Question

From time to time, Congress has raised the minimum wage. Some people suggested that a government subsidy could help employers finance the higher wage. This exercise examines the economics of a minimum wage and wage subsidies. Suppose the supply of lowskilled labor is given by \\[ L^{s}=10 w \\] where \(L^{\mathrm{s}}\) is the quantity of low-skilled labor (in millions of persons employed each year), and \(w\) is the wage rate (in dollars per hour). The demand for labor is given by \\[ L^{D}=80-10 w \\] a. What will be the free-market wage rate and employment level? Suppose the government sets a minimum wage of \(\$ 5\) per hour. How many people would then be employed? b. Suppose that instead of a minimum wage, the government pays a subsidy of \(\$ 1\) per hour for each employee. What will the total level of employment be now? What will the equilibrium wage rate be?

Short answer

The free-market wage rate is $4 per hour, and the employment level is 40 million workers. When the minimum wage set by the government is $5 per hour, the number of people employed is 30 million. If a subsidy of $1 per hour is given for each employee, the total employment level would then be 45 million, and the equilibrium wage rate would be $4.5 per hour.

Step-by-step solution

Set Up the Equations

The supply of labor equation is given by \(L^s = 10w\), and the demand for labor is given by \(L^d = 80 - 10w.\) The equilibrium would occur when \(L^s = L^d\), i.e., when the quantity supplied of labor equals the quantity demanded.

Find the Free-Market Wage Rate and Employment Level.

To find the free-market wage rate and employment level, set the supply equation equal to the demand equation and solve for \(w\). This gives: \(10w = 80 - 10w.\) Solving this equation gives the free-market wage rate. To solve for the quantity of low-skilled labor, plug the value of \(w\) into the equation of \(L^s\) or \(L^d\).

Find The Employment Level After Minimum Wage Increase

Suppose the government sets a minimum wage of $5 per hour. To find how many people would then be employed, plug in \(w = 5\) into supply and demand equations and solve for \(L^s\) and \(L^d.\) The amount of labor demanded would be the number of people employed, as the number supplied would exceed the number demanded.

Determine The Effect Of The Subsidy

Suppose that instead of a minimum wage, the government pays a subsidy of $1 per hour for every employee. The new wage rate perceived by the employers would be \(w - 1.\) Plug this into the demand equation to find the new demand. Find the new equilibrium by setting the new labor demand equal to the labor supply and solve for \(w\) and \(L^d\).

Key concepts

Labor Supply and Demand

Understanding labor supply and demand is fundamental in economics, as it describes the relationship between the availability of workers and the need for their services. The labor supply curve shows how many workers are willing to work at different wage rates, while the demand curve shows how many workers employers want to hire at those wages. In our example, with the supply equation \( L^{s}=10 w \) and the demand equation \( L^{D}=80-10 w \), we can visualize two lines on a graph with wage rate on the vertical axis and quantity of labor on the horizontal axis.

When the wage rate rises, the quantity of labor supplied increases since more individuals are willing to work for higher pay. This behavior is represented by the upward slope of the supply curve. Conversely, as the wage rate increases, the quantity of labor demanded decreases because employers may not find it cost-effective to hire at higher wages, leading to the downward slope of the demand curve.

In the absence of any interventions such as a minimum wage or a subsidy, the labor market will reach an equilibrium point where the amount of labor supplied equals the amount demanded. At this point, the labor force is fully employed at the equilibrium wage. Any deviation from this equilibrium due to external factors can result in surpluses or shortages of labor.

Government Wage Subsidy

A government wage subsidy is a financial incentive provided to employers, where the government pays a part of the wages of their employees. This can encourage employers to hire more workers or to pay higher wages without having to bear the full cost. When we look at the exercise where the government proposes a subsidy of \( \$1 \) per hour for each worker, we see that this effectively reduces the wage rate that employers feel they are paying. If the market wage is \( w \), with the subsidy, the perceived wage rate from the employer's standpoint becomes \( w - 1 \).

This intervention shifts the labor demand curve to the right, indicating that at every given actual wage rate, employers are willing to hire more workers than before since they are now paying less for each hour of work. As a result, the new equilibrium is established at a point where more employment occurs at a lower 'effective' wage cost to employers. In economic terms, the wage subsidy partially internalizes the benefit of hiring additional workers to the employer and can lead to an overall increase in employment.

Employment Level Equilibrium

The employment level equilibrium occurs at the point where the quantity of labor supplied is equal to the quantity of labor demanded. At this juncture, theoretically, all workers who want to work at the current wage can find employment, and employers can hire the amount of labor they need. In the context of our exercise, the free-market equilibrium is disturbed when a minimum wage is introduced. By setting a minimum wage of \( \$5 \) per hour, the government imposes a wage rate that is potentially above the equilibrium rate, which can lead to a surplus of labor, or unemployment, as there will be more workers willing to work at that rate than there are jobs available.

However, when considering a government wage subsidy, the equilibrium can shift to accommodate more employment without creating a labor surplus. If the subsidy is sufficient, it may stimulate demand for labor to the point that it matches or exceeds the increased supply caused by a higher minimum wage, thus maintaining or even increasing employment levels. It balances the need for fair wages for workers and the cost-effective utilization of labor by businesses, addressing the overall economic goal of maximizing employment while minimizing involuntary unemployment.