Deriving Conditional Factor Demands

What Conditional Factor Demands Describe

A firm choosing inputs in the long run can adjust all of its productive resources. If it has a particular output target, it can ask: what quantities of capital and labor produce that output at the lowest expenditure, given their prices? The answers are called conditional factor demands. They are conditional because the output level is specified rather than chosen as part of the decision.

The earlier material on cost minimization establishes the basic comparison: at an interior optimum, the marginal rate of technical substitution matches the ratio of input prices. Here, we solve that condition together with the production requirement to obtain explicit demand functions. The model is a standard Cobb–Douglas technology; other technologies can produce different formulas or corner solutions.

Definition: Given an output target \(q>0\), a production function \(F(K,L)\), and positive input prices \(r\) and \(w\), conditional factor demands \(K^c(q,r,w)\) and \(L^c(q,r,w)\) are the capital and labor quantities that minimize \(rK+wL\) subject to \(F(K,L)\geq q\). Here \(K\) denotes capital, \(L\) labor, \(r\) the price per unit of capital, and \(w\) the wage per unit of labor.

Deriving the Cobb–Douglas Conditional Demands

Assume the production function is \(F(K,L)=A K^\alpha L^\beta\), where \(A>0\) and \(\alpha,\beta>0\). Assume the firm can choose any positive quantities of both inputs, both input prices are positive, and the output target is positive. In this model, more of either input raises output, and the cost-minimizing solution for a positive target is interior.

Because both inputs are productive and have positive prices, the output constraint binds at the minimum. If the firm produced more than \(q\), it could reduce an input slightly, lower expenditure, and still meet the target. Thus the firm minimizes \(rK+wL\) subject to \(A K^\alpha L^\beta=q\).

Let \(s=\alpha+\beta\), and let \(\lambda\) be the multiplier on the production constraint. The first-order conditions for the Lagrangian \(rK+wL+\lambda(q-AK^\alpha L^\beta)\) are

r=\lambda\alpha\frac{q}{K}, \qquad w=\lambda\beta\frac{q}{L}, \qquad A K^\alpha L^\beta=q.

Dividing the first condition by the second eliminates the multiplier. It gives the optimal input ratio, and substituting that ratio into the production constraint gives the quantities of both inputs.

\frac{r}{w}=\frac{\alpha L}{\beta K} \quad\Longrightarrow\quad \frac{K}{L}=\frac{\alpha w}{\beta r}\equiv t.

Set \(K=tL\). The production constraint becomes \(A t^\alpha L^{\alpha+\beta}=q\). Solving for labor and then multiplying by \(t\) to obtain capital yields the conditional demands.

L^c(q,r,w)= \left(\frac{q}{A}\right)^{1/s} \left(\frac{\alpha w}{\beta r}\right)^{-\alpha/s}, \qquad K^c(q,r,w)= \left(\frac{q}{A}\right)^{1/s} \left(\frac{\alpha w}{\beta r}\right)^{\beta/s}.

Interpretation: Holding prices fixed, both conditional demands rise with the output target \(q\). The input ratio \(K^c/L^c=\alpha w/(\beta r)\) rises when labor becomes more expensive relative to capital, and falls when capital becomes more expensive relative to labor. These are demands for inputs conditional on producing \(q\); they do not determine which output level the firm will choose.

The derivation uses the Cobb–Douglas form and its positive exponents. It does not apply unchanged to perfect substitutes, fixed-proportions technologies, or technologies with boundary optima. For those cases, the optimal input choice must be solved using the relevant production function and may not satisfy an interior tangency condition.

Worked Applications

Worked Example: A Target Requiring Much More Labor

A hypothetical workshop has production function \(q=2K^{0.25}L^{0.75}\). It must produce \(q=20\). Capital costs \(r=1\) per unit and labor costs \(w=48\) per unit. Find the conditional demands and check both the production target and the input-ratio condition.

Method. Use the Cobb–Douglas conditional-demand formulas with \(A=2\), \(\alpha=0.25\), \(\beta=0.75\), and \(s=1\). The price-and-technology ratio is \(t=\alpha w/(\beta r)=(0.25\times48)/(0.75\times1)=12/0.75=16\). Also, \(q/A=20/2=10\).

Since \(s=1\), the demand formulas give \(K^c=10(16)^{0.75}\) and \(L^c=10(16)^{-0.25}\). Because \(16^{0.75}=8\) and \(16^{-0.25}=1/2\), the demands are \(K^c=80\) and \(L^c=5\).

Production check. Since \(80=16\times5\), \(80^{0.25}5^{0.75}=16^{0.25}5^{0.25}5^{0.75}=2\times5=10\). Therefore, output is \(2\times10=20\), exactly the target.

Optimal-ratio check. The chosen ratio is \(K^c/L^c=80/5=16\), equal to \(t\). The input expenditure is \(rK^c+wL^c=1(80)+48(5)=320\). Common mistake avoided: reversing the input-price ratio would suggest using less capital when labor becomes relatively expensive. The derived ratio correctly gives \(K/L=\alpha w/(\beta r)\), so the workshop substitutes toward capital.

Worked Example: Equal Production Exponents and Unequal Prices

A hypothetical processor uses \(q=3K^{0.5}L^{0.5}\). It must produce \(q=60\). Capital costs \(r=2\) per unit and labor costs \(w=8\) per unit. Find the conditional demands and verify that the marginal condition is satisfied.

