Advanced Macroeconomics I:
Models with money
Krzysztof Makarski
1
The Basic MIU Model
Agenda
• Basic Money-in-the-utility Model
• Superneutrality of money.
• Literature: Walsh, Chp. 3.
Introduction.
• The two most important questions in the literature on monetary policy are:
– what is the optimal inflation level, or how should optimal monetary policy be conducted?
– can monetary policy be used to stabilize an economy?
• We start with the first one.
1.1
Set up
Preliminaries.
• Extends the Ramsey by,
• putting money in the utility function (short cut).
• Intuition behind: Money facilitates transactions.
• Why short cut? Because it is not so easy without one and we still get some insight.
Households
• There is mass one of agents, everybody is the same which gives rise to the representative household
which plays a role of a stand-in agent making decisions for all households in the economy. We assume
that the population growth n = 0.
• The per-period utility function of households is u(ct , mt ) with u being increasing and concave in both
t
consumption c and real money m, where mt = M
Pt , Mt −nominal money and Pt − price level.
• Note we denote nominal variables with capital letters and real with small.
• To ensure existence of a monetary equilibrium (money is held), it is often assumed that for some m̄,
um (ct , m̄) = 0 and um (ct , mt ) < 0 for mt > m̄ .
• The goal of representative household is to maximize:
W =
∞
X
t=0
1
β t u(ct , mt )
(1.1)
• Households face the following budget constraint
ct + x t +
Mt
Bt
Wt
Rk,t
Rt−1 Bt−1
Mt−1
+
≤
nt +
kt−1 + τt +
+
Pt
Pt
Pt
Pt
Pt
Pt
and the law of motion for capital
kt = xt + (1 − δ)kt−1
(Beware: kt here denotes capital that was invested in t but can be operated in t + 1, different notation
Rk,t
t
then before) combining we get and using wt ≡ W
Pt and rk,t ≡ Pt
ct + kt +
Mt
Bt
Rt−1 Bt−1
Mt−1
+
≤ wt nt + (rk,t + 1 − δ)kt−1 + τt +
+
Pt
Pt
Pt
Pt
To express it in per capita terms note mt ≡
Mt
Pt ,
bt ≡
Bt
Pt ,
and πt =
Pt
Pt−1
(1.2)
therefore we get
ct + kt + mt + bt ≤ wt nt + (rk,t + 1 − δ)kt−1
+ τt +
ct + kt + mt + bt ≤ wt nt + (rk,t + 1 − δ)kt−1 + τt +
Rt−1 Bt−1 Pt−1
Mt−1 Pt−1
+
Pt−1
Pt
Pt−1 Pt
Rt−1
mt−1
bt−1 +
πt
πt
• Additionally households are subject to non-Ponzi game condition bt ≥ −b̄ where b̄ is a sufficiently large
number.
• Note labor is exogenous and since nt ∈ [0, 1] trivially we have nt = 1. Also m−1 , R−1 b−1 , k−1 are given
• Notation: kt −capital, bt −bonds, wt − real wage, nt −labor, rk,t −real rental rate on capital, Rt −nominal
interest rate, πt −gross inflation, τt −government transfers.
Firms
Firms rent capita and labor to produce output. While making their decisions they seek the maximal
possible profits. Each firm is too small to impact prices, thus they take prices as given. In each period, the
representative firm solves the following problem
max
(yt ,nt ,kt−1 )
yt − rk,t kt−1 − wt nt
α
sub. to yt = Akt−1
n1−α
t
Central bank
• Helicopter drops at the gross rate θt
s
Pt τt = Mts − Mt−1
s
.
where Mts = θt Mt−1
• Or, in real terms
τt =
where mst =
Mts
Pt
=
s
θt Mt−1
Pt−1
Pt
Pt−1
=
M s Pt−1
ms
Mts
− t−1
= mst − t−1
Pt
Pt Pt−1
πt
s
θt Mt−1
Pt−1
Pt−1
Pt
=
θt
s
πt mt−1
2
Market clearing
• Markets clear when demand and supply are equal. In the money market demand for money mt must
be equal to supply of money (decided by central bank) Mt
mt = mst
(1.3)
• Since government does not issue debt, in a closed economy the total private debt is equal to zero, thus
the aggregate bond holdings are equal to zero
bt = 0
• Furthermore, since the labor supply is always 1, the labor market clearing condition is
nt = 1
• Capital rental market clearing condition notationally.
