# 程序代写代做代考 finance Lecture 8: Fiscal Policy

Lecture 8: Fiscal Policy

Patrick Macnamara

ECON60111: Macroeconomic Analysis University of Manchester

Fall 2020

Government Budget Constraint

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Dynamics of Debt

• Evolution of government debt:

D ̇ (t) = [G(t) − T(t)]+r(t)D(t) (1)

primary deficit

• Definition of terms:

• G(t) = real government purchases at time t

• T(t) = real government tax revenues at time t • D(t) = real government debt at time t

• r(t) = real interest rate at time t

• For what follows, let’s also define R(t) = t r (τ)dτ as the 0

continuously compounded interest over the period [0,t].1

1This specification accommodates the fact that r(t) may change over time. If r(t) is constant at ̄r, then R(t) = ̄rt.

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Government Budget Constraint

• Start with law of motion for debt:

D ̇ (t) = [G(t) − T(t)] + r(t)D(t)

• Multiply both sides by e−R(t); integrate from t = 0 to t = s:

s −R(t) ̇ s −R(t) s −R(t) e D(t)dt = e [G(t)−T(t)]dt+ e

r(t)D(t)dt

000

• Use integration by parts to replace LHS:

s −R(t) ̇ −R(s) s

• Then we have

−R(t)

e D(t)dt = D(s)e − D(0) + e 00

r(t)D(t)dt Details

D(s)e

−R(s)

= D(0) +

s 0

e

−R(t)

[G(t) − T(t)]dt

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Government Budget Constraint

• From the dynamics of debt, we have (see previous slide):

−R(s)

s 0

−R(t)

[G(t) − T(t)]dt (2)

D(s)e

• Let’s also impose no-Ponzi-game condition:

= D(0) +

e

lim e−R(s)D(s) ≤ 0 (3) s→∞

In the limit, the present value of the government debt ≤ 0. • Take limit of (2) as s → ∞, imposing (3):

∞ −R(t) ∞ −R(t) e G(t)dt ≤ −D (0) + e

T(t)dt (4)

00

PDV of gov’t purchases PDV of taxes

This is the government’s B.C.; it incorporates debt choices the government must make given the path of G(t) and T(t).

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Implications

• Re-write government’s budget constraint, (4):

0

(in present value) to offset its initial debt.

• Nevertheless, the following examples can satisfy the budget constraint:

• Positive but constant value of D

(i.e., government never pays off its debt)

• D grows at a constant rate, less than the real interest rate (i.e., government debt grows without bound)

See the following slides for more details.

∞ −R(t) e

[T(t)−G(t)]dt ≥D(0) Government must run primary surpluses large enough

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Example: Constant Debt

• Suppose r(t) = r > 0 (i.e., r is constant)

• Suppose D(t) = D > 0 (i.e., D is constant) • For D to be constant, we need

D ̇ =G−T+rD=0 ⇒ T−G=rD>0 Government’s primary surplus is constant and positive.

• Plug T − G = rD back into the government’s B.C.:

D(0) ≤

D ≤ rD

[T(t) − G(t)]dt ∞ −rt

∞ −rt e

0

0

e dt = D

=1/r

⇒ Government budget constraint is satisfied.

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Example: Permanently Growing Debt

• Suppose r(t) = r is constant and D grows at a constant rate, with D ̇ /D = γ < r.
• For D to be growing at rate γ, we need: D ̇ G − T + rD
D = D = γ ⇒ T − G = (r − γ)D
Primary surplus must always be positive, and proportional to
the amount of outstanding debt.
• PlugT−G=(r−γ)D=(r−γ)D(0)eγt backintothe government’s B.C.:
0
∞ −rt
e [T(t) − G(t)]dt
D(0) ≤
D(0) ≤ (r − γ)D(0)
∞ 0
dt = D(0) ⇒ Government budget constraint is satisfied.
−(r−γ)t
e
=1/(r−γ)
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Example: Economic Growth
• Suppose r(t) = r is constant and output Y grows at a constant rate, with Y ̇ /Y = g.
• ̇ Suppose the debt-to-GDP ratio is constant, (D/Y ) = 0
• ̇ ̇ ̇ Since(D/Y)/(D/Y)=D/D−Y/Y =0:
D ̇ Y ̇ G − T + rD
D=Y=g ⇒ D =g ⇒ T−G=(r−g)D
Debt grows at constant rate g, D/Y constant, but...
• if r > g, government must run primary surpluses that

increase with D, government B.C. satisfied

(see previous slide).

