sellers, constant marginal cost, no externalities, and no other market distortions. These assumptions matter: if marginal social benefit or marginal social cost differs from demand or private marginal cost, the surplus comparison must account for that difference.

Definition: Consumer surplus is buyers’ willingness to pay, summed over units purchased, minus their total expenditure. Producer surplus is revenue minus variable cost. With no fixed cost or other transfers, producer surplus equals profit. Total surplus is consumer surplus plus producer surplus. Deadweight loss is the total surplus available under the efficient benchmark minus total surplus in the outcome being evaluated.

For an inverse demand curve $P(Q)$, total willingness to pay for $Q$ units is the area under that curve, represented by $\int_0^Q P(q)\,dq$. With constant marginal cost $c$ and no externalities, efficient output satisfies $P(Q)=c$: the value of the last unit to buyers equals the cost of producing it. A monopolist instead selects output to maximize profit, so its marginal-revenue condition is $MR(Q)=c$, provided the optimum is an interior solution.

A Short Derivation for Linear Demand

For inverse demand $P(Q)=a-bQ$, where $a>c$ and $b>0$, revenue is price times quantity. Differentiating revenue with respect to output gives the marginal-revenue curve:

TR(Q)=(a-bQ)Q=aQ-bQ^2,\qquad MR(Q)=a-2bQ.

With constant marginal cost $c$, the interior monopoly optimum solves $a-2bQ=c$. The competitive benchmark solves $a-bQ=c$. Subtracting the two conditions shows why the monopolist produces less: marginal revenue is below price for every positive output, so the monopolist stops before price falls to marginal cost.

Q_M=\frac{a-c}{2b},\qquad P_M=\frac{a+c}{2}; \qquad Q_C=\frac{a-c}{b},\qquad P_C=c.

The monopoly outcome leaves units between $Q_M$ and $Q_C$ unproduced even though buyers’ willingness to pay for each of those units exceeds marginal cost. The resulting deadweight loss is the area between demand and marginal cost over that quantity range. In this linear, constant-cost model it is a triangle. The result does not automatically apply to nonlinear curves or to settings with externalities; there, surplus must be calculated from the relevant curves rather than by using the triangle shortcut.

Worked Example: Monopoly, Competition, and Deadweight Loss

A hypothetical firm faces inverse demand $P(Q)=100-2Q$, with $Q$ measured in units and price in dollars per unit. Its total cost is $C(Q)=20Q$, so marginal cost is constant at $20$. Find the monopoly outcome, the competitive benchmark, and the welfare measures in each case.

Method and assumptions. Use $MR=MC$ for the unregulated single-price monopoly and $P=MC$ for the competitive benchmark. Demand is linear, cost is constant, there are no fixed costs or externalities, and all units have the same price.

Monopoly quantity and price. Total revenue is $TR=(100-2Q)Q=100Q-2Q^2$, so $MR=100-4Q$. Set $MR=MC$: $100-4Q=20$, hence $4Q=80$ and $Q_M=20$. Substituting into demand gives $P_M=100-2(20)=60$.

Monopoly profit. Revenue is $60(20)=1{,}200$ dollars and cost is $20(20)=400$ dollars. Profit is $1{,}200-400=\$800$.

Monopoly consumer surplus and total surplus. Consumer surplus is the triangle between the demand curve and the price, from zero to 20 units: $\tfrac12(20)(100-60)=\$400$. Total surplus is $400+800=\$1{,}200$.

Competitive benchmark. Set demand price equal to marginal cost: $100-2Q=20$, giving $Q_C=40$ and $P_C=\$20$. Consumer surplus is $\tfrac12(40)(100-20)=\$1{,}600$. With price equal to marginal cost and no fixed costs, producer surplus and profit are zero. Total surplus is $\$1{,}600$.

Deadweight loss and check. The lost quantity is $40-20=20$. At $Q=20$, demand price is $60$, which is $40$ dollars above marginal cost; at $Q=40$, the gap is zero. Thus deadweight loss is $\tfrac12(20)(40)=\$400$. This also checks against the total-surplus difference, $1{,}600-1{,}200=\$400$. The method avoids the common mistake of using the monopoly price–cost margin as the triangle’s height; the height is the demand–marginal-cost gap at monopoly output.

Uniform Pricing Across Two Markets

When customers are separated into distinct markets, the firm may be able to set a different price in each one. Third-degree price discrimination means charging different prices to identifiable groups for the same product, with prices chosen to maximize profit. In the standard model, the groups must be distinguishable, resale between them must be prevented or sufficiently limited, and the firm must face different demand conditions across groups.

