Building a Long-Run Cost Function

From Input Choices to Minimum Cost

In the long run, a firm can adjust all of its production inputs. For a specified output target, it chooses the input combination that meets that target at the lowest expenditure. The resulting long-run cost function summarizes that minimum expenditure as output and input prices vary. It does not describe one fixed input plan; the plan can change when prices or the output target change.

Suppose a firm uses capital $K$ and labor $L$, with rental price $r$ per unit of capital and wage $w$ per unit of labor. Let $K^c(q,r,w)$ and $L^c(q,r,w)$ denote the cost-minimizing conditional factor demands for producing target output $q$. The word “conditional” means that these input demands are conditional on producing that specified output, rather than choices made while deciding how much to produce.

Definition: The long-run cost function $c(q,r,w)$ is the minimum expenditure required to produce output $q$, given input prices $r$ and $w$. If the conditional factor demands are $K^c(q,r,w)$ and $L^c(q,r,w)$, then

$$ c(q,r,w)=rK^c(q,r,w)+wL^c(q,r,w). $$

This expression assumes the firm can choose both inputs, input prices are positive, and the stated demands attain the minimum cost of meeting the output target.

The construction is direct: identify the cost-minimizing quantities, multiply each quantity by its price, and add the expenditures. The conditional demands carry the production technology and the firm’s optimal input choices into the cost function. If a demand is piecewise—for example, because the least-cost input changes at a price threshold—the cost function may also be piecewise.

A Reliable Construction Method

The conditional factor demands come from the firm’s cost-minimization problem: minimize expenditure subject to producing at least the target output. Once those demands are known, constructing the cost function is an accounting step. Keep the target $q$ and the input prices attached to the conditional demands; do not treat an input quantity as a price or substitute a profit-maximizing output choice.

Record the output target and prices.
Identify $q$, the rental price $r$, and the wage $w$. Check that the prices are per unit of the corresponding inputs.

Write the conditional demands.
Use the cost-minimizing $K^c(q,r,w)$ and $L^c(q,r,w)$ for the technology and price conditions in the problem.

Substitute into expenditure.
Calculate $rK^c+wL^c$, simplifying only after both input expenditures are included.

Check the result.
Verify the chosen quantities produce the target, and compare the cost expression with the expenditures at those quantities.

Worked Construction with Substitutable Inputs

The first example shows how conditional demands can vary with both input prices. The technology is hypothetical and is chosen to make the price response visible. Let output be $q=(\sqrt K+\sqrt L)^2$, with $K,L\geq 0$ and positive input prices. For a target $q>0$, positive prices mean the cost-minimizing firm will not use more input than is needed to meet the target.

Worked Example: Substituting Conditional Demands into Expenditure

A hypothetical producer uses capital and labor with production function $q=(\sqrt K+\sqrt L)^2$. Its target is $q=36$, the rental price is $r=\$4$ per unit of capital, and the wage is $w=\$1$ per unit of labor. Find the conditional demands and the minimum long-run cost.

Method. First obtain the cost-minimizing demands for this technology, then substitute them into total expenditure. Set $x=\sqrt K$ and $y=\sqrt L$. The target requires $x+y\geq\sqrt{36}=6$. Since both prices are positive, the least-cost choice has $x+y=6$, so minimize $4x^2+y^2$ subject to $x+y=6$.

At an interior minimum, the marginal expenditure per unit of $x$ equals that per unit of $y$ along the constraint. Equivalently, the Lagrange conditions give $8x=\lambda$ and $2y=\lambda$, so $y=4x$. With $x+y=6$, we get $5x=6$, $x=1.2$, and $y=4.8$. Thus $K^c=x^2=1.44$ and $L^c=y^2=23.04$.

Now calculate each expenditure: capital costs $4(1.44)=\$5.76$, and labor costs $1(23.04)=\$23.04$. Therefore, $c(36,4,1)=\$5.76+\$23.04=\$28.80$.

Check. The input quantities produce $(\sqrt{1.44}+\sqrt{23.04})^2=(1.2+4.8)^2=36$. The expenditure agrees with the substituted cost expression. The mistake to avoid is treating $x,y$ as the input quantities: they are square roots of the inputs, so the actual quantities are $K=x^2$ and $L=y^2$.

Why the Cost Function Has These Properties

The definition itself gives a useful justification for several general properties. For any feasible input bundle $(K,L)$, expenditure at prices $(r,w)$ is the linear expression $rK+wL$. The cost function selects the smallest such expenditure among all bundles that can produce the target. This minimum operation explains how input-price changes affect cost.

Theorem: For a fixed production technology and a feasible target output, the minimum cost is weakly increasing in either input price and homogeneous of degree one in both input prices together. In other words, raising an input price while holding the other price fixed cannot reduce minimum cost, and multiplying both prices by a positive number multiplies minimum cost by the same number.

To justify the first statement, raise the rental price from $r$ to $r'>r$, holding $w$ fixed. Every feasible bundle has $K\geq0$, so its new expenditure $r'K+wL$ is at least its old expenditure $rK+wL$. Taking the minimum over the same feasible bundles cannot produce a lower result. The same argument applies to an increase in $w$.

For price homogeneity, multiply both prices by $t>0$. Each feasible bundle’s expenditure becomes $trK+twL=t(rK+wL)$. Because the same positive factor multiplies the expenditure of every feasible bundle, the bundle with minimum expenditure remains a minimum, and the minimum itself is multiplied by $t$. Thus $c(q,tr,tw)=t\,c(q,r,w)$.

These properties are useful checks, not replacements for finding the conditional demands. A proposed formula that falls when a single input price rises, or fails to scale proportionally when both prices scale together, signals an error under the stated assumptions. They do not, by themselves, tell us how much of each input the firm uses.

