Cost Minimization and Productive Efficiency

Why the Input Mix Matters

A firm may be able to produce the same quantity of output using different combinations of labor and capital. Those combinations need not cost the same: labor and capital have prices, and the production technology determines how much of each input is needed. Cost minimization asks which feasible combination produces a specified output at the lowest total cost.

The question is about the input mix, not the amount of output the firm should choose. A firm that minimizes the cost of producing a target is productively efficient for that target: it does not use a more expensive feasible combination to make the same output. The target is taken as given throughout this tutorial.

Definition: Let $L$ denote labor, $K$ denote capital, and let $f(L,K)$ be the firm’s production function. If labor costs $w$ per unit and capital costs $r$ per unit, total input cost is $C=wL+rK$. For a required output $\bar q$, cost minimization is the problem of choosing $L$ and $K$ to minimize $wL+rK$, subject to $f(L,K)\geq \bar q$ and any input constraints. Here $r$ is the price of capital services, such as a rental rate; it is not necessarily an economy-wide interest rate.

The output constraint is usually binding at a minimum when both input prices are positive and producing more than $\bar q$ has no value in the stated problem. If a proposed input bundle produces more than the target, the firm may be able to reduce an input and lower cost while still meeting the target.

The Marginal Rate of Technical Substitution

Suppose the production function is differentiable, and define the marginal products of labor and capital as $MP_L=\partial f/\partial L$ and $MP_K=\partial f/\partial K$. Each marginal product measures the change in output from a small increase in that input while holding the other input fixed. When both marginal products are positive, a small change in inputs that holds output constant satisfies

MP_L\,dL+MP_K\,dK=0.

Rearranging gives $-dK/dL=MP_L/MP_K$. This ratio is the marginal rate of technical substitution of labor for capital, abbreviated MRTS. With this convention, it measures how many units of capital can be reduced when labor increases by one unit, while output is kept unchanged. It is a local rate of substitution, not a claim that one unit of labor can always replace the same amount of capital.

An isocost line lists combinations with equal cost. Solving $C=wL+rK$ for $K$ gives a line with slope $-w/r$: replacing one unit of labor with $w/r$ units of capital leaves cost unchanged. When the least-cost choice is an interior point on a smooth isoquant, the isoquant and isocost have the same slope there. That condition yields the central input-choice result.

Theorem: At an interior cost-minimizing input choice with positive marginal products, differentiable technology, and positive input prices, the marginal rate of technical substitution equals the input-price ratio: $$ \frac{MP_L}{MP_K}=\frac{w}{r}. $$ Equivalently, the marginal product per dollar spent is equal across inputs: $MP_L/w=MP_K/r$. This tangency condition is not a general rule for corner solutions; it also does not by itself guarantee a global minimum when the technology or feasible set is nonconvex.

Why the Condition Holds

The condition can be derived directly from the cost-minimization problem. For an interior solution, use a Lagrange multiplier $\lambda$ on the required-output constraint. The first-order conditions are $w=\lambda MP_L$ and $r=\lambda MP_K$. Since both input prices are positive, dividing the first condition by the second eliminates $\lambda$.

\frac{w}{r} = \frac{\lambda MP_L}{\lambda MP_K} = \frac{MP_L}{MP_K}.

There is also a useful economic interpretation. If $MP_L/w>MP_K/r$, a dollar spent on labor produces more additional output than a dollar spent on capital. A small reallocation of spending toward labor can then increase output for the same cost. That indicates the original bundle cannot be an interior optimum. Conversely, if capital produces more output per dollar, spending should shift toward capital. At an interior optimum, neither reallocation can improve the result, so the two marginal products per dollar must be equal.

The equality is necessary for a smooth interior optimum under these conditions, but a candidate solution still needs to be checked. For example, if the production technology has convex isoquants and the input prices are positive, the cost-minimization problem has the usual convex structure, so a valid tangency can identify a global minimum. With nonconvex isoquants, there may be multiple local candidates; compare feasible costs and examine boundary choices as well.

A Practical Method

A reliable solution separates the technology condition from the output constraint. First derive the MRTS, then equate it to the input-price ratio if an interior solution is appropriate. Use the resulting input ratio together with the production function to find the actual quantities. Finally, calculate cost and verify that both the output target and the input ratio are satisfied.

Write the objective and constraint.
Minimize $wL+rK$ subject to $f(L,K)\geq\bar q$, along with any nonnegativity or other input restrictions.

