Consider the Cobb-Douglas Production function: P(L, K) = 13L0.9K0.1 Find the total units of production when L=19 units of labor and K=10 units of capital are invested. (Give your answer to three (3) decimal places, if

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### Understanding the Cobb-Douglas Production Function

The Cobb-Douglas production function is a particular functional form of the production function. It is widely used in economics to represent the relationship between two or more inputs (typically physical capital and labor) and the amount of output produced. The general form of the Cobb-Douglas production function is:

\[ P(L, K) = 13L^{0.9}K^{0.1} \]

where:
- \( L \) represents the quantity of labor,
- \( K \) represents the quantity of capital,
- \( P(L, K) \) represents the total production or output,
- \( 13 \) is a constant that represents total factor productivity,
- \( 0.9 \) and \( 0.1 \) are the output elasticities of labor and capital, respectively.

### Problem Statement

We are given the specific values for labor and capital and need to calculate the total production. The problem is stated as follows:

**Find the total units of production when \( L = 19 \) units of labor and \( K = 10 \) units of capital are invested. (Give your answer to three (3) decimal places, if necessary.)**

#### Solution

To find the total production, we will substitute the given values of \( L \) and \( K \) into the Cobb-Douglas production function.

**Mathematical Steps:**

1. Substituting the values \( L = 19 \) and \( K = 10 \) into the production function:
\[ P(19, 10) = 13 \times 19^{0.9} \times 10^{0.1} \]

2. Calculating the exponents:
\[ 19^{0.9} \approx 16.109 \]
\[ 10^{0.1} \approx 1.259 \]

3. Multiplying these results by the constant:
\[ P(19, 10) = 13 \times 16.109 \times 1.259 \approx 263.170 \]

Therefore, the total units of production given \( L = 19 \) and \( K = 10 \) is approximately:

\[ \boxed{263.170 \text{ units}} \]
Transcribed Image Text:### Understanding the Cobb-Douglas Production Function The Cobb-Douglas production function is a particular functional form of the production function. It is widely used in economics to represent the relationship between two or more inputs (typically physical capital and labor) and the amount of output produced. The general form of the Cobb-Douglas production function is: \[ P(L, K) = 13L^{0.9}K^{0.1} \] where: - \( L \) represents the quantity of labor, - \( K \) represents the quantity of capital, - \( P(L, K) \) represents the total production or output, - \( 13 \) is a constant that represents total factor productivity, - \( 0.9 \) and \( 0.1 \) are the output elasticities of labor and capital, respectively. ### Problem Statement We are given the specific values for labor and capital and need to calculate the total production. The problem is stated as follows: **Find the total units of production when \( L = 19 \) units of labor and \( K = 10 \) units of capital are invested. (Give your answer to three (3) decimal places, if necessary.)** #### Solution To find the total production, we will substitute the given values of \( L \) and \( K \) into the Cobb-Douglas production function. **Mathematical Steps:** 1. Substituting the values \( L = 19 \) and \( K = 10 \) into the production function: \[ P(19, 10) = 13 \times 19^{0.9} \times 10^{0.1} \] 2. Calculating the exponents: \[ 19^{0.9} \approx 16.109 \] \[ 10^{0.1} \approx 1.259 \] 3. Multiplying these results by the constant: \[ P(19, 10) = 13 \times 16.109 \times 1.259 \approx 263.170 \] Therefore, the total units of production given \( L = 19 \) and \( K = 10 \) is approximately: \[ \boxed{263.170 \text{ units}} \]
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