Microeconomics Problem Set: Elasticity and Tax Incidence Guidance

Q1a. Graph the demand and supply curves, and calculate the equilibrium price and quantity which clear the market.

Background

Topic: Market Equilibrium

This question tests your understanding of how to find the equilibrium price and quantity in a competitive market using linear demand and supply equations.

Key Terms and Formulas

• Demand Curve: $Q_D = 110 - 10P$

• Supply Curve: $Q_S = 40 + 4P$

• Equilibrium: The point where $Q_D = Q_S$

Step-by-Step Guidance

1. Set the demand and supply equations equal to each other to find the equilibrium price: $110 - 10P = 40 + 4P$

2. Combine like terms to isolate $P$ on one side of the equation.

3. Solve for $P$ to find the equilibrium price.

4. Substitute the equilibrium price back into either the demand or supply equation to find the equilibrium quantity.

5. To graph, plot both equations on a Price-Quantity axis and mark the intersection point as the equilibrium.

Try solving on your own before revealing the answer!

Final Answer:

Equilibrium price is $P = 5.00$, and equilibrium quantity is $Q = 60$.

At this price, quantity demanded equals quantity supplied, clearing the market.

Q1b. Suppose the government imposes a $1.00 tax on producers for each unit sold. Show this tax on your graph, and calculate the effect on the equilibrium price and quantity.

Background

Topic: Tax Incidence and Market Adjustment

This question examines how a per-unit tax on producers shifts the supply curve and affects equilibrium price and quantity.

Key Terms and Formulas

• Tax on Producers: Shifts the supply curve vertically upward by the amount of the tax.

• New Supply Curve: $Q_S = 40 + 4(P - 1)$ or equivalently $Q_S = 36 + 4P$

• New Equilibrium: Set $Q_D = Q_S$ with the new supply equation.

Step-by-Step Guidance

1. Adjust the supply equation to account for the $1.00$ tax: $Q_S = 40 + 4(P - 1)$.

2. Simplify the new supply equation: $Q_S = 36 + 4P$.

3. Set the new supply equation equal to the demand equation: $110 - 10P = 36 + 4P$.

4. Combine like terms and solve for the new equilibrium price paid by consumers ($P_C$).

5. Substitute this price into the demand equation to find the new equilibrium quantity.

Try solving on your own before revealing the answer!

Final Answer:

New equilibrium price paid by consumers is $P_C = 5.29$, and new equilibrium quantity is $Q = 57.1$.

The supply curve shifts left/upward, resulting in a higher price for consumers and a lower quantity sold.

Q1c. What is the total tax collected?

Background

Topic: Tax Revenue

This question tests your ability to calculate the total tax revenue collected by the government after a per-unit tax is imposed.

Key Terms and Formulas

• Total Tax Collected: $\text{Tax per unit} \times \text{Quantity sold after tax}$

Step-by-Step Guidance

1. Identify the per-unit tax amount ($1.00$) and the new equilibrium quantity (from part b).

2. Multiply the per-unit tax by the new equilibrium quantity: $\text{Total Tax} = 1.00 \times Q_{\text{after tax}}$

Try solving on your own before revealing the answer!

Final Answer:

Total tax collected is $57.10$.

This is found by multiplying the $1.00$ tax by the new equilibrium quantity ($57.1$ units).

Q1d. How is the burden of the tax shared between buyers and sellers?

Background

Topic: Tax Incidence

This question explores how the burden of a tax is divided between consumers and producers, depending on the changes in prices they pay and receive.

Key Terms and Formulas

• Consumer Price ($P_C$): Price buyers pay after tax.

• Producer Price ($P_P$): Price sellers receive after tax ($P_C$ minus tax).

• Consumer Share: $P_C - P_0$

• Producer Share: $P_0 - P_P$

Step-by-Step Guidance

1. Recall the original equilibrium price ($P_0$) and the new price paid by consumers ($P_C$) from previous parts.

2. Calculate the price received by producers after tax: $P_P = P_C - 1.00$.

3. Find the consumer's share of the tax: $P_C - P_0$.

4. Find the producer's share of the tax: $P_0 - P_P$.

Try solving on your own before revealing the answer!

Final Answer:

Consumers pay $0.29$ of the tax per unit, and producers pay $0.71$ per unit.

This is determined by comparing the changes in prices before and after the tax for both groups.

Q1e. Calculate the price elasticity of demand between $5$and $6$, and the price elasticity of supply between $5$and $6$. How do these values relate to the incidence of tax in part d?

Background

Topic: Price Elasticity and Tax Incidence

This question tests your ability to calculate elasticity using the midpoint formula and interpret how elasticity affects tax incidence.

Key Terms and Formulas

• Price Elasticity of Demand ($E_d$): $E_d = \frac{\%\ \text{change in quantity demanded}}{\%\ \text{change in price}}$

• Price Elasticity of Supply ($E_s$): $E_s = \frac{\%\ \text{change in quantity supplied}}{\%\ \text{change in price}}$

• Midpoint Formula: $E = \frac{\Delta Q / \bar{Q}}{\Delta P / \bar{P}}$

Step-by-Step Guidance

1. Find the change in quantity demanded and supplied between $P = 5$ and $P = 6$ using the demand and supply equations.

2. Calculate the average quantity and average price for each interval.

3. Compute the percentage change in quantity and price using the midpoint formula.

4. Divide the percentage change in quantity by the percentage change in price to find elasticity for both demand and supply.

5. Interpret how the relative elasticities affect the division of the tax burden (the side with lower elasticity bears more of the tax).

Try solving on your own before revealing the answer!

Final Answer:

Price elasticity of demand is $1.00$ (unit elastic), and price elasticity of supply is $0.35$ (inelastic) between $5$and $6$.

Because supply is less elastic than demand, producers bear a larger share of the tax burden.