Understanding the Production Function and Marginal Products

The question asks us to find the marginal product of labour ($\text{MP}_L$) and the marginal product of capital ($\text{MP}_K$) for a given production function. A production function shows the maximum output that can be produced with a given set of inputs, typically labour (L) and capital (K).

The given production function is:

\(Q = 10 L^{0.8} K^{0.2}\)

Here, Q represents the quantity of output, L represents the units of labour, and K represents the units of capital. This specific form of production function is a Cobb-Douglas production function.

Calculating the Marginal Product of Labour ($\text{MP}_L$)

The marginal product of labour ($\text{MP}_L$) is the additional output produced when one more unit of labour is employed, holding all other inputs (like capital) constant. Mathematically, it is the partial derivative of the production function with respect to labour (L).

Let's calculate $\text{MP}_L$:

\(\text{MP}_L = \frac{\partial Q}{\partial L}\)

\(\text{MP}_L = \frac{\partial}{\partial L} (10 L^{0.8} K^{0.2})\)

Treating K as a constant, we apply the power rule of differentiation to L:

\(\text{MP}_L = 10 \times 0.8 \times L^{(0.8 - 1)} \times K^{0.2}\)

\(\text{MP}_L = 8 \times L^{-0.2} \times K^{0.2}\)

We can rewrite \(L^{-0.2}\) as \(\frac{1}{L^{0.2}}\). So, the expression becomes:

\(\text{MP}_L = 8 \frac{K^{0.2}}{L^{0.2}}\)

This can be further written as:

\(\text{MP}_L = 8 \left(\frac{K}{L}\right)^{0.2}\)

So, the marginal product of labour is \(8(K/L)^{0.2}\).

Calculating the Marginal Product of Capital ($\text{MP}_K$)

The marginal product of capital ($\text{MP}_K$) is the additional output produced when one more unit of capital is used, holding all other inputs (like labour) constant. Mathematically, it is the partial derivative of the production function with respect to capital (K).

Let's calculate $\text{MP}_K$:

\(\text{MP}_K = \frac{\partial Q}{\partial K}\)

\(\text{MP}_K = \frac{\partial}{\partial K} (10 L^{0.8} K^{0.2})\)

Treating L as a constant, we apply the power rule of differentiation to K:

\(\text{MP}_K = 10 \times 0.2 \times L^{0.8} \times K^{(0.2 - 1)}\)

\(\text{MP}_K = 2 \times L^{0.8} \times K^{-0.8}\)

We can rewrite \(K^{-0.8}\) as \(\frac{1}{K^{0.8}}\). So, the expression becomes:

\(\text{MP}_K = 2 \frac{L^{0.8}}{K^{0.8}}\)

This can be further written as:

\(\text{MP}_K = 2 \left(\frac{L}{K}\right)^{0.8}\)

So, the marginal product of capital is \(2(L/K)^{0.8}\).

Comparing Calculated Marginal Products with Options

Now, let's compare our calculated results for $\text{MP}_L$ and $\text{MP}_K$ with the statements given in options A, B, C, and D.

From the comparison, we find that statement A ($\text{MP}_L = 8(K/L)^{0.2}$) is true, and statement C ($\text{MP}_K = 2(L/K)^{0.8}$) is true. Statements B and D are false.

Analyzing the Answer Choices

We need to find the option that correctly identifies the true statements. The truth values we found are: A (True), B (False), C (True), D (False).

• Option 1: A and D are true. This is false because D is false.
• Option 2: A and C are true. This is true because A is true and C is true.
• Option 3: B and C are true. This is false because B is false.
• Option 4: B and D are true. This is false because both B and D are false.

Therefore, the correct answer choice is the one stating that A and C are true.

Revision Table: Key Concepts

Additional Information: Cobb-Douglas Production Function

The production function \(Q = 10 L^{0.8} K^{0.2}\) is an example of a Cobb-Douglas production function, which has the general form \(Q = A L^\alpha K^\beta\), where A is a constant representing technology, \(\alpha\) is the output elasticity of labour, and \(\beta\) is the output elasticity of capital.

• In this specific function, \(A=10\), \(\alpha=0.8\), and \(\beta=0.2\).
• For a Cobb-Douglas function \(Q = A L^\alpha K^\beta\), the marginal products are:
  • \(\text{MP}_L = \alpha A L^{\alpha-1} K^\beta = \alpha A \left(\frac{K}{L}\right)^\beta\)
  • \(\text{MP}_K = \beta A L^\alpha K^{\beta-1} = \beta A \left(\frac{L}{K}\right)^\alpha\)
• Let's check if our calculations match the general formulas with the specific values:
  • \(\text{MP}_L = 0.8 \times 10 \times L^{0.8-1} K^{0.2} = 8 L^{-0.2} K^{0.2} = 8 (K/L)^{0.2}\). This matches.
  • \(\text{MP}_K = 0.2 \times 10 \times L^{0.8} K^{0.2-1} = 2 L^{0.8} K^{-0.8} = 2 (L/K)^{0.8}\). This also matches.
• The sum of the exponents, \(\alpha + \beta\), indicates the returns to scale. In this case, \(0.8 + 0.2 = 1\), which means the production function exhibits constant returns to scale. If inputs are doubled, output will also double.