Equalizing Marginal Costs Across Firms

Why Allocate Output by Marginal Cost?

When several firms can produce the same good, a planner or contracting organization may need to meet a specified total output target at the lowest possible cost. The key comparison is not which firm has the lowest total cost at its current output, but how much the total cost would change if that firm produced one more unit. That change is its marginal cost.

This tutorial studies a fixed-output cost-minimization problem. It does not determine how much total output should be produced; it asks how a given target should be divided among firms. The central result is that firms producing positive amounts at an interior minimum have equal marginal costs. If their marginal costs differ, output can generally be shifted toward the lower-marginal-cost firm and total cost reduced.

Definition: Let firm $i$ produce $q_i$ units at total cost $C_i(q_i)$, where $q_i\geq 0$. For a required total output $Q$, a cost-minimizing allocation solves $$ \min_{q_1,\ldots,q_n}\ \sum_{i=1}^{n} C_i(q_i) \quad\text{subject to}\quad \sum_{i=1}^{n}q_i=Q,\qquad q_i\geq 0. $$ If $C_i$ is differentiable, the firm's marginal cost at output $q_i$ is $MC_i(q_i)=C_i'(q_i)$.

The basic equal-marginal-cost rule applies when costs are differentiable, the total output target is fixed, and the firms being compared can adjust output in either direction by a small amount. Convex cost functions are a standard sufficient condition for the rule to identify a global cost minimum: marginal cost does not decrease as output rises. With strictly convex costs, the minimizing allocation is unique. If output must be an integer, or a firm has a binding capacity constraint, the simple equality needs adjustment.

The Equal-Marginal-Cost Condition

Suppose two firms both produce positive output and neither is constrained at a boundary. If firm 1 has a higher marginal cost than firm 2, transfer a small amount of production from firm 1 to firm 2. Total output stays fixed, but total cost falls. Therefore such an allocation cannot minimize cost.

More formally, consider an allocation where firms 1 and 2 produce positive amounts. Reduce $q_1$ by a small $\varepsilon>0$ and increase $q_2$ by the same amount. The change in total cost, divided by $\varepsilon$, approaches the difference between the receiving firm's marginal cost and the giving firm's marginal cost:

\lim_{\varepsilon\to 0^+} \frac{C_1(q_1-\varepsilon)+C_2(q_2+\varepsilon)-C_1(q_1)-C_2(q_2)}{\varepsilon} = -MC_1(q_1)+MC_2(q_2).

At a minimum, this feasible transfer cannot lower cost. Applying the same argument in the reverse direction gives the opposite inequality. Together, the inequalities require equality.

Theorem: At an interior cost-minimizing allocation of a fixed output target, all firms producing positive output have the same marginal cost: $$ MC_i(q_i)=\lambda \quad\text{for every active firm }i, $$ where “active” means $q_i>0$, and $\lambda$ is the common marginal cost. For a differentiable convex-cost problem with no capacity limits, a firm producing zero output can remain inactive when its marginal cost at zero is at least $\lambda$. This boundary condition prevents a small transfer of production to that firm from reducing total cost.

The inactive-firm condition follows from the same transfer logic. If an inactive firm has marginal cost at zero below $\lambda$, shifting a small amount of output from an active firm to it lowers cost. Conversely, if its marginal cost at zero exceeds $\lambda$, starting production there would raise cost at the margin. Do not require equality for a firm at zero output: it cannot reduce output below zero to make the reverse transfer.

Write the target.
Set the sum of firms' outputs equal to the required total $Q$.

Differentiate each cost function.
Compute $MC_i(q_i)=C_i'(q_i)$ for every firm.

Equalize active marginal costs.
Set the marginal costs equal for firms provisionally assumed to produce positive output, and combine this equation with the output target.

Check feasibility and boundaries.
Confirm that outputs are nonnegative, sum to $Q$, and that any inactive firm's marginal cost at zero is not below the common marginal cost.

Worked Applications

Worked Example: Allocating Output Between Two Firms

Two firms must produce a total of 12 units during a production period. Their total costs, in dollars, are $C_1(q_1)=10q_1+q_1^2$ and $C_2(q_2)=4q_2+2q_2^2$. Assume outputs are continuous and nonnegative, both cost functions are differentiable and strictly convex, and there are no capacity limits or fixed costs. Find the cost-minimizing allocation and verify it. The method is equalizing marginal costs; a common mistake is to divide output equally without considering the cost functions.

