**Educational Exercise: Understanding Cobb-Douglas Production Function****Problem Statement:**Suppose an economy's output is determined by a Cobb-Douglas production function:\[ Y = AK^{\alpha}L^{1-\alpha} \]where $ A = 8 $ and $ \alpha = 0.2 $. All other things remaining unchanged, if labor input increases by 5 percent, output will rise by:- $ \circ $ 1%- $ \circ $ 4%- $ \circ $ 2%- $ \circ $ 5%**Explanation:**Given the Cobb-Douglas production function:\[ Y = AK^{\alpha}L^{1-\alpha} \]where:- $ Y $ is the total production (the real value of all goods produced in the economy).- $ A $ is the total factor productivity.- $ K $ is the amount of capital used.- $ L $ is the labor input.- $ \alpha $ is the output elasticity of capital, and $ (1 - \alpha) $ is the output elasticity of labor.**Solution Approach:**1. **Understanding Output Elasticity**: In the Cobb-Douglas function, $ (1 - \alpha) $ represents the elasticity of output with respect to labor. This means if labor input (L) changes by a certain percentage, the output (Y) will change by $ (1 - \alpha) $ times that percentage. 2. **Given Parameters**: - $ \alpha = 0.2 $ (thus, $ 1 - \alpha = 0.8 $) - Labor input increase = 5% 3. **Calculation**: If labor input increases by 5%, the percentage change in output $ \Delta Y $ can be calculated by: \[ \Delta Y = (1 - \alpha) \times \text{percentage change in } L \] \[ \Delta Y = 0.8 \times 5\% \] \[ \Delta Y = 4\% \]**Answer Options:**- $ \circ $ 1%- $ \circ $ 4%- $ \circ $ 2%- $ \circ $ 5%Thus, if labor input increases by 5

Production function refers to the mathematical equation the depicts the relationship between the…

Suppose an economy's output is determined by a Cobb-Douglas production function Y = AKL¹ where A-8 and a-0.2. All other things remaining unchanged, if labour input increases by 5 percent, output will rise by: O 1% O 4% Ⓒ2% O 5%

Suppose an economy's output is determined by a Cobb-Douglas production function Y = AKL¹ where A-8 and a-0.2. All other things remaining unchanged, if labour input increases by 5 percent, output will rise by: O 1% O 4% Ⓒ2% O 5%

ENGR.ECONOMIC ANALYSIS

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ISBN:9780190931919

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Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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Concept explainers

Isocost and Isoprofit Curve Analysis

An isocost curve in the production theory that is a crucial part of economics depicts different combinations of factor inputs at a given market cost. In other words, this shows all the combinations of production factors that cost the same for the firm to employ. Usually, this can be depicted as a downward sloping straight line when the firms act as the price-takers in the factor markets. The slope of this mainly depicts the relative prices of different production factors.

Question

**Educational Exercise: Understanding Cobb-Douglas Production Function**

**Problem Statement:**

Suppose an economy's output is determined by a Cobb-Douglas production function:

\[ Y = AK^{\alpha}L^{1-\alpha} \]

where $ A = 8 $ and $ \alpha = 0.2 $. All other things remaining unchanged, if labor input increases by 5 percent, output will rise by:

- $ \circ $ 1%
- $ \circ $ 4%
- $ \circ $ 2%
- $ \circ $ 5%

**Explanation:**

Given the Cobb-Douglas production function:

\[ Y = AK^{\alpha}L^{1-\alpha} \]

where:
- $ Y $ is the total production (the real value of all goods produced in the economy).
- $ A $ is the total factor productivity.
- $ K $ is the amount of capital used.
- $ L $ is the labor input.
- $ \alpha $ is the output elasticity of capital, and $ (1 - \alpha) $ is the output elasticity of labor.

**Solution Approach:**

1. **Understanding Output Elasticity**: In the Cobb-Douglas function, $ (1 - \alpha) $ represents the elasticity of output with respect to labor. This means if labor input (L) changes by a certain percentage, the output (Y) will change by $ (1 - \alpha) $ times that percentage.

2. **Given Parameters**:
- $ \alpha = 0.2 $ (thus, $ 1 - \alpha = 0.8 $)
- Labor input increase = 5%

3. **Calculation**:
If labor input increases by 5%, the percentage change in output $ \Delta Y $ can be calculated by:

\[ \Delta Y = (1 - \alpha) \times \text{percentage change in } L \]
\[ \Delta Y = 0.8 \times 5\% \]
\[ \Delta Y = 4\% \]

**Answer Options:**
- $ \circ $ 1%
- $ \circ $ 4%
- $ \circ $ 2%
- $ \circ $ 5%

Thus, if labor input increases by 5

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