# Prof. Bryan Caplan

bcaplan@gmu.edu

http://www.gmu.edu/departments/economics/bcaplan

Econ 103

Spring, 2000

1.  Suppose the supply-and-demand equations for the market for textbooks are:

S:  Q= -10 + 50PS

D:  Q= 200 - 3PD

a.  Solve for the equilibrium price and quantity without taxes.

To find P, solve S=D: -10+50P=200-3P.  Then P=210/53.  To get Q, just substitute equilibrium P into either equation to get Q=188.

b.  Suppose the government passes a law saying that booksellers must pay a \$3 tax on each textbook.  What happens to the sticker price ("price tag") of books?  The market quantity?

Sellers take home \$3 less than the sticker price, but buyers pay the sticker price.  So S&D look like:

S:  Q= -10 + 50(P-3)

D:  Q= 200 - 3P

Setting S=D, we find that the sticker price P=360/53, and market quantity is Q=180.

c.  Suppose the government passes a law saying that bookbuyers must pay a \$3 sales tax on each textbook.  What happens to the sticker price of books?  The market quantity?

Sellers take home the sticker price, but buyers pay the sticker price plus \$3.  So S&D look like:

S:  Q= -10 + 50P

D:  Q= 200 - 3(P+3)

Setting S=D, we find that the sticker price P=201/53, and market quantity is Q=180.

d.  Show that consumers pay the same total price per book regardless of how the tax is collected.  In other words, show that the sticker price in (b) equals the sticker price plus the tax in (c).

Sticker price in (b) is 360/53.

Sticker price plus tax in (c) is 201/53+3.  This is equal to 201/53+159/53=360/53, precisely the sticker price in (b)!

e.  How much revenue does the government collect from the tax?

The government collects tQ, where t is the tax and Q is the after-tax quantity.  (You can only collect sales taxes on units actually sold!)  So revenue=\$3*180=\$540.

f.  Using S&D diagrams, shade the government's total tax revenue.  Then use a different kind of shading to indicate the deadweight costs of the tax.

2.  Suppose the supply-and-demand equations for the market for insulin are:

S:  Q= -10 + 5PS

D:  Q= 200

a.  Solve for the equilibrium price and quantity without taxes.

P=210/5.  Q=200.

b.  If the government imposes a tax on insulin manufacturers of t/liter.  How much does the sticker price of insulin increase?

Now:

S:  Q= -10 + 5(P-t)

D:  Q= 200

Solving for P:

P=(210/5)+t

c.  How much revenue does the government raise?  Are there any deadweight costs of the tax?

The government raises 200t.  There are no deadweight costs because demand is perfectly inelastic; there aren't any units of insulin that people would have bought untaxed that they refuse to buy with a tax.

3.  Suppose the supply-and-demand equations for the market for automobiles are:

S:  Q= -10 + 45PS

D:  Q= 2000 - 5PD

a.  If the government imposes a tax of \$100/car, what fraction of that \$100 tax do consumers ultimately pay?  What fraction do producers ultimately pay?

Let's assume that sellers legally pay the tax (we now know that it doesn't really matter, but we've got to do it one way or the other).  Then:

S:  Q= -10 + 45(P-t)

D:  Q= 2000 - 5P

Solving for P:

P=(2010/50)+(45/50)t

Remember that P is the sticker price that buyers pay.  The coefficient on t is 45/50.  Thus, we can see that for every \$t in tax, buyers pay 45/50, or 90%.  Sellers thus pay the remaining 5/50, or 10%.

b.  How what is the smallest tax that would reduce the equilibrium quantity to zero?  Show the revenue raised and deadweight costs of this tax on a supply-and-demand diagram.

One easy way to solve this problem: get an equation for Q as a function of t.  Then set Q=0 and solve for t.  So first, plug in our equation for P into the demand equation (you could also have chosen the supply equation, but it would more work):

Q=2000-5*[(2010/50)+(45/50)t]

Simplifying:

Q=1799-4.5t

Setting Q=0 gives us:

0=1799-4.5t

Solving shows that t=400, which is the smallest tax sufficient to completely close this market.  Revenue raised is thus 0!  All of the consumers' and producers' surplus in the unregulated market is the deadweight cost of the tax, and this diagram shows:

4.  Suppose that:

S:  Q= a + bPS

D:  Q= c - dPD

Prove that for any per-unit tax, the fraction sellers pay is d/(b+d), while the fraction demanders pay is b/(b+d).

