Can Deficits Finance Themselves?
George-Marios Angeletos1
1

Chen Lian2

Northwestern and NBER

2UC Berkeley and NBER
3

MIT and NBER

June 6, 2024

Christian Wolf3

How Are Deficits Financed? [r > g ]Question: how are fiscal deficits, e.g., transfers to households, financed?
Basic answer: Fiscal adjustment: raise tax/cut spending in the futureThis paper: Self-financing in NK with finite lives/liquidity constraints [break Ricardian Equivalence]
Deficit ⇒ Keynesian boom ⇒ tax base ↑ and debt erosion (P0 ↑)• improve budget without tax rate adjustment

Q: How important is such self-financing? Can there ever be full self-financing?

How Big Can “Self-financing” Be? [r > g ]Environment: finite lives (or liquidity constraints) + nominal rigidities [OLG-NK, HANK. . . ]Policy: full delayed fiscal adjustment promised at future date H + monetary policy “neutral” (fix E [r ]) or mildly active
Main result: as fiscal adjustment is delayed more, converge to full self-financing• Monotonicity: as H increases, the actual required future tax hike gets smaller and smaller• Limit: the future tax hike vanishes, i.e., we converge to full self-financing• Split depends on price rigidities. [All via tax base ↑ if rigid, all via prices ↑ if approx. flexible.]How Big Can “Self-financing” Be? [r > g ]Environment: finite lives (or liquidity constraints) + nominal rigidities [OLG-NK, HANK. . . ]Policy: full delayed fiscal adjustment promised at future date H + monetary policy “neutral” (fix E [r ]) or mildly active
Main result: as fiscal adjustment is delayed more, converge to full self-financing• Monotonicity: as H increases, the actual required future tax hike gets smaller and smaller• Limit: the future tax hike vanishes, i.e., we converge to full self-financing• Split depends on price rigidities. [All via tax base ↑ if rigid, all via prices ↑ if approx. flexible.]

Intuition: finite-lives/liq. constraints: “discount” far-future tax & front-loaded Keynesian cross
Practical relevance: holds in many environments & quantitatively powerful[general AD (incl. HANK), active monetary policy, investment, distortionary taxation, . . . ]Outline
1

Environment: OLG-NK

2

Equilibrium Characterization

3

Self-financing of Fiscal Deficits

4

Extensions & Generality

5

Quantitative Analysis

6Conclusion

Households and Firms
Continuum of perpetual youth consumers with survival rate ω [ω = 1 : RANK; ω < 1 : proxy for HANK, later]#"∞k
Et ∑ (β ω) u(Ci,t +k ) − v (Li,t +k ) ,k =0

Invests in actuarially fair annuitiesAi,t +1 =It Ai,t + Pt · Wt Li,t + Qi,t −Ci,t − Ti,t + Transfer to Newborns ,ω|{z}|{z}Yi,tannuity

where transfer to newborns makes sure that all cohorts have the same C in steady state [r > g ].
Tax and transfer
Ti,t =

τy Yi,t| {z }distonary income tax+Tt|{z}lump sum tax/transfer

Firms as in textbook NK model: standard NKPC [in log: πt = κyt + β Et [πt+1 ]]Policy, Market Clearing, and Log-Linearization
Government budget [no Gt , Tt is real tax/transfer]1
Bt +1 = Bt − Pt TtIt(plus no Ponzi)

and define Dt = Bt /Pt as real value of public debt outstanding.
Market clearing
ZYt =andCi,t diZ

Ai,t di = Bt .

Initial condition
Ai,0 = B0 .

Log-linearization: a lower case capture log-deviations from steady statess

ss = 0][with the exception of fiscal variables, e.g., dt = dtY−D
ss , to accommodate DMonetary Policy
Baseline: no monetary accommodation [expected real rate in variant to debt & deficit]
rt ≡ it − Et [πt +1 ] = 0

Extension: different degrees of monetary accommodation
rt = φ yt• φ < 0 : an “accommodative” monetary authority• φ > 0 : leans against the wind [Taylor principle holds]

Baseline (φ ≈ 0) consistent with IRFs to identified fiscal shocks [Ramey; Caldara & Kamps; Wolf]

Fiscal Policy
Baseline: Markovian Fiscal Policy [extension of Leeper (1991)]
Ti,t =

τy Yi,t| {z }

distonary income tax+ T̄ + τd (Dt + Et ) − Et ,|{z}lump sumor after (log-)linearization and aggregation
tt =

τy yt
|{z}

tax base adjustment+ τd · (dt + εt ) −|{z}fiscal adjustment

εt|{z}(1)

i.i.d. deficit shock• τy > 0 : self financing through endogenous adjustment in tax base• τd ∈ [0, 1] : a lower τd captures delay in fiscal adjustment (lump sum)

Variant: a Non-Markovian FP with delayed full fiscal adjustment(τy yt − εt t < H initially no fiscal adjustment
tt =dt t ≥ H eventually full fiscal adjustment (lump sum)• High H, similar to low τd , captures delay in fiscal adjustment(2)Outline
1

Environment: OLG-NK

2

Equilibrium Characterization

3

Self-financing of Fiscal Deficits

4

Extensions & Generality

5

Quantitative Analysis

6Conclusion

Aggregate Demand
Optimal consumption + aggregation + rt = 0"

ct = (1 − β ω) ×| {z }MPCat + E t|{z}|wealth#∑ (β ω) (yt+k − tt+k ) ,∞kk=0{z}post-tax income• ω < 1 : (i) elevated MPC; (ii) discounting future y & t, breaking Ricardian Equiv.