• The goods market clearing condition
ct + xt = yt
which together with
kt = (1 − δ)kt−1 + xt
gives
ct + kt = yt + (1 − δ)kt−1
Competitive equilibrium
Definition 1. A competitive equilibrium is an endogenous allocation {ct , nt , kt , yt , mt , bt }∞
t=0 and endogenous
prices {rk,t , wt , πt , Rt }∞
t=0 satisfying
• {ct , nt , kt , mt , bt }∞
t=0 solves the consumer problem given prices
W =
∞
X
max
{ct ,nt ,kt ,mt ,bt }∞
t=0
β t u(ct , mt )
t=0
sub. to ct + kt + mt + bt ≤ wt nt
Rt−1
mt−1
bt−1 +
πt
πt
nt ∈ [0, 1], m−1 , R−1 b−1 , k−1 given
+ (rk,t + 1 − δ)kt−1 + τt +
• for all t, (nt , kt−1 , yt ) solves the producer problem given prices
max
yt − rk,t kt−1 − wt nt
(yt ,nt ,kt−1 )
α
sub toyt = Akt−1
lt1−α
• Central bank monetary policy satisfies
τt = mst −
where mst =
mst−1
πt
θt
s
πt mt−1 .
• markets clear
mt
=
mst
(1.4)
bt
=
0
(1.5)
nt
=
1
(1.6)
ct + kt
=
yt + (1 − δ)kt−1
(1.7)
3
1.2
Solution
Solving HHs problem
• Recall
W =
∞
X
β t u(ct , mt )
t=0
sub. to ct + kt + mt + bt ≤ wt nt
Rt−1
mt−1
bt−1 +
πt
πt
nt ∈ [0, 1], m−1 , R−1 b−1 , k−1 given
+ (rk,t + 1 − δ)kt−1 + τt +
• Lagrangian
L = β 0 u(c0 , m0 ) + ...β t u( ct , mt ) + β t+1 u(ct+1 , mt+1 ) + ... −
h R−1
m−1 − λ0 c0 + k0 + m0 + b0 − w0 n0 − (rk,0 + 1 − δ)k0 − τ0 −
+ ...+
b−1 −
π0
π0
Rt−1
mt−1 + λt ct + kt + mt + bt − wt nt − (rk,t + 1 − δ)kt−1 − τt −
bt−1 −
+
πt
πt
+ λt+1 ct+1+ kt+1 + mt+1 +bt+1−wt+1 nt+1−(rk,t+1 + 1 − δ) kt −
− τt+1 −
−
∞
hX
∞
i X
mt Rt
+ ... =
β t u(ct , mt )−
bt −
πt+1
πt+1
t=0
λt (ct + kt + mt + bt − wt nt − (rk,t + 1 − δ)kt−1 − τt −
t=0
• FOCs
ct :β t uc (t) − λt = 0
1
=0
πt+1
kt : − λt + λt+1 (rk,t+1 + 1 − δ) = 0
Rt
=0
bt : − λt + λt+1
πt+1
mt :β t um (t) − λt + λt+1
and the transversality conditions (TVCs)
lim λt kt
=
0
lim λt bt
=
0
t→∞
t→∞
Simplifying we get and defining the real interest rate rt ≡ rk,t + 1 − δ
β t uc (t) = λt
1
= λt
β t um (t) + λt+1
πt+1
λt = λt+1 rt+1
Rt
λt = λt+1
πt+1
4
Rt−1
mt−1 i
bt−1 −
)
πt
πt
Substituting for λt = β t uc (t) and λt+1 = β t+1 uc (t + 1)
β t um (t) +
β t+1 uc (t + 1)
= β t uc (t)
πt+1
β t uc (t)
= rt+1
β t+1 uc (t + 1)
Rt
β t uc (t)
=
t+1
β uc (t + 1)
πt+1
• Canceling out β and using the fact that the left hand sides of the two last equations are the same we
get the tree equations characterizing the solution to HHs problem
– money-consumption choice (after division by uc (t)
um (t) βuc (t + 1) 1
+
uc (t)
uc (t) πt+1
– Euler
=
1
(1.8)
uc (t)
= rt+1 = rk,t + 1 − δ
βuc (t + 1)
(1.9)
– Fisher equation
rt+1 =
Rt
πt+1
(1.10)
plus two TVCs
Solving Firm’s Problem.
