Cost Functions From Production Functions

From Production to Variable Cost

A production function describes the output a firm can produce from its inputs. A cost function answers a different question: how much must the firm spend on inputs to produce a specified output? In the short run, some inputs are fixed. If labor is the only variable input, the basic conversion is to find the labor required for each output level and multiply by the wage.

The key step is inversion. A production function gives output as a function of labor; to construct variable cost, we need labor as a function of output. This may require algebra, and if the production function is not one-to-one over its entire domain, we must also identify the economically relevant branch.

Definition: In a short-run model with fixed capital $\bar K$ and variable labor $L$, let output be $q=f(L;\bar K)$. If labor costs $w$ dollars per unit, and $L(q)$ is the labor requirement for producing $q$ on the relevant feasible branch, then variable cost is $$ VC(q)=wL(q). $$ Variable cost includes the cost of the variable input. It does not include fixed cost. If fixed cost is $FC$, total cost is $TC(q)=FC+VC(q)$.

This formula assumes the input price $w$ is constant over the output levels being considered, labor is the only variable input, and the firm can obtain the required labor at that price. The fixed input is held constant as output varies. If there are several variable inputs, their costs must all be included; multiplying labor alone by the wage would then be incomplete.

The production function also determines which output levels are feasible. An algebraic expression for $L(q)$ is not enough by itself: it must correspond to a permitted labor level and produce the target output. When output first rises and later falls as labor increases, the same output may correspond to two labor levels. In the usual short-run cost calculation, use the relevant production branch, often the increasing branch that achieves the target with less labor.

Why Inverting the Function Gives Cost

The conversion follows directly from the definition of an input bill. Suppose a firm wants output $q$. The production function tells us which labor amount can produce that output. Once that amount is known, the wage bill is the price per unit of labor times the number of labor units. Substitution gives variable cost as a function of output.

q=f(L;\bar K) \quad\Longrightarrow\quad L=L(q) \quad\Longrightarrow\quad VC(q)=wL(q).

This derivation does not require a particular functional form. It does require that the selected labor requirement actually produces $q$, that the input price is measured in dollars per unit of labor, and that other variable expenses are absent from the model. The units provide a useful check: dollars per labor unit multiplied by labor units gives dollars.

Hold fixed inputs constant.
Identify the short-run production function and the variable input. Do not vary the fixed input while solving for labor.

Solve for the input requirement.
Set the production function equal to the target output and solve for the feasible labor amount or relevant inverse branch.

Price the input.
Multiply the required labor by the wage to obtain variable cost. Add fixed cost only if the question asks for total cost.

Worked Examples

Worked Example: A Square-Root Production Function

A workshop has fixed equipment and produces output according to $q=5\sqrt{L}$, where $L$ is labor-hours per week and $q$ is batches produced per week. Labor costs $w=\$18$ per labor-hour. Fixed equipment costs $\$240$ per week. Find the variable-cost function and the variable and total costs of producing 40 batches.

Method: invert the production function, then apply the wage. The production function has nonnegative labor and output, so solve for $L$ on that domain:

q=5\sqrt{L} \quad\Longrightarrow\quad \sqrt{L}=\frac{q}{5} \quad\Longrightarrow\quad L(q)=\left(\frac{q}{5}\right)^2=\frac{q^2}{25}.

The labor bill is $\$18$ per labor-hour. Therefore,

VC(q)=18\left(\frac{q^2}{25}\right) =\frac{18}{25}q^2 =\$0.72q^2.

For $q=40$ batches, the required labor is

L(40)=\frac{40^2}{25}=\frac{1{,}600}{25}=64 \text{ labor-hours}.

Thus variable cost is $64\times \$18=\$1{,}152$. Total cost includes fixed equipment cost: $\$1{,}152+\$240=\$1{,}392$. Check by substituting $L=64$ into production: $5\sqrt{64}=5(8)=40$ batches. The calculation is consistent. A common mistake is to report $\$1{,}152+\$240$ as variable cost; the $\$240$ is fixed cost, so it belongs only in total cost.

Worked Example: Choosing the Feasible Branch

A small plant has fixed machinery and a production function $q=10L-L^2$, where $L$ is labor units per shift. The plant operates on the range $0\leq L\leq5$, where output increases with labor. Labor costs $w=\$30$ per unit. Fixed cost is $\$240$ per shift. Find the labor requirement and variable cost for $q=16$, then find total cost.

Method: solve the production equation and use the specified increasing branch. Set output equal to 16:

16=10L-L^2 \quad\Longrightarrow\quad L^2-10L+16=0 \quad\Longrightarrow\quad (L-2)(L-8)=0.

