Fiscal Policies October 28, 2009

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Fiscal Policies
October 28, 2009
Static Endowment Model
• Representative agent has preferences
 ( ) = (1 − ) ln() +  ln()
where  denotes government purchases and 0 ≤  ≤ 1 is a preference
parameter measuring how much  is valued relative to 
• For these preferences,
  =
µ

1−
¶Ã !


• Assume that people have an exogenous productivity  and one unit of time
supplied inelastically; so that GDP is  = (1)
• Resource constraint is given by  +  = 
• How would a “planner” allocate resources in this economy?
max {(1 − ) ln() +  ln() :  +  = }

• Optimal allocation satisfies
µ

1−
¶Ã !

= 1

+ = 
• Solve for
∗ = (1 − )
 ∗ = 
• Notice:  and 1 −  can be interpreted as expenditure shares
• An increase in  (productivity) implies an increase in both ∗ and  ∗ (pure
wealth effect)
• An increase in  (war episode?) implies an increase in  ∗ and a decrease
in ∗ (100% crowding out of private consumption)
Static Production Model
• Can an increase in the demand for government purchases () lead to an
increase in GDP?
• Representative agent has preferences
 (  ) = (1 − ) [ln() +  ln()] +  ln()
where 0 ≤  ≤ 1 and   0 are preference parameters
• For these preferences,
µ

  =
1−
¶Ã !


µ ¶

and  = 

• Assume that people have exogenous productivity  and one unit of time
• Time can be divided between employment () and leisure ();  +  = 1
• Resource constraint is given by  +  =  (note: GDP now given by
 = )
• How would a planner allocate resources in this economy?
max {(1 − ) [ln() +  ln(1 − )] +  ln() :  +  = }

• Optimal allocation satisfies
µ

1−
µ
¶Ã !


= 1
¶

= 

1−
 +  = 
• Solution is
Ã
!
1
1 + (1 − )
Ã
!
1−
∗
 =

1 + (1 − )
Ã
!

∗
 =

1 + (1 − )
∗ =
• with GDP equal to  ∗ = ∗ = (1 + (1 − ))
• So yes: an increase in  can (or at least, should) lead to an increase in
employment and output
• Whether it will or not depends on how  is financed
• Analysis to this point is consistent with benevolent government that has
ability to lump-sum tax
• To see this, consider the following...
• Government levies lump-sum tax  on household sector (take  as exogenous for the moment)
• Household choice problem is
max {(1 − ) [ln( −  ) +  ln(1 − )] +  ln()}

• Solution satisfies
µ
 − 

1−
¶
=
• Now, assume that government sets  =  (GBC) and chooses  to maximize household utility;
max {(1 − ) [ln( − ) +  ln(1 − )] +  ln()}

• Solution satisfies
µ

1−
¶Ã
 − 

!
=1
• The equilibrium allocation is identical to the optimal allocation (verify)
Distortionary Taxes
• What if the government does not have the ability (or willingness) to lumpsum tax?
• Assume that GBC is now  =   (a flat income tax)
• Optimal fiscal policy becomes more complicated (in an interesting and
realistic way)
• Start with household’s problem, viewing  as a parameter
• Household’s budget constraint is  = (1 −  )(1 − ); i.e., slope is −(1 −
 )
• So household optimization implies
µ


1−
¶
= (1 −  )
 = (1 −  )
• Combining these two conditions...
Ã
which simplifies to...
(1 −  )

1−
µ
!
= (1 −  )
¶


=1
1−
• Now, this is rather interesting...we can solve for equilibrium level of employment ̂ = 1(1 + )
• Does not depend on  ! (or  for that matter)
• What is going on here? Substitution and wealth effects on employment
choice exactly cancel
• How does this compare to the optimal level of employment ∗?
̂ =
µ
¶
1

