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Question 2: Consider the production function given by Y = min(L,5K). The firm has a fixed amount of capital, =2, and the pries of output, labor and capital are P, wk, and wl respectively. a. Graph the production function

Question 2: Consider the production function given by Y = min(L,5K). The firm has a fixed amount of capital, =2, and the pries of output, labor and capital are P, wk, and wl respectively. a. Graph the production function

ENGR.ECONOMIC ANALYSIS
ENGR.ECONOMIC ANALYSIS
14th Edition
ISBN:9780190931919
Author:NEWNAN
Publisher:NEWNAN
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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**Question 2:** Consider the production function given by \( Y = \min(L, 5K) \). The firm has a fixed amount of capital, \( K=2 \), and the prices of output, labor, and capital are \( P \), \( w_k \), and \( w_l \) respectively. a. **Graph the production function** - To graph this production function, plot labor (\( L \)) on the x-axis and output (\( Y \)) on the y-axis. - Since \( K \) is fixed at 2, the production function becomes \( Y = \min(L, 10) \). - The graph will have a kink at \( L = 10 \). For \( L < 10 \), \( Y = L \). For \( L \geq 10 \), \( Y = 10 \). - The graph is a 45-degree line from the origin up to \( L = 10 \), where it becomes a horizontal line. b. **Explain how to find the optimal output and labor when the prices are unknown. Show all necessary lines and all possible cases.** - The optimal choice depends on the comparison between \( w_k \) and \( w_l \). - If \( L < 10 \), the firm employs less labor, so the marginal cost of labor \( w_l \) determines the cost structure. - If \( w_k < w_l \), increasing \( K \) will be more cost-effective, suggesting a need to examine the budget for capital increase. - If \( w_l < w_k \), the firm should focus on hiring labor up to the point where marginal cost equals marginal revenue. - For prices that equate, \( L = 10 \) should be evaluated, ensuring \( w_k \times 2 = w_l \times Y \). - The firm should choose the level of \( L \) such that the cost per unit of capital \( w_k \) and labor cost \( w_l \) balance with the price of output \( P \), optimizing their production efficiency.
Transcribed Image Text:**Question 2:** Consider the production function given by \( Y = \min(L, 5K) \). The firm has a fixed amount of capital, \( K=2 \), and the prices of output, labor, and capital are \( P \), \( w_k \), and \( w_l \) respectively. a. **Graph the production function** - To graph this production function, plot labor (\( L \)) on the x-axis and output (\( Y \)) on the y-axis. - Since \( K \) is fixed at 2, the production function becomes \( Y = \min(L, 10) \). - The graph will have a kink at \( L = 10 \). For \( L < 10 \), \( Y = L \). For \( L \geq 10 \), \( Y = 10 \). - The graph is a 45-degree line from the origin up to \( L = 10 \), where it becomes a horizontal line. b. **Explain how to find the optimal output and labor when the prices are unknown. Show all necessary lines and all possible cases.** - The optimal choice depends on the comparison between \( w_k \) and \( w_l \). - If \( L < 10 \), the firm employs less labor, so the marginal cost of labor \( w_l \) determines the cost structure. - If \( w_k < w_l \), increasing \( K \) will be more cost-effective, suggesting a need to examine the budget for capital increase. - If \( w_l < w_k \), the firm should focus on hiring labor up to the point where marginal cost equals marginal revenue. - For prices that equate, \( L = 10 \) should be evaluated, ensuring \( w_k \times 2 = w_l \times Y \). - The firm should choose the level of \( L \) such that the cost per unit of capital \( w_k \) and labor cost \( w_l \) balance with the price of output \( P \), optimizing their production efficiency.
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The production function represents the relationship between the output and inputs. The function is complement when both inputs are used in fixed proportion. The shape of isoquant is L shaped. 

 

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