Question

Suppose a monopsonist faces the production function \(Q=20 L-0.5 L^{2}\) and a labour supply \(L^{S}=w-5\). This means that the wage that the monopsony must pay is \(w=L+5\). Assume that the price of the product is \(£ 1\). Find the labour demand that maximizes the profits of the monopsony. What about the wage?

Short answer

The optimal labor demand is 5 workers, and the wage is £10.

Step-by-step solution

Understanding the Monopsonist's Costs

A monopsonist is the sole buyer of labor, which means it determines the wage based on the number of workers it hires. Here the wage function given is \(w = L + 5\). Since the supply function of labor is \(L^{S} = w - 5\), we can rewrite it as \(w = L + 5\).

Determine the Monopsonist's Total Revenue

The production function is given by \(Q = 20L - 0.5L^2\). The revenue is determined by the price of each unit. Since the price is \(£ 1\), the total revenue \(TR\) is \(TR = Q\). So, \(TR = (20L - 0.5L^2)\).

Calculate the Monopsonist's Total Cost

The total cost based on wages is \(TC = wL = (L+5)L = L^2 + 5L\). This represents how much the monopsonist spends on labor.

Maximizing Profits by Setting Marginal Revenue Equal to Marginal Cost

To maximize profit, we set the derivative of total revenue (\(MR\)) equal to the derivative of total cost (\(MC\)). First, differentiate total revenue: \(MR = \frac{d(20L - 0.5L^2)}{dL} = 20 - L\). Next, differentiate total cost and find \(MC\): \(MC = \frac{d(L^2 + 5L)}{dL} = 2L + 5\). Set \(MR = MC\), so \(20 - L = 2L + 5\).

Solve for Optimal Labor Demand

Solve the equation from Step 4: \(20 - L = 2L + 5\). Rearrange terms to form \(15 = 3L\), which simplifies to \(L = 5\).

Determine the Wage

Using \(w = L + 5\), and the \(L\) found in Step 5, substitute \(L = 5\). Therefore, \(w = 5 + 5 = 10\).

Key concepts

Labour Supply

In a monopsonistic market, the company is the only one hiring labor, allowing it to set wages lower than they might be in a competitive market. The exercise presents a simple labor supply function, \( L^{S} = w - 5 \). This function describes how the amount of labor workers are willing to supply changes with wages. In this function:

• \( L \) represents the labor supply, or the number of workers.
• \( w \) is the wage rate.
• The constant \(-5\) shows that at a wage of zero, the supply of labor would be negative (which isn't practical in real life, but helps in showing the function's intercept).

Break down the equation to determine how much labor will be supplied for any given wage. The equation is rearranged to show the wage function: \( w = L + 5 \). This formula tells us that as more workers are hired, the wage must increase by the amount of labor plus 5, to ensure willing labor participation.

Production Function

The production function provided, \( Q=20 L-0.5 L^{2} \), illustrates how much output (\( Q \)) is produced given a certain amount of labor (\( L \)). This function highlights the concepts of diminishing returns, which are typical as more labor is added:

• The term \( 20L \) signifies that for each unit of labor, the firm initially produces 20 units of output.
• The \(-0.5L^2\) term indicates that as more labor is employed, each additional worker contributes less to overall output than the previous one.

This phenomenon is known as diminishing marginal returns to labor. Initially, workers are highly productive; however, after a certain point, adding more workers provides less additional output because the incremental workers have less capital or resources to work with efficiently.

Profit Maximization

Profit maximization for a monopsonist involves setting marginal revenue equal to marginal cost. To find maximum profit, we begin with calculating:

• Total Revenue (TR): Derived from the production function multiplied by the price per unit, which is \( £1 \). Here, \( TR = Q = 20L - 0.5L^2 \).
• Marginal Revenue (MR): The change in total revenue resulting from selling one more unit of output. Differentiating \( TR \) with respect to \( L \), we get \( MR = 20 - L \).
• Total Cost (TC): Labor cost calculated as \( TC = wL = (L+5)L = L^2 + 5L \).
• Marginal Cost (MC): The change in total cost when hiring one more unit of labor. Differentiating \( TC \) gives \( MC = 2L + 5 \).

We achieve maximum profit by setting \( MR = MC \), resulting in \( 20 - L = 2L + 5 \), solving this gives optimal labor demand.

Marginal Cost

In economics, marginal cost refers to the additional cost incurred from producing one more unit of output. In the context of a monopsonist, marginal cost is particularly related to hiring extra labor.For this exercise, the marginal cost (\( MC \)) of hiring an additional unit of labor is obtained by differentiating the total cost function: \( TC = L^2 + 5L \). Hence, \( MC = \frac{d(L^2 + 5L)}{dL} = 2L + 5 \).Having a higher MC than in a perfectly competitive labor market is typical because the monopsonist must raise wages to attract more workers, affecting all current workers as well. Therefore, \( MC \) is pivotal in comparison to Marginal Revenue, for the firm to determine how many workers to employ and ensures the business is maximizing its profits efficiently.