money, co-existence of assets, inflation: an olg ... · money neutrality note that p t+1 p t = 1+n...
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Outline Previously Competing Assets Inflation
Money, Co-existence of Assets, Inflation:An OLG Interpretation
Timothy Kam
Research School of EconomicsAustralian National University
May 1, 2018
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Outline Previously Competing Assets Inflation
Outline
1 Previously
2 Competing Assets
3 InflationNeutralityNon-SuperneutralityPareto Efficiency?Superneutrality
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Outline Previously Competing Assets Inflation
Roadmap
Study competitive equilibrium allocation in simple endowmentOLG economy.
Competitive allocation is Pareto inefficient.
Introduce fiat money.
Derive equilibrium demand for money.
Is money essential? Does it improve allocations/welfare in aPareto sense? Yes.
Under some condition, possible for monetary (competitive)equilibrium to replicate Pareto application.
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Outline Previously Competing Assets Inflation
Previous Endowment OLG Economy
Agents endowed with 1 unit of good when young.
No endowment when old.
Perishable goods (equivalently, net return to storage r = −1),so no storage.
No uncertainty, perfect foresight.
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Outline Previously Competing Assets Inflation
Competing Assets
Agents endowed with 1 unit of good when young.
No endowment when old.
Storage exists: (equivalently, net return to storage r 6= −1).
No uncertainty, perfect foresight.
Two cases: r < n and r > n.
Stored goods: kt.
Now M and k are competing stores of value – vehicles fortransferring resources intertemporally.
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Outline Previously Competing Assets Inflation
Agent’s decision problem
Agent young at t solves
max ln(ctt) + β ln(ctt+1)
such that
ctt ≤ 1− kt −Mt
Pt
and
ctt+1 ≤ (1 + r)kt +PtPt+1
Mt
Pt.
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Outline Previously Competing Assets Inflation
The Karush-Kuhn-Tucker (KKT) FONCs for kt and MtPt
are,respectively:
− 1
ctt+ β(1 + r)
1
ctt+1
< 0 if kt = 0
= 0 if kt > 0
,
and,
− 1
ctt+ β
PtPt+1
1
ctt+1
< 0 if Mt
Pt= 0
= 0 if MtPt
> 0
,
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Outline Previously Competing Assets Inflation
What the KKT conditions say:
Preference set is strictly convex on the set of allocation(ctt, c
tt+1)
This is implied by Inada conditions for U(ctt, ctt+1)
Our log utility example satisfies these conditions
Then we have that any optimal choice (ctt, ctt+1) is a strictly
non-zero bundle.
Which implies saving (so, either kt > 0 or MtPt
> 0).
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Outline Previously Competing Assets Inflation
Comparing the FONCs gives that if
PtPt+1
< 1 + r
then ...
Mt
Pt= 0 and kt > 0.
In words: If money earns a better (worse) rate of return thanthe storage technology, then real money balances will be held,and none of the storage technology will be.
If the two rates are equal, then the agent is indifferentbetween the two.
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Outline Previously Competing Assets Inflation
Discussion
Long-standing problem with monetary models.
Competing assets: asset with dominating rate of returnsurvives existence problem.
If both assets have same rate of return, then indeterminacy inthe composition of these assets held.
How to have a determinate distribution of, and relative price,for these assets?
More microeconomic foundations from information economics:e.g. asymmetric information re: asset quality; limitedcommitment to repaying. Beyond the scope of our study here.
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Outline Previously Competing Assets Inflation
Inflation
So far, we have assumed a constant nominal money supply H.
Now, allow money growth Ht+1 = (1 + σ)Ht.
We’ll see that at steady state, we will have gross inflationPt+1
Pt= σ − n.
Suppose that new money is given to the old agents via lumpsum transfer, Tt, at time t.
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Outline Previously Competing Assets Inflation
Now agent young at t solves
max ln(ctt) + β ln(ctt+1)
such that
ctt ≤ 1− kt −Mt
Pt
and
ctt+1 ≤ (1 + r)kt +PtPt+1
Mt
Pt+Tt+1
Pt+1.