• if r < g, government can run primary deficits forever; but
no-Ponzi game condition is violated.
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U.S. Federal Surplus/Deficit
5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9
-10 1960
Total Surplus Primary Surplus
1970 1980
1990 2000 2010 2020 Year
Source: Congressional Budget Office (CBO)
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Surplus (Pct. of GDP)
Sustainability of U.S. Fiscal Policy
• U.S. federal government has run frequent primary deficits over last 4 decades (especially since 2008)
Average primary deficit 1980–2019, 1.1% of GDP
• Will future taxes be enough to meet future promises regarding government spending – i.e., is there a fiscal imbalance?
• Auerbach (1997) measure of “fiscal gap”.
• By what constant fraction of GDP would taxes have to be
increased for the government budget constraint to be
satisfied (given projections of future taxes and spending)?
• Fiscal gap ∆ is the solution to:
0 YPROJ(t)
∞ PROJ TPROJ(t)−GPROJ(t)
e−R (t) +∆ YPROJ(t)dt=D(0)
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Sustainability of U.S. Fiscal Policy
• Auerbach and Gale (2017): U.S. Fiscal Gap ranges from 5.4% to 9.3%2
• This is large. For comparison, federal tax revenues are only about 18% of GDP.
• Large projected increases in federal spending on Social Security, Medicare and Medicaid (due to demographic changes and medical progress)
• However, there are large uncertainties in these projections.
• Large adjustments needed to satisfy budget constraint.
e.g., spending reductions, tax increases, default
• For comparison purposes, consider some European fiscal
gaps:3 UK 4.3%, France 0.2%, Germany 2.2%
2Estimates differ depending on projections for future Medicare spending. 3Source: Table 4.2, EC’s Debt Sustainability Monitor 2019
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Ricardian Equivalence
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Ricardian Equivalence
• How should the government choose between taxes and bonds, for a given path of government spending?
• Consider the household’s budget constraint in the Ramsey-Cass-Koopmans model, with taxes:
∞ −R(t) ∞ −R(t) e C(t)dt ≤K(0)+D(0)+ e
[W(t)−T(t)]dt where D(0) is the bond holdings of the household and
00
K (0) + D(0) is its initial wealth.4
• Suppose government B.C., (4), holds with equality:
∞ −R(t) ∞ −R(t) e T(t)dt = e
00
G(t)dt +D(0) (5)
4We assume number of households H = 1 and total population L(t) = 1. 14/33
Ricardian Equivalence: Consumption
• Use government B.C., (5), to re-write household’s B.C.: ∞ −R(t) ∞ −R(t) ∞ −R(t)
e C(t)dt ≤ K(0)+ e W(t)dt− e G(t)dt 000
Household’s budget constraint depends only on the present value of government purchases.
• It’s reasonable to assume that taxes do not enter directly into household utility.
• Therefore, for a given path of government spending, G(t), path of taxes T(t) does not matter for consumption.
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Ricardian Equivalence: Capital
• Household is indifferent between holdings of capital and government bonds because both offer the same return.
• Therefore, let A(t) = K(t) + D(t).
• Law of motion for total household assets, A(t):
A ̇ ( t ) = r ( t ) A ( t ) + W ( t ) − T ( t ) − C ( t )
• Using A ̇ = K ̇ + D ̇ :
r[K+D]+W−T−C=K ̇ +[G−T]+rD
A ̇ D ̇ ⇒ K ̇ = rK + W − G − C
Government purchases, not taxes, affect capital accumulation.
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Ricardian Equivalence: Summary
• This result is known as Ricardian equivalence.
For a given pattern of government purchases, the timing of taxes and the method of financing does not affect consumption or capital accumulation.
• To see the logic, suppose the government lowers taxes T today, increasing household’s after tax-income W − T
• Government then issues more debt, since D ̇ = (G − T ) + rD. • Government B.C. implies future taxes must increase.
• Household increases savings (in the government bond),
A ̇ = rA + W − T − C.
• Household saves the tax cut (to pay higher future taxes),
leaving consumption and capital accumulation unaffected.
