Monetary Policy, 22/3 2018

Henrik JensenDepartment of EconomicsUniversity of Copenhagen

Plan for today:

1. Public budget accounting, inflation and debt

2. Equilibrium seigniorage

Literature: Walsh (Chapter 4, pp. 137—162)

c© 2018 Henrik Jensen. This document may be reproduced for educational and research purposes, as long as the copies contain this notice and are retained for personaluse or distributed free.

Introductory remarks• Inflation or the nominal interest rate have been viewed as a tax on household’s resources in theprevious lectures.

—In particular, through the erosion of real money balances

—This inflation tax has been shown to have implications for private-sector behavior (money demand,consumption, labor supply etc.)

• The flip-side of the “tax-coin”has been ignored: All proceedings from the tax have been returned aslump-sum transfers. I.e., the government budget includes no spending or government debt servicing

• One ignores central, and highly relevant, questions like:

—Can inflation be used as a means of financing public expenditures, deficits and debt?

—Will monetary and fiscal policy interact in ways that qualify some of the conclusions reached sofar from the flex-price models?

—Will fiscal policy (e.g., deficit creation) have implications for monetary policy and thus, e.g.,inflation?

—Can inflation always “save the day”for a public budget, or can it ultimately “ruin the day”?

—Is money creation always inflationary? (Particular relevant in currenttimes of “quantitative eas-ing”)

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Seigniorage across (some) countries, 1960-1999

Source: Aisen and Veiga (2005)

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Continued

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Public budget accounting, inflation and debt

The core link between monetary and fiscal policy is the public budget constraint

• The fiscal branch of government (the Treasury) satisfy the following budget identity in nominal terms

Gt + it−1BTt−1 = Tt +

(BTt −BT

t−1

)+ RCBt (4.1)

Superscript T denotes total public debt; RCBt is receipts from the central bank

• The central bank’s budget identity may be written(BMt −BM

t−1

)+ RCBt = it−1B

Mt−1 + (Ht −Ht−1) (4.2)

SuperscriptM denotes public debt held by the central bank;Ht is “high-poweredmoney”– the mon-etary base (the central bank’s own liabilities: Currency in circulation and reserves held by commercialbanks)

• Let Bt = BTt −BM

t be public debt held by the private sector. The two budget identities are combinedto the consolidated budget identity

Gt + it−1Bt−1 = Tt + (Bt −Bt−1) + (Ht −Ht−1) (4.3)

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• Expressed relative to total nominal income Ptyt, and ignoring population and output growth:

gt + rt−1bt−1 = tt + (bt − bt−1) +Ht −Ht−1

Ptyt; (4.4’)

lower-case letters are variables relative to total nominal income and

rt−1 ≡1 + it−1

1 + πt− 1

is the ex post real interest rate

• Define the ex ante real interest rate as

rt−1 ≡1 + it−1

1 + πet− 1,

=⇒ gt + rt−1bt−1 = tt + (bt − bt−1) +(1 + rt−1) (πt − πet)

1 + πtbt−1 +

Ht −Ht−1

Ptyt(4.5’)

The last term of (4.5’) is seigniorage, the real proceedings from issuing non-interest-bearing debt:Base money

• Seigniorage relative to total nominal income is

st ≡Ht

Ptyt− Ht−1

Ptyt= ht −

1

1 + πtht−1 = ht − ht−1 +

πt1 + πt

ht−1 (4.6’)

• Even in steady state, st > 0 when π > 0, as h is a real liability for the government;Inflation “taxes”its real value

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• What if inflation is zero? Are there no revenue from seigniorage? Yes, by issuing non-interest bearingdebt (= money) instead of interest bearing debt (=bonds), the government saves interest paymentsfor given total debt

To see this, rewrite budget identity, defining dt ≡ bt + ht as total liabilities:

gt + rt−1 (dt−1 − ht−1) = tt + (dt − dt−1) +(1 + rt−1) (πt − πet)

1 + πt(dt−1 − ht−1) +

πt1 + πt

ht−1

gt + rt−1dt−1 = tt + (dt − dt−1) +(1 + rt−1) (πt − πet)

1 + πtdt−1 +

[πt − (1 + rt−1) (πt − πet)

1 + πt+ rt−1

]ht−1

gt + rt−1dt−1 = tt + (dt − dt−1) +(1 + rt−1) (πt − πet)

1 + πtdt−1 +

it−1

1 + πtht−1 (4.8)

• When constraint is formulated in terms of total liabilities, the steady-state seigniorage is

s =i

1 + πh (4.9)

