Profit Maximization in Office Chair Production: A Managerial Analysis

Introduction

As a manager of a major office furniture manufacturer, it is crucial to make informed decisions to optimize profits. This involves understanding the production function and cost structure. In this scenario, we have an economist’s report providing insight into the production function for a particular line of office chairs. We will explore the short-run and long-run aspects of profit maximization, taking into account capital, labor, and market conditions.

Short-Run Profit Maximization

In the short run, the company has spent $9,000 on 9 units of capital (K) and is unable to change this fixed investment. The production function is given as Q = 2(K)^1/2 (L)^1/2, where K represents capital and L is labor. The rental rate for capital is $1,000 per unit, and the wage rate is $120 per labor. The market price for the chair is $400.

To maximize profits, the company must determine the optimal level of labor input (L) given the fixed capital (K). Profit maximization occurs where marginal cost (MC) equals marginal revenue (MR). In this case, MR is the market price of the chair ($400).

First, calculate the marginal product of capital (MPK) and the marginal product of labor (MPL):

MPK = (L/K)^1/2 MPL = (K/L)^1/2

With K = 9 units and the rental rate for capital at $1,000 per unit, we can calculate the total capital cost as K * Rental Rate = 9 * $1,000 = $9,000.

Now, we can set up the profit maximization equation:

MR = MC $400 = MPL * Wage Rate $400 = (9/L)^1/2 * $120

Solving for L:

(9/L)^1/2 = $400 / $120 (9/L)^1/2 = 3.33

L = (9/3.33)^2 ≈ 6.48 units of labor

Now, we can calculate the corresponding output (Q) and profit:

Q = 2(K)^1/2 (L)^1/2 Q = 2(9)^1/2 (6.48)^1/2 Q ≈ 38.93 units of chairs

Profit = Total Revenue – Total Cost Profit = (Q * Price) – (Capital Cost + Labor Cost) Profit = (38.93 * $400) – ($9,000 + (6.48 * $120)) Profit ≈ $15,572 – $9,777.6 Profit ≈ $5,794.4

 Long-Run Profit Maximization

In the long run, the firm has the flexibility to choose any combination of capital (K) and labor (L). Would the firm choose the same input levels as in part (a)?

No, the firm would not necessarily choose the same input levels as in part (a) in the long run. In the short run, the firm was constrained by the fixed investment in capital (9 units). However, in the long run, the firm can adjust both capital and labor to maximize profits. To optimize profit, the firm would compare the marginal product of capital (MPK) and the marginal product of labor (MPL) to their respective costs, considering various combinations of K and L. The optimal input combination would be where MPK equals the rental rate for capital, and MPL equals the wage rate.

Conclusion

In the short run, given a fixed capital investment, the profit-maximizing level of labor input was approximately 6.48 units, resulting in a profit of approximately $5,794.4. In the long run, the firm would analyze various combinations of capital and labor to find the input levels that maximize profit, considering the relationship between MPK, MPL, and input costs. Making informed decisions about input allocation is essential for any company to achieve long-term profitability and sustainability in the competitive market.