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Chapter 3: Problem 6

Low-skilled workers operate in a competitive market. The labor supply is \(Q^{S}=10 W\) (where \(W\) is the price of labor measured by the hourly wage) and the demand for labor is \(Q^{D}=\) \(240-20 W . Q\) measures the quantity of labor hired (in thousands of hours). a. What is the equilibrium wage and quantity of low-skilled labor working in equilibrium? b. If the government passes a minimum wage of $$\$ 9$$ per hour, what will the new quantity of labor hired be? Will there be an excess demand or excess supply of labor? How large? c. What is the deadweight loss of a $$\$9$$ minimum wage? d. How much better off does the $$\$9$$minimum wage make low-skilled workers (in other words, how much does producer surplus change), and how much worse off are employers? e. How do your answers to (c) and (d) change if the minimum wage is set at 11 rather than at $$\$ 9$$?

Short Answer

Expert verified
Equilibrium wage is $8 with 80,000 hours. A $9 wage reduces hours to 60,000, with 30,000 excess supply and a 10,000 hour deadweight loss. A $11 wage results in 90,000 excess supply, a 90,000 hour deadweight loss, and 360,000 in total losses.

Step by step solution

01

Set Supply Equal to Demand

To find the equilibrium wage and quantity, set the supply equation equal to the demand equation: \( Q^S = Q^D \), which means \( 10W = 240 - 20W \).
02

Solve for Equilibrium Wage

Rearrange and solve the equation for \(W\): \[ 10W + 20W = 240 \] \[ 30W = 240 \] \[ W = \frac{240}{30} = 8 \].The equilibrium wage is $8 per hour.
03

Solve for Equilibrium Quantity

Substitute \(W = 8\) into either the supply or demand equation to find \(Q\): Using the supply equation, \( Q^S = 10W = 80 \). The equilibrium quantity of labor is 80,000 hours.
04

Calculate New Quantity at $9 Wage

Substitute \(W = 9\) into the supply and demand equations to find the new quantities: \( Q^S = 10 \times 9 = 90 \) (90,000 hours).\( Q^D = 240 - 20 \times 9 = 60 \) (60,000 hours). The new quantity of labor hired will be 60,000 hours.
05

Determine Excess Supply

Excess supply is calculated by \( Q^S - Q^D \): \( 90 - 60 = 30 \). There will be an excess supply of 30,000 hours.
06

Calculate Deadweight Loss for $9 Wage

The deadweight loss (DWL) is the area of the triangle formed between the quantities with the equilibrium and minimum wage: Use the formula: \[ DWL = \frac{1}{2} \times (Q^* - Q_{min}) \times (W_{min} - W^*) \] \[ DWL = \frac{1}{2} \times (80 - 60) \times (9 - 8) = 10 \] DWL is 10,000 hours.
07

Calculate Changes in Surplus for $9 Wage

The surplus change for workers (producer surplus) is the rectangle with the base as excess supply and the height as the difference in wage: \[ Producer \text{ } Surplus \text{ } Change = Excess \text{ } Supply \times (W_{min} - W^*) = 30 \times (9 - 8) = 30 \] Employers lose this and the DWL, totaling 40,000.
08

Calculate Changes for $11 Wage

For a wage of $11, first calculate new quantities: \( Q^S = 110 \) and \( Q^D = 20 \) which gives an excess supply of 90,000. DWL is \[ DWL = \frac{1}{2} \times (80 - 20) \times (11 - 8) = 90 \].Surplus change: For workers: \[ 90 \times (11 - 8) = 270 \].Employers lose 90 + 270 = 360.