Worked Examples at Fixed Proportions and at a Corner

When the technology fixes the input proportions, changing relative prices changes the cost of that required bundle but not its proportions. With perfect substitutes, by contrast, the cost-minimizing choice may be a corner: the firm uses only the input with the lower cost per unit of output. These cases make clear why conditional demands must match the technology and prices.

Worked Example: Cost with Fixed Input Proportions

A hypothetical workshop uses a fixed-proportions process requiring $K=2q$ units of capital and $L=3q$ units of labor to produce $q$ units of output. Its target is $q=15$, the rental price is $r=\$6$, and the wage is $w=\$2$. Find its minimum cost.

Method. Use the fixed-proportions conditional demands: $K^c=2q$ and $L^c=3q$. At $q=15$, these give $K^c=2(15)=30$ and $L^c=3(15)=45$. The process requires these quantities together, so the firm cannot reduce cost by replacing one input with the other.

Capital expenditure is $6(30)=\$180$. Labor expenditure is $2(45)=\$90$. Therefore, $c(15,6,2)=\$180+\$90=\$270$.

Check. The bundle meets the target because $K/2=30/2=15$ and $L/3=45/3=15$. Substituting the demands directly gives $6(2\cdot15)+2(3\cdot15)=180+90=270$. A common error is to add the input quantities, $30+45$, as though their units were interchangeable; cost is the sum of price times quantity for each distinct input.

Worked Example: Comparing Cost per Unit of Output

A hypothetical packaging process produces $q=3K+L$. The rental price of capital is $r=\$6$ per unit and the wage is $w=\$3$ per unit. Find the least-cost input bundle and cost for target output $q=24$.

Method. Compare input cost per unit of output, not the input prices alone. One unit of capital costs $\$6$ and produces 3 units of output, so its cost per output unit is $6/3=\$2$. One unit of labor costs $\$3$ and produces 1 unit of output, so its cost per output unit is $\$3$. Capital is the cheaper way to produce output.

The conditional demands are therefore $K^c=24/3=8$ and $L^c=0$. Capital expenditure is $6(8)=\$48$, while labor expenditure is $3(0)=\$0$. Hence $c(24,6,3)=\$48$.

Check and trap. The chosen bundle produces $3(8)+0=24$. The incorrect comparison $r=6>w=3$, followed by choosing labor because its raw price is lower, ignores the different output contributions of the inputs. Choosing only labor would require $L=24$ and cost $3(24)=\$72$, which is greater than $\$48$. If the two costs per output unit were equal, any feasible mix along the production requirement would have the same expenditure.

Interpreting the Result and Avoiding Common Errors

A long-run cost function answers a specific question: the minimum expenditure needed for a stated output at stated input prices. If output or a price changes, the conditional demands may change too. Therefore, a cost calculation at one set of conditions should not be reused at another set without checking the demands and recomputing expenditure.

Do not omit an input’s expenditure. Even if one conditional demand is zero at a corner, include its price and quantity when writing the expenditure expression; its contribution is then zero.
Do not confuse input price with cost per output unit. For technologies with different productivity across inputs, compare price per unit of output each input can produce.
Do not assume input proportions are fixed. That is true for fixed-proportions technologies, but not generally. Conditional demands may respond to relative prices.
Do not confuse a cost function with a production function. The production function maps inputs into output; the cost function maps an output target and input prices into minimum expenditure.
Do not infer the shape of cost as output changes from price homogeneity. Price homogeneity describes proportional changes in input prices. How cost varies with output depends on the technology and is a separate question.

For a hypothetical business decision, the cost function can help compare the minimum resource expenditure associated with alternative output targets or input-price conditions. The result is only as appropriate as the technology and prices assumed: it is not a claim about any actual firm or industry. In production analysis, it also separates two questions that are easy to mix up: which bundle is cheapest for a fixed output, and which output the firm ultimately chooses.

Key takeaway: First determine the cost-minimizing conditional input demands for the target and price conditions. Then substitute those quantities into $rK+wL$, verify production and expenditure, and use price monotonicity and homogeneity as consistency checks.

Check Your Understanding

For each problem, use the stated production technology and prices to find or interpret the long-run cost. All scenarios are hypothetical.

A fixed-proportions producer requires $K=4q$ and $L=q$. For $q=12$, $r=\$3$, and $w=\$5$, find the conditional demands and minimum cost. (Answer: $K^c=48$, $L^c=12$, and $c=\$204$.)
A producer has $q=(\sqrt K+\sqrt L)^2$. For $q=25$, $r=\$3$, and $w=\$2$, its conditional demands are $K^c=\frac{w^2q}{(r+w)^2}$ and $L^c=\frac{r^2q}{(r+w)^2}$. Find both demands and the minimum cost. (Answer: $K^c=4$, $L^c=9$, and $c=\$30$.)
A process produces $q=2K+3L$, with $r=\$8$ and $w=\$9$. For target $q=18$, which input is least costly per output unit, what are the conditional demands, and what is minimum cost? (Answer: labor, because $9/3=\$3$ per output unit versus $8/2=\$4$; $K^c=0$, $L^c=6$, and $c=\$54$.)
Suppose a cost function is $c(q,r,w)=q\frac{rw}{r+w}$. At $q=10$, $r=\$6$, and $w=\$4$, find cost. Then find the cost if both input prices rise by 50 percent while output is unchanged. (Answer: initial cost $=\$24$; after the price change, cost $=\$36$.)
A firm can produce a fixed target using either of two inputs, and every feasible input bundle has nonnegative quantities. Explain why raising only the rental price cannot reduce minimum cost, even if the firm changes its input mix. (Answer: Every feasible bundle’s expenditure weakly rises because its capital quantity is nonnegative; minimizing over the same feasible bundles therefore cannot yield a lower cost.)

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