It is positive for i > 0– represents saved interest on b

Hence, (4.9) takes into account the composition of public liabilities

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• By accounting, a change in s or s requires offsetting changes in either taxes, spending or debt

• How such monetary changes affect fiscal policy depends on the “definition of fiscal policy”

—If it is in terms of fixed spending and interest-rate bearing debt, changes in s and the offsettingchanges in taxes are monetary policy

—If it is in terms of fixed spending and total liabilities, changes in s and offsetting taxes andcomposition of liabilities are monetary policy

—So which seigniorage definition is appropriate, depends on how fiscal policy is conducted

• Note implication of simple version of budget identity (ignoring unanticipated inflation, and assumingrt = r > 0)

gt + rbt−1 = tt + bt − bt−1 + st,

the “solvency requirement”:

(1 + r) bt−1 +

∞∑i=0

gt+i

(1 + r)i=

∞∑i=0

tt+i + st+i

(1 + r)i(4.10’)+(4.11’)

holds when limi→∞ (1 + r)−i bt+i = 0 – no “Ponzi games”

If the government has initial debt, it must at some future date run surplus(es), tt+i + st+i > gt+i,generated through taxes or seigniorage

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• This raises the issue of whether government debt or deficits will ultimately create seigniorage andpotentially inflation

This will depend on the “fiscal-monetary regime”

—If fiscal policy is “dominant”(or “active”), monetary policymust be “passive,”and secure solvency

—If monetary policy is “dominant”, fiscal policy must secure solvency

• Hence, in regimes of fiscal dominance, it may be the case that debt and deficits will be inflationary

• Also, it may be the case that monetary contractions (e.g., aimed at reducing inflation) will reduceseigniorage revenues, increasing deficits and debt, which ultimately requires increased seigniorage,and thus, inflation in the future (Sargent and Wallace’s “Unpleasant Monetarist Arithmetic”)

• This emphasizes that treating money as independent of fiscal policy, could be misleading, as monetarypolicy changes could very well be the result of changes in fiscal policy

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A simple model to prove the point

• Assume government spending is zero. The budget constraint then becomes

(1 + rt−1) bt−1 = tt + bt + st (4.15)

• Let the present value of taxes cover a fraction of current government liabilities (1 + rt−1) bt−1:

Tt ≡∞∑s=t

(1

1 + rs

)s−tts = ψ (1 + rt−1) bt−1, 0 < ψ ≤ 1

—For ψ = 1, any debt is fully backed by taxes over time (this is sometimes referred to as aRicardian fiscal policy – or, non-dominance in fiscal policy)

—For ψ < 1 only a fraction is backed. I.e., some fiscal dominance is present

• The present value of taxes is the bounded solution to the forward-looking difference equation

Tt = tt +1

1 + rtTt+1

Hence,

Tt = tt +1

1 + rtψ (1 + rt) bt = tt + ψbt

• From the assumption about Tt one gets

ψ (1 + rt−1) bt−1 = tt + ψbt

Note that with ψ = 1 one gets the government budget constraint with st = 0. So, for ψ < 1,seigniorage is necessary

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• Now consider the households’budget constraint (yt is endowment)

yt + (1 + rt−1) bt−1 +mt−1

1 + πt− tt = ct + mt + bt

• Inserting the expression for tt from the government budget constraint one gets

yt + (1− ψ) (1 + rt−1) bt−1 +mt−1

1 + πt= ct + mt + (1− ψ) bt

—For ψ = 1, “debt disappears”. In general equilibrium, it plays no role for determination of theprice level; only the money stock matters

—For ψ < 1, debt will matter for asset demand and price determination

• To exemplify the role of debt for prices (and inflation), consider a MIU preference specification forhouseholds, where the per-period utility function is given by

u (ct,mt) = ln ct + δ lnmt, δ > 0,

The households then maximize life-time discounted utility subject to the budget constraint

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Deriving the money demand function and consumption-Euler equation.Material not for lecturing, but for reading (it is essentially deriving what Walsh stateson p. 150).

• Maximization characterized by

V (bt−1,mt−1) = max {ln ct + δ lnmt + βV (bt,mt)}

where maximization is over c, m, and b and subject to the budget constraint. As usual, use thebudget constraint to eliminate bt to obtain an unconstrained maximization problem over c and m

• First-order condition w.r.t. c:1

ct=

β

1− ψVb (bt,mt) . (i)

• First-order condition w.r.t. m:δ

mt+ βVm (bt,mt) =

β

1− ψVb (bt,mt) . (ii)

• Relationships between partial derivatives of the value function by the envelope theorem:

Vb (bt−1,mt−1) = β (1 + rt−1)Vb (bt,mt) , (iii)

Vm (bt−1,mt−1) =β

(1 + πt) (1− ψ)Vb (bt,mt) . (iv)

• Forward (iii) and multiply by β/ (1− ψ) on both sides to get:

β

1− ψVb (bt,mt) =β2

1− ψ (1 + rt)Vb (bt+1,mt+1) .