The right-most term being the new real money balances.
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Outline Previously Competing Assets Inflation
Denote:
g as gross deflation rate,m =M/P as real money demand, andt = T/P as new real money balance.
We can rewrite this as
maxkt,mt
{ln(1− kt −mt)
+ β ln[(1 + r)kt + (1 + gt)mt + tt+1]
}.
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Outline Previously Competing Assets Inflation
As before, the Karush-Kuhn-Tucker (KKT) FONCs for kt and MtPt
are, respectively:
− 1
ctt+ β(1 + r)
1
ctt+1
< 0 if kt = 0
= 0 if kt > 0
,
and,
− 1
ctt+ β
PtPt+1
1
ctt+1
< 0 if Mt
Pt= 0
= 0 if MtPt
> 0
,
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Outline Previously Competing Assets Inflation
Steady State
In a steady state, mt+1 = mt, for all t, and so gross inflationrate is
Pt+1
Pt=
Nt
Nt+1
Ht+1
Ht
=1
1 + n(1 + σ) ≈ σ − n, for (σ, n) small.
Alternatively, in terms of the gross return on money (i.e.,deflation), at steady state,
PtPt+1
=1 + n
1 + σ
so that g ≈ n− σ.
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Outline Previously Competing Assets Inflation
Special Case
Assume that 1+n1+σ ≥ 1 + r: Money weakly dominates storage
in RoR.Consumers’ FONCs imply that:
kt = 0,ctt+1
βctt> 1 + r,
mt > 0, and
1
ctt=
1
1−mt
=β(1 + gt)
(1 + gt).mt + (1 + gt)Tt+1
Pt
=β.(1 + gt)
ctt+1
=β
mt +Tt+1
Pt
.
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Outline Previously Competing Assets Inflation
Now, since
Tt+1 =σHt
Nt,
then
Tt+1
Pt= σmt.
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Outline Previously Competing Assets Inflation
Getting back to the FONC above gives that
1
1−mt=
β
mt + σmt
so that in steady state, real money balance is
mt =β
1 + σ + β,
consumption for young agent is
ctt =1 + σ
1 + σ + β,
and, for old agent is
ctt+1 =(1 + n)β
1 + σ + β.
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Outline Previously Competing Assets Inflation
Money Neutrality
Note that
Pt+1
Pt=
(1 + n
1 + σ
)−1
.
For constant population growth rate, n,
Pt+1
Pt∝ 1 + σ =
Ht+1
Ht.
Prices will adjust at the same rate as money supply growth.
Proposition
If a monetary equilibrium exists, then money supply growth has noreal effects in the short run – i.e. Money is neutral.
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Outline Previously Competing Assets Inflation
Discussion
What does the money neutrality imply for “real-world”monetary policy?
In monetary policy really neutral in the “real world”?
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Outline Previously Competing Assets Inflation
Non-superneutrality
Recall, we showed that if a monetary equilibrium exists (inlog-utility model), then in steady state, we have consumptionfor young agent is
ctt =1 + σ
1 + σ + β,
and, for old agent is
ctt+1 =(1 + n)β
1 + σ + β.
Proposition
Money is not super-neutral — there are real effects on long-runallocations.
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Outline Previously Competing Assets Inflation
(In)Efficiency of Monetary Equilibrium
Monetary equilibrium is no longer Pareto Optimal if σ > 0.
To see this, consider the steady state equilibrium with kt = 0.
The FONCs are
ctt+1
ctt> 1 + r
and
ctt+1
βctt=
1 + n
1 + σ.
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Outline Previously Competing Assets Inflation
Exercise
What happens if money earns interest? Before, RoR on money isgross deflation. Now suppose the t+ 1 budget constraint for agentt is
ctt+1 = (1 + σ)Mt
Pt.PtPt+1
+ (1 + r).kt
and everything else is as before. Show that money is nowsuper-neutral.