If marginal cost is the same for both markets and the firm can price separately, the profit-maximizing condition is $MR_A=MR_B=MC$. This is not the same as setting prices equal. Under a single uniform price, the firm instead adds the two markets’ quantities at each possible price to form total demand, then chooses its price and total quantity. The allocation of sales between markets follows from that common price.

Worked Example: Separate-Market Prices Versus One Uniform Price

A hypothetical seller faces demand $P_A=90-Q_A$ in market A and $P_B=70-2Q_B$ in market B. Marginal cost is constant at $\$10$ per unit. Assume the markets are identifiable and resale is prevented. Compare separate-market pricing with a single uniform price, including total surplus.

Method and assumptions. First apply $MR=MC$ separately to each market. Then aggregate quantities at a common price for the uniform-price outcome. Treat demand as willingness to pay and assume no externalities or fixed costs.

Separate prices. In market A, total revenue is $90Q_A-Q_A^2$, so $MR_A=90-2Q_A$. Setting this equal to 10 gives $Q_A=40$, and $P_A=90-40=\$50$. In market B, total revenue is $70Q_B-2Q_B^2$, so $MR_B=70-4Q_B$. Setting this equal to 10 gives $Q_B=15$, and $P_B=70-2(15)=\$40$.

Total profit is $(50-10)(40)+(40-10)(15)=1{,}600+450=\$2{,}050$. Consumer surplus is $\tfrac12(40)(90-50)=\$800$ in A and $\tfrac12(15)(70-40)=\$225$ in B. Total surplus is $2{,}050+800+225=\$3{,}075$.

Uniform price. At a price below $\$70$, both markets buy positive quantities: $Q_A=90-P$ and $Q_B=(70-P)/2$. Thus total quantity is $Q=125-1.5P$, or $P=(250-2Q)/3$. Revenue is $PQ=(250Q-2Q^2)/3$, giving $MR=(250-4Q)/3$. Setting this equal to 10 gives $250-4Q=30$, so $Q=55$ and $P=(250-110)/3=140/3$, or $\$46.67$ to the nearest cent.

At that price, $Q_A=90-140/3=130/3$ and $Q_B=(70-140/3)/2=35/3$; these sum to 55. Profit is $(140/3-10)(55)=\$2{,}016.67$. Consumer surplus is $\tfrac12(130/3)^2+\tfrac12(35/3)(70/3)=\$1{,}075.00$. Total surplus is $\$3{,}091.67$, so uniform pricing produces $\$16.67$ more total surplus here, despite producing less profit.

Welfare check. Efficient quantities satisfy demand price equal to 10: $Q_A=80$ and $Q_B=30$. The corresponding total surplus is $\$3{,}200+\$900=\$4{,}100$, calculated as the area under each demand curve less variable cost. Deadweight loss is therefore $4{,}100-3{,}075=\$1{,}025$ with separate prices and $4{,}100-3{,}091.67=\$1{,}008.33$ with a uniform price. The method avoids assuming that higher monopoly profit must mean higher total welfare: separate pricing changes the allocation of units between markets as well as the prices.

When Can the Firm Discriminate?

Definition: First-degree price discrimination means charging each buyer, or each unit purchased by a buyer, that buyer’s willingness to pay. Second-degree price discrimination means offering a menu of prices or product versions so buyers select options themselves; the seller does not directly observe each buyer’s type. Third-degree price discrimination means charging different prices to groups the seller can identify, based on differences in demand.

These labels describe different information and pricing mechanisms, not simply different numbers of prices. A seller needs detailed information about each buyer’s willingness to pay for first-degree pricing, and must be able to make the resulting individualized charges. A menu can support second-degree discrimination only if the options differ in a way that can induce different types to select different options; merely listing the identical product at two prices will not work, because buyers choose the lower price. Third-degree pricing requires observable group membership and limited arbitrage across groups.

A useful feasibility check is to ask, in order: Can the seller identify each customer’s willingness to pay? If not, can it offer a menu that customers sort themselves into? If neither is available, can it identify relevant groups and keep their purchases separate? These checks establish whether a form of discrimination is possible; they do not by themselves show that it is profitable. The seller must still compare the revenue and cost consequences of the available options.

Worked Example: Personalized Prices and the Information Needed

A hypothetical seller has two potential buyers, each of whom wants at most one unit. Buyer H values a unit at $\$50$, buyer L at $\$30$, and marginal cost is $\$10$. Consider a single uniform price and, separately, perfect individualized pricing. Then assess what information would be needed for second- or third-degree pricing.

Method and assumptions. For a uniform price, compare prices that serve both buyers with prices that serve only buyer H. For perfect individualized pricing, assume the seller knows each buyer’s willingness to pay and can charge that amount. Buyers purchase when value is at least price; there are no fixed costs or resale.