• Result differs from usual economic intuition – e.g., U.S. cut taxes in 2008, with the objective of increasing consumption.
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Ricardian Equivalence in Practice
• Does Ricardian equivalence hold in practice? If it doesn’t hold, are there large departures from Ricardian equivalence?
• Turnover in population (e.g., overlapping generations model) is one reason Ricardian equivalence might not hold.
• Some of the future tax burden associated with a bond issue today are borne by individuals who are not alive yet.
• Yet, intergenerational links can cause a series of households with finite lifetimes to behave as one household with an infinite horizon.
• Households live long enough that most of the future taxes will be levied during the lifetime of individuals alive today.
• Even still, due to the permanent income hypothesis (PIH), a temporary change in after-tax income due to a tax cut today, would have a small effect on consumption.
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Ricardian Equivalence and the PIH
• The issue of whether Ricardian equivalence is a good approximation depends on whether the PIH is a good description of consumption behavior.
• Consider tax cut / bond issue today, which is financed by higher taxes in future
• While path of after-tax income changes, present value of after-tax income is the same ⇒ consumption unaffected if PIH holds.
• We saw in lecture 4 on consumption that there are significant empirical deviations from the PIH
• Let’s consider two examples which cause the PIH to fail.
1. Liquidity/borrowing constraints. 2. Precautionary-savings motive.
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Liquidity Constraints
• Suppose the government lowers taxes today and issues a bond to finance the tax cut.
• Suppose households either cannot borrow or borrow at a higher interest rate than the government.
• In this case, household consumption could increase.
• Government is essentially borrowing on behalf of the
household – lower taxes today in return for higher taxes in
the future.
• This benefits constrained households, who would increase
consumption.
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Precautionary Savings
• Suppose the government lowers taxes today and issues a bond to finance the tax cut.
• With lump-sum taxes, the tax cut still has no effect on the household’s budget constraint.
• However, taxes usually depend on income (and are progressive)
• Distortionary income taxes generate an insurance benefit: low taxes if future income is low, higher taxes if future income is high.
• An increase in taxes in the future (but only for high income states) will increase the insurance benefit, reducing the incentive to save ⇒ consumption increases today.
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Tax Smoothing
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Tax Smoothing
• When Ricardian equivalence holds, the actual time path of taxes is indeterminate.
• Therefore, what actually determines the deficit?
• We’ll develop a model, due to Barro (1979), in which deficits
are chosen optimally.
• Assume there are distortions from raising revenue.
• Before, we have considered lump-sum taxes.
• However actual taxes are a function of income – hence they
distort behavior and cause inefficiencies.
• Barro assumes these distortions increase more than
proportionally with the amount of revenue raised.
• This creates an incentive for the government to smooth taxes over time.
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Tax Smoothing Under Certainty
• Assume time is discrete.
• There is no uncertainty (for now).
• Path of output (Yt) and government purchases (Gt) are exogenously given.
• Interest rate (r) is exogenous and constant.
• There is some initial value of government debt, D0.
• Government will choose the path of taxes (Tt) to minimize the present value of the distortion costs created by taxes, subject to the government’s budget constraint.
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Costs of Distortions
• Assume the distortion costs from raising taxes Tt are given by
Ct Tt
Y =f Y , f(0)=f′(0)=0, f′′(·)>0

