A Comparative Advantage Approach to Government

Debt Maturity

Robin Greenwood Sam Hanson Jeremy C. Stein
Introduction

I How should the govt. manage the maturity structure of its debt?

I Tax-smoothing (Barro ë79; Lucas and Stokey í83; Bohn í90): Want to

smooth taxes over time since distortionary costs are convex in taxes

I Key theme: If future interest rates are uncertain, debt should be long to

insulate taxes from ìreÖnancing riskî

I Trade-o§ view articulated by debt management practitioners:

I Lawrence Summers: ìI think the right theory is that one tries to [borrow]

short to save money but not [so much as] to be imprudent with respect

to rollover risk. Hence there is certain tolerance for [short term] debt but

marginal debt once [total] debt goes up has to be more long term.î

I Postulated trade-o§ between ìrollover riskî and ìcheapî short-term debt

I Does this trade-o§ view make sense?

I Doesnít make sense if ìcheapnessî is compensation for risk

I This paper: Could make sense if consumers/investors value short-term

ìmoney-likeî securities
Introduction

A Trade-o§ Model of Government Debt Maturity

I Government: Raises taxes and issues debt to Önance a one-time

expenditure (or an accumulated deÖcit)

I Standard tax-smoothing motive due to convex distortionary costs

I New twist: households derive greater monetary/liquidity services from

short-term debt

I Absent money demand, govt. opts for longer-term debt

I Eliminates reÖnancing risk (i.e., govt. needs to raise taxes when short

rates rise) which enables govt. to perfectly smooth taxes

I With money demand, optimally tilts towards short-term debt and

incurs some reÖnancing risk

I Central trade-o§: Govt. tries to satisfy money demand for short-term

debt, but is limited by tax-smoothing costs of uncertain reÖnancing

I Trade-o§s appear to be reáected in U.S. government maturity

choices over time
Introduction

Adding Private-sector Money Creation

I Add private-sector banks who can also engage in money-creation

I Banks want to issue short-term, safe debt because it is cheap

I Caballero & Krishnamurthy ë08: Responding to a global shortage, US

Önancial sector tried to manufacture ìrisklessî assets pre-crisis

I Gorton ë10, Gorton & Metrick ë09: Money creation by unregulated

shadow banking system

I Banking sector response to cheapness may be socially excessive

I Stein ë12: Excessive private money creation makes the system too

vulnerable to crises

I Short-term debt leads to costly Öre sales in bad states, since banks must

liquidate assets to repay

I Private banks issue too much short-term debt because they do not fully

internalize these Öre-sale costs
Introduction

Plannerís Problem

I If households demand short-term safe debt, who should supply it?

I It is costly for both government and banks to create short-term

money-like claims, but banks may not fully internalize these costs

I Comparative advantage approach: If government has the lowest social

cost of supplying money, it should tilt towards more short-term

I First best: Marginal social cost of government money creation = social cost

of private money creation = social beneÖt of money creation.

I Second best: Directly regulating private money creation may be

costly/di¢ cult, so a more robust solution may be to reduce the temptation:

I Second best: government partially crowds out excessive private

creation by tilting further towards short-term debt

I Goal is to a§ect the relative price of long- vs. short-term debt, reducing

incentives for private money creation

I Adds a regulatory dimension to the governmentís debt-maturity choice

I Our analysis here is prescriptive rather than descriptive
Stylized Facts

Demand for safe securities

I Krishnamurthy and Vissing-Jorgensen ë12 argue that money-like

securitiesó i.e., liquid securities with absolute safety of nominal cash

áowó such as U.S. Treasuries embedded a convenience yield: have

lower yields than they would in standard asset-pricing models

I IdentiÖcation: Downward-sloping demand for monetary services means

that AAA-UST spread is high when Debt/GDP is low

I This paper: short-term safe securities (e.g., T-bills) are especially

money-like: even greater liquidity and absolute safety of nominal return

since have almost no interest rate risk

I Presumably, these attributes are what make T-bills so attractive to

money-market investors.
Stylized Facts

Liquidity premium for short-term T-bills

I T-bill curve is extremely steep at front-end

I Compare T-bills to Ötted UST curve from Gurkaynak, Sack, & Wright ë07

I Plot avg. spread of the w-week bill to curve z

(w)