The algebra gives $L=2$ or $L=8$. The operating range is $0\leq L\leq5$, so $L=8$ is not feasible. The relevant labor requirement is $L=2$. Inverting on the stated branch gives $L(q)=5-\sqrt{25-q}$, since this expression lies between 0 and 5 for the attainable outputs on that branch. At $q=16$, $L(16)=5-\sqrt{9}=2$.

Variable cost is

VC(16)=wL(16)=\$30(2)=\$60.

Total cost is $\$60+\$240=\$300$. Check the production amount: $10(2)-2^2=20-4=16$. The common mistake is to choose $L=8$ merely because it solves the quadratic. That labor level is outside the stipulated operating range; the production equation must be solved together with its domain.

Worked Example: A Cubic Input Requirement

A packaging line with fixed equipment has production function $q=2L^{1/3}$, where $L$ is labor-hours per day and $q$ is crates per day. The wage is $w=\$7.50$ per labor-hour. Find the variable-cost function and the labor and variable cost required for 4 crates. Assume $L\geq0$ and no other variable input costs.

Method: isolate labor before applying the wage. Divide by 2 and cube both sides:

\frac{q}{2}=L^{1/3} \quad\Longrightarrow\quad L(q)=\left(\frac{q}{2}\right)^3 =\frac{q^3}{8}.

Consequently, the variable-cost function is

VC(q)=\$7.50\left(\frac{q^3}{8}\right) =\$0.9375q^3.

At $q=4$ crates, required labor is $L(4)=4^3/8=64/8=8$ labor-hours. Variable cost is $8\times \$7.50=\$60$. Check the output: $2(8^{1/3})=2(2)=4$ crates. The cost-function calculation also gives $\$0.9375(4^3)=\$0.9375(64)=\$60$. A common error is to use $q/2$ as the labor requirement; $q/2$ is $L^{1/3}$, so it must be cubed to recover labor.

Interpreting the Result Correctly

A variable-cost function summarizes the input bill as output changes while the short-run production conditions remain fixed. It is not simply the production function with a price attached: the production function maps labor into output, so it must first be inverted to map output into labor. The shape of the resulting variable-cost function depends on that inverse and on the input price.

Keep the following distinctions clear when interpreting a calculation:

Production relation: $q=f(L;\bar K)$ describes output from labor with fixed capital.
Input requirement: $L(q)$ gives the labor amount corresponding to a target output on the selected feasible branch.
Variable cost: $VC(q)=wL(q)$ prices the variable input only.
Total cost: $TC(q)=FC+VC(q)$ adds fixed cost when it is requested.

The inverse may be valid only over a particular output range. For example, a quadratic production function can have two labor solutions, while a production schedule or restricted operating range may permit only one. Always check the production equation, the labor domain, and the output range before pricing the input. If the wage changes with labor or output, the simple product $wL(q)$ no longer describes the input bill without modification.

Key takeaway: To obtain short-run variable cost from a one-variable production function, solve for the relevant feasible input requirement at each output level and multiply by the input’s price. Keep fixed cost separate unless the question asks for total cost.

Check Your Understanding

For each problem, identify the relevant labor requirement before calculating cost. Treat each production function as a short-run relation with all other inputs fixed.

A print shop has $q=6\sqrt{L}$, with labor measured in hours and a wage of $\$20$ per hour. Find the labor and variable cost required for $q=30$. (Answer: $L=25$ hours; $VC=\$500$.)
A plant produces $q=8\sqrt{L}$. Labor costs $\$9$ per unit, and fixed cost is $\$120$. For $q=24$, find variable cost and total cost. (Answer: $L=9$; $VC=\$81$; $TC=\$201$.)
A workshop’s production function is $q=15L-\tfrac12L^2$, with operation restricted to $0\leq L\leq15$. Labor costs $\$11$ per unit. Find the labor requirement and variable cost for $q=112$. (Answer: $L=14$; $VC=\$154$.)
A line produces $q=3L^{1/3}$, and labor costs $\$16$ per hour. Find the variable-cost function and the variable cost of producing $q=9$. (Answer: $VC(q)=\$16(q/3)^3$; at $q=9$, $L=27$ and $VC=\$432$.)
A producer has $q=10L-\tfrac12L^2$, operates on $0\leq L\leq10$, and pays $\$18$ per labor unit. Fixed cost is $\$125$. For $q=32$, find the feasible labor requirement, variable cost, and total cost. (Answer: $L=4$; $VC=\$72$; $TC=\$197$.)

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