Why Test the Marginal Unit?
A competitive market outcome can be checked one unit at a time: does the value buyers place on the marginal unit match the cost of the resources needed to produce it? If buyers value an additional unit more than it costs to make, expanding output can create a gain. If that unit costs more than buyers value it, contracting output can avoid a loss. This comparison is the central test for a one-good outcome.
The test concerns the allocation of a single good. It is not enough to observe a market price or to confirm that buyers and sellers are willing to trade at that price. We must also know that the quantity traded clears the market, and that the price-taking assumptions link buyers’ marginal willingness to pay and firms’ marginal cost to that market price.
The Competitive Efficiency Test
Assume a single market for a divisible good, price-taking consumers and firms, no taxes or other wedges between the price buyers pay and the price sellers receive, and no effects on people outside the market. Assume also that inverse demand is a valid measure of marginal willingness to pay and inverse supply represents marginal cost. These assumptions let us use the market curves to test the allocation; without them, the curves may not measure all relevant benefits and costs.
Write marginal willingness to pay at quantity \(Q\) as \(MB(Q)\), and marginal cost as \(MC(Q)\). For a positive, interior quantity in a smooth market, the competitive equilibrium clears the market and satisfies the following condition:
Here \(Q_D(P)\) is quantity demanded at price \(P\), \(Q_S(P)\) is quantity supplied, and \(Q^*\) and \(P^*\) are the equilibrium quantity and price. Market clearing means the amount buyers want equals the amount firms supply. The equality between marginal willingness to pay, price, and marginal cost follows from the model’s price-taking behavior: buyers’ marginal choice is guided by the market price, while firms’ marginal production choice is also guided by that price.
The equality is a useful test, but it is not a complete test in every setting. It applies to an interior choice with a divisible good and appropriately measured curves. At a boundary, or with indivisible units, efficiency can instead be expressed as a comparison of gains and costs from the available changes in quantity.
Worked Example: Checking a Linear Competitive Equilibrium
A market has inverse demand \(P=96-3Q\) and inverse supply \(P=12+Q\), where price is in dollars per unit and quantity is in hundreds of units. Find the competitive equilibrium and test whether its marginal willingness to pay equals marginal cost.
Method. In a competitive equilibrium, quantity demanded equals quantity supplied. Since both curves give the price associated with a quantity, set them equal:
Subtract 12 from both sides and add \(3Q\) to both sides:
Substitute \(Q^*=21\) into either curve to find the price:
Check the supply curve independently: \(12+21=33\), so both curves give the same price. At \(Q=21\), marginal willingness to pay is \(96-3(21)=33\) dollars per unit, and marginal cost is \(12+21=33\) dollars per unit. Thus the equilibrium clears the market and satisfies \(MB(Q^*)=P^*=MC(Q^*)\).
Common mistake to avoid. Solving only the demand curve for price does not establish an equilibrium. The supply curve must give the same price at the same quantity. Here, the independent supply check confirms market clearing.
Why Equality Is the Interior Efficiency Condition
The comparison can be justified by considering total gains from trade as output changes. Under the assumptions above, the marginal gain from one more unit is its marginal benefit minus its marginal cost. For smooth curves, the total gains from producing \(Q\) units, relative to producing none, can be written as:
The integrand is the gain from the marginal unit at quantity \(q\). By the fundamental theorem of calculus, the rate at which total gains change as output increases is:
If \(MB(Q)>MC(Q)\), a small increase in output raises total gains. If \(MB(Q)<MC(Q)\), a small decrease in output raises total gains. Therefore, at a smooth interior maximum, neither a small increase nor a small decrease should improve the outcome. That requires:
This condition is necessary for an interior maximum. For it to identify the best quantity rather than another stationary point, the surrounding marginal-benefit and marginal-cost values must also support a maximum. A common sufficient pattern is that \(MB(Q)-MC(Q)\) is positive below the crossing and negative above it, as with downward-sloping demand and upward-sloping supply.
Read inverse demand as marginal willingness to pay and inverse supply as marginal cost only when the model assumptions make those interpretations valid.
At the proposed price and quantity, verify that quantity demanded equals quantity supplied.
At an interior quantity for a divisible good, check whether \(MB(Q)\), the price, and \(MC(Q)\) are equal.
If marginal benefit exceeds marginal cost, more output can create gains; if marginal cost exceeds marginal benefit, less output can avoid losses.
Diagnosing Output Away from the Efficient Quantity
A proposed outcome need not be a competitive equilibrium. To test it, evaluate marginal willingness to pay and marginal cost at the proposed quantity, not at a different quantity or merely at the stated price. A positive gap \(MB(Q)-MC(Q)\) indicates that the marginal unit creates a net gain; a negative gap indicates that producing the marginal unit uses resources worth more than the benefit it provides.
Worked Example: Measuring the Gain from Expanding Output
A market’s inverse demand is \(P=75-Q\), and its inverse supply, representing marginal cost, is \(P=15+2Q\). An output restriction holds production at \(Q=14\). Find the unrestricted competitive quantity, test the restricted quantity, and calculate the gains available from expanding output to the competitive quantity. Price is in dollars per unit and quantity is in units.
Step 1: Find the competitive quantity. Set marginal willingness to pay equal to marginal cost:
At this quantity, the price is \(75-20=55\) dollars per unit. The supply curve confirms it: \(15+2(20)=55\).
Step 2: Test the restricted quantity. At \(Q=14\), marginal willingness to pay is \(75-14=61\) dollars per unit, while marginal cost is \(15+2(14)=43\) dollars per unit. The gap is \(61-43=18\) dollars per unit. Since marginal benefit exceeds marginal cost, expanding output from 14 can increase total gains.
