# 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
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) 8/33 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. 9/33 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) 10/33 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) 11/33 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 12/33 Ricardian Equivalence 13/33 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. 15/33 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. 16/33 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. 17/33 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. 18/33 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. 19/33 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. 20/33 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. 21/33 Tax Smoothing 22/33 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. 23/33 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. 24/33 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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• 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 Gt (gov’t spending expected to rise), t Yt+1 Yt
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′ b􏰗b′
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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