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• Then use (i) to obtain the consumption Euler equation (the Keynes-Ramsey rule):1

ct= β (1 + rt)

1

ct+1

ct+1 = β (1 + rt) ct (v)

• Then use (iv) on (ii) to getδ

mt+

β2

(1 + πt+1) (1− ψ)Vb (bt+1,mt+1) =

β

1− ψVb (bt,mt)

δ

mt+

β

(1 + πt+1)

1

ct+1=

1

ctδ

mt+

1

(1 + πt+1) (1 + rt)

1

ct=

1

ct

where the last two equations follows from applying (iv) and (i), and finally (v).

• From this the money demand relationship follows as:

mt = δ

[1− 1

(1 + πt+1) (1 + rt)

]−1

ct

mt = δ

[1− 1

(1 + it)

]−1

ct

and thus

mt = δ

(1 + itit

)ct (vi)

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• Note that this follows immediately from the general characterization of optimal moneydemand from the MIU approach in Chapter 2

um (ct,mt)

uc (ct,mt)=

it1 + it

• This indeed givesδ/mt

1/ct=

it1 + it

,

and thus the money demand function (vi)

End of material not for lecturing

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• Definewt ≡ mt + (1− ψ) bt

as the “total”real wealth net of taxes: Real money balances plus non-tax-backed government debt

Household budget constraint then becomes

yt + (1 + rt−1)wt−1 −it−1

1 + πtmt−1 = ct + wt

• Remarkit−1

1 + πtmt−1,

which is the deduction from total net wealth arising from the fact that real money pays no interest

• Use the lagged money demand function and consumption-Euler equation derived above,

mt−1 = δ

(1 + it−1

it−1

)ct−1, ct = β (1 + rt−1) ct−1

yt + (1 + rt−1)wt−1 −it−1

1 + πtδ

1 + it−1

it−1ct−1 = ct + wt

yt + (1 + rt−1)wt−1 − δ (1 + rt−1) ct−1 = ct + wt

yt + (1 + rt−1)wt−1 −δ

βct = ct + wt

• Use that yt = ct in equilibrium and examine the steady state:

wss =δ

βrssyss =

M ss + (1− ψ)Bss

P ss

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• The price level is thus determined as

P ss =βrss

δyss[M ss + (1− ψ)Bss] (4.18)

Government debt matters for the price level when ψ < 1, i.e., when there is some fiscaldominance

• Empirics (Resende and Rebei, 2008):

—Canada, ψ ≈ 0.99.

—US, ψ ≈ 0.96.

—South Korea, ψ ≈ 0.78.

—Mexico, ψ ≈ 0.63

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Equilibrium seigniorage

• What can seigniorage achieve in terms of financing given deficits? Anything? Or are there limits?

• What are the inflationary implications of relying on seigniorage? Can hyperinflations result fromseigniorage collection?

Are there limits to collection of seigniorage?

• Yes!

—On the one hand, higher inflation and nominal interest rates, increase seigniorage for given realmoney balances

—On the other hand, real money balances will fall as inflation and nominal interest rates increase(a money demand response)

—Hence, as the inflation tax goes up, the tax base is going down

• This is shown formally in a MIU model

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• It suffi ces to find a money-demand function from the usual first-order condition:um (ct,mt)

uc (ct,mt)=

it1 + it

≡ Υt

Specific form of per-period utility function:

u (ct,mt) = ln ct + mt (B −D lnmt) , B > D > 0

• Resulting money demand relationB −D lnmt −D

1/ct= Υt

leading tomt = Ae−Υt/Dct, A ≡ e(B−D)/D. (4.24’)

A standard specification of money demand

• [Often replaced by Cagan’s (1956) version:

mt = Ke−απet , α > 0,

in studies of hyperinflations (where assumption is that real interest rate and consumption/outputfluctuates relatively little), or for steady-state/long-run analyses, where by the Fisher relationshipinflation and nominal interest rate moves one-for-one]

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• Steady-state seigniorage is [from (4.9)]

s =i

1 + πm = (1 + r)

i

1 + im = (1 + r) Υm

• Using the money demand function one gets

s = (1 + r) ΥA exp[− Υ

Dc

]Assume r and c are invariant to monetary policy in steady state

—For Υ close to zero, seigniorage is close to zero; this will be a characteristic of relatively lowinflation

