Suppose That A Firms Production Function Is Q = 10L1/2K1/2. The Cost Of A Unit Of Labor Is $20 And The

Suppose That A Firms Production Function Is Q = 10L1/2K1/2. The Cost Of A Unit Of Labor Is $20 And The firm aims to optimize production costs while maintaining a desired output level. Understanding the relationship between inputs, costs, and output is essential for effective decision-making in microeconomics. In this article, we will explore the production function, derive the cost minimization problem, analyze the firm's input choices, and discuss the implications for business strategy.

Understanding the Production Function Q = 10L^1/2K^1/2

What Does the Production Function Represent?

The production function Q = 10L^1/2K^1/2 describes the relationship between the inputs—labor (L) and capital (K)—and the resulting output (Q). It indicates that output increases with input quantities but at a diminishing rate, characteristic of the square root function.

Properties of the Production Function

• Cobb-Douglas Form: This function is a specific case of the Cobb-Douglas production function, which is widely used due to its simplicity and ability to model returns to scale.
• Returns to Scale: To assess whether the firm experiences increasing, decreasing, or constant returns to scale, analyze the sum of the exponents:

\[ 1/2 + 1/2 = 1 \]

Since the sum equals 1, the production function exhibits constant returns to scale. Doubling inputs doubles output, providing predictability in scaling operations.

• Marginal Products:


• Marginal Product of Labor (MP_L):

\[
MP_L = \frac{\partial Q}{\partial L} = 10 \times \frac{1}{2}L^{-1/2}K^{1/2} = 5 \frac{K^{1/2}}{L^{1/2}}
\]

• Marginal Product of Capital (MP_K):

\[
MP_K = \frac{\partial Q}{\partial K} = 10 \times \frac{1}{2}K^{-1/2}L^{1/2} = 5 \frac{L^{1/2}}{K^{1/2}}
\]

These marginal products show how additional units of labor or capital influence output, declining as input quantities increase.

Cost Structure and Optimization

Input Costs

Given the cost of labor per unit, the firm faces:

• Labor Cost (w): $20 per unit
• Capital Cost (r): (Assuming as given or to be determined; if not specified, we can analyze generally or assume a specific value)

The total cost (C) of inputs is:

\[
C = wL + rK
\]

The goal is to produce a given output level Q at the minimum possible cost.

Formulating the Cost Minimization Problem

To minimize costs for a given output Q, the firm solves:

\[
\min_{L,K} C = 20L + rK
\]

subject to:

\[
Q = 10L^{1/2}K^{1/2}
\]

This is a constrained optimization problem that can be approached using methods like Lagrangian multipliers.

Deriving the Cost-Minimizing Input Combination

Setting Up the Lagrangian

Define the Lagrangian:

\[
\mathcal{L} = 20L + rK + \lambda \left( Q - 10L^{1/2}K^{1/2} \right)
\]

where λ is the Lagrange multiplier.

First-Order Conditions

Differentiate with respect to L, K, and λ:

1. With respect to L:

\[ \frac{\partial \mathcal{L}}{\partial L} = 20 - \lambda \times 10 \times \frac{1}{2} L^{-1/2} K^{1/2} = 0 \] \[ 20 = 5 \lambda \frac{K^{1/2}}{L^{1/2}} \]

1. With respect to K:

\[ \frac{\partial \mathcal{L}}{\partial K} = r - \lambda \times 10 \times \frac{1}{2} K^{-1/2} L^{1/2} = 0 \] \[ r = 5 \lambda \frac{L^{1/2}}{K^{1/2}} \]

1. With respect to λ:

\[ Q - 10 L^{1/2} K^{1/2} = 0 \]

Solving for Input Ratios

Dividing the first equation by the second:

\[
\frac{20}{r} = \frac{K^{1/2}/L^{1/2}}{L^{1/2}/K^{1/2}} = \frac{K^{1/2}}{L^{1/2}} \times \frac{K^{1/2}}{L^{1/2}} = \frac{K}{L}
\]

Thus:

\[
\frac{K}{L} = \frac{20}{r}
\]
or equivalently:

\[
K = \frac{20}{r} L
\]

This ratio indicates the optimal proportion of capital to labor to minimize cost for producing output Q.

Calculating the Optimal Input Quantities

Expressing K in Terms of L

From the input ratio:

\[
K = \frac{20}{r} L
\]

Substitute into the production function:

\[
Q = 10 L^{1/2} \left( \frac{20}{r} L \right)^{1/2}
\]

Simplify:

\[
Q = 10 L^{1/2} \times \left( \frac{20}{r} \right)^{1/2} L^{1/2} = 10 \left( \frac{20}{r} \right)^{1/2} L
\]

Solve for L:

\[
L = \frac{Q}{10} \left( \frac{r}{20} \right)^{1/2}
\]

Similarly, K:

\[
K = \frac{20}{r} L = \frac{20}{r} \times \frac{Q}{10} \left( \frac{r}{20} \right)^{1/2}
\]

Simplify:

\[
K = 2 \times \frac{Q}{r} \times \left( \frac{r}{20} \right)^{1/2} = \frac{2Q}{r} \times \frac{\sqrt{r}}{\sqrt{20}} = \frac{2Q}{\sqrt{20} \times \sqrt{r}}
\]

Summary of Optimal Inputs:

\[
L^ = \frac{Q}{10} \left( \frac{r}{20} \right)^{1/2}
\]

\[
K^ = \frac{2Q}{\sqrt{20} \times \sqrt{r}}
\]

Note: These formulas depend on the capital cost r, which can be specified or estimated based on market conditions.

Calculating the Minimum Cost

Using the optimal input quantities, total cost becomes:

\[
C_{min} = 20 L^ + r K^
\]

Substitute the expressions:

\[
C_{min} = 20 \times \frac{Q}{10} \left( \frac{r}{20} \right)^{1/2} + r \times \frac{2Q}{\sqrt{20} \times \sqrt{r}}
\]

Simplify:

\[
C_{min} = 2Q \left( \frac{r}{20} \right)^{1/2} + 2Q \frac{\sqrt{r}}{\sqrt{20}}
\]

Note that:

\[
\left( \frac{r}{20} \right)^{1/2} = \frac{\sqrt{r}}{\sqrt{20}}
\]

Thus,

\[
C_{min} = 2Q \times \frac{\sqrt{r}}{\sqrt{20}} + 2Q \times \frac{\sqrt{r}}{\sqrt{20}} = 4Q \times \frac{\sqrt{r}}{\sqrt{20}}
\]

Or,

\[
C_{min} = 4Q \frac{\sqrt{r}}{\sqrt{20}}
\]

This formula provides the minimum cost for producing output Q given input prices.

Implications for Business Strategy

Cost Efficiency and Input Choices

By understanding the relationship between input costs and output, firms can adjust their input proportions to achieve cost efficiency. For example, if the cost of capital (r) increases, the optimal amount of capital K decreases,