tt

where Ct is the cost of distortions in period t.

• These assumptions imply:

• f′(T/Y)>0whenT/Y >0.

• Ct/Yt rises more than proportionally with Tt/Yt.

• Be careful with the notation here – following the Romer textbook, Ct is the distortion cost, not consumption!

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Government’s Problem

• The government’s problem:

∞ 1 Tt min t Ytf

{Tt}∞t=0 t=0 (1+r) Yt

Ct

subject to

∞ Tt =D +∞ Gt (6)

(1+r)t 0 (1+r)t t=0

t=0

• Government minimizes present value of distortion costs

subject to its budget constraint.

• Equation (6) is the discrete-time analogue of the continuous-time budget constraint we saw earlier (see Equation (4)).5

5Equation (6) can be derived by combining individual period-t constraints, Dt+1/(1 + r) = Dt + Gt − Tt and the no-Ponzi game condition, limT→∞DT/(1+r)T =0.

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Solving the Model

• Set up the Lagrangian:

∞ 1 Tt ∞ Gt−Tt

L= Yf +λD+

(1+r)t t Y 0 t=0 t

• FOC:

∂L Ytf′Tt 1 λ

t=0

(1+r)t T

=YtYt− =0⇒λ=f′t ∂Tt (1+r)t (1+r)t Yt

• This condition holds for all t. Hence:

Tt Tt+1 Tt Tt+1

f′ =f′ ⇒ = (7) Yt Yt+1 Yt Yt+1

Since f ′′(·) > 0, the tax rate is constant over time. • To find the tax rate, use (7) with the government’s B.C.

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Implications

• What’s the implication of this model for the deficit?

• Letτ=T0/Y0 =T1/Y1 =T2/Y2 =···

be the constant tax rate

• The government’s primary deficit, as a percentage of GDP:

Gt − Tt = Gt − τ

Yt Yt

Government runs a primary deficit when Gt /Yt is high, and a primary surplus when Gt/Yt is low.

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Adding Uncertainty

• Now assume that path of Gt and Yt are uncertain.

• The government’s problem is now

∞ 1 Tt

minE Yf

0 (1+r)t t Yt

t=0 subject to

Dt+1 =Dt +Gt −Tt forallt 1+r

• Due to uncertainty, we use the government’s period-t budget constraint rather than the infinite-horizon budget constraint.

• This budget constraint is the discrete-time analogue to the continuous-time equation we saw earlier

(D ̇ = (G − T) + rD, see Equation (1)).

• Government chooses the tax Tt in each period, after learning the value of Yt and Gt.

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Solving the Model with Uncertainty

• Set up the Lagrangian, starting from time t:

L =Et

∞ 1 Tt+s

s=0

(1 + r)s Yt+sf

Y + t+s

Et

λ t + s (1+r)s

∞ s=0

D t + s + 1 Dt+s +Gt+s −Tt+s − 1+r

• FOC: ∂L

Tt Yt

Tt Yt

⇒ λt=f′

∂D =−1+r+Et 1+r =0 ⇒ λt=Et[λt+1]

∂Tt ∂L

=f′

−λt=0

λt • Therefore, we get:

f′

λt+1

t+1

Tt Yt

Tt+1

=Et f′

Yt+1

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Solution with Uncertainty

• To summarize, with uncertainty, we get:

Tt+1

Tt Yt

f′

There can be no predictable changes in the marginal

distortion costs.

• Assume f (·) is a quadratic.6 Then:

Tt Tt+1

=Et f′

Yt+1

Y=Et Y (8) t t+1

There cannot be predictable changes in the tax rate ⇒ the tax rate follows a random walk.

6For example, if f (x) = a x2. 2

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Implications

• Suppose D0 = 0 and suppose Gt /Yt follows a random walk:

Gt+1 Gt Et Y =Y

t+1 t

• Then a balanced-budget policy, Tt /Yt = Gt /Yt will also

follow a random walk, satisfying (8)

• In this case, the primary deficit is always zero.

• This suggests that deficits and surpluses arise when the Gt/Yt is expected to change.

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Implications

• Suppose for example E Gt+1 < Gt . t Yt+1 Yt
That is, government spending is expected to fall.
• Tomorrow’s expected primary deficit (as a % of GDP):
Gt+1 Tt+1 Gt+1 Tt Gt Tt Et Y −Y =Et Y −Y

the primary deficit is expected to increase.

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Appendix

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Using Integration By Parts

• If u = u(t) and v = v(t), then the integration by parts formula states that

b′ bb′

u(t)v (t)dt = [u(t)v(t)]a − u (t)v(t)dt

aa

• Use u(t) = e−R(t) and v′(t) = D ̇ (t), implying

u′(t) = −r(t)e−R(t) and v(t) = D(t) • Then re-write:

s

s s

e−R(t) D ̇ (t)dt = e−R(t) D(t)

−r(t)e−R(t) D(t)dt

0

u(t)

0

− D(s) − D(0) +

′

u(t) −R(s)

v(t)

v(t) 0

u′(t) −R(t)

v(t) r(t)D(t)dt

Back

= e

s 0

e

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