Step 3: Calculate the gains from expansion. The marginal-benefit-minus-marginal-cost gap is \(75-Q-(15+2Q)=60-3Q\). It falls from 18 dollars at \(Q=14\) to zero at \(Q=20\). With linear curves, the gains from expanding are the area of a triangle with base \(20-14=6\) units and height 18 dollars per unit:
The check is consistent: the gap declines linearly to zero at the competitive quantity, so it is positive over the expansion from 14 up to 20 and zero at 20. The restriction leaves 54 dollars of gains unrealized in this model. Common mistake to avoid: treating the price at a proposed quantity as proof of efficiency. The relevant test is the difference between marginal benefit and marginal cost at that quantity.
Indivisible Units and Boundary Cases
For an indivisible good, the marginal unit cannot be divided into an arbitrarily small change. Exact equality between marginal willingness to pay and marginal cost is therefore not required. Instead, compare the value and cost of each unit that could be added or removed. Produce a unit when its willingness to pay is at least its marginal cost, and do not produce the next unit if its cost exceeds its willingness to pay.
Worked Example: Testing an Indivisible-Unit Allocation
A small market has the following willingness to pay for successive indivisible units and marginal costs of producing them. Buyers are ordered from highest to lowest willingness to pay, and units are produced in order from lowest to highest cost.
| Unit | Willingness to pay (dollars) | Marginal cost (dollars) | Net gain if traded (dollars) |
|---|---|---|---|
| 1 | 50 | 14 | 36 |
| 2 | 42 | 22 | 20 |
| 3 | 34 | 30 | 4 |
| 4 | 26 | 38 | -12 |
| 5 | 18 | 46 | -28 |
For unit 1, the net gain is \(50-14=36\) dollars. For unit 2, it is \(42-22=20\) dollars. For unit 3, it is \(34-30=4\) dollars. These three units each create a positive gain. Unit 4 would create \(26-38=-12\) dollars, and unit 5 would create \(18-46=-28\) dollars. Therefore, the efficient quantity is three units.
At a price of 32 dollars, buyers want the first three units because their willingness to pay is 50, 42, and 34 dollars; they do not want unit 4, valued at 26 dollars. Sellers offer the first three units because their costs are 14, 22, and 30 dollars; they do not offer unit 4, which costs 38 dollars. Quantity demanded and quantity supplied are both three. The outcome clears the market and the allocation is efficient, even though the third unit’s willingness to pay, 34 dollars, does not equal its cost, 30 dollars. Common mistake to avoid: imposing the smooth-curve equality condition on indivisible units. Here the correct test is whether each traded unit has willingness to pay at least as high as its cost, and whether the next unit would have a net gain.
There is also a boundary qualification for divisible goods. If output cannot fall below zero, an efficient outcome at \(Q=0\) need not have \(MB(0)=MC(0)\): producing the first unit must not offer a positive net gain. If output is capped at a maximum \(\bar Q\), the best feasible quantity can be \(\bar Q\) even when \(MB(\bar Q)>MC(\bar Q)\), because the model does not permit further production. In either case, equality is an interior condition, not a rule to apply mechanically at a boundary.
Common Exam Traps
- Checking the price but not the quantity. A price is not enough. Verify market clearing and evaluate marginal benefit and marginal cost at the same quantity.
- Reversing the direction of the gap. If \(MB>MC\), expand output; if \(MC>MB\), contract it. A common wrong conclusion is that a positive \(MC-MB\) calls for expansion. In fact, when \(MC-MB>0\), the marginal unit costs more than it is worth, so reducing output can improve the allocation.
- Treating every equality as proof of a maximum. Equality is the interior first-order condition. Check the surrounding values or the direction of the gap to determine whether the quantity is a maximum.
- Using the test when the curves omit relevant effects. The test relies on the stated model assumptions. If the market price does not capture all relevant marginal benefits and costs, equality of the private curves alone does not establish that the outcome is efficient.
Check Your Understanding
For each problem, state the relevant marginal comparison and interpret the result.
- A market has inverse demand \(P=90-2Q\) and inverse supply \(P=18+Q\). Find the competitive quantity and price, and verify the marginal condition. (Answer: \(Q^*=24\), \(P^*=42\), and \(MB=MC=42\).)
- Inverse demand is \(P=64-Q\), and marginal cost is \(MC=16+2Q\). Output is held at \(Q=12\). Find the competitive quantity and the gains available from expanding from 12 to that quantity. (Answer: \(Q^*=16\); marginal benefit at 12 is 52, marginal cost is 40, and expansion gains are \(\frac{1}{2}(4)(12)=24\) dollars.)
- For two indivisible units, willingness to pay is 40 dollars for unit 1 and 29 dollars for unit 2. Their respective costs are 12 dollars and 31 dollars. What quantity maximizes gains from trade, and what are the net gains on the units traded? (Answer: Trade 1 unit; its net gain is \(40-12=28\) dollars, while unit 2 would lose \(31-29=2\) dollars.)
- Inverse demand is \(P=120-3Q\), and marginal cost is \(MC=24+Q\). If output is \(Q=30\), compare marginal benefit and marginal cost. Find the unrestricted competitive quantity and the gains from reducing output to it. (Answer: At \(Q=30\), \(MB=30\) and \(MC=54\), so output is too high; \(Q^*=24\), and reducing output gains \(\frac{1}{2}(6)(24)=72\) dollars.)
- Marginal benefit is \(MB=70-2Q\), marginal cost is \(MC=10+Q\), and capacity is limited to \(\bar Q=18\). Find the unconstrained crossing and the efficient feasible quantity. Explain why equality does not hold at the feasible outcome. (Answer: The unconstrained crossing is \(Q=20\); the efficient feasible quantity is \(Q=18\), where \(MB=34>MC=28\), because production cannot expand further.)