A is apples and Z is all other goods.

1

• A is apples and Z is all other goods.
• A tax on apples rotates the budget constraint
  inwards and leads to a new equilibrium at pt. a.
• But a tax on Apples is inefficient compared to a
  tax on all goods (consumption tax).
• A tax on all goods shifts the budget constraint
  inwards (parallel to the original budget
  constraint). To be fair we compare the tax on
  apples with an equal revenue consumption tax
  this means that the consumption tax line must
  pass through pt. a.
• Notice the consumer can now consume at pt. b on
  the higher yellow indifference curve.
• Intuition for inefficiency of apple tax
  consider extreme case where the tax causes no
  apples to be consumed. Harm is done to the
  consumers but how much revenue is raised?

The Public Finance Argument for a Wide Tax Base
2

• Note that we can reinterpet Z and A from the
  previous slide in different ways. For example, A
  could be money income and Z leisure in which case
  the theory says that an efficient tax would
  include leisure, i.e. a lump sum tax. This is
  the broadest interpretation and the one that BB
  focus on but the logic is identical.
• Key to the public finance approach is to find the
  tax that minimizes deadweight loss holding
  revenue constant.
• But what kind of assumption is that!
• Do you think that taxes would have been the same
  without the 16th Amendment to the Constitution
  the income tax amendment?
• When government has access to different tax bases
  we should assume that the amount of revenue the
  government will take will change.
• BB assume a Leviathan government. Whatever the
  tax base is Leviathan will seek to maximize
  revenues just like a monopolist.
• BBs assumption is analytically convenient.
• True for most governments in the history of the
  world.
• Necessary to understand why citizens may want to
  constraint government. It is not an objection to
  the Leviathan assumption, for example, to argue
  that the modern US government is not a Leviathan.
  To the extent that this is true it is true
  because of constitutional constraints,
  constitutional constraints that we want to
  explain.

3

• First implication of the Leviathan model. Lump
  sum taxes are the worst! If Leviathan can lump
  sum tax, Leviathan will take everything beyond
  bare subsistence.
• What happens if Leviathan is constitutionally
  constrained to money-income taxation only? What
  is the maximum that Leviathan can take?

4
L is leisure and Y is money income. When
Leviathan cannot tax L the maximum tax revenue is
Max. Tax. Why? Taxpayers are much better off
than when lump-sum taxation is available. Note
that Max. Tax is available only with a regressive
tax. Leviathan must tax at the rate given by the
slope of the Im indif. Curve. Interesting result
Regressive taxes are revenue maximizing!
(Application to IO.) What happens if Leviathan is
restricted to a proportional (flat tax) on money
income?
La
Im
Ya
Ym
Max. Tax
5

• A proportional tax rotates the budget line
  inwards. Thus find the price consumption
  curve, all the equilibrium pts for any tax.
• Now find the proportional tax that raises the
  most revenue.
• The most revenue that can be raised is Ya-Yp.
• With tax rate (Ya-Yk)/Yk.
• To avoid clutter we dont show this but the
  revenue raised by the proportional tax will be
  less than the revenue raised by the optimally
  regressive tax.
• Thus the tax constitution may want to include
  restrictions on regressivity.

La
PCC
k
Ya
Yp
Yk
6
Optimal Commodity Taxation The Public Finance
View

• As usual the public finance perspective says
  optimal commodity taxation occurs when dead
  weight loss is minimized for a given revenue
  constraint.
• Under some simplifying assumptions it can be
  shown that this implies that commodity tax rates
  should be set proportional to inverse
  elasticities. The Ramsey Rule.
• First we derive an expression for deadweight loss
  in the simplified case where supply is perfectly
  elastic (constant returns to scale) and when
  demand curves are independent.

DWL½ ?P?Q ½ (tax amount)?Q ½ (tax
rate)P?Q e(?Q/Q)/(?P/P) Thus, ?Q
e?P/PQ e(tax rate)Q So DWL(½ tax
rateP)(e(tax rate)Q) Or DWL½ (tax rate)²
epQ
DWL
Tax Amount ?P
S
?Q
7
Traditional View (cont)

• DWL½ (tax rate)² epQ
• Notice that DWL increases with the square of the
  tax rate so DWL is increasing at an increasing
  rate. Thus to minimize dead weight loss we will
  want to spread the tax across many commodities.
• Also note that DWL is larger the higher is e so
  we will want to tax low elasticity commodities
  (elt1) more than high elasticity commodities
  (egt1).
• We can make the last point a bit more precise.
  We want to minimize DWL subject to a revenue
  constrant. E.g. consider two commodities then we
  want to
• Min. (½ (t1)² e1p1Q1) (½ (t2)² e2p2Q2)
• s.t. (ReqRev - t1p1Q1 - t2p2Q2 0)

8

• From a Lagrangian
• L½ (t1)² e1p1Q1 ½ (t2)² e2p2Q2 ?(ReqRev -
  t1p1Q1 - t2p2Q2)
• FOC
• t1 e1 p1Q1- ?p1Q10
• t2 e2 p2Q2- ?p2Q20
• Or
• t1 ?/e1
• t2 ?/e2
• The inverse elasticity version of the Ramsey
  rule.

9
Optimal Commodity Taxation The Public Choice
Perspective

• The public finance view assumes that government
  has a fixed revenue constraint and that it will
  tax commodities using the Ramsey rule to minimize
  dead weight loss.
• If instead government is a Leviathan it will act
  like a monopolist it will tax low elasticity
  commodities more than high elasticity commodities
  (for the same reason that a monoplist charges
  more for goods with low elasticity demands) but
  it will maximize revenues not minimize dead
  weight loss.
• It follows that it may be desirable to prevent
  Leviathan from taxing goods with inelastic
  demands! The opposite conclusion of the public
  finance perspective.

10
Should we have a Balanced Budget Amendment?

• Public Finance To minimize dead weight loss you
  want to spread taxes across many commodities.
  The same idea holds over time to minimize dead
  weight loss you want to spread taxes across many
  time periods. Thus, the conventional public
  finance view says that debt finance can be a good
  things especially for extraordinary expenditures
  such as wars.
• (There is also a Keynesian argument that budget
  deficits are necessary to stimulate the economy
  in times of recession.)
• But wake up! We have had budget deficits almost
  continually since 1969. Deficits appear not to
  be used to smooth taxes but to reduce the
  apparent costs of spending (Democrats) or tax
  decreases (Republicans).
• Budget deficits can also occur without fiscal
  illusion. Imagine a time of ideological
  division. A government that rules today may not
  rule tomorrow. By spending today and running up
  a debt, todays government can constraint
  tomorrows government. More spending on defense
  today, for example, can mean less social spending
  tomorrow and vice-versa.
• Prediction ideological divisive times and
  countries will see larger budget deficits.
• Note that from behind a veil of ignorance all
  voters, even ideologically divided voters, would
  want a balanced budget rule.
• What to do about occassional high expenditures?