Three-Input Cost Minimization with One Fixed Input

A production function can contain three inputs even when a firm can adjust only two of them today. That is the short-run problem studied here: labor $L$ and materials $M$ are variable, while capital $K$ is fixed at $\overline K$.

The useful lesson is not simply that a three-dimensional picture can be reduced to two dimensions. It is that fixing one input changes both the feasible production set and the cost-minimization problem. Once that setup is explicit, the conditional demands for labor and materials follow from the same marginal-product-per-dollar logic used in the familiar two-input model.

This article is an educational microeconomics example. It is not business, investment, accounting, or operational advice. Real production choices can involve contracts, uncertainty, indivisibilities, capacity limits, and input-quality differences that this model omits.

1. Set up the short-run problem

Let output be produced by

$$ q=f(L,K,M), $$

and let the input prices be:

• $w>0$: price of one unit of labor,
• $r>0$: rental or user cost of one unit of capital,
• $m>0$: price of one unit of materials.

Total input cost is

$$ C=wL+rK+mM. $$

In the short run, take capital as fixed: $K=\overline K>0$. For a target output $\overline q>0$, the firm chooses only $L$ and $M$:

$$ \min_{L,M\ge 0}\; wL+mM+r\overline K \quad\text{subject to}\quad f(L,\overline K,M)\ge \overline q. $$

The term $r\overline K$ is constant with respect to $L$ and $M$. Therefore it does not change which variable-input bundle minimizes cost. It still belongs in total short-run cost after the variable-input problem is solved:

$$ C_{SR}(\overline q;\overline K)=r\overline K+VC(\overline q;\overline K). $$

Calling the capital payment “fixed” within this model does not prove that every real-world capital payment is legally unavoidable or economically sunk. It means only that $K$ is not a choice variable for the decision horizon being modeled.

2. Why the problem becomes two-dimensional

With three freely varying inputs, an isoquant for a fixed output is generally a surface in $(L,K,M)$ space. Fixing $K=\overline K$ restricts the firm to a plane. The intersection of that plane with the production surface is a one-dimensional isoquant curve in the two-dimensional $(L,M)$ plane.

Likewise, the full isocost equation

$$ C=wL+rK+mM $$

becomes, on the fixed-capital plane,

$$ C-r\overline K=wL+mM. $$

For a given amount of variable cost, this is a straight line in an $(L,M)$ diagram. Its slope, with $L$ on the horizontal axis and $M$ on the vertical axis, is $-w/m$.

This geometric reduction is exact; no missing image is needed to use it. The remaining task is to find the least-cost point on the target-output isoquant.

3. Interior first-order condition

Suppose the reduced production function is differentiable and the least-cost solution uses positive amounts of both variable inputs. Define the marginal products

$$ MP_L=\frac{\partial f(L,\overline K,M)}{\partial L}, \qquad MP_M=\frac{\partial f(L,\overline K,M)}{\partial M}. $$

At an interior optimum, the marginal rate of technical substitution equals the input-price ratio:

$$ \frac{MP_L}{MP_M}=\frac{w}{m}. $$

An equivalent and often more intuitive form is

$$ \frac{MP_L}{w}=\frac{MP_M}{m}. $$

The last equation says that the final currency unit spent on either variable input must add the same marginal output. If labor generated more marginal output per currency unit than materials, the firm could move spending toward labor while preserving output and lowering cost.

Tangency is not a universal shortcut. It describes an interior solution under suitable regularity and convexity conditions. A corner solution can arise when an input is not essential, while fixed-proportion or other non-smooth technologies may have a kink rather than a differentiable tangency. In those cases, compare feasible boundaries or use the relevant constrained-optimization conditions directly.

4. Cobb-Douglas reduction with fixed capital

Consider the three-input Cobb-Douglas technology

$$ q=A L^{\alpha}K^{\beta}M^{\gamma}, $$

where $A>0$, $\alpha>0$, $\beta>0$, and $\gamma>0$. After fixing capital,

$$ q=A\overline K^{\beta}L^{\alpha}M^{\gamma}. $$

It is convenient to define the short-run productivity constant

$$ B=A\overline K^{\beta}>0, $$

so the reduced production function is

$$ q=B L^{\alpha}M^{\gamma}. $$

For positive $L$ and $M$,

$$ MP_L=\alpha\frac{q}{L}, \qquad MP_M=\gamma\frac{q}{M}. $$

Substituting these marginal products into the interior condition gives

$$ \frac{\alpha}{\gamma}\frac{M}{L}=\frac{w}{m}. $$

Therefore the cost-minimizing input ratio is

$$ \boxed{\frac{M^*}{L^*}=\frac{\gamma w}{\alpha m}}. $$

This ratio has a sensible interpretation. A higher labor price $w$ shifts the mix toward materials; a higher material price $m$ shifts it toward labor. Larger output elasticities also make the corresponding input more productive at the margin.

