Why Price Can Differ Across Markets
A monopolist may sell the same product to distinct groups at different prices when it can identify each group, prevent resale between groups, and respond to each group’s demand separately. For example, a hypothetical software provider might set different prices for two customer categories that cannot readily transfer access to one another. The key decision is not to maximize revenue in each market independently: the firm must account for the cost of serving the combined quantity.
This pricing method is called third-degree price discrimination. The central result is that, when the firm serves a market at an interior optimum, that market’s marginal revenue equals the marginal cost of total production. Because marginal revenue depends on each group’s demand, optimal prices can differ even though the firm faces one common marginal cost.
The Multi-Market Profit-Maximization Rule
Let \(q_i\) be quantity sold in market \(i\), and let \(p_i(q_i)\) be that market’s inverse demand: the price buyers there will pay for quantity \(q_i\). Total output is \(Q=\sum_i q_i\). If the firm’s total cost is \(C(Q)\), its profit is
Assume inverse demands and cost are differentiable, and first consider a solution with positive output in every market being served. Increasing \(q_i\) by a small amount changes revenue in market \(i\) at the rate \(MR_i\), its marginal revenue. It also increases total output, so cost rises at the rate \(MC(Q)=C'(Q)\). At an interior profit maximum, these marginal changes must balance in each served market:
This does not mean that each market has its own separate cost curve. With a common cost function depending on total output, the marginal cost is the same for all markets at a given combined quantity. The firm allocates sales across markets until their marginal revenues are equal to that common marginal cost.
The boundary condition comes from considering a small increase from zero in a market while holding other quantities fixed. If \(MR_i(0)<MC(Q)\), that small increase reduces profit. This observation alone says nothing conclusive about all larger quantities when marginal revenue and marginal cost can change in unusual ways. To establish a global optimum, use assumptions that make the profit-maximization problem concave, or compare the candidate with all feasible alternatives.
Why Elasticity Affects the Price
For market \(i\), define the point price elasticity of demand in absolute value as \(\varepsilon_i=-(dq_i/dp_i)(p_i/q_i)\), where \(p_i\) and \(q_i\) are evaluated at the same point on the demand curve. For a downward-sloping demand curve, \(\varepsilon_i\) is positive. Marginal revenue can be written in terms of this elasticity.
The first equality follows by differentiating revenue \(R_i(q_i)=p_i(q_i)q_i\) using the product rule. For the second, the elasticity definition gives \(dq_i/dp_i=-\varepsilon_i q_i/p_i\), so \(dp_i/dq_i=-p_i/(\varepsilon_i q_i)\). Substitution yields \(MR_i=p_i-p_i/\varepsilon_i\).
At an interior optimum with positive marginal cost, \(MR_i=MC>0\). The elasticity relationship then implies \(\varepsilon_i>1\) and gives the optimal price formula below. It applies to an interior solution with positive marginal cost; it is not a formula for a market the firm does not serve, or for a zero-marginal-cost case.
With the same positive marginal cost in all served markets, a market with lower elasticity has a smaller denominator and therefore a higher price. Equivalently, the firm charges a higher markup over marginal cost to the group whose demand is less responsive to price. This comparison applies when the groups are both served at interior optima under the stated model. It does not imply that a market with any given elasticity must be served: the firm must still consider feasible quantities and any market-specific costs.
Worked Example: Two Linear Markets and Constant Marginal Cost
A hypothetical supplier sells a digital service in two separable markets. In market A, inverse demand is \(p_A=100-2q_A\); in market B, it is \(p_B=80-q_B\). Total cost is \(C(Q)=20Q\), where \(Q=q_A+q_B\). Find the optimal quantities and prices, and verify the common marginal-revenue condition.
Method. Derive marginal revenue separately in each market, set each equal to the common marginal cost, and then obtain price from the relevant inverse demand. The common mistake avoided here is to apply the same demand curve, or the same price, to both markets.
Revenue in A is \(R_A=(100-2q_A)q_A=100q_A-2q_A^2\). Therefore \(MR_A=100-4q_A\). Revenue in B is \(R_B=(80-q_B)q_B=80q_B-q_B^2\), so \(MR_B=80-2q_B\). Since \(C(Q)=20Q\), marginal cost is \(MC=20\).