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Key Concepts

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

Equilibrium Wage
An equilibrium wage is the hourly pay rate at which the quantity of labor supplied equals the quantity demanded. In a competitive market, this balance ensures there is neither a shortage nor a surplus of labor.
To find the equilibrium wage, set the labor supply and demand equations equal: \( Q^S = Q^D \). In our example, the supply equation is \( Q^S = 10W \) and the demand equation is \( Q^D = 240 - 20W \). By solving \( 10W = 240 - 20W \), you get an equilibrium wage \( W \) of $8 per hour.
At this wage, both employers find the exact amount of workers they need, and workers find enough jobs to match their hourly rate desires.
Minimum Wage
A minimum wage is a legally mandated lowest hourly rate that employers can pay workers. It is designed to ensure a basic standard of living for all employees. However, setting a minimum wage can also upset the balance in a labor market.
  • When the minimum wage is set above the equilibrium wage, it can cause a reduction in the quantity of labor demanded by employers while increasing the quantity supplied by workers.
  • For example, when a $9 minimum wage is implemented (greater than the $8 equilibrium wage), fewer hours of labor are demanded (60,000 hours) compared to what is supplied (90,000 hours).
This discrepancy between supply and demand leads to excess supply of labor, which means some workers are unable to find jobs.
Excess Supply
Excess supply occurs when the quantity of labor supplied exceeds the quantity of labor demanded. This typically happens when the minimum wage is set above the equilibrium wage.
In our specific case, a minimum wage of $9 caused the supply of labor to increase to 90,000 hours, while demand dropped to 60,000 hours.
This results in 30,000 hours of excess supply.
  • This situation reflects a surplus of labor, meaning more workers are unable to find jobs at the new minimum wage.
  • Excess supply can lead to unemployment because the number of job seekers exceeds the number of available positions.
Deadweight Loss
Deadweight Loss (DWL) is a measure of lost economic efficiency when the equilibrium condition is not achieved. This can occur with price controls, such as minimum wages, where the market is prevented from reaching its natural balance.
When the government sets a minimum wage above the equilibrium wage, fewer workers are hired than would be in a free market, resulting in DWL.
Calculating DWL involves finding the area of the triangle formed between:
  • The equilibrium quantity of labor (80,000 hours) and the new quantity with the minimum wage (60,000 hours).
  • The difference in wages (\(9 - \)8).
The formula for DWL is \[ DWL = \frac{1}{2} \times (Q^* - Q_{min}) \times (W_{min} - W^*) \]. In our example, DWL amounts to 10,000 hours, representing the lost potential for trade between workers willing to work for the equilibrium wage and the employers willing to hire them.

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Most popular questions from this chapter

Black markets are markets where items are sold in violation of government rules and regulations. a. Suppose that the government imposes a $$ 1\( per loaf price ceiling on bread sales, well below its free-market price. Explain how the existence of a black market for bread could potentially improve society's well-being. b. Suppose that the government establishes a \)\$ 20$ per bushel price floor in the market for corn. Explain how the existence of a black market for corn could potentially improve society's wellbeing.

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The demand for ice cream is given by \(Q^{D}=20-2 P\), measured in gallons of ice cream. The supply of ice cream is given by \(Q^{S}=4 P-10\). a. Graph the supply and demand curves, and find the equilibrium price and quantity of ice cream. b. Suppose that the government legislates a $$ 1\( tax on a gallon of ice cream, to be collected from the buyer. Plot the new demand curve on your graph. Does demand increase or decrease as a result of the tax? c. As a result of the tax, what happens to the price paid by buyers? What happens to the price received by sellers? How many gallons of ice cream are sold? d. Who bears the greater burden of the tax? Can you explain why this is so? e. \)\quad\( Calculate consumer surplus both before and after the tax. f. \)\quad$ Calculate producer surplus both before and after the tax. g. How much tax revenue did the government raise? h. How much deadweight loss does the tax create?

Consider the following fiscal scheme designed to directly transfer welfare from coffee drinkers to coffee vendors: The government will impose a $$ 1.00\( tax, collected from buyers, for each cup of coffee sold. The government will then subsidize coffee vendors $$ 1.00 for each cup of coffee sold. a. What will happen to the equilibrium price of coffee? b. What will happen to the equilibrium quantity of coffee? c. How will the outcome of this scheme differ from one in which the government collects a \)\$ 1.00$ tax for each cup of coffee sold and divides the total tax collections equally among all coffee vendors? (It is safe to assume that all coffee vendors are identical.)

Consider the demand for broadband Internet service, given as follows: \(Q^{D}=224-4 P,\) where \(Q\) is the number of subscribers in a given area (in hundreds) and \(P\) is the price in dollars per month. This demand relationship is illustrated in the diagrambelow. Assume that the price of broadband service is $$ 25\( per month. Determine the following, paying particular attention to the units in which quantity is denominated: a. The total number of subscribers at that price b. The total amount paid by subscribers for broadband service, area \)B\( c. The consumer surplus received by subscribers, \)\operatorname{area} A\( d. The total value to consumers of the broadband service they received, areas \)A\( and \)B$
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