Method. Here \(A=3\), \(\alpha=\beta=0.5\), and \(s=1\). First calculate \(q/A=60/3=20\). The optimal ratio is \(t=(0.5\times8)/(0.5\times2)=4\).

The formulas give \(K^c=20(4)^{0.5}=20(2)=40\) and \(L^c=20(4)^{-0.5}=20(1/2)=10\).

Production check. \(3(40)^{0.5}(10)^{0.5}=3\sqrt{400}=3(20)=60\).

Marginal-condition check. For this production function, \(MP_K=1.5\sqrt{L/K}\) and \(MP_L=1.5\sqrt{K/L}\). At \(K=40,L=10\), these are \(MP_K=1.5(1/2)=0.75\) and \(MP_L=1.5(2)=3\). Thus \(MP_K/MP_L=0.75/3=0.25=r/w=2/8\), as required at an interior cost minimum. Expenditure is \(2(40)+8(10)=160\). Common mistake avoided: setting \(K/L=r/w\) would ignore the production exponents. The correct ratio includes both exponents: \(K/L=\alpha w/(\beta r)\).

Worked Example: Output Scaling with Nonconstant Exponents

A hypothetical producer has technology \(q=K^{0.5}L^{0.25}\), with a target of \(q=8\). Capital costs \(r=10\) per unit and labor costs \(w=5\) per unit. Find the conditional demands, then determine the demands if the target falls to \(q=1\) while prices and technology stay fixed.

Method. The parameters are \(A=1\), \(\alpha=0.5\), \(\beta=0.25\), and \(s=0.75\). The optimal ratio is \(t=(0.5\times5)/(0.25\times10)=2.5/2.5=1\). For the target \(q=8\), the formulas reduce to \(K^c=L^c=(8)^{1/0.75}=8^{4/3}\). Since \(8^{1/3}=2\), this is \(2^4=16\). Thus \(K^c=16\) and \(L^c=16\).

Production and price-ratio checks. Output is \(16^{0.5}16^{0.25}=4\times2=8\). The input ratio is \(16/16=1=t\). The marginal products are \(MP_K=0.5q/K=4/16=0.25\) and \(MP_L=0.25q/L=2/16=0.125\), so their ratio is \(2=r/w=10/5\).

For a target of \(q=1\), the same formula gives \(K^c=L^c=1^{4/3}=1\). Reducing the target from \(8\) to \(1\) reduces each input from \(16\) to \(1\). This illustrates the output dependence in the derived formula; it does not mean that input quantities must change in direct proportion to output. Common mistake avoided: assuming input use scales one-for-one with output without checking the exponent \(1/s\).

Reading the Demands and Avoiding Common Errors

The formulas separate the influence of the output target from the influence of relative prices. Both inputs contain the factor \((q/A)^{1/s}\), so a larger target raises both conditional demands. The ratio \(K^c/L^c\), by contrast, depends on input prices and production exponents, not on the target. Within this model, a higher wage relative to the price of capital shifts the optimal mix toward capital; a higher capital price relative to the wage shifts it toward labor.

The exponent \(1/s\) matters when comparing target changes. If \(s=1\), multiplying the target by a given factor multiplies both demands by that same factor. If \(s\ne1\), the input quantities instead change by the target factor raised to \(1/s\). This is a property of the conditional-demand formulas, not a separate assumption about input prices.

Do not confuse the conditional target with an output choice. The firm is minimizing input expenditure for a specified \(q\); these equations do not decide whether producing that output is profitable.
Keep capital and labor prices attached to the right inputs. The ratio is \(\alpha w/(\beta r)\), not its reciprocal.
Use the production constraint to check the quantities. A correct input ratio alone does not ensure the specified output is produced.
Check that the model supports an interior solution. The formulas rely on positive Cobb–Douglas exponents and positive prices; they are not universal formulas for all production technologies.

Key takeaway: Conditional factor demands solve two requirements at once: the chosen inputs must produce the specified output, and their ratio must reflect both relative input prices and the production exponents. Checking the target and the ratio is an efficient way to detect algebra and interpretation errors.

Check Your Understanding

For each problem, derive or apply the conditional demands and use the production target as a check.

A firm has \(q=5K^{0.5}L^{0.5}\), must produce \(q=100\), and faces \(r=4,w=1\). Find \(K^c\) and \(L^c\). (Answer: \(K^c=10,\ L^c=40\).)
A producer has \(q=2K^{0.25}L^{0.75}\), target \(q=16\), and prices \(r=2,w=24\). Find both conditional demands, rounded to three decimals. (Answer: \(K^c\approx22.627,\ L^c\approx5.657\).)
A firm has \(q=4K^{0.4}L^{0.6}\), target \(q=80\), and prices \(r=5,w=15\). Find the optimal input ratio and both demands, rounded to three decimals. (Answer: \(K^c/L^c=2,\ K^c\approx30.314,\ L^c\approx15.157\).)
A producer has \(q=K^{0.5}L^{0.25}\), target \(q=8\), and prices \(r=10,w=5\). Calculate the conditional demands and verify production. (Answer: \(K^c=16,\ L^c=16\); output is \(8\).)
A firm has \(q=\sqrt{KL}\) and target \(q=40\). Capital costs \(r=2\) and labor costs \(w=8\). Find the demands, then find them again if the wage falls to \(w=4\), with the target and capital price unchanged. (Answer: Initially \(K^c=80,\ L^c=20\); after the wage falls, \(K^c=40\sqrt{2}\approx56.569,\ L^c=20\sqrt{2}\approx28.284\).)

What You'll Learn