t = y

(w)

t  yb

(w)

t

from ë83-ë09

I Weíre controlling for the general shape of the yield curve, so probably a lower

bound on the average liquidity premium of short-term T-bills
Stylized Facts

Liquidity premium varies with quantity of T-bills

I ìMoneyî premium is low when quantity of outstanding T-bills is

large

I Plot spread of 4-week bill to the curve (z

(4)

t

) versus (BILLS/GDP)t

I Positive relationship, but series are persistent. And endogenous govt. supply

response to money demand shocks will reduce coe¢ cient (e.g., fall of í08).
Stylized Facts

Exploit seasonal variation in supply of T-bills

I Large seasonal variation in the supply of Treasury bills

I Driven by the seasonal áuctuations in tax receipts: plausibly unrelated to

business cycle conditions or shocks to money demand

I Pattern became much stronger in early 1990s

I First stage: Regress 4-week change in bill supply on week-of-year dummies:

∆4(Bills/GDP)t = c + ∑

52

w =2 d

(w)1fweek(t) = wg + ∆4vt

.
Stylized Facts

Exploit seasonal variation in supply of T-bills (Cont.)

I Regress 4-week changes in z-spreads on 4-week changes in T-bill supply.

∆4z

(n)

t = a

(n) + b(n)

 ∆4(Bills/GDP)t + ∆4ε

(n)

t

Instrument for change in T-bill supply with week-of-year dummies.
Stylized Facts

Government Debt Maturity and Debt/GDP

I When Debt/GDP increases, govt. debt maturity rises (ρ = 0.71):

I This is not mechanical: the maturity of govt. debt issuance rises when

Debt/GDP rises.
Stylized Facts

Crowding Out in the Maturity Dimension

I Greenwood, Hanson, Stein (ë10): When government shortens its maturity

structure, Örms issue longer-term.

I Financial money creation is particularly responsive to supply of ST USTs.

I Estimate PrivateMoneyt/GDPt= a + b  Xt+ut for Xt= Dt/GDPt

and Xt= D

S

t /GDPt and Önd b < 0.

I R

2

is much higher when focus in on short-term govt. debt.
Trade-o§ Model of Government Debt Maturity

Basic Set-Up

I Households have linear preferences over consumption at t = 0, 1, 2.

U = C0 + E [C1 + βC2] + v (M0)

I Households have a deterministic income of 1 each period

I ReÖnancing risk: β = Random discount rate between time 1 and 2 with

E[β] = 1. Becomes known at t = 1.

I v (M0) = Utility from money services at t = 0: v

0 > 0 and v00  0.

Only derive utility from riskless, short-term debt at t = 0

I Households can transfer wealth between periods by purchasing

government bonds:

I B0,1 : ST bonds issued at t = 0, due at t = 1; P0,1 = 1 + v

0

(M0)

I B0,2 : LT bonds issued at t = 0, due at t = 2; P0,2 = 1

I B1,2 : ST bonds issued at t = 1, due at t = 2; P1,2 = β

I Some notation:

I D = B0,1 + B0,2: Scale of initial government borrowing

I S = B0,1/D: Short-term share of government debt
Trade-o§ Model of Government Debt Maturity

Government and Household Budget Constraints

I Government Önances a one-time expenditure G at t = 0

I Government budget constraint: Uses = Sources

t = 0: G = τ0+B0,1P0,1+B0,2P0,2

t = 1: B0,1= τ1+B1,2P1,2

t = 2: B1,2+B0,2= τ2

I Distortionary costs of taxes: Captured through a convex function of the tax

rate, (1/2) τ

2

, which induces a tax-smoothing motive

I Household consumption: Substitute in government budget constraint:

C0= 1  τ0  (1/2) τ

2

0  B0,1P0,1B0,2P0,2

C1= 1  τ1  (1/2) τ

2

1 + B0,1B1,2P1,2

C2= 1  τ2  (1/2) τ

2

2 + B1,2+B0,2

=1  (1/2) τ

2

0G

=1  (1/2) τ

2

1

=1  (1/2) τ

2

2
Trade-o§ Model of Government Debt Maturity

Social Plannerís Objective Function

I The social planner maximizes

U = C0 + E [C1 + βC2] + v (M0)

subject to the governmentís budget constraint

I Planner values monetary services from short-term debt

I Planner wants taxes to be low and smooth over time
Trade-o§ Model of Government Debt Maturity

Solution without Money Demand

I Without money demand, terms involving v () disappear

I Bond prices: P0,1 = P0,2 = 1 and P1,2 = β is realized at t = 1.