1+
Ã
1
1 + (1 − )
!
= ∗
• There will be “underemployment” in the competitive equilibrium
• Private consumption will satisfy
 = (1 −  )̂
• Now, impose the GBC  =  ̂ Note: since ̂ does not depend on   
is an increasing function of  (no Laffer curve here)
• So private consumption will satisfy
 = ̂ − 
• Government chooses  to solve
max {(1 − ) [ln(̂ − ) +  ln(1 − ̂)] +  ln()}

• Solution must satisfy
µ

1−
¶Ã
̂ − 

• Solving, we have...
̂ = ̂
!
=1
• It follows that
̂ = ; ̂ = (1 − )̂ and ̂ = ̂
• If government is constrained to finance  with distortionary income tax,
then optimal tax rate is  (share of expenditure desired by households)
• An exogenous increase in  (an increase in the demand for government
purchases) implies a corresponding increase in the tax rate
• In this model, the increase in tax rate has no employment or output effects
(SE = WE)
• Therefore, the desired increase in  must come entirely from a corresponding decline in  (100% crowding out again)
Summary
• If  must be supplied by a government (rather than a market) and if
 must be financed with distortionary tax, then equilibrium  less than
socially desirable 
• Sometimes, this is called a “second-best” solution (the socially desirable
outcome is called “first-best”)
• In either case, an exogenous increase in  implies an increase in  and a
decrease in  (crowding out effect)
• Under first-best, less than 100% crowding out because GDP expands (under second-best, 100% crowding out because GDP unaffected)
Business Cycle Implications
• Imagine that cycle is generated by exogenous changes in 
• In both cases, optimal (first or second best) policies suggest that  should
be procyclical
• Conventional wisdom suggests that  should be countercyclical
• Not clear why this should be the case
• E.g., imagine “irrational” decline in  — this may be corrected with distortionary subsidy on leisure (financed by lump-sum tax) and holding 
constant
Dynamic Endowment Model
• Preferences are given by
(1 1 2 2) = (1−1) ln(1)+1 ln(1)+ [(1 − 2) ln(2) + 2 ln(2)]
• For simplicity, assume  =  and assume that government purchases
(1 2) are exogenous (it is better, but more complicated, to think of 
as endogenous, responding to exogenous changes in  )
• For these preferences,
21 =
1 2
 1
• Households have an exogenous endowment (1 2)
• Exogenous interest rate  (small open economy)
• Assume that government finances expenditure programs with lump-sum
taxes ( 1  2)
• Hence, household lifetime budget constraint is

( −  2)
1 + 2 ≤ (1 −  1) + 2


• Define  ≡ 1 + −12 and  ≡  1 + −1 2; and rewrite LBC as

1 + 2 ≤  − 

• Note:  −  equals lifetime disposable wealth (after-tax wealth)
• This should look familiar: we’ve already studied this model for the case in
which  = 0
• Solution must satisfy
1 2
= 
 1

1 + 2 =  − 

• This implies solution (1  2 ) is invariant to timing of taxes (important)
• Now, GBC is
2
2
1 +
= 1 +


• Define  ≡ 1 + −12; so that GBC is  = 
• Substitute  =  into household optimality conditions to derive equilibrium
Ã
!
1
( − )
1+
Ã
!

2 =
 ( − )
1+
1 =
• Implication: (1  2 ) depends only on  (=  ) and not on timing of
spending/taxes
NIPA
• Private sector saving is 1 = 1 −  1 − 1

• Government sector saving is 1 =  1 − 1

• So, domestic sector saving is 1 + 1 = 1 − 1 − 1
A Deficit-Financed Tax Cut
• A policy frequently advocated during recession
• Basic idea is hold government spending fixed and cut household taxes;
increasing household disposable income, thereby increasing consumer demand
• What are the effects of such a policy according to our model economy?
• Deficit-financed tax cut means hold (1 2) fixed and cut current taxes;
i.e., ∆ 1  0


• Since 1 =  1 − 1; this means ∆1 = ∆ 1  0 (i.e., a decrease in the
surplus, or an increase in the deficit)
• GBC implies
∆1 +
∆2
∆ 2
= ∆ 1 +