For an interior optimum, set \(MR_A=20\) and \(MR_B=20\):
The corresponding prices are \(p_A=100-2(20)=60\) and \(p_B=80-30=50\). Total output is \(Q=20+30=50\). The marginal-revenue check gives \(MR_A=100-4(20)=20\) and \(MR_B=80-2(30)=20\), equal to marginal cost in both markets.
As an additional check, total revenue is \(60(20)+50(30)=1{,}200+1{,}500=2{,}700\). Total cost is \(20(50)=1{,}000\), so profit is \(2{,}700-1{,}000=1{,}700\). The solution is consistent with each market having positive output and with both marginal-revenue conditions.
The elasticity comparison also explains the price difference. For inverse demand \(p=a-bq\), the absolute elasticity is \(\varepsilon=p/(bq)\). Thus \(\varepsilon_A=60/[2(20)]=1.5\), while \(\varepsilon_B=50/[1(30)]=5/3\). Demand is less elastic in A at the chosen quantities, and the firm charges the higher price there.
When Marginal Cost Depends on Total Output
With nonlinear total cost, do not solve each market using an assumed marginal cost before finding combined output. Each market’s marginal revenue depends on its own quantity, while the common marginal cost depends on the sum of all market quantities. A convenient approach is to call the common marginal cost \(m\), express each market’s quantity in terms of \(m\), and then impose \(m=C'(Q)\).
Worked Example: Two Markets with Increasing Marginal Cost
A hypothetical producer faces inverse demands \(p_A=90-q_A\) and \(p_B=70-q_B\). Its total cost is \(C(Q)=10Q+Q^2/4\), where \(Q=q_A+q_B\). Find the interior allocation, the prices, and the resulting profit.
Method. Express both quantities in terms of their common marginal cost and solve the total-output consistency condition. The mistake to avoid is setting each market’s marginal revenue equal to a different marginal cost just because the quantities differ.
The marginal revenues are \(MR_A=90-2q_A\) and \(MR_B=70-2q_B\). The marginal cost is \(MC(Q)=10+Q/2\). Let the common marginal revenue and marginal cost be \(m\). The two interior conditions imply
Now impose \(m=10+Q/2\). Substituting \(Q=80-m\) gives \(m=10+(80-m)/2=50-m/2\), so \(3m/2=50\) and \(m=100/3\). Therefore \(q_A=(90-100/3)/2=85/3\) and \(q_B=(70-100/3)/2=55/3\). Their sum is \(Q=140/3\), and \(MC=10+(140/3)/2=100/3\), as required.
Prices are \(p_A=90-85/3=185/3\) and \(p_B=70-55/3=155/3\). The marginal-revenue check is \(MR_A=90-2(85/3)=100/3\) and \(MR_B=70-2(55/3)=100/3\). Both equal the calculated marginal cost.
For the profit check, total revenue is \((185/3)(85/3)+(155/3)(55/3)=15{,}725/9+8{,}525/9=24{,}250/9\). Cost is \(10(140/3)+(140/3)^2/4=4{,}200/9+4{,}900/9=9{,}100/9\). Profit is \(15{,}150/9=1{,}683.33\), rounded to the nearest cent. The solution has positive quantities in both markets and satisfies the common marginal condition.
Corner Solutions and a Necessary Caution
An unconstrained calculation may produce a negative quantity for one market. Since quantities cannot be negative, that result indicates a boundary candidate, not a sale of negative units. At a candidate where a market is excluded, check the marginal condition at zero and the optimality of the remaining allocation. Concavity is particularly useful: if the profit function is concave, the appropriate interior and boundary first-order conditions are sufficient for a global maximum.
Worked Example: Checking Whether to Serve a Market
A hypothetical firm considers markets A and B, with inverse demands \(p_A=15-q_A\) and \(p_B=50-q_B\). Total cost is \(C(Q)=5Q+Q^2/2\). Determine whether the optimum serves market A as well as B.
Method. First optimize the allocation when only B is served. Then evaluate the boundary condition for A and check that the objective is concave. The mistake to avoid is treating a marginal comparison as a global conclusion without checking the model’s curvature.
If only B is served, \(Q=q_B\), \(MR_B=50-2q_B\), and \(MC=5+q_B\). Setting them equal gives \(50-2q_B=5+q_B\), so \(q_B=15\). The corresponding price is \(p_B=50-15=35\), and marginal cost is \(MC=5+15=20\).