Step 1: Find marginal costs. Differentiating gives $MC_1(q_1)=10+2q_1$ and $MC_2(q_2)=4+4q_2$. At an interior minimum, these must be equal, and $q_1+q_2=12$.

10+2q_1=4+4q_2,\qquad q_1+q_2=12.

Step 2: Solve the equations. The first equation gives $2q_1-4q_2=-6$, or $q_1=2q_2-3$. Substitute this into the output constraint: $2q_2-3+q_2=12$, so $3q_2=15$, $q_2=5$, and $q_1=7$. Both outputs are positive.

Step 3: Check marginal costs and total cost. The marginal costs are $MC_1(7)=10+2(7)=24$ and $MC_2(5)=4+4(5)=24$. The total cost is $C_1(7)+C_2(5)=(70+49)+(20+50)=119+70=189$ dollars. As a boundary check, producing all 12 units at firm 1 costs $C_1(12)=120+144=264$ dollars; producing them all at firm 2 costs $C_2(12)=48+288=336$ dollars. The interior allocation has lower cost, and strict convexity makes it the unique minimum. The output check is $7+5=12$.

Worked Example: Checking Whether a Firm Should Remain Inactive

Three firms can produce a total of 6 units. Their costs are $C_1(q_1)=q_1^2$, $C_2(q_2)=2q_2+q_2^2$, and $C_3(q_3)=20q_3+q_3^2$, in dollars per production period. Assume outputs are continuous and nonnegative, costs are differentiable and strictly convex, and there are no capacities or fixed costs. Determine whether all three firms should produce. The method is to solve for the active firms and then apply the zero-output marginal-cost check.

Step 1: Find marginal costs. They are $MC_1(q_1)=2q_1$, $MC_2(q_2)=2+2q_2$, and $MC_3(q_3)=20+2q_3$. Start by testing firms 1 and 2 as active and firm 3 as inactive. Equalize the active marginal costs and impose $q_1+q_2=6$:

2q_1=2+2q_2,\qquad q_1+q_2=6.

Step 2: Solve and check the inactive firm. The first equation gives $q_1=q_2+1$. Therefore $q_2+1+q_2=6$, so $q_2=2.5$ and $q_1=3.5$. Their common marginal cost is $2(3.5)=7$, also $2+2(2.5)=7$. Firm 3's marginal cost at zero is $MC_3(0)=20$, which is greater than 7. Starting production at firm 3 would not lower cost at the margin, so it is consistent for that firm to remain inactive.

Step 3: Calculate and verify total cost. The total cost is $C_1(3.5)+C_2(2.5)+C_3(0)=3.5^2+(2(2.5)+2.5^2)+0=12.25+(5+6.25)=23.5$ dollars. The output check is $3.5+2.5+0=6$. The inactive firm is not required to have marginal cost 7; its marginal cost at zero is 20. Requiring equality for all three firms would be the boundary-condition error.

Worked Example: Using a Cost Comparison to Detect a Poor Allocation

A total of 10 units must be produced by two firms with costs $C_1(q_1)=5q_1+\tfrac12 q_1^2$ and $C_2(q_2)=q_2+q_2^2$, measured in dollars. Assume continuous, nonnegative output, differentiable strictly convex costs, no capacities, and no fixed costs. Compare the proposed allocation $(q_1,q_2)=(6,4)$ with the cost-minimizing allocation. The method is to use marginal costs to find the minimum and then compute total costs; a common mistake is to assume that a feasible allocation is efficient just because it meets the target.

Step 1: Test the proposed allocation. The marginal costs are $MC_1(q_1)=5+q_1$ and $MC_2(q_2)=1+2q_2$. At $(6,4)$, they are $MC_1(6)=11$ and $MC_2(4)=9$. Since firm 1's marginal cost is higher, shifting a small amount of output from firm 1 to firm 2 reduces total cost.

Step 2: Find the minimum-cost allocation. Set the marginal costs equal and use $q_1+q_2=10$:

5+q_1=1+2q_2,\qquad q_1+q_2=10.

The first equation gives $q_1=2q_2-4$. Substitution into the target gives $2q_2-4+q_2=10$, so $3q_2=14$, $q_2=14/3$, and $q_1=16/3$. Both are positive. The common marginal cost is $5+16/3=31/3$, which agrees with $1+2(14/3)=31/3$.