• Recall the firm problem
max
yt − wt nt − rk,t kt−1
(yt ,nt ,kt−1 )
sub to yt = F (kt−1 , nt )
• We construct Lagrangian
L = yt − wt nt − rk,t kt−1 − $t (yt − F (kt−1 , nt ))
• FOCs
yt :1 − $t = 0
nt : − wt + $t Fn (kt−1 , nt ) = 0
kt : − rk,t + $t Fk (kt−1 , nt ) = 0
• Finally Substituting $t = 1 we get
wt
= Fn (kt−1 , nt )
(1.11)
rk,t
= Fk (kt−1 , nt )
(1.12)
plus the production function
yt = F (kt−1 , nt )
5
(1.13)
Equilibrium.
• Equilibrium is characterized by three equations from the HH problem (1.8)-(1.10) (plus TVCs), three
equations from the firm problem (1.11)-(1.13); and four market clearing conditions (1.4)-(1.7).
• Next we find equations characterizing equilibrium allocation (we partially eliminate prices).
– First, we use nt = 1.
– Second, we substitute for rk,t from (1.12) into Euler (1.9) to get
uc (t)
= rt+1 = Fk (kt , nt+1 ) + 1 − δ = fk (kt ) + 1 − δ
βuc (t + 1)
define f (kt ) ≡ F (kt , 1), then fk (kt ) = Fk (kt , 1)
uc (t)
= fk (kt ) + 1 − δ
βuc (t + 1)
(1.14)
(compare with equation in the text below equation 2.12 in Walsh)
– Next take (1.8)
um (t) βuc (t + 1) 1
+
uc (t)
uc (t) πt+1
=
1
and substitute from (1.9) to get (compare with 2.12 in Walsh)
um (t)
uc (t)
=
1−
1
1
=1−
πt+1 rt+1
πt+1 (fk (kt ) + 1 − δ)
(1.15)
(recall rt+1 ≡ fk (kt ) + 1 − δ). Or, substituting further from (1.10)
um (t)
uc (t)
=
1−
1
Rt − 1
=
Rt
Rt
– Finally, substituting from the the production function yt = f (kt−1 ) into the market clearing
conditions
ct + kt = yt + (1 − δ)kt−1 = f (kt−1 ) + (1 − δ)kt−1
(1.16)
– Additionally, we have the Fisher equation (compare with 2.13 in Walsh)
rt+1 =
Rt
πt+1
(1.17)
– Note finding π would require specific utility functions so we leave it.
Cashless economy.
• Equilibrium is characterized by:
– three equations from the HH problem (1.8)-(1.10) (plus TVCs):
um (t) βuc (t + 1) 1
+
uc (t)
uc (t) πt+1
=
1
uc (t)
= rt+1 = rk,t+1 + 1 − δ
βuc (t + 1)
rt+1 =
plus two TVCs.
6
Rt
πt+1
(1.18)
– three equations from the firm problem (1.11)-(1.13)
wt
=
fn (kt−1 , nt )
rk,t
=
fk (kt−1 , nt )
yt = F (kt−1 , nt )
– and four market clearing conditions (1.4)-(1.7).
mt
=
mst
bt
=
0
nt
=
1
ct + kt
=
yt + (1 − δ)kt−1
(1.19)
• So what would change if we removed money? Equations (1.18) would disappear (1.19) and money
would disappear from equation
uc (ct , mt )
= rt+1 = rk,t+1 + 1 − δ
βuc (ct+1 , mt+1 )
• Therefore unless there is some interesting dynamics coming from the fact that uc (c, m) depends on m
removing money does not change the model or its predictions. In particular when u(c, m) = v(c)+φ(m)
you can remove money.
• Recall that while analyzing welfare issues removing money may have an effect since m enters utility
function in MIU models.
1.3
Steady state
Steady state.
• We denote steady state value of xt as x (no time index).
• First, notice that for mt = m for all t, and since
Mt
Pt
θt Mt−1
πt Pt−1
=
we must have π = θ in the steady state.