Check whether an interior solution is plausible.
If the technology is smooth and both inputs are used, calculate $MP_L/MP_K$ and set it equal to $w/r$. If the technology has fixed proportions or a linear isoquant, consider whether the minimum is instead at a corner or kink.

Combine the input condition with the target output.
Solve the resulting equations for $L$ and $K$, retaining only quantities that satisfy the problem’s restrictions.

Verify the result.
Substitute the inputs into the production function, compute total cost, and check whether a boundary or alternative feasible bundle could be cheaper when the technology requires that comparison.

Worked Applications

Worked Example: A Smooth Cobb–Douglas Technology

A hypothetical workshop uses labor and capital according to $q=\sqrt{LK}$. Labor costs $w=\$4$ per unit and capital costs $r=\$1$ per unit. Find the least-cost input bundle that produces $\bar q=20$, assuming both inputs can be varied continuously.

Method. Use the interior MRTS condition and the output target. The production function is smooth, and its isoquants are convex, so the tangency candidate is the cost-minimizing bundle.

Step 1: Find marginal products and the MRTS. Since $q=(LK)^{1/2}$,

MP_L=\frac{1}{2}L^{-1/2}K^{1/2}, \qquad MP_K=\frac{1}{2}L^{1/2}K^{-1/2}, \qquad \frac{MP_L}{MP_K}=\frac{K}{L}.

Step 2: Apply the input-price condition. The price ratio is $w/r=4/1=4$, so $K/L=4$, or $K=4L$.

Step 3: Apply the output target. The target requires $\sqrt{LK}=20$, hence $LK=400$. Substituting $K=4L$ gives $4L^2=400$, so $L^2=100$. Inputs are nonnegative, so $L=10$ and $K=4(10)=40$.

Step 4: Calculate and check cost. Total cost is $C=4(10)+1(40)=\$80$. Output is $\sqrt{10(40)}=\sqrt{400}=20$, exactly the target, and $K/L=40/10=4=w/r$.

Common mistake avoided. The condition is $K/L=w/r$, not $L/K=w/r$, under the stated MRTS convention. Reversing the ratio would give the wrong input mix.

Worked Example: Unequal Output Elasticities

A hypothetical service firm has production function $q=L^{1/3}K^{2/3}$. Labor costs $w=\$12$ per unit and capital costs $r=\$3$ per unit. Find its least-cost inputs for $\bar q=10$, assuming an interior solution.

Method. Derive the marginal-product ratio for this technology, equate it to the price ratio, and solve with the output equation. The Cobb–Douglas isoquants are convex, so a feasible interior tangency gives the least-cost bundle.

Step 1: Calculate the MRTS. Differentiating gives $MP_L=(1/3)L^{-2/3}K^{2/3}$ and $MP_K=(2/3)L^{1/3}K^{-1/3}$. Therefore,

\frac{MP_L}{MP_K} = \frac{(1/3)}{(2/3)}\frac{K}{L} = \frac{K}{2L}.

Step 2: Find the input ratio. The price ratio is $w/r=12/3=4$. Thus $K/(2L)=4$, which gives $K=8L$.

Step 3: Use the output target. Substitute $K=8L$ into the production function:

10=L^{1/3}(8L)^{2/3} =L^{1/3}\,8^{2/3}L^{2/3} =4L.

It follows that $L=10/4=2.5$, and $K=8(2.5)=20$.

Step 4: Calculate and check cost. Cost is $12(2.5)+3(20)=\$30+\$60=\$90$. Substitution gives $q=(2.5)^{1/3}(20)^{2/3}=10$, and the chosen ratio is $K/(2L)=20/5=4=w/r$.

Common mistake avoided. The exponents in the production function matter: here the MRTS includes the factor $1/2$. Using $K/L$ alone would produce an incorrect input ratio.

Worked Example: A Cost-Minimizing Corner

A hypothetical producer can make output $q=2L+K$. Labor costs $w=\$6$ per unit and capital costs $r=\$2$ per unit. Find the least-cost bundle for $\bar q=30$, allowing $L\geq0$ and $K\geq0$.

Method. Compare the cost of generating one unit of output with each input. Because this technology has linear isoquants, a cost minimum need not be an interior tangency.

