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Chapter 16: Problem 5

Carl the clothier owns a large garment factory on an isolated island. Carl's factory is the only source of employment for most of the islanders, and thus Carl acts as a monopsonist. The supply curve for garment workers is given by $$l=80 w$$ where \(l\) is the number of workers hired and \(w\) is their hourly wage. Assume also that Carl's labor demand (marginal revenue product ) curve is given by $$l=400-40 M R P_{l}$$ a. How many workers will Carl hire to maximize his profits, and what wage will he pay? b. Assume now that the government implements a minimum wage law covering all garment workers. How many workers will Carl now hire, and how much unemployment will there be if the minimum wage is set at \(\$ 4\) per hour? c. Graph your results. d. How does a minimum wage imposed under monopsony differ in results as compared with a minimum wage imposed under perfect competition? (Assume the minimum wage is above the market-determined wage.)

Short Answer

Expert verified
Answer: In Carl's monopsony labor market, the imposition of a minimum wage law can lead to an increase in both wages and employment as long as it is between the monopsony wage and the competitive wage. In this case, the minimum wage increased from $2 to $4, while employment increased from 160 to 240 workers. However, there is an excess supply of labor, resulting in 80 unemployed workers. In contrast, in a perfectly competitive labor market, imposing a minimum wage above the market-determined wage would lead to an excess supply of labor (unemployment) and a decrease in employment.

Step by step solution

01

Set up labor supply and demand equations

The labor supply and demand curves are given as follows: Labor supply curve: $$l = 80w$$ Labor demand curve: $$l = 400 - 40MRP_l$$
02

Calculate the marginal cost of labor (MCL)

Now, let's find the expression for the marginal cost of labor (MCL). The MCL is equal to the derivative of the total cost of labor with respect to the number of workers. Here, the total cost of labor is given by the product of the number of workers (l) and the wage rate (w): Total cost of labor: $$C = lw$$ Plugging the labor supply curve into the total cost of labor expression, we get: $$C = 80w^2$$ Taking the derivative of C with respect to l, we get the MCL: $$\frac{dC}{dl} = MCL = \frac{d(80w^2)}{dl} = 160w$$
03

Find the optimal number of workers and hourly wage

To find the number of workers Carl hires to maximize his profits, we need to equate the MCL to the MRP of labor: $$MCL = MRP_l$$ From Step 2, we have: $$160w = MRP_l$$ Using labor demand curve, we can rewrite MRP as: $$MRP_l = 10(400-l)$$ Equating MCL and MRP, we get: $$160w = 10(400-l)$$ Substitute the labor supply curve, $$l=80w$$, into the equation: $$160w = 10(400-80w)$$ Solving for w: $$w = 2$$ Now, solving for l using the labor supply curve: $$l = 80(2) = 160$$ Thus, Carl will hire 160 workers and pay them an hourly wage of $2 to maximize his profits.
04

Calculate the new equilibrium and unemployment after the minimum wage law

Now, let's find the new equilibrium and unemployment when the government implements a minimum wage of $4 per hour. We can use the labor supply curve to find the number of workers who are willing to work at the minimum wage: $$l_{supply} = 80(4) = 320$$ Now, let's find the number of workers Carl is willing to hire at the minimum wage. We use the MRP of labor from Step 3, and substitute w = $4: $$160(4) = 10(400 - l_{demand})$$ Solving for l: $$l_{demand} = 240$$ At the minimum wage of $4, Carl is willing to hire 240 workers. However, 320 workers are willing to work at this wage, resulting in an unemployment of 80 workers (320 - 240).
05

Graph the results

Create a graph with the labor (l) on the horizontal axis and the wage (w) on the vertical axis. 1. Plot the labor supply curve, $$l = 80w$$ 2. Plot the labor demand curve, $$l = 400 - 40MRP_l$$, using the equation found in step 3. 3. Mark the initial equilibrium point (160 workers, $2) as point A. 4. Draw a horizontal line at $4 to represent the minimum wage. 5. Mark the new equilibrium point (240 workers, $4) as point B and the additional 80 unemployed workers as unemployment.
06

Compare the effects of a minimum wage under monopsony and perfect competition

Under monopsony, when a minimum wage is imposed, it can lead to an increase in both wages and employment, as long as the imposed wage is between the monopsony wage and the wage that would prevail under perfect competition. In this case, the minimum wage increased both wages (from $2 to $4) and employment (from 160 to 240 workers). However, in a perfectly competitive labor market where wages are determined by the intersection of labor supply and demand, the imposition of a minimum wage above the market-determined wage leads to an excess supply of labor (unemployment) and a decrease in employment.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Labor Supply and Demand Curves
To start with, let's delve into the concept of labor supply and demand curves. These graphical representations form the backbone of labor market analysis.

