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 $\mathrm{\$}4$ 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 $\mathrm{\$}4$ 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 $4perhour,andwhatisthenewemploymentlevelandthetotalsubsidy?Therequiredsubsidyis\$2.50perhour.Thenewemploymentlevelis150people,andthetotalsubsidypaymentis\$375.c)Whatisthelabordemandandunemploymentlevelsifa\$4perhourminimumwageisset?Thelabordemandata\$4perhourminimumwageis250people.Theunemploymentlevelincreasesto50people.d)Howdothegraphschangeineachsituation?Plottheoriginaldemandandsupplycurvesonagraph,markingtheinitialequilibriumpoint(Eq)at\text{(}(w_1,l_1)=(3,300)$. 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 $4perhourbyahorizontallineatw=4andfindwhereitintersectsthedemandcurve,markingthispoint(D^{\prime })at$(4, 250)$.

Key concepts

Supply and Demand

Supply and demand are two fundamental concepts in economics. They dictate the price and quantity of goods in the market.
In this scenario, labor is the good in focus. The supply function, $l=100w$, shows the relationship between the labor force's willingness to work at different wage rates.
The demand function, $l=-50w+450$, shows how many workers employers are willing to hire at each wage level.
To find the market equilibrium, where supply equals demand, we set these equations equal.

• From the above, equilibrium occurs when both suppliers (workers) and demanders (employers) agree on a wage rate and employment level.
• At this point, there is no excess supply (unemployment) or demand (vacancies).

Finding this balance is key to understanding labor market dynamics.

Government Subsidy

A government subsidy in the labor market is financial support provided to employers. It encourages them to hire more workers than they usually would at given wage levels.
This exercise showcases how a subsidy affects the supply curve.

• When a subsidy is introduced, the supply equation adjusts to account for the new financial support.
• In this example, the government aims for a wage of \(4 by pushing the supply curve rightward.
• To achieve an effective subsidy, solving $100s=250$ reveals that \)2.50 per hour is needed.

Such policy adjustments aim to achieve higher employment, modifying market conditions creatively.

Minimum Wage

A minimum wage is a legally mandated lowest hourly wage that workers must be paid. It is designed to ensure a living wage.
When a minimum wage like $4 per hour, exceeds the equilibrium wage, various impacts emerge.

• Here, the imposed minimum wage of $4 shifts the labor market conditions.
• Employers will only demand 250 workers at this wage rate, based on the demand curve.
• Such dynamics often result in discrepant labor demand and supply, leading to unemployment.

Thus, minimum wages aim to protect workers but must align with market realities to avoid adverse effects.

Unemployment

In labor economics, unemployment refers to the situation where labor supply exceeds labor demand.
This disparity often occurs when wages exceed equilibrium levels, like with a minimum wage enforcement.

• As wages rise to $4, supply remains at 300 but demand dips to 250.
• This gap results in an unemployment figure of $300-250=50$ extra workers wishing to work.
• Unemployment points to inefficiencies where labor isn't fully utilized.

Addressing unemployment often requires nuanced policy interventions balancing wages and job availability effectively.

Equilibrium Wage Rate

The equilibrium wage rate is where labor supply equals labor demand in the market, ensuring full employment without excess supply or demand.
In our scenario, this rate was initially found at $3 per hour.

• It signifies a natural market balance based on current economic conditions.
• An equilibrium wage supports maximum efficiency in workforce allocation.
• When external factors change (like subsidies or minimum wages), reaching this balance again can be complex.

Understanding equilibrium wages helps grasp how labor markets respond to various forces, fostering informed economic planning and decision-making.