Microeconomics, Module 1, “Consumer’s Demand, Producer’s Supply, and Equilibrium”

Homework Assignment

(The attached PDF file has better formatting.)

Homework Problem 1 is a numerical problem to be completed by all candidates. The procedure is described in Landsburg’s textbook and reviewed in the practice problems.

The demand curve is Q = 200 – 20P, and the supply curve is Q = 10P – 10.

A.    What is the equilibrium price and quantity? (These are P and Q at the intersection of the demand and supply curves.)
B.    If a sales tax of $1 per unit is imposed on consumers, what is the revised demand curve? Explain why the demand curve changes but the supply curve does not change. Explain intuitively why the demand curve shift left = shifts down.
C.    With the sales tax of $1 per unit, what is the equilibrium price and quantity? For the price, give both the pre-tax price and the after-tax price.
D.    If an excise tax of $1 per unit is imposed on suppliers, what is the revised supply curve? Explain why the supply curve changes but the demand curve does not change. Explain intuitively why the supply curve shifts left = shifts up.
E.    With the excise tax of $1 per unit, what is the equilibrium price and quantity? For the price, give both the pre-tax price and the after-tax price.

The following addendum is not required for the homework assignment. It is discussed in the Microeconomics Macroeconomics textbooks, but since it uses calculus, it is not worked out in either textbook.

Addendum: The government wishes to find the maximum tax revenue. If it charges a sales tax of $0 per unit, it collect $0 of revenue. If it charges a sales tax of $10 per unit, the after-tax price is at least $10, the quantity demanded is 200 – 20 × 10 = 0, and it collects $0 of revenue. The maximum revenue is collected with a tax between $0 and $10 per unit.

A.    What tax generates the maximum revenue?
B.    What is the quantity Q at the point of maximum tax revenue?
C.    What is the amount of maximum tax revenue?

To solve the problem, let the sales tax per unit be T. Derive a formula for the equilibrium quantity as a function of T with a sales tax of T per unit. Multiply this quantity by T to get the tax revenue. Set the partial derivative of this tax revenue with respect to T equal to zero, and solve for T at the point of maximum tax revenue.

This addendum is not part of the homework assignment, but it helps you understand both Landsburg’s and Barro’s comments on the effects of high tax rates. You may post your solution on the discussion forum.

Homework Problem 2:

You may select either of the following two scenarios for Problem 2. They are similar, using different goods. Your solution should be about two paragraphs, showing that you have read the text. The scenarios are simplified, to create flat or steep demand and supply curves. By slope of the curve, we mean flat or steep, not a numerical slope.

Problem 2A: Gas Taxes

Assume all gas is produced from oil imported from the Middle East.

●    Jacob believes that U.S. drivers can take public transportation (buses, trains) or walk if gas prices are too high; Rachel believes that U.S. drivers would rather pay more for gas than take public transportation or walk.
●    Jacob believes that the supply of oil is fixed, and the amount supplied depends on how much can be pumped out of the ground; Rachel believes that OPEC determines the supply of oil based on potential profit, not on the supply in the ground.

The price of gas is $2.00 a gallon before any taxes. Suppose the government imposes a $1.00 per gallon sales tax on gas.

A.    From Jacob’s perspective, what are the slopes of the demand and supply curves for gas before the tax?
B.    From Rachel’s perspective, what are the slopes of the demand and supply curves for gas before the tax? (For Parts A and B, Jacob believes that a change in price affects the quantity demanded, but not the quantity supplied; Rachel believes that a change in prices affects the quantity supplied, but not the quantity demanded.)
C.    From Jacob’s perspective, what is the effect of the tax on the equilibrium price and quantity? How will price and quantity change, and who will pay most of the tax?
D.    From Rachel’s perspective, what is the effect of the tax on the equilibrium price and quantity? How will price and quantity change, and who will pay most of the tax?

Problem 2B: Cocaine Use

Assume all cocaine is produced in Latin America and imported into the U.S.

●    Jacob believes that U.S. cocaine users are indifferent about the type of drugs which they use; Rachel believes that U.S. cocaine users would rather pay more for cocaine that switch to other drugs (or stop taking drugs).
●    Jacob believes that cocaine is produced only in certain fields in Columbia, and other land cannot be substituted; Rachel believes that many Latin American countries produce cocoa, and almost any land can be used to produce cocaine.

The U.S. government is considering two options to deal with cocaine use:

Option 1:    Set longer prison terms for persons using cocaine, but not for other drugs.
Option 2:    Spray the cocoa fields in Columbia (but not other countries) with chemicals that destroy the crops.

