Question

(This problem is somewhat advanced.) Using symbols, we can write that the marginal product of labor is equal to \(\Delta Q / \Delta L .\) Marginal cost is equal to \(\Delta \mathrm{TC} / \Delta Q .\) Because fixed costs by definition don't change, marginal cost is also equal to \(\Delta \mathrm{VC} / \Delta \mathrm{Q} .\) If jill Johnson's only variable cost (VC) is labor cost, then her variable cost equals the wage multiplied by the quantity of workers hired, or \(w \mathrm{~L}\) a. If the wage Jill pays is constant, then what is \(\Delta V C\) in terms of \(w\) and \(L ?\) b. Use your answer to part (a) and the expressions given for the marginal product of labor and the marginal cost of output to find an expression for marginal cost, \(\Delta \mathrm{TC} / \Delta \mathrm{Q},\) in terms of the wage, \(w,\) and the marginal product of labor, \(\Delta Q / \Delta L\) c. Use your answer to part (b) to determine Jill's marginal cost of producing pizzas if the wage is \(\$ 750\) per week and the marginal product of labor is 150 pizzas. If the wage falls to \(\$ 600\) per week and the marginal product of labor is unchanged, what happens to Jill's marginal cost? If the wage is unchanged at \(\$ 750\) per week and the marginal product of labor rises to 250 pizzas, what happens to Jill's marginal cost?

Short answer

a) \(\Delta VC = w \Delta L\) b) Marginal cost = \(w / (\Delta Q / \Delta L)\) c) If wage is $750 per week and the marginal product of labor is 150, then the marginal cost is $5 per pizza. If wage falls to $600 with unchanged marginal product of labor, the marginal costdecreases to $4 per pizza. If wage remains $750 and the marginal product of labor rises to 250 pizzas, the marginal cost falls to $3 per pizza.

Step-by-step solution

Identify Variable Cost Change

We know that the variable cost (VC) for Jill Johnson is the wage multiplied by the number of workers hired, or \(wL\). According to the problem, the wage is constant, so \( \Delta VC = w \Delta L\) .

Find Expression for Marginal Cost

Using the formula provided, Marginal Cost is \(\Delta TC / \Delta Q \) and also \(\Delta VC / \Delta Q\). Substituting \(\Delta VC\) from step1 formula, we get Marginal cost as \(w \Delta L / \Delta Q\). However, provided \(\Delta Q / \Delta L \) is the Marginal product, so reversing this we get \(w / (\Delta Q / \Delta L)\).

Find Jill's Marginal Cost

Given that the wage \(w\) is $750 per week and the marginal product of labor is 150 pizzas, substitute these values into the formula we obtained in Step 2 to derive: Marginal cost = \(750 / 150 = $5\) per pizza.

Evaluate the Changes

If the wage falls to $600 per week and the marginal product of labor remains unchanged, replacing these values into the formula, we get: Marginal cost = \(600 / 150 = $4\) per pizza. Hence, the marginal cost decreases. If the wage remains at $750 per week and the marginal product of labor rises to 250 pizzas, Marginal cost = \(750 / 250 = $3\) per pizza. So, an increase in the marginal product of labor leads to a decrease in the marginal cost.

Key concepts

Marginal Product of Labor

The marginal product of labor is a fundamental concept in economics. It represents the additional output produced when one more worker is added, while holding other factors constant. This is calculated using the formula \(\Delta Q / \Delta L\). Here, \(\Delta Q\) is the change in quantity of output, and \(\Delta L\) is the change in labor input. The significance of this measure is to gauge efficiency. A high marginal product indicates that labor is being used effectively. Conversely, if the marginal product is low, it may signal inefficiencies or overstaffing. Understanding how labor contributes to production helps businesses optimize operations and decision-making, ensuring resources are allocated efficiently.

Variable Cost Calculations

Variable costs are those expenses that change with the level of production. In Jill's context, the variable cost is primarily the wage paid to workers. As the exercise shows, the change in variable cost, \(\Delta VC\), can be expressed as \(w \Delta L\), where \(w\) is the constant wage rate and \(\Delta L\) is the change in the number of workers. Calculating variable costs is important because it ties directly to production volume. As production increases, so do variable costs, because more labor might be needed. This connection makes managing and predicting costs crucial for maintaining profitability and setting competitive prices.Effective variable cost management can also help businesses adjust to changes in market conditions, such as wage fluctuations.

Cost-Production Relationships

The relationship between costs and production is central to understanding marginal cost. Marginal cost is the additional cost incurred by producing one more unit of output. It is given by \(\Delta \mathrm{TC} / \Delta Q\), which simplifies to \(w / (\Delta Q / \Delta L)\) when dealing with changes solely in labor costs.This formula illustrates how wages and labor productivity directly impact production costs. For instance, in Jill's case, if the wage decreases while productivity remains constant, the marginal cost of producing each pizza decreases. Conversely, if productivity increases while wages remain fixed, marginal costs also decrease.Understanding these dynamics helps businesses make informed decisions about resource allocation, production levels, and pricing strategies.