Analyze this as a tax on sellers.  Then:

S:  Q= a + b(P-t)

D:  Q= c - dP

Solving for P:

a + b(P-t)=c-dP

(b+d)P=-a+c+bt

P=[-a+c+bt]/[b+d]

Note that the coefficient on t simplifies to: b/(b+d).  Since demanders pay the sticker price, this means that when the tax goes up by \$t, the price demanders pay goes up by b/(b+d)*\$t.

If the demanders pay a fraction b/(b+d), we know that suppliers pay the rest, or 1-b/(b+d).  This simplifies to: d/(b+d).

5.  Suppose that:

S:  Q= 10 + 1,000,000PS

D:  Q= 50 - 1,000,000PD

a.  What fraction of any tax paid would suppliers and demanders bear?

From the results for (4), we know that suppliers pay 1,000,000/(1,000,000+1,000,000), or 50%.  Demanders pay the other 50%.

b.  How much revenue would the government raise in this market from a \$.01/unit tax?

The government would raise 0!  Why?  Formally, solve for Q as a function of t, as in problem 3.

Assuming sellers legally pay the tax, P=(40/2,000,000)+.5t.  Then plug this into the demand equation to get:

Q= 50 - 1,000,000[(40/2,000,000)+.5t]

Q=50-20-500,000t

Q=30-500,000t

Plugging in t=.01, we find that Q=-4970!  In practical terms, quantity can't be less than 0, so Q=0.  Thus, tQ=0, and the government earns no revenue.

Both supply and demand are highly elastic in this question.  This means that as taxes rise, both demanders and suppliers substitute easily into other activities.  If both substitute away, no one is left to pay the tax.

5.  In roughly half a page, analyze the incidence of a national sales tax on Internet commerce.  Carefully explain: (a) The main groups involved; (b) The supply and demand elasticities of each group; (c) What this implies about the fraction of the tax each group pays; (d) What this implies about the total revenue the government will collect.

There are two obvious groups to consider: computer-savvy consumers who might buy things over the Internet, and Internet businesses.  Currently Internet sales are basically untaxed.  Consumers are likely to have fairly high demand elasticities; if Internet purchases get more expensive, they can just go to a normal store.  But Internet shopping is more convenient for some consumers, so demand won't be infinitely elastic.  Internet businesses are also likely to have high supply elasticities; if Internet commerce isn't profitable, they will just invest their resources elsewhere.  Moreover, Internet businesses can move overseas and possibly escape U.S. tax law!  This implies that each group will bear an intermediate fraction of sales taxes, though consumers will probably pay more.  It also means that such a tax might raise very little revenue, since both demanders and suppliers will sharply respond to taxes.

6.  In roughly half a page, analyze the incidence of a government ban on ATM fees.  Carefully explain: (a) The main groups involved; (b) The supply and demand elasticities of each group; (c) What this implies about the fraction of the tax each group pays; (d) What this implies about the total revenue transferred from banks to consumers.

The main groups involved are bank customers and banks.  Bank customers' demand curves for ATM access are probably moderately elastic, but not extremely so.  They may use fewer ATM machines if the price goes up, but they won't quit using them entirely.  On the other hand, banks' supply curves are likely to be quite elastic; if a machine quits earning them a profit, they can easily sell off the machine.  A ban on fees can be seen as a big tax on banks; but due to the elasticities, customers will pay most of the tax.  Banks will either charge higher non-ATM fees and restrict ATM access to customers, or else close down ATMs.  This means that the revenue transferred from banks to customers as a result of the ban is minimal.

7.  In roughly half a page, discuss the deadweight costs of the Internet tax OR the ATM fee ban.  Be careful to distinguish deadweight costs from transfers.

I choose to analyze the ATM fee ban.  Remember that deadweight losses are losses that some people suffer that do not correspondingly benefit anyone else.  Every ATM machine that would have been profitable that gets closed down implies deadweight costs of inconvenience for people who have to try harder to find an ATM machine.  If the fee ban means that only a bank's customers can use their machines, these deadweight costs of inconvenience grow further: a patron of NationsBank might be right next to a Chevy Chase machine, but have to spend 10 minutes searching for a NationsBank machine.  The consumer loses valuable time but no one else benefits as a result.