• Second, using (1.14) -(1.17) we get in the steady state:
– Euler
uc (c, m)
= r = fk (k) + 1 − δ
βuc (c, m)
(1.20)
– consumption-money
um (c, m)
uc (c, m)
=
1−
1
1
=1−
π(fk (k) + 1 − δ)
θ(fk (k) + 1 − δ)
(1.21)
or
um (c, m)
uc (c, m)
=
1−
1
R−1
=
R
R
– Finally, we have the market clearing condition
c + k = f (k) + (1 − δ)k
(1.22)
– Additionally, we have the Fisher equation
r=
7
R
θ
(1.23)
Superneutrality of money.
• Note θ does not affect c, k, n (or y) in the steady state.
• Recall, n = 1.
• From (1.20) we get
1
= r = fk (k) + 1 − δ
β
assuming the Cobb-Douglas production function f (k) = zk α we get
1
= zαk α−1 + 1 − δ
β
"
zαβ
k=
1 − β(1 − δ)
1
# 1−α
and it does not depend on θ
• From (1.22) we get
c = f (k) − δk
and since k does not depend on θ so does c.
• So what is affected by θ = π? The answer is real money. First using the Fisher equation note
R = rθ = β −1 θ
and from (1.21)
um (c, m)
uc (c, m)
β −1 θ − 1
θ−β
=
β −1 θ
θ
=
therefore since c is constant the above equation can only be satisfied if m changes with θ.
• Money is superneutral, it affects only nominal variables and real money holdings.
Existence of the steady state.
• Note for the existence of the steady state it is necessary that
um (c, m)
uc (c, m)
β −1 θ − 1
θ−β
=
−1
β θ
θ
=
is satisfied.
• Our assumption that for some m̄, limm→0 um (c, m) = ∞, um (c, m̄) = 0 and um (c, mt ) < 0 for mt > m̄
guarantees that.
• But such a strong conditions are not necessary, existence is satisfied also with weaker conditions (omitted).
• For example steady state exists for u(c, m) = v(c) + log(m).
8
1.4
Optimal policy
Optimal long run monetary policy.
• Consider the planner problem
max
∞
X
{ct ,nt ,mt }∞
t=0
β t u(ct , mt )
t=0
sub. to ct + kt ≤ f (kt−1 ) + (1 − δ)kt−1
• Lagrangian
L=
∞
X
β t u(ct , mt ) −
t=0
∞
X
λt [ct + kt − f (kt−1 ) − (1 − δ)kt−1 ]
t=0
• FOCs
ct :β t uc (t) − λt = 0
mt :β t um (t) = 0
kt : − λt = λt+1 (fk (kt ) − (1 − δ))
Therefore we get
um (t)
uc (t)
uc (t)
βuc (t + 1)
ct + kt
=
0
(1.24)
=
fk (kt ) + 1 − δ
(1.25)
=
f (kt−1 ) + (1 − δ)kt−1
(1.26)
Compare (1.14)-(1.16) with (1.24)-(1.26). These equations are the same if Rt = 1 (net nominal interest
rate is zero - the Friedman rule).
• In the steady state it means (using the Fisher equation R = rθ = β −1 θ) that θ = β and π = β. Usually
it is assumed that β = 0.99 (for quarterly data) therefore in the steady state π = 0.99 (recall it is gross
inflation, net inflation is −1%) which means deflation (equal to −1%).
Welfare costs of inflation.
• Could be computed as the area under the money demand curve at a given positive nominal interest
rate (Figure 2.2, p. 61).
• This is the consumer surplus lost by a positive nominal interest rate. Some estimates indicate that
inflation at 10% corresponds to 1 − 3% of GDP per year.
• Could, as Lucas suggests, be computed as the percentage increase in steady-state consumption needed
to compensate for a suboptimal low real money stock caused by (Rt − 1)/Rt > 0.
– Normalizing css = 1, this implies that the cost of inflation ω(R) is implicitly given as
u(1 + ω(R), m(R)) = u(1, m)
where m is the optimal quantity of real money balances, and m(R) is the money demand function,
mR < 0.
– With specific form of the utility function, and using numbers from estimated money demand
functions, Lucas finds a 10% nominal interest rate is equivalent of around 1.3% lost steady-state
consumption (each year).
– A large number or not?
• Note that MIU model ignores other cost of inflation (e.g., its variability, impact on relative prices, etc.)
9
Non-superneutrality of money.
• Steady state inflation can also affect other variables in the steady state. It is enough to introduce
endogenous labor.