I Solution = Perfect tax-smoothing

I τ0 = τ1 = τ2 = G/3, B0,1 = B0,2 = G/3, and D = (2/3) G

I S = 1/2 and B1,2 (β)  0 for all realizations of β

I Intuition: In the absence of money demand, the govt. perfectly smooths

taxes over time by issuing a long-term ìconsolî bond that makes the same

payment each period. The govt. never rolls over debt at the interim date,

thus fully insulating budget/taxes from uncertain future reÖnancing.
Trade-o§ Model of Government Debt Maturity

Solution with Money Demand

I Prices: P0,1 = 1 + v

0

(B0,1) and P0,2 = 1

I v

0

(B0,1)=Money premium on ST debt

I b  Var [β] /2. First order condition for S:

Marginal tax-smoothing cost

z }| {

Db (S  1/2) =

Marginal beneÖt of money

z }| {

v

0

(SD) +

Marginal tax-lowering beneÖt

z }| {

τ0 



v

0

(SD) +SDv 00 (SD)



Central trade-o§:

1. Tax-smoothing cost: When S > 1/2, must raise taxes when ST rates are

high at t = 1. Smoothing costs are large if D (Debt/GDP) is large or if

uncertainty about ST rates (b) is high

2. Direct money beneÖt: Planner is willing to incur some tax-smoothing costs

to deliver monetary services to households

3. Tax-lowering beneÖt: Can raise revenue by taxing or by selling liquidity

services. If the latter is non-distortionary, this pushes further toward ST
Trade-o§ Model of Government Debt Maturity

Solution with Money Demand (Cont.)

I Ignore tax-lowering beneÖt in what follows for simplicity

I Equiv. to assuming that all ways of raising revenue are distortionary

I Same conclusions if we include this e§ect (Prop. 3 in paper)

I Basic result: In presence of money demand (v

0

() > 0), govt. chooses a

shorter maturity structure (S

> 1/2), trading o§ the increased reÖnancing

risk of more ST debt against the beneÖts of additional monetary services

I Comparative statics for S



:

I Go shorter when future short rates more uncertain

I Go longer when govt. spending and debt are large relative to GDP

I Issue short when money demand is strong (e.g. Fall ë08)
Adding Private-Sector Money Creation

Summary

I Formulation of the private-sector money creation follows Stein ë12

I Continuum of banks borrow from households to invest in real projects

I Issue either ST debt or LT debt

I ST debt is made riskless by liquidating assets in bad state at t = 1

I Since ST debt is riskless, it is cheap: banks can capture money

premium v

0

(M0)

I However, resulting Öre sales reduce the quantity of real investment

I Banks prefer to issue cheap ST debt, even though doing so incurs risk

of Öre sale

I But donít fully internalize the social cost of under-investment in bad state

I ) Socially excessive short-term Önancing (private money creation)
Adding Private-Sector Money Creation

Banks, Investment Projects, and Financing

I For simplicity, assume that banks invest a Öxed amount I at t = 0

I Good state occurs with probability p: Project returns F > I

I Bad state occurs with probability 1  p: Expected output λI  I with

non-zero probability of 0 ) LT debt is not riskless

I Bank can Önance this investment by issuing:

I Risky long-term bonds due at t = 2

I ST riskless bonds with face value MP : Results in savings of MP  v

0

(M0)

relative to long-term
Adding Private-Sector Money Creation

Fire sales

I If the bad state occurs at t = 1, bank must liquidate fraction ∆ of its

assets to pay-o§ short-term bond holders

I ∆ satisÖes MP = ∆kλI where k < 1 is endogenous Öre-sale discount

I Assets purchased by patient investors (PIs) with war chest W

I PIs can buy existing bank assets or invest K in new real projects at t = 1

which return g(K) at t = 2 where g

0 > 0 and g00 < 0

I Fire sales a§ect real investment at t = 1: In the good state, new

investment is K = W ; in the bad state, K = W  M

P

I Imperfect pledgeability: only fraction φ < 1 of returns from new

investments are pledgeable to PIs:

I ) Banks do not fully internalize the social costs of Öre sales

I Equilibrium determination of k: PIs must be indi§erent between

buying existing bank assets and investing in real new projects

Firesale return on existing bank assets

z}|{

1/k =

PRIVATE return on marginal real investment

z }| {

φg

0

(W  M

P

)
Adding Private-Sector Money Creation

Private Market Solution

I Private Market Solution: Banks trade-o§ beneÖts of cheap

short-term debt versus the cost of Öre-sale liquidations, but do not fully

internalize the latter

I Contrast with social planner: planner takes the full cost of Öre sales

into account (i.e. sets φ = 1 in the above), so socially optimal quantity

of private money, M

P

, is less than private market outcome, M

P

.
The Social Plannerís Problem

First Best

I The plannerís objective function is to maximize Utility from money -

Distortionary costs of taxes + NPV of time 1 investment

I Assume planner can directly control private money MP

I Thus, planner chooses 3 variables: MP , D, and S.
The Social Plannerís Problem

First Best (Cont.)

I Comparative advantage principle: At the social optimum, the

marginal social cost of both private and public money creation are set

equal to the marginal social beneÖt of additional money services:

Fire-sale cost

of private money

=

Marginal beneÖt

of money services

=

Tax-smoothing cost

of govt. money

I May be costly/di¢ cult to implement the Örst best outcome via

regulations that limit private money creation

I Private money creation may simply áow from regulated to unregulated

sectors (i.e. the ìshadow banking systemî) in response to heightened

liquidity regulations ... but externality still exists

I Regulation may otherwise create deadweight costs
The Social Plannerís Problem

Second Best Implementation without Direct Regulation

I Suppose that it is impossible or prohibitively costly to directly regulate

private money (will relax this below)

I However, government can still reduce the temptation to engage in

private money creation by issuing more T-bills

I If govt. money creation is MG , equilibrium private money creation is

Internalized Öre-sale cost

of private money

=

Marginal beneÖt

of money services

I DeÖnes a decreasing reaction function of private money based on public

money
The Social Plannerís Problem

Second Best Implementation without Direct Regulation (Cont.)

I First order condition for short-term share, S:

Tax-smoothing cost

of govt money

=

Marginal beneÖt

of money services

+

Crowding out

beneÖt

I ìCrowding-outî term is positive.

I Intuition for ìCrowding-outî beneÖt: The govt. depresses the

premium on money-like claims by issuing more T-bills. This crowds out some

private money creation and reduces under-investment in the bad state.

I ìCrowding-outî beneÖt is linked to the di§. between social and private

investment return in bad state
Magnitude of the "Crowding-out" motive

A back-of-the-envelope calculation

I Money beneÖt:  40 bps based on extreme steepness of front-end of

the yield curve

I Crowding out beneÖt: (1  p) (φ  1) g

0

(W  M

P

) ∂M

P

/∂MG

I Annual probability of a crisis: (1  p) = 5%

I based on Barro and Ursua (2008) and Laeven and Valencia

I Non-pledgeable fraction of investment: (1  φ) = 10%

I chosen somewhat arbitrarily, but seems plausible

I Gross Öre-sale return in bad state: g

0



W  M

P



= 130%

I based Pulvino (1998) and Campbell, Giglio, & Pathak (2011)

I Crowding-out impact: ∂M

P

/∂MG = 100%

I from estimates in Table 2

I Crowding out beneÖt = 0.05 0.10 1.30 1 = 65 bps

Plausibly the same magnitude as money beneÖt
The Social Plannerís Problem

Second Best: Allowing for Direct Regulation of Private Money

I Now suppose the govt. can impose a tax on private money creation at

rate θP

I However, regulation is imperfect/costly:

I Pigouvian taxes create deadweight costs of (Υ/2)θ

2

P

I Reduced-form way of capturing resources banks devote to evasion /

regulatory arbitrage

I Government now has two toolsó ìcrowding-outî by issuing more ST or

direct regulationó both of which are costly to use

I Equilibrium private money creation is pinned down by

v

0

(M

P + MG ) = θP + (1  p)



φg

0



W  M

p



 1



.

