Math Problem Statement

Consider the following production functions. For each case, factor prices w1 and w2 are given by $3 and $5, respectively. f(x1, x2) = 2x1 + 3x2 i) Provide an example/a scenario that describes this technology. ii) Derive the conditional factor demand function for each input. Also, derive the cost function. iii) Consider a short-run period, during which x2 is fixed at 4 when you plan to produce 20 units of output. How many units of the first input are needed? How much cost is incurred

Solution

Solution:

i) Example Scenario for the Technology:

The production function f(x1,x2)=2x1+3x2f(x_1, x_2) = 2x_1 + 3x_2 represents a linear production technology. This can describe a scenario where a factory uses two inputs (e.g., labor x1x_1 and capital x2x_2) to produce a good. The coefficients 2 and 3 indicate the marginal productivity of each input: for every unit increase in x1x_1, output increases by 2 units; for every unit increase in x2x_2, output increases by 3 units.

ii) Derivation of the Conditional Factor Demand and Cost Function:

  • Conditional Factor Demand: To produce qq units of output: q=2x1+3x2q = 2x_1 + 3x_2 Rearrange for x1x_1: x1=q3x22,provided q3x2x_1 = \frac{q - 3x_2}{2}, \quad \text{provided } q \geq 3x_2

    Thus, the conditional demand for x1x_1 is: x1(q,x2)=q3x22x_1(q, x_2) = \frac{q - 3x_2}{2}

    If x1x_1 and x2x_2 are perfect substitutes and both inputs are available, the firm minimizes cost by choosing the input with the lowest cost per unit of productivity (w1MPL1\frac{w_1}{MPL_1} vs. w2MPL2\frac{w_2}{MPL_2}).

    • Input Costs Per Unit of Output:

      • w1/MPL1=3/2=1.5w_1 / MPL_1 = 3/2 = 1.5
      • w2/MPL2=5/31.67w_2 / MPL_2 = 5/3 \approx 1.67

      Since w1/MPL1<w2/MPL2w_1 / MPL_1 < w_2 / MPL_2, the firm will use x1x_1 exclusively unless constrained.

  • Cost Function: If only x1x_1 is used (x2=0x_2 = 0): C(q)=w1x1=3q2=3q2C(q) = w_1 \cdot x_1 = 3 \cdot \frac{q}{2} = \frac{3q}{2}

    If only x2x_2 is used (x1=0x_1 = 0): C(q)=w2x2=5q3=5q3C(q) = w_2 \cdot x_2 = 5 \cdot \frac{q}{3} = \frac{5q}{3}

    Overall Cost Function: C(q)=min(3q2,5q3)C(q) = \min \left(\frac{3q}{2}, \frac{5q}{3}\right)

iii) Short-Run Case:

  • Given: x2=4x_2 = 4, q=20q = 20.

    Substituting into the production function: 20=2x1+3(4)20 = 2x_1 + 3(4) 20=2x1+1220 = 2x_1 + 12 2x1=8    x1=42x_1 = 8 \implies x_1 = 4

    Cost Incurred: Using w1=3w_1 = 3 and w2=5w_2 = 5: Total Cost =w1x1+w2x2\text{Total Cost } = w_1 \cdot x_1 + w_2 \cdot x_2 Total Cost =3(4)+5(4)=12+20=32\text{Total Cost } = 3(4) + 5(4) = 12 + 20 = 32

Final Answers:

  1. Scenario: A factory with two inputs of different marginal productivity producing goods.
  2. Conditional Factor Demand: x1=q3x22x_1 = \frac{q - 3x_2}{2}.
  3. Cost Function: C(q)=min(3q2,5q3)C(q) = \min\left(\frac{3q}{2}, \frac{5q}{3}\right).
  4. Short-Run Inputs: x1=4x_1 = 4; Total cost = $32.

Let me know if you'd like further clarification! Here are some related questions for deeper understanding:


Related Questions:

  1. How would the cost function change if w1w_1 and w2w_2 were reversed?
  2. What happens to the conditional factor demand if q<3x2q < 3x_2?
  3. How would the scenario change if the production function had decreasing marginal returns?
  4. What is the role of factor prices in determining the optimal combination of inputs?
  5. How does fixing x2x_2 in the short run affect cost minimization?

Tip: Always compare input costs relative to their marginal productivity when selecting inputs to minimize costs.

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Math Problem Analysis

Mathematical Concepts

Linear Production Functions
Optimization
Cost Analysis

Formulas

Production function: f(x1, x2) = 2x1 + 3x2
Conditional factor demand: x1 = (q - 3x2) / 2
Cost function: C(q) = min(3q/2, 5q/3)
Total cost: w1 * x1 + w2 * x2

Theorems

Optimization principle: Minimize cost for a given output level
Linear production efficiency: Allocate resources based on marginal productivity

Suitable Grade Level

Undergraduate (Economics or Mathematics)