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 represents a linear production technology. This can describe a scenario where a factory uses two inputs (e.g., labor and capital ) to produce a good. The coefficients 2 and 3 indicate the marginal productivity of each input: for every unit increase in , output increases by 2 units; for every unit increase in , output increases by 3 units.
ii) Derivation of the Conditional Factor Demand and Cost Function:
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Conditional Factor Demand: To produce units of output: Rearrange for :
Thus, the conditional demand for is:
If and are perfect substitutes and both inputs are available, the firm minimizes cost by choosing the input with the lowest cost per unit of productivity ( vs. ).
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Input Costs Per Unit of Output:
Since , the firm will use exclusively unless constrained.
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Cost Function: If only is used ():
If only is used ():
Overall Cost Function:
iii) Short-Run Case:
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Given: , .
Substituting into the production function:
Cost Incurred: Using and :
Final Answers:
- Scenario: A factory with two inputs of different marginal productivity producing goods.
- Conditional Factor Demand: .
- Cost Function: .
- Short-Run Inputs: ; Total cost = $32.
Let me know if you'd like further clarification! Here are some related questions for deeper understanding:
Related Questions:
- How would the cost function change if and were reversed?
- What happens to the conditional factor demand if ?
- How would the scenario change if the production function had decreasing marginal returns?
- What is the role of factor prices in determining the optimal combination of inputs?
- How does fixing 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)
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