I DeÖnes a reaction function M

P

(MG , θP ) with lower private money

when MG or θP is high
The Social Plannerís Problem

Second Best: Allowing for Direct Regulation of Private Money (cont.)

I First order condition for S

Tax-smoothing cost

z }| {

D

b (S  1/2) =

Money beneÖt

z }| {

v

0

(M

P + S

D) +

ìCrowding-outî beneÖt

z }| {

Ω (1  p) (φ  1) g

0

(W  M

P

)

∂M

P

∂MG

where Ω = Υ/ (Υ + j∂M

P

/∂θP j) < 1: with direct regulation,

crowding-out beneÖt is reduced

I Under some conditions, govt. does more crowding out and less

regulation when (i) tax smoothing costs are lower or (ii) when costs of

direct regulation are higher
Conclusion

I Trade-o§ model of optimal government debt maturity: satisfying money

demand vs. tax-smoothing

I Tax-smoothing costs loom larger when the debt is larger relative to GDP

I Government issues more ST when the demand for money is stronger, or

when there is less uncertainty about future short rates

I Extend model to allow for competing private creation of money-like securities

I Comparative advantage principle reigns:

I If there are uninternalized costs associated with private money creation,

government should crowd out private money

I Conclusion holds so long as regulation of private money is

costly/imperfect

I Open questions:

I Implementation: Treasury vs. central bank?

I Model is about ëlongíversus ëshortí, but money premium appears

primarily in the ëvery shortí
Extra Slides 1

Multiple maturities

I Suppose there are multiple maturities of debt: short, medium, and long

I Optimal maturity structure depends on type of shifts to yield curve:

I If interest rate shocks primarily involve parallel shifts in the yield curve:

I Govt. can create a large volume of monetary services without incurring

much reÖnancing risk

I Govt. implements this by pursuing a a ìbarbellî strategy: issues lots of

short- and long-term debt, but little medium term debt

I Govt. keeps the average maturity of debt close to that is the

perfect-smoothing (consol-bond) solution

I If there is a signiÖcant risk that the yield curve can change shape:

I Govt. must incur more reÖnancing risk, so it creates a lower volume of

monetary services

I Govt. pursues less of a barbell strategy

I Govt. shortens the average maturity of debt
Extra Slides 2

Private Money Less Valuable than Public Money

I Suppose money utility given by v (kPMP + MG ) where kP < 1

I First best private money set lower according to

Fire-sale cost

of private money

z }| {

(1  p) (g

0

(W  M

P

)  1)

kP

=

Marginal beneÖt

of money services

z }| {

v

0

(kPM

P + MG

) =

Tax-smoothing cost

of govt. money

z }| {

b (M

G  D

/2)

I Second best condition for S

Tax-smoothing cost

z }| {

D



b (S

  1/2)

=

Money beneÖt

z }| {

v

0

(kPM

P + S

D

) +

ìCrowding-outî beneÖt

z }| {

(1  p) (φ  1) g

0

(W  M

P

)

∂M

P

∂MG

I Summary: Basic forces unchanged but less equilibrium private money
Extra Slides 3

I Table 2: Determinants of Private Money
Extra Slides 4

I Bennett, Garbade, and Kambhu (2000):

I "Minimizing the cost of funding the federal debt is a leading objective of

Treasury debt management policy...In the most extreme form, the

Treasury Department could Önance any current deÖcit, and reÖnance

maturing debt, with frequent sales of large quantities of shorter bills.

This would concentrate Treasury indebtedness in the most liquid sector of

the market: large, short-maturity, and unseasoned discount obligations."

I "...The Treasury has historically chosen to issue at a variety of short,

intermediate, and long maturities. This policy has ancillary beneÖts: ...

it facilitates budget planning because it enhances the predictability of

interest expenses during a Öscal year and over longer intervals."