Using fiscal policy (1) and market clearing"
yt = F1 · (dt + εt ) + F2 · Et+∞#k∑ (β ω) yt+k ,k=0ω)(1−ω)(1−τd )1−ω
and F2 = (1 − β ω) 1 − τy 1−ω(1−τ.
with F1 = (1−β1−ω(1−τd)d)• F1 captures PE effect of debt/deficits on AD⋆

F1 > 0 iff ω < 1 (failure of Ricardian Equiv)⋆deficits are transfer from future generations to current generations• F2 captures GE effect through intertemporal Keynesian cross⋆jointly governed by FP (τd and τy ), and MPC (ω)(3)

Equilibrium Characterization1AD: (3).

2AS: NKPC, πt = κyt + β Et [πt +1 ].

3

Evolution of real value of public debt:
dt +1 = β −1 dt + εt − τd · (dt + εt ) −|{z}fiscal adjustment

τy yt|{z} D ss − ss (πt +1 − Et [πt +1 ]).
|Y{z}self financing: tax baseself financing: debt erosion

TheoremLet ω < 1 and τy > 0. There exists unique bounded eq’m taking the form:
yt = χ (dt + εt ) ,Et [dt +1 ] = ρd (dt + εt ) .

Moreover, χ > 0 (deficits trigger boom) and 0 < ρd < 1 (debt converges to steady state).

(4)Outline
1

Environment: OLG-NK

2

Equilibrium Characterization

3

Self-financing of Fiscal Deficits

4

Extensions & Generality

5

Quantitative Analysis

6Conclusion

Channels of Self Financing
Start with d0 = 0 (steady state) and consider ε0 > 0 (one-time unexpected positive deficit shock)
Gov’s intertemporal budget constraint ⇒debt erosion≡ν εtax base≡νy ε0p 0}|{! z}|{ z+∞+∞ssD
ε0 = τd ε0 + ∑ β k E0 [dk ] + ss (π0 − E−1 [π0 ]) + ∑ τy β k E0 [yk ]|{z}Yk =0k =0deficit{z}{z} ||
self-financing
fiscal adjustment≡ νε0≡ (1 − ν)ε0

where ν ≡ fraction of deficit that is self-financed, contrast with fiscal adjustment.
RANK benchmark (ω = 1): zero self financing, ν = 0 [standard eq’m (φ → 0+ )]Now (ω < 1): full self financing ν → 1 with delayed fiscal adjustment [τd → 0 or H → +∞]The Self Financing ResultTheorem
Suppose that ω < 1 and τy > 0. The self-financing share ν has the following properties.
1.

[Monotonicity] ν increases in the delay of fiscal adjustment (i.e., it is increasing in H anddecreasing in τd ).

2.

[Limit] As fiscal financing is delayed further (i.e., as H → ∞ or τd → 0), there is complete self
financing: ν converges to 1.
• In this limit, self-financing is strong enough to return d to the steady state.
[τd → 0 : limk→∞ Et [dt+k ] → 0; H → ∞ : limH→∞ E0 [dH ] → 0]

The Self Financing Result
TheoremSuppose that ω < 1 and τy > 0. The self-financing share ν has the following properties.
1.

[Monotonicity] ν increases in the delay of fiscal adjustment (i.e., it is increasing in H anddecreasing in τd ).

2.

[Limit] As fiscal financing is delayed further (i.e., as H → ∞ or τd → 0), there is complete self
financing: ν converges to 1.
• In this limit, self-financing is strong enough to return d to the steady state.
[τd → 0 : limk→∞ Et [dt+k ] → 0; H → ∞ : limH→∞ E0 [dH ] → 0]

3.

[Split]. With rigid price (κ = 0), all self-financing occurs through tax base (νy = ν); as prices
become more flexible (a higher κ), more self-financing occurs through debt erosion

A Graphical Illustration [tt = τy yt − εt for t < H and tt = dt for t ≥ H]

A Graphical Illustration [tt = τy yt − εt for t < H and tt = dt for t ≥ H]

A Graphical Illustration [tt = τy yt − εt for t < H and tt = dt for t ≥ H]

A Graphical Illustration [tt = τy yt − εt for t < H and tt = dt for t ≥ H]

A Graphical Illustration [tt = τy yt − εt for t < H and tt = dt for t ≥ H]

Economic Intuition [Fully Rigid Price, κ = 0]
To illustrate, consider the total adj. of tax base from an ad-hoc static Keynesian cross• Transfer ε at t = 0, static Keynesian cross at t = 0, tax (if needed) at t = 1.