• In our model since labor does not enter utility function nt = 1, exogenous labor.
• It is quite easy in our model, it is enough to include labor in the utility function u(c, l, m), endogenous
labor.
• How to get the IS-LM type effects (expansive monetary policy raises output)? We need nominal
rigidities. Later (next semester).
1.5
Summary
Summary
• Presented MIU model.
• We explicitly analyze effect of money on the economy.
• Inflation is bad (the Friedman rule optimal).
• Money is superneutral if labor is exogenous, and affects (negatively - opposite then in the IS-LM model)
real variables if labor is endogenous.
• We can use these type of models to compute welfare cost of inflation.
• Next: motivation for money in the utility: shopping time and CIA models.
2
The Shopping Time Model
Agenda
• Shopping Time Model.
• Cash-In-Advance Model.
• Superneutralitity of money revisited.
• Welfare Costs of Inflation: Intro.
• Literature: Walsh, Chp. 3.
Introduction
• In MIU models, money provides utility directly in order to secure a demand for money
– Motivation: Approximation to utility from saved time on transactions; liquidity services
• Shopping-time models and cash-in-advance models formalize this by explicitly modeling demand for
money due to transaction motive.
– Helps put restrictions on, e.g., the signs of ucm (·) and ulm (·) under the MIU approach (these signs
determine how labor supply and output react to changes in money growth).
10
2.1
Set up
Assumptions
• Instantaneous utility function
v(c, l)
where l = 1 − n − ns , where l−leisure, ns −shopping time and n−work.
• Suppose consumption c requires transaction services ψ = c which are produced with inputs o real cash
balances m ≡ M/P and shopping time ns
ψ = ψ(m, ns ) = c
where ψm ≥ 0, ψns ≥ 0 and ψmm , ψns ns ≤ 0.
• This is restated in terms of shopping time:
ns = g(c, m)
where gc > 0 and gm ≤ 0.
• Instantaneous utility function:
u(c, n, m) ≡ v(c, 1 − n − g(c, m))
• Note we denote nominal variables with capital letters and real with small.
Households
• There is mass one of agents. We assume that there is no population growth.
• The goal of representative household is to maximize:
W =
∞
X
β t v(ct , 1 − nt − g(ct , mt ))
(1.27)
t=0
• Households face the following budget constraint
ct + kt + mt + bt ≤ wt nt + (rk,t + 1 − δ)kt−1 + τt +
mt−1
Rt−1
bt−1 +
πt
πt
• Additionally nt + g(ct , mt ) ∈ [0, 1] and households are subject to non-Ponzi game condition bt ≥ −b̄
where b̄ is a sufficiently large number.
• Note m−1 , R−1 b−1 , k−1 are given.
Firms
Firms rent capita and labor to produce output. While making their decisions they seek the maximal
possible profits. Each firm is too small to impact prices, thus they take prices as given. In each period, the
representative firm solves the following problem
max
yt − rk,t kt−1 − wt nt
(yt ,nt ,kt−1 )
α
sub. to yt = Akt−1
n1−α
t
11
Central bank
• Helicopter drops at the gross rate θt
s
Pt τt = Mts − Mt−1
s
where Mts = θt Mt−1
.
• Or, in real terms
τt =
where mst =
Mts
Pt
=
s
θt Mt−1
Pt−1
Pt
Pt−1
=
M s Pt−1
ms
Mts
− t−1
= mst − t−1
Pt
Pt Pt−1
πt
s
θt Mt−1
Pt−1
Pt−1
Pt
=
θt
s
πt mt−1
Market clearing
• Markets clear when demand and supply are equal. In the money market demand for money mt must
be equal to supply of money (decided by central bank) Mt
mt = mst
(1.28)
• Since government does not issue debt, in a closed economy the total private debt is equal to zero, thus
the aggregate bond holdings are equal to zero
bt = 0
• Market clearing conditions for labor and capital rental markets are included notationally (by using the
same symbols in HH and firm problems).
• The goods market clearing condition
ct + xt = yt
which together with
kt = (1 − δ)kt−1 + xt
gives
ct + kt = yt + (1 − δ)kt−1
Competitive equilibrium
Definition.
