Question

An enterprising entrepreneur purchases two firms to produce widgets. Each firm produces identical products, and each has a production function given by \\[ q=\sqrt{k_{i} l_{i}}, \quad i=1,2 \\] The firms differ, however, in the amount of capital equipment each has. In particular, firm I has \(k_{1}=25\) whereas firm 2 has \(k_{2}=100 .\) Rental rates for \(k\) and \(l\) are given by \(w=v=\$ 1\) a. If the entrepreneur wishes to minimize short-run total costs of widget production, how should output be allocated between the two firms? b. Given that output is optimally allocated between the two firms, calculate the short-run total, average, and marginal cost curves. What is the marginal cost of the 100 th widget? The 125 th widget? The 200 th widget? c. How should the entrepreneur allocate widget production between the two firms in the long run? Calculate the long-run total, average, and marginal cost curves for widget production. d. How would your answer to part (c) change if both firms exhibited diminishing returns to scale?

Short answer

a. Allocate such that the marginal costs of each firm are equal. b. Total, average, and marginal costs depend on \( l_1, l_2 \); calculate accordingly for given widgets. c. In the long run, optimize using equal input allocation. d. Consider adjustments due to diminishing returns affecting labor and cost functions.

Step-by-step solution

Understanding the Production Functions

Both firms have production functions given by \( q = \sqrt{k_i l_i} \) where \( i = 1, 2 \). Firm 1 has \( k_1 = 25 \) and Firm 2 has \( k_2 = 100 \). The rental rates for capital \( k \) and labor \( l \) are both given as \( w = v = 1 \), so the cost for both is \( C_i = w l_i + v k_i \).

Equalizing Marginal Costs in Short Run

To minimize the short-run total costs, we equalize the marginal costs of production between the two firms. The marginal product of labor for firm \( i \) is \( \text{MPL}_i = \frac{\partial q}{\partial l_i} = \frac{1}{2}\sqrt{\frac{k_i}{l_i}} \). Equating the marginal costs for each firm gives \( \frac{1}{2}\sqrt{\frac{k_1}{l_1}} = \frac{1}{2}\sqrt{\frac{k_2}{l_2}} \). Substitute \( k_1 = 25 \) and \( k_2 = 100 \) to find the labor usage \( l_1 \) and \( l_2 \).

Solving for Labor and Output Allocation

From the derived equation, we find \( \frac{l_2}{l_1} = \frac{k_2}{k_1} = 4 \). Let \( q_1 = \sqrt{k_1 l_1} \) and \( q_2 = \sqrt{k_2 l_2} \). Since they produce the same product, the total output \( Q = q_1 + q_2 \); solve these equations to determine \( q_1 \) and \( q_2 \) for any given \( Q \).

Calculating Short-run Cost Functions

With \( l_2 = 4l_1 \), the total cost for production in the short run becomes \( TC = l_1 + 25 + 4l_1 + 100 = 5l_1 + 125 \). Express the costs in terms of output \( Q \), and derive the total, average, and marginal cost functions. Short-run average cost (SAC) is \( \frac{TC}{Q} \), and short-run marginal cost (SMC) is the derivative \( \frac{d(TC)}{dQ} \).

Calculating Marginal Costs for Specific Widgets

Use the derived SMC to find the marginal costs of the 100th, 125th, and 200th widgets by substituting these values into the SMC expression.

Long-run Production Allocation

In the long run, both firms can adjust their capital \( k_i \) and labor \( l_i \) fully. Allocations where \( \sqrt{k_1 l_1} = \sqrt{k_2 l_2} \) are equalized in the long run, leading to economies of scale. Using \( w = v \), derive the optimal combined cost functions.

Economies of Scale Effect

If diminishing returns to scale are present, the marginal product diminishes as more is produced, affecting allocation. Examine how such adjustments would affect the production and cost equations defined previously.

Key concepts

Production Function

A production function is a mathematical formula that describes how input resources, like capital and labor, convert into output goods or services. For the two firms producing widgets, the production function used is given by \( q = \sqrt{k_i l_i} \). Here, \( q \) is the quantity of output produced by each firm, \( k_i \) denotes the amount of capital, and \( l_i \) represents the labor utilized by firm \( i \). This equation tells us how the product output rises when more resources are thrown into the mix.

In simpler terms, this equation shows the relationship between inputs (capital and labor) and output (widgets). Since the firms have different amounts of capital (\( k_1 = 25 \) for Firm 1 and \( k_2 = 100 \) for Firm 2), they effectively have different capabilities to produce output even if labor is expanded similarly. This production function illustrates the concept of how resources transform into products, which is the cornerstone of understanding production efficiency.

Marginal Cost

Marginal cost refers to the change in total production cost that comes from producing one additional unit of output. In the context of our widget-producing firms, understanding marginal costs helps in determining how to allocate resources to minimize costs as production scales up. To find it, you calculate the additional cost incurred when production is increased by one extra widget.

The exercise involves equalizing the marginal costs for both firms in the short run. This means, for cost-effective production, each firm's additional cost per widget produced should be the same. It ensures that both firms are producing efficiently without over-relying on one or the other. Essentially, the marginal cost plays a crucial role in strategic planning and decision-making since it directly impacts pricing and overall profitability.

When calculating specific costs such as those for the 100th, 125th, or 200th widget, you use the derived short-run marginal cost function. This process helps firms anticipate future costs associated with increasing production output.

Economies of Scale

Economies of scale describe the cost advantages that firms experience when they increase the scale of production. As production expands, the average costs per unit may decrease, especially when a company can employ efficiencies gained from scaling up. In widget production, achieving economies of scale means strategically allocating resources between the two firms to maximize these cost advantages.

In the long run, where firms can adjust all inputs like capital and labor, the company aims to spread its fixed costs over a higher output level, thereby reducing the average costs. By analyzing cost functions, entrepreneurs can identify the most efficient levels of production that allow firms to benefit from economies of scale. This concept is vital since operating on the right scale can give competitive advantages in pricing while optimizing for long-term growth.

Diminishing Returns to Scale

Diminishing returns to scale occurs when increasing all input factors does not lead to proportional increases in output. This often means that, after a certain point, each additional input contributes less and less to overall production. This concept is especially important if the firms have used up their easy efficiency gains and cannot maintain the same output growth through increased input alone.

In the exercise provided, if both firms experience diminishing returns to scale, they no longer maintain as much of a productivity boost from additional inputs. When planning for the long term, entrepreneurs must consider how adding more resources impacts output. This recognition helps avoid overinvesting in more inputs without achieving meaningful gains in production.

Consideration of diminishing returns is crucial for long-term decision-making as it addresses when and how much to invest in additional resources. Balancing the initial benefits from scaling up with the reality of diminishing returns allows firms to maintain cost-effective and productive operations.