The Case for Open-Market Purchases in a Liquidity Trap
Alan J. Auerbach and Maurice Obstfeld
2005
Keywords: monetary policy
[WHAT ROLES DOES A SIMULTANEOUS TAX CUT PLAY HERE? ARE TAXES
AUTOMATICALLY CUT TO MATCH THE SEIGNIORAGE SO AS TO MAINTAIN
NEUTRAL DEFICIT?]
Prevalent thinking: perfect substitutability of money and short term bonds at ZLB. However, if
short term rate are expected to be positive at some future point, than
1. “Trading money for interest-bearing public debt reduces future debt-service requirements
and hence the distortions of the requisite taxes,” even if such operations were to be
perceived as totally ineffective at influencing current prices or output.
2. Moreover, “The same reasoning implies that credibly permanent open-market operations
will be beneficial as a stabilization tool as well, even when the economy is expected to
remain mired in a liquidity trap for some time. That is, […] a monetary expansion that
markets perceive to be permanent will affect prices and, in the absence of fully flexible
prices, output as well.”
For a flexible price model with monopolistic competition and distortionary taxes, even with
short-term rates at the ZLB, open market operations can counteract deflationary tendencies,
reducing the real value of public debt and thus the burden of future taxes. Conditions:
1. Long term nominal interest rates must be positive at some horizon [i.e., natural rate is
expected to be positive at some horizon]
2. Central bank must be able to credibly carry out permanent expansions of the money
supply
I.
Model setup
A. Households
Households maximize utility over consumption and labor
-
Specific functional form of utility (unless otherwise noted):
-
Household’s real wealth at beginning of period t, before payment of interest
-
Household wealth at period t+1: (Omega is profits from firms)
-
Define real interest rate rt = (1+it)/[Pt/Pt-1]-1
Evolution of real wealth
Money demand comes from a cash in advance constraint
- Let tau be the consumption tax
- This is always binding, unless it=0
- Thus, it is always true [by Kuhn-Tucker conditions] that
𝑖 𝑡 𝑀𝑡 = 𝑖𝑡 (1 + 𝜏𝑡 )𝑃𝑡 𝐶𝑡
- Rewrite evolution of wealth as
-
Because Tt = τCt
-
Solve forward and impose transversality condition to get the lifetime budget constraint:
Utility maximization problem
- Maximize utility subject to the above budget constraint
- MRS condition:
-
Euler equation
B. Firms and price-setting
Continuum of producers with monopolistic competition
- Ct above is a composite good
Nominal price stickiness: staggered price setting
- Each producer must set a nominal price that is maintained over two periods
- Two classes of goods: C1 (has prices set in odd periods) and C2 (has prices set in even
periods)
-
Let cti(z) be a type z good of class i in period t
Production function:
Y=L
Goods enter utility via Cobb-Douglas preferences:
1/2 1/2
𝐶𝑡 = 2 𝐶𝑡1 𝐶𝑡2
Composites are a CES function of individual goods
𝜌−1
𝐶𝑡𝑖 =
1
[∫0 𝑐𝑖𝑡 (𝑧) 𝜌
𝑑𝑧]
𝜌
𝜌−1
rho>1, i=1,2
Define a price index
1/2 1/2
𝑃𝑡 = 2𝑃𝑡1 𝑃𝑡2
Where
1
𝑃𝑖𝑡 =
1
1−𝜌
[∫0 𝑝𝑡𝑖 (𝑧)1−𝜌 ]
II.
Flexible-price case
Nominal interest rates at zero for a finite period; thus CIA constraint not binding in this period
𝑖𝑡 = 0
∀𝑡 = 0,1, … , 𝑇 − 1
𝑖𝑡 > 0
∀𝑡 ≥ 𝑇
Under flexible prices, monetary policy can still affect the price level before date T
- “That is, any prospect of future nominal interest rates above zero, no matter how remote,
implies that the economy cannot be in a monetary-policy trap. Indeed it suffices, one can
show, that there be some future state of nature, occurring with any positive probability, in
which it > 0.”