∞
A competitive equilibrium is an endogenous allocation {ct , nt , kt , yt , mt , bt }∞
t=0 and endogenous prices {rk,t , wt , πt , Rt }t=0
satisfying (given exogenous (k−1 , m−1 , R−1 b−1 ))
• {ct , nt , kt , mt , bt }∞
t=0 solves the consumer problem given prices
W =
max
{ct ,nt ,kt ,mt ,bt }∞
t=0
∞
X
β t u(ct , mt , nt )
t=0
sub. to ct + kt + mt + bt ≤ wt nt
mt−1
Rt−1
bt−1 +
πt
πt
nt + g(ct , mt ) ∈ [0, 1], m−1 , R−1 b−1 , k−1 given
+ (rk,t + 1 − δ)kt−1 + τt +
Definition 1. cont.
12
• for all t, (nt , kt−1 , yt ) solves the producer problem given prices
max
yt − rk,t kt−1 − wt nt
(yt ,nt ,kt−1 )
α
sub toyt = Akt−1
lt1−α
• Central bank monetary policy satisfies
τt = mst −
mst−1
πt
Definition 1. cont.
• markets clear
= mst
(1.29)
bt
=
0
(1.30)
nt
=
1
(1.31)
ct + kt
=
yt + (1 − δ)kt−1
(1.32)
mt
2.2
Solution
Solving HHs problem
• Recall
W =
∞
X
β t u(ct , mt , nt )
t=0
sub. to ct + kt + mt + bt ≤ wt nt
mt−1
Rt−1
bt−1 +
πt
πt
nt ∈ [0, 1], m−1 , R−1 b−1 , k−1 given
+ (rk,t + 1 − δ)kt−1 + τt +
• Lagrangian
L = β 0 u(c0 , m0 , n0 ) + ...β t u( ct , mt , nt ) + ... −
h R−1
m−1 − λ0 c0 + k0 + m0 + b0 − w0 n0 − (rk,0 + 1 − δ)k0 − τ0 −
b−1 −
+ ...+
π0
π0
Rt−1
mt−1 + λt ct + kt + mt + bt − wt nt − (rk,t + 1 − δ)kt−1 − τt −
bt−1 −
+
πt
πt
+ λt+1 ct+1+ kt+1 + mt+1 +bt+1−wt+1 nt+1−(rk,t+1 + 1 − δ) kt −
− τt+1 −
−
∞
hX
∞
i X
mt Rt
bt −
+ ... =
β t u(ct , mt )−
πt+1
πt+1
t=0
λt (ct + kt + mt + bt − wt nt − (rk,t + 1 − δ)kt−1 − τt −
t=0
13
Rt−1
mt−1 i
bt−1 −
)
πt
πt
• FOCs
ct :β t uc (t) − λt = 0
mt :β t um (t) − λt + λt+1
1
πt+1
=0
nt :β t un (t) + λt wt = 0
kt : − λt + λt+1 (rk,t+1 + 1 − δ) = 0
Rt
bt : − λt + λt+1
=0
πt+1
and the transversality conditions (TVCs)
lim λt kt
=
0
lim λt bt
=
0
t→∞
t→∞
Simplifying we get and defining the real interest rate rt ≡ rk,t + 1 − δ
β t uc (t) = λt
1
β t um (t) + λt+1
= λt
πt+1
β t un (t) = −λt wt
λt = λt+1 rt+1
Rt
λt = λt+1
πt+1
Substituting for λt = β t uc (t) and λt+1 = β t+1 uc (t + 1)
β t um (t) +
β t+1 uc (t + 1)
πt+1
β t un (t)
− t
β uc (t)
t
β uc (t)
β t+1 uc (t + 1)
β t uc (t)
β t+1 uc (t + 1)
= β t uc (t)
= wt
= rt+1
=
Rt
πt+1
• Canceling out β and using the fact that the left hand sides of the two last equations are the same we
get the tree equations characterizing the solution to HHs problem
– money-consumption choice (after division by uc (t)
um (t) βuc (t + 1) 1
+
uc (t)
uc (t) πt+1
=
1
(1.33)
– consumption-labor choice
−
– Euler
un (t)
= wt
uc (t)
uc (t)
= rt+1 = rk,t+1 + 1 − δ
βuc (t + 1)
(1.34)
(1.35)
– Fisher equation
rt+1 =
plus two TVCs .
14
Rt
πt+1
(1.36)
Equilibrium.
