Double Dividend
© P. Berck 2008
Sources
• Goulder, Parry, Burtraw. Rand 1997
• Fullerton. AER 1997
• Fullerton and Metcalf. NBER wp 6199
1997
Pictures
mc
tl
Private mc
X.0
L.0
Labor Demand
Dirty good
Income tax distorts labor market while externality distorts goods market
GPB Model
• 3 Goods
• Dirty X
• Clean Y
• Leisure H
• Dirty good externality
• PPF: T=X+Y+H
• Producer prices are
all 1.
• T – H is labor
• Taxes
• tX for X
• tl for T-H, labor
• Gov’t revenue
• TR= tl(T-H) + txX
• Given back to
consumer lump sum.
• Is constant
Consumer Problem
• Consumer problem max U(X,Y,H)
• s.t. (1+tx)X + Y =(1-tl)(T-H) + TR
• good X costs more than good Y
• labor (T-H) is taxed at rate tl
• foc: Ux=(1+tx) l; UY= l; UH=(1-tl) l
l is marginal utility of income
• Demands are X(tx,tl), Y(), H().
• Write X(t), Y(t), H(t) for short.
Consumer prices for Goods
•
•
•
•
Approx 1/(1-tl) as 1+tl
Budget constraint is then
(1+tl)(1+tx)X + (1+tl) Y =(T-H) + TR (1+tl)
So is equivalent to a tax on both goods
and a subsidy on TR
• (not appealing, but shows that Y really
isn’t “untaxed”.
More setup
• Gov Rev Constraint + Budget imply PPF
• just substitute for TR in budget
• Demands Equations satisfy Budget by
construction
• So only one equation remains
• TR= tl(T-H(t)) + txX(t)
• Taking the total derivative and rearranging
give
Effect of tax increase on x
dX
H
x  tx
 tl
dtl
dt x
t x

H
dt x
T  H  tl
t x
Social Problem
• U() + V(Q(X))
• utility plus negative contribution from dirty
good.
• V doesn’t enter into consumer choice
because it is aggregate X, not individual X
that impairs breathing
Change in utility
• D = 1/l V’ Qx
• Num of M is (1+t) – 1
times lost hours;
partial equilib welfare
loss
• Denom is partial
equilib increase in tax
rev from increase in
labor tax
M
H
tl
tl
H
T  H  tl
tl
Intermediate Steps
dU
dX
dY
dH
 U x  V ' Qx 
 UY
 UH
dt x
dt x
dt x
dt x
Now substitute: l (1+tx) for Ux and so on.
And D l for V’Qx (and note the sign reversal! My error, their
error?
And totally differentiate the ppf to get: dY/dtx = - dH/dtx- dX/dtx
Putting this together with the definition of M gives the final
expression on
The next slide
 dX
1 dU
  D  tx   
l dt x
 dt x

dX 
M  X  tx

dt x 

H
(1  M )tl
t x



Comments
• Empirical applications are via CGE’s, which
have lots of other things in them.
• When one raises a tax on labor it is equivalent to
taxing both goods, to tx is the difference in the
tax rate between the two goods with tl
normalized to one.
• A standard doesn’t have the revenue recycling
effect, cause there is no revenue.
• The pigouvian tax is probably not the right tax,
though one can argue for too low or too high,
depending on parameters. Goulder says too
high.
The Dual
• DM notation.
• PPF: p’y = 0
• (sign of work is negative, of goods positive)
• Simple version has p fixed
• Budget: q’x = 0
• gov’t budget: R= p’z = (q-p)’x
• Treat z as fixed
• 3 equations
Down to one eq.
• R = (q-p)’x
• Let x = x(q-p) = x(t), the demand equation.
• x(t) always satisfies x(t)’q = 0.
• R = t’x(t)
Feasible tax Variation
t ' x
dt j
ti

t ' x
dti
t j
• W = V(q) – Dxi(q)
• Indirect utility less damage
 a is the marginal util of income
• dV/dt = dV/dq = - a x by Roy’s identity
What happens when only taxes i and j are
perturbed. Like tax on dirt up and labor down.
xi xi dti
V  a [ xi  x j ]  D( 
)
dti
ti ti dt j
dt j
• Double dividend means first term is nonzero and original tax system is nonoptimal.
•
•
•
•
t = q-p
p is constant, so can write
q(t) = q(t+p) as the demand system
q(t) satisfies budget constraint by
construction
1 Equation left
Direct Approach
• Form the indirect utility function
• IN(X(t),Y(t),H(t))= IN(t)
• Use Roy’s identity to get
• dIN/dtx = -aX –aH dtl/dtx
• Adding the Pigou term
• dU/dtx = -aX –aH dtl/dtx + V’Qx dX/dtx
• Here the dwl in X market decreases by aX
dX; in labor market by –aH dtl/dtx dX.