In a labor market, the supply curve reflects the relationship between wage rates and the number of workers willing to work at those rates. Typically, as wages increase, more individuals are inclined to offer their labor. In the case of Carl's island factory, the labor supply curve is directly proportional to the wage rate, as shown by the equation given in the original problem, which implies that the number of workers willing to work increases uniformly with wage increases.

The demand curve for labor, on the other hand, illustrates how the number of workers that firms are ready to employ varies with the wage rate. Under normal circumstances, employers want to hire more workers at lower wages. For Carl's factory, their willingness to hire decreases as the Marginal Revenue Product (MRP) of labor increases.
Marginal Revenue Product (MRP)
The Marginal Revenue Product (MRP) of labor is a crucial concept that is often misunderstood by students. It measures the additional revenue generated by employing one more worker. MRP is critical for employers because it helps them decide how many workers to hire.

In our example, Carl's demand curve for labor is a decreasing function of MRP, meaning that as he hires more workers, the additional revenue each one brings in goes down. This is a common situation since, after a certain point, adding more workers can lead to a lower increase in output due to factors like limited equipment or space. MRP bridges the gap between the input of labor and output revenue, guiding employers like Carl in their hiring decisions.
Minimum Wage Law
Minimum wage laws are implemented by governments to set the lowest legal wage that workers can be paid. The goal is to ensure a minimum standard of living for workers.

In Carl's scenario, the minimum wage law altered the dynamics of the labor market on the island. With the mandate of a \( \$4 \) per hour minimum wage, the labor supply exceeded the demand. This discrepancy between the number of workers willing to work at this rate and the number of workers Carl wants to hire at this increased expense can lead to unemployment. This is because while the law ensures that workers get paid more, it also means that employers like Carl may not find it profitable to hire as many workers as are available at this new, higher wage.
Unemployment in Labor Market
Unemployment in the labor market can occur for various reasons, such as technological changes, market downturns, and indeed, government regulation like minimum wage laws. In our example, a minimum wage set above the equilibrium wage that Carl would normally pay leads to 'surplus' labor. This surplus is mathematically represented by the excess of the labor supply over the labor demand at the new wage rate.

The exercise already showed us how to calculate the unemployment in Carl's factory with the minimum wage: a difference of 80 workers arises between those willing to work (320) and those that Carl is willing to employ (240) at the minimum wage. This simplified model is one of the ways economists gauge the potential impact of policy decisions on employment.
Perfect Competition vs Monopsony
Understanding the distinction between perfect competition and a monopsony in labor markets is crucial for students studying economics. In a perfectly competitive labor market, there are many employers, and none of them is large enough to influence wage rates; hence, the market wage rate is determined by the intersection of the labor supply and demand curves.

A monopsony like Carl's, where a single buyer (employer) substantially controls the market, creates a different dynamic. As the sole employer, Carl has the power to set wages below the competitive level, leading to less employment and lower wages than in a competitive market. When a minimum wage is introduced in a monopsony setting, it can actually increase employment if it is set between the monopsony wage and the competitive wage, because it forces the employer to pay a higher wage, which aligns closer to the competitive level and encourages more workers to join the labor force. This is in stark contrast with a perfectly competitive market, where minimum wages above the market wage can lead to unemployment by creating a labor surplus.

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Most popular questions from this chapter

A welfare program for low-income people offers a family a basic grant of \(\$ 6,000\) per year. This grant is reduced by \(\$ 0.75\) for each \(\$ 1\) of other income the family has. a. How much in welfare benefits does the family receive if it has no other income? If the head of the family earns \(\$ 2,000\) per year? How about \(\$ 4,000\) per year? b. At what level of earnings does the welfare grant become 0 ? c. Assume the head of this family can earn \(\$ 4\) per hour and that the family has no other income. What is the annual budget constraint for this family if it does not participate in the welfare program? That is, how are consumption \((c)\) and hours of leisure ( \(h\) ) related? d. What is the budget constraint if the family opts to participate in the welfare program? (Remember, the welfare grant can only be positive. e. Graph your results from parts (c) and (d). f. Suppose the government changes the rules of the welfare program to permit families to keep 50 percent of what they earn. How would this change your answers to parts (d) and (e)? g. Using your results from part (f), can you predict whether the head of this family will work more or less under the new rules described in part (f)?

