Question

Suppose demand for labor is given by \\[ l=-50 w+450 \\] and supply is given by \\[ l=100 w \\] where \(l\) represents the number of people employed and \(w\) is the real wage rate per hour. a. What will be the equilibrium levels for \(w\) and \(l\) in this market? b. Suppose the government wishes to increase the equilibrium wage to \(\$ 4\) per hour by offering a subsidy to employers for each person hired. How much will this subsidy have to be? What will the new equilibrium level of employment be? How much total subsidy will be paid? c. Suppose instead that the government declared a minimum wage of \(\$ 4\) per hour. How much labor would be demanded at this price? How much unemployment would there be? d. Graph your results.

Short answer

The equilibrium wage (w) is $3 per hour, and the equilibrium employment level (l) is 300 people. b) How much is the subsidy needed to raise the wage to \(4 per hour, and what is the new employment level and the total subsidy? The required subsidy is $2.50 per hour. The new employment level is 150 people, and the total subsidy payment is $375. c) What is the labor demand and unemployment levels if a $4 per hour minimum wage is set? The labor demand at a $4 per hour minimum wage is 250 people. The unemployment level increases to 50 people. d) How do the graphs change in each situation? Plot the original demand and supply curves on a graph, marking the initial equilibrium point (Eq) at \((w_1, l_1)=(3, 300)\). Next, plot the adjusted supply curve with the subsidy (Shifted Supply) and the new equilibrium point (Eq') at \((w_2, l_2)=(4, 150)\). Lastly, draw a horizontal line at w=4, representing the minimum wage level, and mark the intersection point with the demand curve as (D') at \)(4, 250)$.

Step-by-step solution

Find the equilibrium levels for w and l

To find the equilibrium wage and the number of people employed, set the supply equation equal to the demand equation: \\[ 100w = -50w + 450 \\] Now, solve for \(w\): \\[ 150w = 450 \\ w = \dfrac{450}{150} = 3 \\] Next, find the equilibrium number of people employed \(l\) by substituting the value of \(w\) into either equation. Let's use the supply equation: \\[ l = 100w = 100(3) = 300 \\] The equilibrium levels for \(w\) and \(l\) are $3 and 300, respectively.

Calculate the subsidy to increase the wage to \(4 per hour

In order to raise the equilibrium wage to \)4 per hour, the government needs to offer a subsidy to employers. Denote the subsidy by \(s\). The new supply equation becomes: \\[ l = 100(w - s) \\] Now, we want \(w\) to be equal to \(4\). So, we rewrite the equation as: \\[ l = 100(4 - s) \\] Now, set the demand equation and the adjusted supply equation equal to each other: \\[ -50w + 450 = 100(4 - s) \\] Substitute \(w = 4\): \\[ -200 + 450 = 400 - 100s \\ 100s= 250 \\ s = 2.50 \\] The needed subsidy is $2.50 per hour.

Determine the new equilibrium for employment and total subsidy

Substitute the subsidy value back into the adjusted supply equation: \\[ l = 100(4 - 2.5) = 150 \\] The new equilibrium level of employment is \(l = 150\). To calculate the total subsidy paid, multiply the subsidy rate by the number of people employed: \\[ Total\ Subsidy = s \cdot l = 2.50 \cdot 150 = 375 \\] The total subsidy to be paid by the government is $375.

Determine labor demand and unemployment with a \(4 per hour minimum wage

Calculate the labor demand at a minimum wage of \)4 per hour by using the demand equation: \\[ l = -50(4) + 450 = 250 \\] The labor demand at a $4 per hour wage is 250 people. To determine the level of unemployment, subtract the labor demand from the original equilibrium level of employment: \\[ Unemployment = l - Labor \ Demand = 300 - 250 = 50 \\] The unemployment level with a $4 per hour minimum wage is 50 people.

Graph the results

To graph the results, plot the original demand and supply curves, marking the initial equilibrium point (Eq) at \((w_1, l_1)=(3, 300)\). Plot the adjusted supply curve with the subsidy (Shifted Supply) and the new equilibrium point (Eq') at \((w_2, l_2)=(4, 150)\). Finally, denote the minimum wage level of \(4 per hour by a horizontal line at w=4 and find where it intersects the demand curve, marking this point (D') at \)(4, 250)$.