y = MPC · ydisp and ydisp = (1 − τy )y + ε =⇒ y =MPC×ε1 − (1 − τy )MPC

• $1 increase in transfer leads to $MPC increase in AD• $1 increase in AD leads to $(1 − τy ) GE increase in post-tax income• $(1 − τy ) increase in post-tax income lead to $MPC × (1 − τy ) increase in ADτ yτ MPC

Self-financing through tax base adjustment: ν ≡ yε = 1−(1y−τy )MPC is increasing in the MPC• t = 1 tax hike needed: R(1 − ν)εFull self-financing would require MPC = 1, giving y = τ1y × ε.
[Hint: Dynamic: cumulative MPC = 1]

Economic Intuition [Fully Rigid Price, κ = 0]

Our th’m: features of static model have analogues in dynamic economy
1. Static: expected “future” tax hike does not affect “current” spending behavior=⇒ Dynamic: discount (ω < 1) =⇒ far future H-tax’s impact on short-run consumption vanishes[IKC matrix: income change at t + ℓ has a vanishing effect on t consumption: limℓ→∞ β −ℓ Mt,t+ℓ = 0]

Economic Intuition [κ = 0, PE effect of transfer-and-tax vector M ·tt PE , with t PE = −1, · · · , β −H ]Economic Intuition [Fully Rigid Price, κ = 0]

Our th’m: features of static model have analogues in dynamic economy
1. Static: expected “future” tax hike does not affect “current” spending behavior=⇒ Dynamic: discount (ω < 1) =⇒ far future H-tax’s impact on short-run consumption vanishes[IKC matrix: income change at t + ℓ has a vanishing effect on t consumption: limℓ→∞ β −ℓ Mt,t+ℓ = 0]

2. Static: “current” transfer & additional GE income are fully spent currently (MPC → 1)=⇒ Dynamic: front-loaded MPCs (ω < 1) =⇒ cumulative short-run MPCs approach 1 far before H[IKC matrix: income change at t + ℓ has a vanishing effect on t consumption: limℓ→∞ β −ℓ Mt,t+ℓ = 0]=⇒ Transfer receipt (and higher-order GE income) is fully spent before the tax hike at H=⇒ Thus debt stabilizes on its own before H, and tax hike at H is not needed.

Economic Intuition [κ = 0, PE and GE effect of tax-and-transfer vector]

The Role of Nominal Rigidities, κ > 0

A simple rescaling of the perfect rigid price case κ = 0
From NKPC, self financing through debt erosion proportional to tax base expansion+∞

π0 − E−1 [π0 ] = κ · NPV(y ) = κ · ∑ β k E0 [yk ]k =0Split between sources of self financing:τytax base: νy =
νD ss
τy + κ Yss

ss& debt erosion: νp =D
κYssssDτy + κ YssνOutline
1

Environment: OLG-NK

2

Equilibrium Characterization

3

Self-financing of Fiscal Deficits

4

Extensions & Generality

5

Quantitative Analysis

6Conclusion

Extensions & Generality
Monetary policy:• full self-financing remains to hold with mildly active MP when the Taylor principle holds• partial self-financing with very active MP

More general aggregate demand• Discounting + front-loaded MPCs

Fiscal policy• full self-financing result unaffected if far-ahead fiscal adjustment is distortionary• result applies with little change to gov’t purchases instead of transfers

Allow for investment, limit result unaffected [same IKC among consumers]

Outline
1

Environment: OLG-NK

2

Equilibrium Characterization

3

Self-financing of Fiscal Deficits

4Extensions & Generality

5

Quantitative Analysis

6Conclusion

Model & Calibration Strategy
Key targets: (i) consumer spending behavior [iMPCs] & (ii) fiscal adjustment speed

Model: generalize demand block to OLG-spender hybrid[Why? disentangles level & slope of dynamic MPC profile, consistent with evidence.]• Results based on full-blown one-asset HANK similar

Calibration strategy• Match evidence on iMPCs to lump-sum income receipt in Fagereng-Holm-Natvik[Later: other calibration targets, behavioral models, and a full-blown HANK model. . . ]• Consider range of τd consistent with literature on fiscal adjustment estimation[Galí-López-Salido-Vallés, Bianchi-Melosi, Auclert-Rognlie, . . . ]• Flat NKPC [Hazell-Herreno-Nakamura-Steinsson]; steeper NKPC [later]

Quantitative Relevance of Self-financing

Outline
1

Environment: OLG-NK

2

Equilibrium Characterization

3

Self-financing of Fiscal Deficits

4

Extensions & Generality

5

Quantitative Analysis

6

Conclusion

Conclusion

Key: delayed fiscal adjustment ⇒ strong self-financing (esp. from tax base adjust.)
Implications:
1

Theory: grounded in a failure of Ricardian equivalence + nominal rigidities[consistent with Taylor principle & promise to return d to SS]

2

Practice: self-sustaining stimulus may be less implausible than commonly believed

Our analysis here is entirely positive, not normative.
• If start at an efficient SS, self-financing stimulus never optimal• If output is inefficiently low, self-financing stimulus can be a benefitial stabilization tool