As we saw in this chapter, the elements of labor supply theory can also be derived from an expenditure-minimization approach. Suppose a person's utility function for consumption and leisure takes the Cobb-Douglas form \(U(c, h)=c^{\alpha} h^{1-\alpha}\) Then the expenditure-minimization problem is $$\operatorname{minimize} c-w(24-h) \text { s.t. } U(c, h)=c^{\alpha} h^{1-\alpha}=\bar{U}$$ a. Use this approach to derive the expenditure function for this problem. b. Use the envelope theorem to derive the compensated demand functions for consumption and leisure. c. Derive the compensated labor supply function. Show that \(\partial l^{c} / \partial w>0\) d. Compare the compensated labor supply function from part (c) to the uncompensated labor supply function in Example \(16.2(\text { with } n=0) .\) Use the Slutsky equation to show why income and substitution effects of a change in the real wage are precisely offsetting in the uncompensated Cobb- Douglas labor supply function.

Universal Fur is located in Clyde, Baffin Island, and sells high-quality fur bow ties throughout the world at a price of \(\$ 5\) each. The production function for fur bow ties \((q)\) is given by \(q=240 x-2 x^{2}\) where \(x\) is the quantity of pelts used each week. Pelts are supplied only by Dan's Trading Post, which obtains them by hiring Eskimo trappers at a rate of \(\$ 10\) per day. Dan's weekly production function for pelts is given by $$x=\sqrt{l}$$ where \(l\) represents the number of days of Eskimo time used each week. a. For a quasi-competitive case in which both Universal Fur and Dan's Trading Post act as price-takers for pelts, what will be the equilibrium price \(\left(p_{x}\right)\) and how many pelts will be traded? b. Suppose Dan acts as a monopolist, while Universal Fur continues to be a price-taker. What equilibrium will emerge in the pelt market? c. Suppose Universal Fur acts as a monopsonist but Dan acts as a price-taker. What will the equilibrium be? d. Graph your results, and discuss the type of equilibrium that is likely to emerge in the bilateral monopoly bargaining between Universal Fur and Dan.

A family with two adult members seeks to maximize a utility function of the form $$U\left(c, h_{1}, h_{2}\right)$$ where \(c\) is family consumption and \(h_{1}\) and \(h_{2}\) are hours of leisure of each family member. Choices are constrained by $$c=w_{1}\left(24-h_{1}\right)+w_{2}\left(24-h_{2}\right)+n$$ where \(w_{1}\) and \(w_{2}\) are the wages of each family member and \(n\) is nonlabor income. a. Without attempting a mathematical presentation, use the notions of substitution and income effects to discuss the likely signs of the cross- substitution effects \(\partial h_{1} / \partial w_{2}\) and \(\partial h_{2} / \partial w_{1}\) b. Suppose that one family member (say, individual 1 ) can work in the home, thereby converting leisure hours into consumption according to the function $$c_{1}=f\left(h_{1}\right)$$ where \(f^{\prime}>0\) and \(f^{\prime \prime}<0 .\) How might this additional option affect the optimal division of work among family members?

The theory developed in this chapter treats labor supply as the mirror image of the demand for leisure. Hence, the entire body of demand theory developed in Part 2 of the text becomes relevant to the study of labor supply as well. Here are three examples. a. Roy's identity. In the Extensions to Chapter 5 we showed how demand functions can be derived from indirect utility functions by using Roy's identity. Use a similar approach to show that the labor supply function associated with the utility-maximization problem described in Equation 16.20 can be derived from the indirect utility function by $$l(w, n)=\frac{\partial V(w, n) / \partial w}{\partial V(w, n) / \partial n}$$ Illustrate this result for the Cobb-Douglas case described in Example 16.1 b. Substitutes and complements. A change in the real wage will affect not only labor supply, but also the demand for specific items in the preferred consumption bundle. Develop a Slutsky-type equation for the cross-price effect of a change in \(w\) on a particular consumption item and then use it to discuss whether leisure and the item are (net or gross) substitutes or complements. Provide an example of each type of relationship. c. Labor supply and marginal expense. Use a derivation similar to that used to calculate marginal revenue for a given demand curve to show that \(M E_{l}=w\left(1+1 / e_{l, w}\right)\)
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