Step 3: Compare total costs and verify output. At the minimum, $C_1(16/3)=5(16/3)+\tfrac12(16/3)^2=80/3+128/9=368/9$. Also, $C_2(14/3)=14/3+(14/3)^2=42/9+196/9=238/9$. The total is $606/9=202/3$, or approximately 67.33 dollars, rounded to two decimals. At the proposed allocation, the cost is $C_1(6)+C_2(4)=(30+18)+(4+16)=68$ dollars. The proposed allocation costs about 0.67 dollars more. The output checks are $16/3+14/3=10$ at the minimum and $6+4=10$ at the proposal.

Interpreting the Condition and Avoiding Traps

Equal marginal costs describe the allocation of a fixed production target, not a requirement that firms have equal total costs, average costs, or output levels. Firms with different cost curves will generally produce different quantities even when their marginal costs match. The condition is useful because a small reallocation between firms leaves total output unchanged, isolating whether the production mix can be made less costly.

The derivation also explains the direction of a cost-reducing transfer. If firm $i$ has $MC_i(q_i)>MC_j(q_j)$, move a small amount $\varepsilon$ of output from $i$ to $j$. The approximate change in total cost is

\Delta C\approx \bigl(MC_j(q_j)-MC_i(q_i)\bigr)\varepsilon<0.

The inequality is negative, so the transfer lowers cost. When the two marginal costs are equal, no small transfer between those active firms changes cost to first order. With convex costs, this first-order condition is sufficient for a global minimum when the boundary conditions also hold.

Do not equalize total costs. The relevant comparison is the derivative of each cost function at its assigned output.
Do not ignore the output constraint. A candidate allocation must add to the required total.
Do not force an inactive firm into the equality. At zero output, check whether its marginal cost at zero is at least the active firms' common marginal cost.
Do not overlook constraints. If a firm is at a capacity limit, it cannot accept an additional small transfer; equality may not hold for that firm.
Do not confuse this rule with choosing total production. The target $Q$ is given in this problem; deciding the efficient total is a different question.

Key takeaway: For a given total output, allocate production across unconstrained active firms until their marginal costs are equal. Then verify that outputs are feasible and that any firm left at zero would not reduce cost by beginning production. This fixed-output cost-minimization result is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Check Your Understanding

For each problem, assume continuous nonnegative output, differentiable strictly convex cost functions, and no capacity limits or fixed costs unless stated otherwise.

Two firms must produce 10 units. Their costs are $C_1(q_1)=2q_1+q_1^2$ and $C_2(q_2)=8q_2+2q_2^2$, in dollars. Find the minimizing outputs and common marginal cost. (Answer: $q_1=23/3,\ q_2=7/3$, with common marginal cost $52/3$ dollars per unit.)
Two firms must produce 8 units. Their costs are $C_1(q_1)=3q_1+q_1^2$ and $C_2(q_2)=9q_2+\tfrac12q_2^2$, in dollars. Find the minimizing outputs and verify the common marginal cost. (Answer: $q_1=14/3,\ q_2=10/3$, with common marginal cost $37/3$ dollars per unit.)
Three firms must produce 5 units. Their costs are $C_1(q_1)=4q_1+q_1^2$, $C_2(q_2)=2q_2+\tfrac32q_2^2$, and $C_3(q_3)=15q_3+q_3^2$, in dollars. Find the minimizing outputs, including any zero output, and check the inactive firm's marginal cost. (Answer: $q_1=13/5,\ q_2=12/5,\ q_3=0$; the active marginal costs are $46/5$, while $MC_3(0)=15$ dollars per unit.)
At a proposed allocation, firm A produces 4 units with marginal cost 13 dollars per unit, and firm B produces 6 units with marginal cost 8 dollars per unit. Total output must stay fixed, and both firms can adjust. Which direction of transfer lowers cost, and what is the approximate cost change from transferring 0.1 unit? (Answer: Transfer 0.1 unit from A to B; the approximate change is $(8-13)(0.1)=-0.50$ dollars.)
At an interior cost minimum, three active firms have marginal costs 12, 12, and 15 dollars per unit. Can this allocation be a minimum under the stated unconstrained assumptions? State the adjustment indicated by the marginal costs. (Answer: No; transfer a small amount from the third firm, whose marginal cost is 15, to either firm with marginal cost 12.)

What You'll Learn