A.    From Jacob’s perspective, what are the slopes of the demand and supply curves for cocaine before the tax?
B.    From Rachel’s perspective, what are the slopes of the demand and supply curves for cocaine before the tax? (For Parts A and B, Jacob believes that a change in price affects the quantity demanded, but not the quantity supplied; Rachel believes that a change in prices affects the quantity supplied, but not the quantity demanded.)
C.    From Jacob’s perspective, what is the effect of Option 1 (longer prison terms) on the equilibrium price and quantity? How will price and quantity change?
D.    From Rachel’s perspective, what is the effect of Option 1 (longer prison terms) on the equilibrium price and quantity? How will price and quantity change?
E.    From Jacob’s perspective, what is the effect of Option 2 (aerial spraying) on the equilibrium price and quantity? How will price and quantity change?
F.    From Rachel’s perspective, what is the effect of Option 2 (aerial spraying) on the equilibrium price and quantity? How will price and quantity change?

Homework assignments 2A and 2B show that public policy issues, like the effects of taxes on gas consumption and drug use, depend on the shapes of supply and demand curves.

Problem 3 (not required), is an insurance application of the material in Module 1. It is not required homework; you may post your solution on the discussion forum (or not work through the problem at all). This problem shows how Landsburg’s perspective can be used to analyze the effects of premium taxes on insurance demand and supply.

In the town of East Oshkosh, auto insurance is required by law (compulsory). There are no auto insurance agents in East Oshkosh, and drivers buy insurance from agents located in other towns and selling to all consumers in the state, so the price of insurance in East Oshkosh won’t affect the quantity demanded.

Permanent life insurance is bought only by wealthy consumers for the tax benefits, who buy coverage only if the returns are greater than the returns from other investments. Only one life insurance agent, a resident of East Oshkosh, sells permanent life insurance to consumers in the town. (Implication: the price of insurance has a great effect on the quantity demanded.)

The town of East Oshkosh wishes to raise tax revenues. To avoid harming less wealthy citizens (including the life insurance agent), it imposes a 10% sales tax on consumers of life insurance and a 10% excise tax on agents selling auto insurance in the town.

A.    What are the slopes of the demand and supply curves for auto insurance?
B.    What are the slopes of the demand and supply curves for permanent life insurance?
C.    What is the effect of the life insurance sales tax on the equilibrium price and quantity? Who pays the tax: the agent or the consumers?
D.    What is the effect of the auto insurance excise tax on equilibrium price and quantity? Who pays the tax: the agents or the consumers?

The point of this exercise is that the choice of sales tax vs excise tax has no effect; the shapes of the supply and demand curves determine who pays the tax.

For question 1, does the excise tax really move the supply curve up? It seems like the opposite should happen.

It does move up.  Keep these things in mind:

1) the x-axis is quantity, y-axis is actual price (not the price recieved by the supplier.)

2) the quantity depends on the price.  Assume the tax is $1.  So if the non-tax supply curve goes through the point (100, $9), then after the tax curve will go through the point (100, $10). 

To see this, note: at a price of $10, the suppliers are going to get $9, so we look at the old curve and see this point corresponds to the quantity 100.

Question: Can you explain why the supply curve shifts up with an excise tax?

Answer: Suppose the supply curve is Y = 20 + X, or P = 20 + Q. We re-write this as Q = P – 20: given any price, the firm produces this price minus 20. If the firm pays an excise tax of $2 per good, it really gets P – 2, so it produces P – 2 – 20 = P – 22. This curve is P = 22 + Q, or an upward shift of 2.

I'll be the eager beaver and make the first post of the Fall '06 Microecon course. I decided (too much time on my hands, I guess) to do the addendum and optional problem 3 in the Module 1 Homework, so I thought I'd post my answers here. Agreement or disagreement with my answers is welcome, of course.

For the addendum after problem 1, I have the following answers: A) $4.50.   B) 30.   C) $135.00.

For the not-required problem 3, my bottom-line answer is that the city will not accomplish its goal with this tax approach. My specific answers are: A) For auto, the demand curve is a straight up-and-down vertical line [slope infinity]. The supply curve is a typical, positively sloped line.   B) For life, the demand curve has a very flat, slightly negative slope [imagine a line with slope of something like -.01]. The supply curve is a more typical, positively sloped line.   C) Life insurance sales tax will shift the equilibrium quantity and price to both be lower. The market price for life insurance drops by almost the whole amount of the tax. Agent pays almost all of the tax.   D) Auto insurance excise tax will cause the equilibrium price to go up by the full amount of the tax, while the equilibrium quantity stays the same. The market price for auto insurance goes up by the amount of the tax. Consumers pay the tax.