• Equilibrium is characterized by four (one more than previously - endogenous labor) equations from the
HH problem (1.33)-(1.36) (plus TVCs), three equations from the firm problem (the same as previously)
wt
= Fn (kt−1 , nt )
rk,t
= Fk (kt−1 , nt )
(1.38)
= F (kt−1 , nt )
(1.39)
yt
(1.37)
and four market clearing conditions (1.4)-(1.7).
• Next we find equations characterizing equilibrium allocation (we partially eliminate prices).
– First, we substitute for wt from (1.37) into (1.34) to get
−
un (t)
= wt = Fn (kt−1 , nt )
uc (t)
(1.40)
– Second, we substitute for rk,t from (1.38) into Euler (1.35) to get
uc (t)
= rt+1 = Fk (kt , nt+1 ) + 1 − δ
βuc (t + 1)
(1.41)
– Next take (1.33)
um (t) βuc (t + 1) 1
+
uc (t)
uc (t) πt+1
=
1
and substitute from (1.41) to get
um (t)
uc (t)
=
1−
1
1
=1−
πt+1 rt+1
πt+1 (Fk (kt , nt+1 ) + 1 − δ)
(recall rt+1 ≡ Fk (kt , nt+1 ) + 1 − δ). Or, substituting further from (1.36)
um (t)
uc (t)
=
1−
1
Rt − 1
=
Rt
Rt
(1.42)
– Finally, substituting from the production function yt = F (kt−1 , nt ) into the market clearing
conditions
ct + kt = yt + (1 − δ)kt−1 = F (kt−1 , nt ) + (1 − δ)kt−1
(1.43)
– Additionally, we have the Fisher equation
rt+1 =
Rt
πt+1
(1.44)
– Note finding π would require specific utility functions so we leave it.
2.3
Superneutrality of Money
Steady state.
• We denote steady state value of xt as x (no time index).
• First, notice that for mt = m for all t, and since
Mt
Pt
=
• Second, using (1.37) -(1.44) we get in the steady state:
15
θt Mt−1
πt Pt−1
we must have π = θ in the steady state.
– consumption-labor choice
−
– Euler
un (c, m, n)
= Fn (k, n)
uc (c, m, n)
uc (c, m, n)
= r = Fk (k, n) + 1 − δ
βuc (c, m, n)
(1.45)
(1.46)
– consumption-money
um (c, m, n)
uc (c, m, n)
=
1−
1
1
=1−
π(Fk (k, n) + 1 − δ)
θ(Fk (k, n) + 1 − δ)
(1.47)
or
um (c, m, n)
uc (c, m, n)
=
1−
1
R−1
=
R
R
– Finally, we have the market clearing condition
c + k = F (k, n) + (1 − δ)k
(1.48)
– Additionally, we have the Fisher equation
r=
R
θ
(1.49)
Superneutrality of Money revisited.
• Simplifying notice that if F (k, n) exhibits constant returns to scale, i.e. λF (k, n) = F (λk, λn) for any
λ > 0, then Fk (λk, λn) = F (k, n) and Fn (λk, λn) = F (k, n). Therefore from (1.46) we get
k
1
= r = Fk ( , 1) + 1 − δ
β
n
defining f ( nk ) ≡ F ( nk , 1) we get
thus
k
n
k
Fk ( , 1) = β −1 − (1 − δ)
n
= const. is constant and does not depend on monetary policy.
(1.50)
• Next dividing (1.48) by n
k
c
+
n n
c
n
1
k
k
k
F (k, n) + (1 − δ) = F ( , 1) + (1 − δ)
n
n
n
n
k
k
= f ( ) + (1 − δ)
n
n
k
k
k
= φ( ) ≡ f ( ) − δ
n
n
n
=
which together with the fact that
policy.