At the excluded-market boundary, \(MR_A(0)=15\), which is below the marginal cost of 20 at the B-only allocation. The derivative of profit with respect to \(q_A\) at that boundary is therefore \(15-20=-5\). A small increase in A sales, holding B output fixed, reduces profit.
For this model, the boundary conclusion is globally valid. Each market’s revenue is a concave quadratic in its own quantity, and \(C(Q)=5Q+Q^2/2\) is convex. Consequently, profit is strictly concave in \((q_A,q_B)\). The B-only allocation satisfies the interior condition for B and the boundary condition for A, so it is a global maximum. The resulting profit is revenue \(35(15)=525\) minus cost \(5(15)+15^2/2=75+112.5=187.5\), or \(337.5\).
Without concavity or a direct global comparison, a marginal comparison at zero is not enough to conclude that a market should never be served. For instance, with demand \(p=10-q\) and cost \(C(q)=12q-2q^2\), marginal revenue at zero is 10 and marginal cost at zero is 12. Yet at \(q=2.5\), price is \(7.5\), revenue is \(7.5(2.5)=18.75\), and cost is \(12(2.5)-2(2.5)^2=30-12.5=17.5\). Profit at that quantity is \(18.75-17.5=1.25\). Here marginal cost falls with output, so the initial marginal comparison does not establish that all positive quantities are unprofitable.
Interpreting the Result and Avoiding Exam Traps
Third-degree price discrimination is an allocation problem as well as a price-setting problem. The firm compares demand conditions across markets while accounting for the cost of total output. A few checks help keep the reasoning precise:
- Use each market’s own demand curve. Derive \(MR_i\) from that market’s revenue, not from total revenue treated as if there were one price.
- Use common marginal cost when cost depends on total output. Calculate \(Q\) across all served markets before evaluating \(C'(Q)\).
- Check quantities and boundaries. A negative interior quantity is infeasible. Test the boundary and use concavity or a global profit comparison to support exclusion.
- Interpret elasticity at the chosen point. A less elastic served market has a higher optimal price when the common marginal cost is positive and both markets are at interior optima.
- Keep the model’s assumptions visible. If buyers can resell across markets, the separate-price outcome may not be sustainable. The model also abstracts from legal or regulatory restrictions, which must be considered separately in real pricing decisions.
The main economic logic is that the firm shifts sales toward markets where an additional unit produces higher marginal revenue, while reducing sales where marginal revenue is lower, until the served markets’ marginal revenues match the cost of the combined output. Elasticity explains how that common marginal-revenue target translates into different prices.
Check Your Understanding
Work each problem by identifying the market-specific marginal revenues and the common marginal cost. Answers are included for checking.
- A hypothetical seller faces \(p_1=72-2q_1\) and \(p_2=60-q_2\), with total cost \(C(Q)=12Q\). Find both optimal quantities and prices. (Answer: \(q_1=15,\ p_1=42;\ q_2=24,\ p_2=36\).)
- Two markets have inverse demands \(p_A=100-2q_A\) and \(p_B=70-q_B\), and total cost is \(C(Q)=8Q+Q^2/2\). Find the interior quantities and prices. (Answer: \(MC=272/7;\ q_A=107/7,\ p_A=486/7;\ q_B=109/7,\ p_B=381/7\).)
- At an interior optimum, two served markets have absolute demand elasticities \(1.5\) and \(3\), and common marginal cost is 18. Find each market’s price and state which price is higher. (Answer: Prices are 54 and 27, respectively; the less elastic market has the higher price.)
- Markets A and B have inverse demands \(p_A=12-q_A\) and \(p_B=50-q_B\), with total cost \(C(Q)=5Q+Q^2/2\). Find the optimal allocation and explain the boundary decision. (Answer: \(q_A=0,\ q_B=15,\ p_B=35\); the B-only allocation is globally optimal because profit is concave and \(MR_A(0)=12<MC=20\).)
- For one market, demand is \(p=14-q\) and cost is \(C(q)=16q-3q^2\). Compare marginal revenue and marginal cost at zero with profit at \(q=3\). What does this show about using the zero-output comparison alone? (Answer: \(MR(0)=14<MC(0)=16\), but at \(q=3\), revenue is \(11(3)=33\), cost is \(48-27=21\), and profit is 12; the initial comparison alone does not establish that every positive quantity is unprofitable.)