Proof
- Write the Euler equation in terms of wages rather than prices by equalizing (3) and (4)
-
Thus, for all t < T–1
-
Thus, at the ZLB, wages (and prices) rise if (or are constant if equals to, etc)
𝑘𝑡+1
>1
𝑘𝑡
- From Euler equation (4) and binding CIA constraint (2), we have
𝛽𝑡+1
-
From (5) and (7), the wage at t > T–1 evolves according to
-
Thus, once we have an expression for wT-1, (6) and (8) provide us the whole path of
wages
Write Euler equation (4) for dates T-1 and T; substitute CIA constraint (2) at T; substitute
𝑤
labor-consumption MRS (3) at T-1. Thus 𝑘 𝑇−1 = 𝑀𝑇 /𝛽𝑇
-
Solve forward (6) and (8), and substitute to get:
-
(note they are the same for t=T-1)
𝑇−1
Thus, a permanent increase in the money supply today Δ𝑀0 can still affect the nominal wage
level:
- Assume that all future money growth rates remain fixed, i.e. Mt+1/Mt unchanged for all
t>0
- MT rises by a factor of 1 + Δ𝑀0 /𝑀0 (as do all money-supply levels after t=0)
- [Numerical examples (mine): money supply at t=0 is 100. Money growth rate is 2% for
all periods except the first one. In the first period, increase the money supply 10% to 110.
At t=10, money supply will be 110*(1.02)^(10)=134.089, instead of
100*(1.00)*1.02^10=124.337
- Note 124.337*1.1=134.089
- More simply, imagine a stable money supply, except for a one period increase of 10%]
- Equation (9) demonstrates that – despite a zero nominal interest rate – all future nominal
wage rates, yes, including those for dates 0 through T-1, will be scaled up by 1 +
Δ𝑀0 /𝑀0
It can be shown that monopolists charge a fixed percentage markup over marginal cost, which is
wage, so that 𝑃𝑡 = 𝜌𝑤𝑡 /(𝜌 − 1). Thus this increase in the nominal wage level increases the price
level.
During a a period of zero short-term interest rates, the price level is governed by the money stock
expected for the first date on which interest rates turn positive.
- Intuition: (“disarmingly simple”)
- On date T, money stock determines the price level in the conventional way
- As long as the interest rate is zero, the price level moves toward that price level at a rate
governed by the household’s rate of time preference
- By raising the money stock permanently, the future price level must go up; that
necessitates upward shift in the time path of the price level
“The dependence of nominal prices on expectations of money supplies on future positiveinterest-rate dates, rather than on the current money supply, is the essence of the liquidity trap”
- “The central bank can affect today's equilibrium by changing expectations of future
money supplies. In our analysis, the central bank changes those expectations simply by
changing the money stock's level immediately and allowing ‘base drift’”
However, even fixing money supplies after the liquidity trap ends, the current price level is not
independent of the central bank’s control: as shown below, a sufficiently large reduction in the
current money supply, given future moneysupplies, could end the liquidity trap by forcing the
short term nominal rate up from zero
- [Wut??]
Note this is essentially a dynamic extension of Krugman’s two period model
- Allows for really long liquidity traps
- Even allows for indefinitely long liquidity traps: replace everything with expectations; as
long as there is still some small probability of eventually exiting the liquidity trap, the
central bank can still affect the current price level
- (If only very small probability of exiting a la Japan, takes extremely large permanent
increase in money supply today)
III.
Welfare analysis of open market purchases under flexible prices
“Consider the impact on welfare of an open- market operation, as measured by a policy index
psi”
-
-
(makes use of fact that under flexible prices Ct=Lt
Note the expression = 0 if there is no seigniorage (i=0), no taxes (𝜏=0), and no markup
(P/w = 1): i.e. with no distortions, allocation is Pareto optimal
Assume 𝜏 is set to be constant over time; psi holds Mt+1/Mt constant in the future; thus all
𝑑𝑈
Let 𝑑𝜀0 = Δ
With some more manipulations, the dollar value of an open market operation:
This is the “standard result” that marginal deadweight loss equals the product of the
wedges due to taxes, seignoriage, and price markups over marginal cost
𝑑𝐶
For flexible prices, Mt+1/Mt fixed in the future, and a constant tax rate 𝜏, we have 𝑑𝜀𝑡 =
𝑡
−𝐶𝑡
𝑑𝑙𝑛(1+𝜏)
𝑑𝜀
, so we have
-
-
Summation equals the present value of revenue from taxes and seigniorage plus
monopoly profits
So the welfare effect equals that total minus the percentage change in (1 + 𝜏)
“The welfare effect of an open market operation equals the present value of tax revenues,
seigniorage, and noncompetitive rents, multiplied by minus the percent change in the
term (1 + 𝜏), roughly the absolute reduction in the tax rate itself.”
If we make utility CRRA, then (12’) is the same multiplied by 𝛾, the intertemporal
elasticity of consumption; less sensitive preferences (how much you like to smooth) lead
to less sensitive welfare change
IV.