• Or, as (1 2) is being held fixed, this implies
∆ 1 +
∆ 2
=0

• And since ∆ 1  0 this implies ∆ 2  0 (current deficit implies future
taxes)
• Note that this policy implies ∆ = 0 (since ∆ = 0)
• Implication: deficit-financed tax cut has no effect on desired consumption
(1  2 )
• In fact, the timing of taxes is irrelevant (budget deficits do not matter)
• Put differently, taxes and deficits (future taxes) are equivalent ways to
finance a given spending program (1 2)
• This result is known as the Ricardian Equivalence Theorem (RET)
• Note: RET does not imply government spending does not matter (get this
straight in your head)
• Note: timing of taxes does matter for composition of domestic saving
(between private and government sectors)

• We already know that ∆1 = ∆ 1  0 (government issues bonds to pay
for tax cut)
• Since 1 = 1 −  1 − 1 , we have ∆1 = −∆ 1  0 (private sector saves
the entire tax cut by purchasing these government bonds)
• Government bonds are then used by households to pay for anticipated
higher future taxes
• Note that while the timing of taxes may affect the composition of domestic
saving, it does not affect its level (therefore, this policy will not affect the
trade balance either)
• In a closed economy version of this model (trade balance zero, endogenous
interest rate), such a policy has no influence on the equilibrium interest
rate
• Interesting theoretical result, but what assumptions does it depend on and
does it have empirical support?
• Assumptions: [1] Perfect financial markets; [2] forward-looking, long-lived
(or altruistic) agents; [3] lump-sum taxation
• Assumption [3], in particular, appears to be grossly violated in reality; [1]
and [2] surely do not hold exactly either
• And yet, perhaps surprisingly, it appears very difficult to reject RET in
available data (perhaps assumptions are reasonably good approximations?)
• Cautious lesson: deficit-financed tax-cut not likely to be as “stimulative”
as one might initially have imagined (say, based on Keynesian consumption
function)
Keynesian: 1 =  + (1 −  1)
Our Theory:  = 1(1 + ) and  = (2 −  2)
Dynamic Production Model
• Preferences are now
(1 1 1 2 2 2) = (1 − 1) (ln(1) +  ln(1)) + 1 ln(1)
+ [(1 − 2) (ln(2) +  ln(2)) + 2 ln(2)]
• You get the idea...we can generalize more and more, building on what we
have learned earlier
• We can simplify, without loss of generality, depending on the questions we
want to explore
• Question: how should a transitory increase in desired government purchases be financed?
• The real world event I have in mind is the U.S. entry into WW2 (late 1941)
• I like to think of this “exogenous” event as a temporary increase in the
demand for government output (in particular, military goods and services)
• Ideally, we can model this as ∆1  0 ∆2 = 0
• But it will be simpler to assume  = 0 and treat (1 2) as exogenous;
so experiment is now ∆1  0 ∆2 = 0
• Note: this simplification will not affect the answer to the question at hand
(but may not be suitable for other questions)
• And so, write preferences like this
 (1 1 2 2) = ln(1) +  ln(1) +  [ln(2) +  ln(2)]
• Note: you’ve seen these before (Re: discussion on Intertemporal Substitution of Labor Hypothesis)
• If government can lump-sum tax, then Ricardian Equivalence Theorem
holds (believe me)
• Implication is that it does not matter how the government chooses to
finance ∆1  0; it may choose ∆ 1 = ∆1 or ∆ 2 = ∆1
• In either case, the effect is to decrease after-tax household wealth
• Therefore, a decrease in the demand for all normal goods (1 1 2 2);
which implies an increase in (1 2) and (1 2)
— Note: a transitory increase in  leads to a persistent increase in 
• What if lump-sum taxes are not available? Then Ricardian Equivalence
Theorem fails: timing of taxes will matter
• It turns out to be optimal to “smooth” taxes across time (smooths out
distortions)
• Implication: at least some of ∆1  0 should be financed with debt
• Indeed, this is exactly what we saw in the U.S. during WW2
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