Question

If a firm is producing where \(M P_{1} / w_{1}>M P_{2} / w_{2},\) what can it do to reduce costs but maintain the same output?

Short answer

To reduce costs while maintaining output, increase usage of input 1 and decrease usage of input 2.

Step-by-step solution

Understanding the Problem

We are given that the marginal product of input 1 divided by its wage, \(\frac{MP_1}{w_1}\), is greater than the marginal product of input 2 divided by its wage, \(\frac{MP_2}{w_2}\). This implies that input 1 is currently more cost-effective than input 2 for producing additional output.

Analyzing the Ratio

The ratio \(\frac{MP}{w}\) shows the additional output per unit cost of an input. Higher values mean the firm is getting more output per dollar spent for that input compared to other inputs. Thus, the firm is currently getting more output per dollar for input 1 than for input 2.

Action Plan: Adjust Input Usage

To maintain the same output while reducing costs, the firm should use more of input 1 and less of input 2. This adjustment will align the marginal rate of technical substitution (MRTS) with the ratio of the input prices \(\left(\frac{w_1}{w_2}\right)\).

Why the Adjustment Works

By increasing input 1 and decreasing input 2, the firm moves closer to an optimal production point where \(\frac{MP_1}{w_1}\) equals \(\frac{MP_2}{w_2}\). This ensures that the firm achieves the same level of output at a lower cost, as it uses inputs more efficiently.

Key concepts

Marginal Product of Inputs

The concept of the marginal product of inputs is essential in understanding how efficiently a firm uses its resources. The marginal product of an input, denoted as \(MP\), refers to the additional output produced by one extra unit of that input, assuming all other inputs remain constant. In simpler terms, it's the extra goods we get by adding one more unit of a resource, like labor or materials.

This concept helps firms determine which input gives them more bang for their buck. For instance, if the marginal product of labor (\(MP_L\)) is high, adding more workers results in significantly increased output. However, it's crucial to weigh this against the cost of the input.

By examining the ratio of \(\frac{MP}{w}\) for each input, where \(w\) represents the cost (like wages), firms can pinpoint which inputs are more cost-effective. A higher ratio indicates better efficiency since it shows more output per dollar spent on that input. This assessment is a vital stepping stone in ensuring competitive production costs.

Technical Substitution

Technical substitution involves swapping out one input for another while maintaining the same level of output. It's like mixing ingredients to achieve the same flavor in cooking using different components.

In the context of production, the key is understanding the concept of Marginal Rate of Technical Substitution (MRTS). MRTS tells us the rate at which a firm can substitute one input for another without affecting production levels. Mathematically, it's expressed as \(\frac{MP_1}{MP_2}\). When a firm realizes that one resource is cheaper or more productive, they might use more of it.

The firm will adjust input levels to align MRTS with the ratio of input costs, \(\frac{w_1}{w_2}\). This balance ensures that the firm is not wasting resources on less efficient inputs. By achieving this balance, they enhance productivity and reduce costs, which are central goals in competitive markets.

Optimal Production Point

The optimal production point is where a firm achieves maximum efficiency in its use of resources for production. This is the sweet spot where costs are minimized without sacrificing output quality or quantity.

To reach this point, firms must ensure that the cost per unit of marginal product for each input is equal. In mathematical terms, this occurs when \(\frac{MP_1}{w_1} = \frac{MP_2}{w_2}\).

Realizing the optimal production point might require shifting the quantities of inputs used. For a firm with \(\frac{MP_1}{w_1} > \frac{MP_2}{w_2}\), it should increase the use of input 1 and decrease input 2. This adjustment brings the ratios inline and minimizes expenses while maintaining output levels.

Ultimately, achieving the optimal production point not only reduces costs but also boosts a firm's ability to remain competitive in the market.