The sticky-price case
- [Monopolistically competitive firms result in a markup of prices over marginal cost.]
- Even anticipated inflation is beneficial; anticipated deflation is even bad
- This is because it can be shown that the markup is a decreasing function of the inflation
rate when (1 + 𝜋)𝛽 < 1 (and an increasing function of inflation when (1 + 𝜋)𝛽 > 1
1
- And since welfare is decreasing in the markup, optimal inflation is 𝜋 = 𝛽 − 1 > 0
When calculating welfare, it is no longer true that L=C.
- Relative price distortion between two sectors lowers consumption C below the cost of
production L = C1 + C2 whenever prices are not equal (??)
- Noting Lt + Ct1 + Ct2 and Cti = .5 (Pti/Pt)^(-1) Ct, and letting 𝛿𝑡 = 𝑃𝑡1 /𝑃𝑡2 , and then letting
1+𝛿
𝜙(𝛿) = 2𝛿1/2, and noting 𝜙 ′ (𝛿) = (𝛿 1\2 − 𝛿 −1\2 )/4𝛿, and that 𝜙(𝛿) = 𝜙(1/𝛿)
- Assume beta constant
- As before, if interest rate is zero, even with sticky prices 𝛿 = 𝛽 will hold, i.e. relative
prices, like the price level, are falling at the rate of time preference
- A small enough unexpected monetary expansion will push delta closer to one; but too
large an expansion will (temporarily) worsen the price distortion
- There is also the second unanticipated inflation effect on welfare through the output
channel as wellas through the effect on average markup
-
Blah
V.
Quantitative estimates of welfare gains
- Under flexible prices, “The welfare effect of an open market operation equals the present
value of tax revenues, seigniorage, and noncompetitive rents, multiplied by minus the
percent change in the term (1 + 𝜏), roughly the absolute reduction in the tax rate itself.”
- Under sticky prices, different in only transitory ways
- Calibrated to Japanese economy, a one percent permanent increase in the monetary base
increases output by 5%
VI.
Simulating various policy changes
- If we now take into account sticky prices
- Solution using backwards solving algorithm in numerical simulations
-
-
-
-
“A level shift in the money stock at the initial date, with no subsequent change in money
growth rates, will not affect the number of periods during which the liquidity trap
applies”
Suppose discount rate greater than one (1/.97) until period five, at which it falls below
one (1/1.02)
Suppose also kt, variations in which can be thought of as the same as variations in a
productivity parameter, falls at a rate of 0.05 until reaching 1 at period 5, when it remains
constant
“If productivity is relatively low now but is expected to be higher in the future, then
inflation will be lower to make room for the real balances needed to support higher
income level”
Money stock starts at 1.0 in period 0
Money growth at steady 2%:
- Liquidity trap until period five shift
One-time increase in money stock at period zero
- “Alternative paths for period-0 increases in the money stock of 1 percent and 10 percent.
The welfare gains from these policies are, respectively, 0.09 percent and 0.84 percent of
output per year”
Unannounced increase in the money stock can bring the economy out of the liquidity trap
immediately
- Money growth of 4% instead of 2% in periods 2 through 5
- Inflation of 2-3%
- Welfare gain is .66% of output per year
VII. Adding long-term bonds
- “composition of government debt is of no consequence” for a one-time increase in money
supply
- However, for an increase in the growth rate, there is an effect
- “Thus, along any equilibrium path, the composition of government debt is of no
consequence. The effects of an unexpected policy change, however, would depend on the
maturity structure of debt outstanding at the time of the policy announcement. Whereas
one-period bonds' nominal value is unaffected by a change in the path of future short-
term rates (because, by assumption, their coupons are reset each period), bonds of longer
duration will experi- ence nominal price changes in the opposite di- rection of changes in
future short-term rates. This provides a second channel through which an open market
monetary expansion can reduce the real value of government liabilities (money plus
debt), in addition to that induced by a price level increase”
VIII. Adding financial intermediaries
- “Until now, we have ignored the existence of financial intermediaries, assuming that the
money stock equals the monetary base, and hence is directly controllable by the central
bank. With a banking sector, the ability of the central bank to increase the money stock
depends on its success at getting the banking sector to translate increases in reserves into
increases in the ultimate money stock, through lending.”
- “One may show by the same logic used above that banks, like other agents in the private
sector, should react immediately to a credible increase in the monetary base. Thus, a
failure by banks to respond may reflect skepticism the central bank will carry through on
its commitment to maintain its increase in the monetary base.”
