Conditional factor demand tells you how much of each input a cost-minimizing firm will use to produce a given output level. It is “conditional” on output: the firm must produce exactly q units and wants to do so at minimum cost, given input prices. This page shows how to calculate conditional factor demand and explains why it depends on both input prices and output level.

1. The Cost-Minimization Problem

For two inputs, the firm’s problem is:

\[ \min_{x_1,x_2} \; C = w_1 x_1 + w_2 x_2 \] subject to \[ f(x_1,x_2) = q. \]

• \(x_1, x_2\) = quantities of inputs 1 and 2
• \(w_1, w_2\) = input prices
• \(f(\cdot)\) = production function
• \(q\) = required output level

The solution gives the conditional factor demands: \[ x_1 = x_1(w_1,w_2,q), \quad x_2 = x_2(w_1,w_2,q). \]

2. Why Conditional Factor Demand Depends on Input Prices

Input prices determine the slope of the isocost line: \[ \text{slope of isocost} = -\frac{w_1}{w_2}. \] The firm chooses the tangency between the isoquant (technology) and the isocost line (cost).

At the optimum: \[ \frac{MP_1}{MP_2} = \frac{w_1}{w_2}, \] where \(MP_i\) is the marginal product of input \(i\).

Because the right-hand side contains prices, the optimal input mix changes when prices change.

Diagram (Isoquant–Isocost)

x2
^
|        q isoquant
|       /
|      /
|     /   isocost: C = w1 x1 + w2 x2
|    /
+------------------> x1

Changing \(w_1/w_2\) rotates the isocost line and shifts the tangency point, changing the conditional factor demands.

3. Why Conditional Factor Demand Depends on Output Level

The isoquant for output \(q\) is the set of input bundles that can produce exactly \(q\). Higher output means an isoquant farther from the origin, requiring more inputs.

x2
^   q2 isoquant
|  q1 isoquant
| q0 isoquant
+------------------> x1

As \(q\) increases, the firm moves to higher isoquants, so the cost-minimizing input bundle scales up. The mix of inputs is determined by prices; the scale is determined by \(q\).

4. Example 1 – Cobb–Douglas Technology

Consider the production function: \[ q = x_1^{1/2} x_2^{1/2}. \]

Step 1: Set up the Lagrangian

\[ \mathcal{L} = w_1 x_1 + w_2 x_2 + \lambda \left(q – x_1^{1/2} x_2^{1/2}\right). \]

Step 2: First-order conditions

\[ \frac{\partial \mathcal{L}}{\partial x_1} = w_1 – \lambda \frac{1}{2} x_1^{-1/2} x_2^{1/2} = 0 \] \[ \frac{\partial \mathcal{L}}{\partial x_2} = w_2 – \lambda \frac{1}{2} x_2^{-1/2} x_1^{1/2} = 0 \]

Step 3: Ratio of FOCs (tangency condition)

Divide the first FOC by the second: \[ \frac{w_1}{w_2} = \frac{x_2^{1/2} x_1^{-1/2}}{x_1^{1/2} x_2^{-1/2}} = \frac{x_2}{x_1}. \] So: \[ x_2 = \frac{w_1}{w_2} x_1. \]

Step 4: Use the production constraint

\[ q = x_1^{1/2} x_2^{1/2} = x_1^{1/2} \left(\frac{w_1}{w_2} x_1\right)^{1/2} = x_1 \left(\frac{w_1}{w_2}\right)^{1/2}. \]

Solve for \(x_1\): \[ x_1 = q \left(\frac{w_2}{w_1}\right)^{1/2}. \] Then: \[ x_2 = \frac{w_1}{w_2} x_1 = q \left(\frac{w_1}{w_2}\right)^{1/2}. \]

Conditional factor demands

\[ x_1(w_1,w_2,q) = q \left(\frac{w_2}{w_1}\right)^{1/2}, \quad x_2(w_1,w_2,q) = q \left(\frac{w_1}{w_2}\right)^{1/2}. \]

• Depend on prices via the ratio \(w_1/w_2\).
• Depend on output linearly via \(q\).

Numerical Example

• \(w_1 = 4\), \(w_2 = 1\), \(q = 10\)

\[ x_1 = 10 \left(\frac{1}{4}\right)^{1/2} = 10 \cdot \frac{1}{2} = 5 \] \[ x_2 = 10 \left(\frac{4}{1}\right)^{1/2} = 10 \cdot 2 = 20 \]

If \(q\) doubles to 20 (same prices): \[ x_1 = 20 \cdot \frac{1}{2} = 10,\quad x_2 = 20 \cdot 2 = 40. \] Doubling output doubles both inputs.

5. Example 2 – Price Changes and Substitution

Keep \(q = 10\), but change prices.

Case 1: \(w_1 = 4\), \(w_2 = 1\)

\[ x_1 = 10 \left(\frac{1}{4}\right)^{1/2} = 5,\quad x_2 = 20. \]

Case 2: \(w_1 = 1\), \(w_2 = 4\)

\[ x_1 = 10 \left(\frac{4}{1}\right)^{1/2} = 20,\quad x_2 = 10 \left(\frac{1}{4}\right)^{1/2} = 5. \]

Same technology, same output, different prices → the firm substitutes toward the relatively cheaper input.

6. Geometric Summary

• Input prices change the slope of the isocost line → change the cost-minimizing input mix.
• Output level shifts the isoquant outward → scales up the cost-minimizing bundle.

x2
^
|   higher q isoquant
|  /
| /   lower q isoquant
+------------------> x1

Conditional factor demand functions \(x_i(w,q)\) capture both substitution (price-driven) and scale (output-driven) effects.

7. Link to the Cost Function and Shephard’s Lemma

The cost function is: \[ C(w_1,w_2,q) = \min_{x_1,x_2} \{ w_1x_1 + w_2x_2 : f(x_1,x_2)=q \}. \]

Shephard’s Lemma: \[ x_i(w_1,w_2,q) = \frac{\partial C(w_1,w_2,q)}{\partial w_i}. \]

Because the cost function depends on input prices and output, its partial derivatives (conditional factor demands) must also depend on input prices and output.

8. Key Takeaways

• Conditional factor demand is derived from a cost-minimization problem with a fixed output level.
• It depends on input prices because the optimal input mix equates MRTS to the price ratio.
• It depends on output level because higher output requires more inputs (higher isoquants).
• In many technologies (e.g., Cobb–Douglas), conditional factor demands scale linearly with output and adjust in proportion to relative prices.