5. Closed-form conditional input demands

Define

$$ c=\frac{\gamma w}{\alpha m}, $$

so that $M^*=cL^*$. A cost-minimizing bundle with positive input prices will meet the output constraint exactly; producing more than $\overline q$ would waste costly inputs. Substitute $M=cL$ into the constraint:

$$ \overline q=B L^{\alpha}(cL)^{\gamma} =Bc^{\gamma}L^{\alpha+\gamma}. $$

Solving for the conditional demands yields

$$ \boxed{ L^*(\overline q;\overline K,w,m) =\left(\frac{\overline q}{Bc^{\gamma}}\right)^{\!1/(\alpha+\gamma)} } $$

and

$$ \boxed{M^*=cL^*.} $$

The minimized variable and total short-run costs are then

$$ VC^*=wL^*+mM^*, $$$$ \boxed{C_{SR}^*=r\overline K+wL^*+mM^*.} $$

Notice two separate effects of fixed capital. The quantity $\overline K$ changes productive capacity through $B=A\overline K^\beta$, and its payment $r\overline K$ is added after the variable-input choice. Because $\overline K$ is fixed, neither $r$ nor the fixed payment appears in the optimal $M/L$ ratio.

6. Worked numerical example

Suppose

$$ A=1,\quad \overline K=16,\quad \alpha=\beta=\gamma=\frac12, $$

and input prices are

$$ w=20,\quad m=5,\quad r=3. $$

The firm must produce $\overline q=80$.

Step 1: Reduce the technology

Because

$$ B=A\overline K^{\beta}=1\times16^{1/2}=4, $$

the short-run production constraint is

$$ 80=4L^{1/2}M^{1/2}=4\sqrt{LM}. $$

Step 2: Find the least-cost input ratio

The optimal ratio is

$$ \frac{M^*}{L^*} =\frac{\gamma w}{\alpha m} =\frac{(1/2)(20)}{(1/2)(5)}=4. $$

Thus $M^*=4L^*$.

Step 3: Meet the output target

Substitute $M=4L$ into production:

$$ 80=4\sqrt{L(4L)}=4\sqrt{4L^2}=8L. $$

With nonnegative inputs, this gives

$$ L^*=10,\qquad M^*=40. $$

Step 4: Verify output and cost

The output check is

$$ q=1\times10^{1/2}\times16^{1/2}\times40^{1/2} =4\sqrt{400}=80. $$

Variable cost is

$$ VC^*=20(10)+5(40)=200+200=400. $$

Fixed capital cost is

$$ r\overline K=3(16)=48, $$

so total short-run cost is

$$ \boxed{C_{SR}^*=400+48=448.} $$

The equal spending on labor and materials in this particular example is not a general rule. It occurs because $\alpha=\gamma$. In fact, the first-order condition implies $wL^*/(mM^*)=\alpha/\gamma$ for this Cobb-Douglas problem.

7. A reusable solution workflow

For any three-input short-run cost-minimization exercise:

1. Identify which inputs are variable and which are fixed during the stated horizon.
2. Substitute the fixed input into the production function.
3. Remove the fixed payment from the variable-cost objective, but remember to add it back to total cost.
4. Check whether an interior differentiable solution is plausible.
5. If it is, set marginal products per currency unit equal.
6. Combine the resulting input ratio with the binding output constraint.
7. Verify the candidate bundle by substituting it into both production and cost.
8. Check boundaries or kinks whenever the technology permits a corner or is non-smooth.

8. Common mistakes

Treating all three inputs as choices

Once $K=\overline K$ is imposed, optimizing over $K$ solves a different, long-run problem.

Dropping fixed cost from the final answer

The constant $r\overline K$ does not affect the minimizing $L$ and $M$, but it remains part of total short-run cost under the stated cost model.

Inverting the price ratio

With $L$ on the horizontal axis and $M$ on the vertical axis, the isocost slope is $-w/m$. The interior condition is $MP_L/MP_M=w/m$, which leads to $M/L=\gamma w/(\alpha m)$ for the reduced Cobb-Douglas function.

Assuming every optimum is a tangency

Tangency is an interior result, not a substitute for checking feasibility, boundaries, or the shape of the technology.

Confusing conditional cost minimization with profit maximization

Here output $\overline q$ is fixed in advance. Choosing the profit-maximizing output requires an output price and a different optimization problem.

Sources and further study

• MIT OpenCourseWare, Principles of Microeconomics: Recitation 5 notes summarizes Cobb-Douglas production, MRTS, and the short-run distinction between fixed and variable cost.
• MIT OpenCourseWare, Principles of Microeconomics: Productivity and Costs develops the isocost/isoquant cost-minimization condition.
• OpenStax, Principles of Economics 3e, Section 7.2 explains production functions and fixed versus variable inputs.
• OpenStax, Principles of Economics 3e, Section 7.3 connects input requirements and factor prices to short-run fixed, variable, and total costs.

Archive note

This page preserves the original 2016 route, publication date, series position, author, and the old platform archive identity. It was rewritten in 2026 because the archived English version ended with repeated image placeholders instead of the promised solution. The replacement supplies the full derivation, a numerical example, explicit assumptions, and source links in accessible text.