- Simple model to show this
- “Why is the equilibrium unaffected by the introduction of banks? Because banks know
that the liquidity trap will end in period T, and that they will cease to have excess
reserves at that date, this ties down their behavior in earlier periods. When the money
stock is increased permanently, this increases the money stock projected for period T, and
the impact of this increase cascades back to the present, following the logic of our
previous analysis.”
IX.
Credibility of permanent money supply expansions
No time inconsistency because fiscal benefits
- Classic time inconsistency problem: “If the nominal interest rate is at zero, however, the
only way to lower the real rate of interest is to convince markets that the price level will
be higher next period than had previously been expected. Credibility is an issue because
the authorities reap the benefit of higher promised prices today, but may be tempted to
renege on their promise in the future when it comes time to create the expected inflation.”
- However, fiscal benefits outweigh benefits of reversal, thus no time inconsistency: “Our
model and simulations suggest, however, that the preceding characterization of an
inflation-averse central bank's credibility problem is unrealistically special and extreme.
The associated fiscal benefits affect the credibility of a permanent money-supply
increase. There is no welfare gain to reversing a permanent money-stock increase (or,
equivalently in our setup, to reneging on a promise to raise the date-T money stock). On
the contrary, such a move would have a negative welfare impact in our model compared
to non-reversal. This finding is consistent with Eggertsson's (2003) observation that when
monetary policymakers internalize the fiscal benefits of monetary expansion, permanent
money-stock increases can become quite credible.”
Even if central bankers are insane and detest inflation
- Suppose central bankers tolerate [−𝜀, 𝜋] for both 𝜀, 𝜋 positive but small
- Suppose deflation at date zero exceeds varepsilon, and on economy’s preannouncement
path inflation at date T, 𝜋0 , is less than pi
-
-
-
“To reduce deflation occurring while the econ- omy is in a liquidity trap, the central bank
will wish to increase the money stock immediately and permanently, even if it places no
weight at all on the additional benefits that this measure might produce for the economy.
One possible equilibrium outcome following such an increase is that markets find the
increase credible. As the resulting increase in inflation will play out over a relatively
short period, the policy will succeed in reducing deflation around period 0 without
increasing inflation around period T. In this equilibrium, the central bank brings the
inflation trajectory within its target range, and the action also yields the fiscal and
macroeconomic benefits to the economy discussed above.
But another potential outcome is that private agents will not find the announced
permanent increase in money credible, in which case prices will not respond
immediately, but only when the economy exits the liquidity trap at period T and the
higher money stock affects prices di- rectly. At that later date, however, the central bank
would be forced to reverse its announced policy if failing to do so leads to an inflation
rate in excess of pi. But, if inflation in period T, pi_0, is not already at the central bank's
ceiling, pi, then this second, Pareto-inferior equilibrium cannot exist. To understand why,
consider first the simple case with flexible prices.
Assume that markets do not believe that the central bank will follow through with a
perm- anent deviation from its pre-announcement money-supply path. Then wages and
prices do not move prior to date T. On that date, however, the central bank has the option
to increase the money supply permanently (relative to the base- line path) by a percentage
x = pi – pi_0 without breaching its upper inflation target. Because the central bank will
wish at least partially to fulfill its promise of a higher money supply at T, the private
sector must rationally believe that at least an x-percent money supply increase is
permanent. That belief implies, however, that the price level (and with it, the entire path
of future prices) will rise by x percent immediately.
This immediate price increase, though, implies that people would be wrong to expect no
further increase in the date T price level. Ex- pected inflation between dates T - 1 and T is
again pi_0 – pi, so there remains room for some unexpected inflation between dates T - 1
and T if people do not believe that a further increase in the initial money stock is
permanent. This raises the initial and subsequent price levels further, again making room
for a bit more inflation between dates T - 1 and T, and so on. By backward induction, the
entire announced per- cent increase in the money stock on date 0 will be viewed as
permanent under our central bank preference assumptions.”
“assumptions. In summary, the equilibrium in which the monetary expansion is
noncredible requires, with flexible prices, that inflation at period T already be at least
equal to the upper limit of the central bank's target range, pi.”
Same applies to sticky prices
Simulation
-
“Our argument rests on the assumption that inflation is not already too high on date 5 and
after. Otherwise, monetary expansion on date 5, no matter how small, would breach the
3-percent limit. That circumstance would in- deed undermine current credibility. Thus, a
very stringent upper limit on the central bank's inflation tolerance ultimately reduces its
ability to fight deflation, for this stance takes away the bank's ability to adopt a credible
increase in the money stock while the economy is in a liquidity trap.”