Uniform price. At $\$30$, both buyers purchase. Revenue is $2(30)=\$60$, cost is $2(10)=\$20$, and profit is $\$40$. Buyer H receives consumer surplus $50-30=\$20$; buyer L receives zero. Total surplus is $\$40+\$20=\$60$. At $\$50$, only H purchases, so profit is $50-10=\$40$, consumer surplus is zero, and total surplus is $\$40$. For prices between $\$30$ and $\$50$, only H buys and profit is less than $\$40$; below $\$30$, both buy but profit is less than $\$40$. Thus either $\$30$ or $\$50$ maximizes uniform-price profit, with different welfare consequences.

Perfect individualized pricing. Charge H $\$50$ and L $\$30$. Both buy, so revenue is $50+30=\$80$, total cost is $2(10)=\$20$, and profit is $\$60$. Consumer surplus is zero for both, while total surplus is $60$. Compared with the uniform price of $\$50$, serving L adds $30-10=\$20$ of surplus; compared with the uniform price of $\$30$, total surplus is unchanged, but $\$20$ of surplus shifts from H to the seller.

Feasibility and check. Individualized prices require the seller to know which buyer has which willingness to pay. If types are hidden but the seller can offer different product versions that induce self-selection, a second-degree menu may be possible; the information given here is insufficient to calculate such a menu’s profit. Third-degree pricing would require an observable group classification and the ability to prevent resale; the two private valuations alone do not provide that classification. The check is that perfect individualized pricing serves both buyers and produces total surplus $(50-10)+(30-10)=\$60$, matching the calculated profit when consumer surplus is zero. The method avoids treating any pair of different prices as proof of a particular degree of discrimination.

Common Mistakes in Integrated Problems

Using price instead of marginal revenue for monopoly output. A monopolist with downward-sloping demand generally has $MR<P$, so setting $P=MC$ finds the efficient or competitive quantity, not the monopoly quantity.
Mixing up surplus measures. Profit is not total surplus. Calculate consumer surplus and producer surplus separately, then add them.
Using the wrong deadweight-loss triangle. For a linear demand curve and constant marginal cost, the triangle spans the output difference between the monopoly and efficient quantities. Its height is the demand–cost gap at the restricted output.
Assuming price discrimination always improves welfare. It can raise profit, but its effect on total welfare depends on how output and the allocation of units change.
Ignoring feasibility. Distinct prices require a workable way to identify groups, elicit self-selection, or learn individual willingness to pay, as well as controls on resale where relevant.

Key takeaway: Keep the firm’s objective and the welfare benchmark separate. Monopoly pricing uses marginal revenue and marginal cost; efficient output compares willingness to pay with marginal cost. Price discrimination changes the information and pricing arrangement, but its welfare effect must still be calculated.

Check Your Understanding

For each problem, state the relevant assumption or method, show the intermediate steps, and compare your result with the answer.

A monopolist faces $P=80-Q$ and constant marginal cost of $\$20$, with no fixed costs or externalities. Find monopoly output and price, competitive output, and deadweight loss. (Answer: Monopoly $Q=30,\ P=\$50$; competitive $Q=60,\ P=\$20$; deadweight loss $=\$450$.)
A firm faces $P=120-3Q$ and constant marginal cost of $\$30$. Find monopoly output and price, efficient output, and deadweight loss. (Answer: Monopoly $Q=15,\ P=\$75$; efficient $Q=30$; deadweight loss $=\$337.50$.)
A seller can separate two markets with demands $P_1=100-2Q_1$ and $P_2=80-Q_2$. Marginal cost is $\$20$ in both markets, and resale is impossible. Find the profit-maximizing quantities and prices under third-degree pricing. (Answer: $Q_1=20,\ P_1=\$60;\ Q_2=30,\ P_2=\$50$.)
Two buyers value one unit at $\$45$ and $\$25$; marginal cost is $\$5$. Find the maximum profit under a uniform price and the profit under perfect individualized pricing. What happens to total surplus under each? (Answer: Uniform-price maximum profit is $\$40$, achieved at either $\$25$ serving both or $\$45$ serving only the high-value buyer; total surplus is respectively $\$60$ or $\$40$. Individualized pricing earns $\$60$ profit and yields total surplus $\$60$.)
A firm cannot observe individual willingness to pay but can verify whether a customer belongs to one of two groups. It can prevent resale, and it can offer two versions of a product that customers may select voluntarily. Which forms of discrimination are potentially feasible, and what additional information is needed to calculate the best menu? (Answer: Third-degree pricing is potentially feasible using the verified groups; second-degree pricing is potentially feasible through a self-selection menu. First-degree pricing is not supported by the stated information. To calculate the best menu, the seller needs each type’s willingness to pay for each version, the relevant costs, and the conditions governing customer selection.)

What You'll Learn