One unit of labor adds $2$ units of output and costs $\$6$, so its cost per output unit is $6/2=\$3$. One unit of capital adds $1$ unit of output and costs $\$2$, so its cost per output unit is $2/1=\$2$. Capital is cheaper per unit of output. Produce the entire target with capital: $L=0$ and $K=30$. This costs $6(0)+2(30)=\$60$, and output is $2(0)+30=30$.

To verify that no feasible bundle is cheaper, any bundle meeting the target satisfies $2L+K\geq30$. Its cost can be written $6L+2K=2(2L+K)+2L$. Since $2L+K\geq30$ and $L\geq0$, the cost is at least $2(30)+0=\$60$. The proposed bundle attains that bound.

Common mistake avoided. Applying the interior tangency condition mechanically is inappropriate here. The marginal products are constant, so the cheaper input per unit of output is used alone; the cost minimum is a corner.

Productive Efficiency and the Limits of the Condition

For a fixed output target, using the least-cost feasible input bundle is a form of productive efficiency: the firm produces that output without avoidable input cost. The interior condition explains how the firm balances technology against input prices. If labor is relatively expensive, the cost-minimizing bundle generally substitutes toward capital to the extent the technology allows; if capital becomes relatively expensive, the incentive runs in the other direction.

The ratio condition is not a recipe for every production problem. With perfect substitutes, as in the previous example, the optimum may be a corner. With fixed-proportion technology, such as a process requiring inputs in a fixed ratio, the least-cost bundle may be at a kink. Input limits can also force a boundary solution. In each case, the firm must compare feasible choices using the cost objective rather than assume a smooth tangency.

It is also important to keep the decision being analyzed precise. Cost minimization takes output as given. It does not determine the firm’s profit-maximizing output, establish market equilibrium, or show that society’s allocation is efficient. Those are distinct questions. For the current problem, the relevant test is whether the chosen feasible input bundle makes the specified output at least cost.

Common Exam Traps

Reversing the ratio. With MRTS defined as $MP_L/MP_K$, the interior condition is $MP_L/MP_K=w/r$. Reversing either ratio changes the implied input mix.
Stopping after the tangency condition. A ratio such as $K=4L$ is not the solution by itself. Use the output target to determine the quantities and then calculate cost.
Treating a necessary condition as a universal solution. A smooth interior tangency may be relevant, but corners, kinks, constraints, or nonconvexities require additional checks.
Confusing least cost with least inputs. The firm minimizes the value $wL+rK$, not necessarily the number of input units. A larger quantity of a cheap input may reduce total cost.
Choosing output instead of input mix. The target $\bar q$ is fixed here. Cost minimization does not answer how much output the firm should produce.

Key takeaway: At a smooth interior cost minimum, the marginal product per dollar is equal across inputs, $MP_L/w=MP_K/r$, or equivalently $MP_L/MP_K=w/r$. Then use the output constraint and check whether the solution is feasible and globally least-cost; boundary and nonsmooth technologies may require a corner or kink solution instead.

Check Your Understanding

For each problem, identify whether the interior condition applies, find the least-cost inputs, and check the output and cost.

A producer has $q=\sqrt{LK}$, with $w=\$6$, $r=\$3$, and target output $\bar q=12$. Find the least-cost bundle and its cost. (Answer: $L=6\sqrt{2}$, $K=12\sqrt{2}$, cost $72\sqrt{2}$ dollars.)
A firm has $q=\sqrt{LK}$, with $w=\$2$, $r=\$8$, and target output $\bar q=10$. Find the least-cost bundle and verify its output. (Answer: $L=20$, $K=5$, cost $\$80$, output $10$.)
A workshop produces $q=3L+2K$. Labor costs $\$9$ per unit, capital costs $\$4$ per unit, and the target is $36$. Which input should be used, and what is the minimum cost? (Answer: Use $L=0$, $K=18$; minimum cost is $\$72$.)
A firm has $q=L^{1/3}K^{2/3}$, with $w=\$6$, $r=\$3$, and target output $4\sqrt[3]{2}$. Find the least-cost input bundle and its cost. (Answer: $L=2$, $K=8$, cost $\$36$.)
A process produces $q=\min(L,K/3)$, with $w=\$5$, $r=\$2$, and target output $15$. Find the least-cost bundle and explain why a smooth tangency condition is not the method. (Answer: $L=15$, $K=45$, cost $\$165$; fixed proportions make the minimum a kink solution.)

What You'll Learn