Microeconomics, Module 1, "Consumer’s Demand, Producer’s Supply, and Equilibrium"

Homework Assignment: Addendum

(The attached PDF file has better formatting.)

Updated: December 21, 2006

{The solution in the posting above is correct. The Jacob - Rachel dialogue below provides the intuition and the method.}

Jacob: How can we grasp the intuition for the maximum tax revenue?

Rachel: Consider the end points. If the tax is $0 per unit, the total tax revenue is zero.

Jacob: At this tax, the equilibrium price is the intersection of the supply and demand curves

200 – 20P = 10P – 10 A 210 = 30P A P = $7.

At P = $7, the quantity demanded is 200 – 20P = 60.

If the tax is $3, the total price paid by consumers is $7 + $3 = $10.

At a total price of P = $7, the quantity demanded is 200 – 20P = 0.

How can the tax giving the maximum revenue be $4.50?

Rachel: Landsburg emphasizes this point. You assume the tax is paid by consumers. Landsburg explains that the incidence of the tax is shared by consumers and producers.

Jacob: Who pays more of the tax?

Rachel: The slope of the demand curve is twice the slope of the supply curve. The curves are linear, so the consumers pay half as much of the tax as producers.

Jacob: Can you show this intuitively?

Rachel: Suppose the tax were $9 per unit. The demand curve becomes 200 – 20P – 180 = 20 – 20P. To find P, we equate:

20 – 20P = 10P – 10 A 30 = 30P A P = $1.

At P = $1, the quantity supplied is 10 × 1 – 10 = 0. The quantity demanded depends on the total price of $1 + $9 = $10. The quantity demanded is zero.

Jacob: Do we infer that at a tax of $9, the equilibrium quantity is zero, so the total tax revenue is zero?

Rachel: Yes. The supply and demand curves are linear, so the tax revenue is symmetric. The maximum tax revenue occurs at the midpoint of 0 and 9, or $4.50. The equilibrium quantity at that point is the midpoint of 0 and 60, or Q = 30. The tax revenue is Q × T = $135.

Jacob: How do we solve this by calculus?

Rachel: The demand curve is Q = 200 – 20P, and the supply curve is Q = 10P – 10. If the tax is T, the demand curve becomes Q = 200 – 20P – 20T. The equilibrium quantity is

200 – 20P – 20T = 10P – 10 A

210 = 30P + 20T A

30P = 210 – 20T A P = 7 – b T

Q = 10P – 10 = 60 – 20T / 3.

The total tax revenue is T × Q = 60T – 20/3 T²

To maximize the tax revenue, we differentiate with respect to T: 60 – 40/3 T.

We set this to zero, to get T = 60 × 3/40 = 4.50.

Jacob: What do we learn from this addendum to the homework assignment?

Rachel: Barro discusses this in depth (the macroeconomics on-line course). There is a point of maximum tax revenue. If the government raises the tax rate above this point, it collects less tax, not more tax.

Jacob: Do countries have tax rates above the maximum point?

Rachel: Several European countries are above the maximum point. If the government would lower the tax rate, it would collect more tax.

Jacob: Do we see this in the United States?

Rachel: Clinton raised the tax rate on wealthy people during his first term. The result was that wealthy people paid less tax, because they worked less.

Jacob: If the tax increase caused less tax revenue, why would anyone support it?

Rachel: Barro did an informal survey of his friends. Barro teaches at Harvard, and many of his colleagues favor higher taxes on wealthy persons. Barro asked: "If a tax increase on the wealthy causes the government to collect less total tax, is it good?"

Jacob: I presume they all said no. If the government gets less tax, it has less money to give to the poor or to public programs. How can this be good?

Rachel: On the contrary: many said yes. One goal of high taxes is to redistribute money from wealthy persons to poor persons (or public programs). But there is another purpose. Many people don’t like income inequality. The best outcome is to raise the income of the poor so everyone is wealthy. If we can’t do this, the second best outcome is to reduce the income of the wealthy.

Jacob: It seems strange that people have this desire.

Rachel: Several European countries have such high tax rates and restrictions on work that no one earns that much after-tax. One might think citizens are upset that their incomes are lower than those in the U.S. or other countries with lower tax rates. But many people are satisfied. They don’t want the high income inequality of the U.S.

Jacob: Do people in the U.S. care less about income inequality?

Rachel: The difference in the U.S. is that most people believe they (or their children) can move up by hard work. They may care about income inequality, but they expect to move up and become part of the high earning group.

In parts C and D of problem 2A, would someone from the NEAS please clarify what it is being asked? Aside from who pays the greater amount of the tax, it seems like the "effect of the tax on the equilibrium price and quantity" and "How will the price and quantity change.." are the same thing. Right?

[NEAS: Correct.]

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