—For Υ very high, seigniorage is also close to zero; this will thus be a characteristic of relativelyhigh inflation

• Hence, with inflation increasing from a low level, seigniorage is increasing, but eventually the fallingmoney demand reduces seigniorage. A maximal amount of seigniorage thus exist:

• An inflation rate π∗ exists for which seigniorage is at a maximum:

—For π > π∗ equilibrium seigniorage is decreasing in π

—For π < π∗ equilibrium seigniorage is increasing in π

—=> A seigniorage “Laffer curve”is faced by the government

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Inflationary implications of relying on seigniorage

• The Laffer-curve property implies that two steady-state inflation rates can finance the same deficit;a high and a low inflation rate

• Stability properties of the two steady states, both accomplishing the same financing target?

—Walsh (2017) exemplify how low-inflation equilibrium is stable but high-inflation equilibrium isunstable

—An increase in finance requirements will then raise inflation, but cause accelerating inflation ifthe economy was in the unstable equilibrium

• Also, the Laffer curve property shows that there are limits as to how much one can finance byseigniorage

—If financing requirement suddenly increases above what is feasible to finance by seigniorage, thegovernment may engage in futile financing attempts by printing money at a faster rate

• In both cases, hyperinflations arise (often defined as monthly inflation rates of +50%), which canonly be stopped by fiscal reform

• Examples: Weimar Republic in the 1920s, Zimbabwe in 2008—9

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• Children using money as toys in Weimar Republic during 1922

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• Money denominations printed within a year in Zimbabwe 2007—8

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Inflation “bubbles”

• Hyperinflations can also be non-fundamental; i.e., occur in isolation of money growth. These arelabelled speculative hyperinflations (or “bubble paths”)

• As an example, let money demand be (now variables are logarithms):

mt − pt = −α (Etpt+1 − pt) , α > 0,

This is rearranged as an expression for the (log of) price level:

pt =1

1 + αmt +

α

1 + αEtpt+1 (4.29)

This is a first-order expectational difference equation in pt (as it depends on its expected future value)

• Money is for simplicity assumed to be constant, mt = θ0

The variable θ0 is the model’s fundamental, and a solution of pt depending only on the fundamentaland parameters is a fundamental solution

• To find this solution, use the method of undetermined coeffi cients:

—Conjecture a form of the solution by undetermined coeffi cients

—Forward the conjecture, take expectations and insert into the expectational difference equation

—Identify the coeffi cients if form of conjecture is correct

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• Here: Conjecture the solution pt = Xθ0, where X is the undetermined coeffi cient

• Forward the conjecture and take expectations: Etpt+1 = Xθ0 and insert into difference equation:

pt =1

1 + αθ0 +

α

1 + αXθ0 =

1 + αX

1 + αθ0

• The undetermined coeffi cient is then identified as pt = Xθ0 = 1+αX1+α θ0 and thereby

X =1 + αX

1 + α

and thus X = 1 Hence, pt = θ0 is the fundamental solution

• However, it is easy to see that infinitely many solutions of the form

pt = θ0 + bubt, bubt ≶ 0

exists forEtbubt+1 =

(1 + α−1

)bubt (bubt will grow with α−1)

• To see this, note that is it fully consistent with the difference equation:

θ0 + bubt =1

1 + αθ0 +

α

1 + αEt [θ0 + bubt+1]

θ0 + bubt =1

1 + αθ0 +

α

1 + α

[θ0 +

(1 + α−1

)bubt

]θ0 = θ0

• Hence, if bubt > 0, we have growing prices even though the money supply is constant

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Summary

• Monetary and fiscal policy are linked through the public budget constraint

• Ignoring this may be valid if governments have access to lump-sum taxation and follow policies thatfully back interest-bearing debt with taxes (no fiscal dominance)

• Otherwise, important channels frommonetary policy to fiscal policy and vice versamay be overlooked,as the financing properties of inflation is ignored

• Also, it is important to stress that an observed change in monetary policy may or may not be dueto fiscal considerations, and therefore have different implications for the real economy depending onthe source of the change

• While a potential financing tool, one must be aware of the dangers of hyperinflation associated withreliance on seigniorage as a means of financing public expenditures

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Plan for next lecture

Wednesday, March 28

1. Inflation and quantitative easing

2. Open-market operations at the zero lower bound for interest rates

Literature: Auerbach and Obstfeld (2005, AER), pp. 110—117, until Section IV)

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