k
n
= const. implies that
c
n
(1.51)
= const. and does not depend on monetary
• Using the fact that F (k, n) exhibits constant returns to scale (homogeneous of degree 1) we have
F (k, n)
=
kFk (k, n) + nFn (k, n)
Dividing by n and using f ( nk ) ≡ F ( nk , 1) we get
1
F (k, n) =
n
k n
F( , ) =
n n
k
F ( , 1) =
n
k
n
Fk (k, n) + Fn (k, n)
n
n
k
k n
n
Fk ( , ) + Fn (k, n)
n
n n
n
k
k
Fk ( , 1) + Fn (k, n)
n
n
16
Therefore
k
k
k
Fn (k, n) = f ( ) − fk ( )
n
n
n
Substituting into (1.45) we get
−
k
k
k
un (c, m, n)
= f ( ) − fk ( )
uc (c, m, n)
n
n
n
(1.52)
• But the particular value of c and therefore n depends on money. From (1.47) using (1.50)
um (c, m, n)
uc (c, m, n)
=
1−
β
θ−β
=
θ
θ
(1.53)
• Note that equations (1.51) using (1.53) determine jointly c, n, m in the steady state and are clearly
affected by the growth rate of money. If for example unm = 0 and ucm > 0 an increase in the growth
rate of money could lower m, c and n, which implies that output could fall.
• Since the model of the shopping time provides the justification for the MIU model we can use it to get
the signs of the derivatives.
• Therefore using our specification of u we can get the signs of the derivatives
uc
=
vc − vl gc
(1.54)
um
=
−vl gm
(1.55)
un
=
−vl
(1.56)
• The marginal utility of leisure is decreasing with money
unm = vll gm > 0
(1.57)
since it increases leisure for given consumption (by reducing shopping time).
• The marginal utility of consumption changes with money in an ambiguous way since
ucm = − vcl gm + vll gc gm − vl gcm ≶ 0
|{z} |{z} |{z} |{z} |{z} |{z} |{z}
?
−
−
+
−
+
(1.58)
?
– − vcl gm is positive if vcl > 0, i.e. money frees up leisure, increasing the marginal utility of
|{z} |{z}
?
−
consumption directly.
– + vll gc gm is positive: More money frees up leisure, which decreases marginal utility of leisure
|{z} |{z} |{z}
−
+
−
=> less utility loss from transaction costs.
– − vl gcm is positive if gcm < 0: More money reduces the marginal transaction costs (in terms
|{z} |{z}
+
?
of lost leisure).
• Therefore, most likely ucm > 0 and higher m will lead to more work, unless consumption and leisure
are strong substitutes, vcl 0 and/or more money increases marginal transactions costs, gcm 0.
• Since, unm > 0 and ucm is most likely positive, −un /uc = vl /uc increases when m falls. Note θ %⇒
1 − β/θ %⇒ um /uc % which can happen if um %⇒ m &. But as m & we have both −un = vl %
(m &⇒ un &⇒ −un %⇒ vl %) and uc &which implies −un /uc = vl /uc & which requires vl to go
down which can be achieved by l % and so n &.
• Hence, lower real money balances reduce employment and output in a shopping time model. Since a
MIU model is equivalent to a shopping time model, and provides a justification for putting money in
the utility (liquidity services of money) we know how and why superneutrality fails
17
2.4
Optimal Monetary Policy.
The optimal monetary policy.
• Take (1.42), substitute from (1.55) and use the above formula for uc
−vl (t) gm (t)
uc (t)
=
1−
1
Rt − 1
=
Rt
Rt
Next since un = −vl
un (t)gm (t)
uc (t)
=
1−
1
Rt − 1
=
Rt
Rt
=
Rt − 1
Rt
using (1.40) we get (see 3.12 in Walsh)
−wt gm (t)
(1.59)
• Since there is no social cost of producing money we should have gm = 0 therefore (1.59) implies optimal
monetary policy satisfies the Friedman rule Rt = 1.
2.5
Summary
Summary
• Motivation for money in the utility.
• Money is not superneutral: higher growth of money lowers real money holdings, consumption, labor
and output.
• Optimal monetary policy satisfies the Friedman rule.
3
Cash-In-Advance Model
3.1
Preliminaries
Environment
• In the real world there are goods that can be bought only with money.
• There are also goods that can be bought with credit.
• We implement that distinction into the model. Two types of goods:
– cash good c1t (that can be bought only with money)
– and credit good c2t that can be bought without using money (you can think of it as goods that
can be bought with a credit card and at the end of the period you pay for this good).
• With two types of goods: credit goods, and cash goods; we have also additional constraint which is
called CIA (cash in advance constraint) which says that cash good can be bought only with money
(this makes consumers to hold money in the model, otherwise consumer would not hold any money).
Timing
• First, households go to the assets market in which they buy money Mt and bonds Bt with their initial
assets holding At .
• Households split and one party goes shopping using money to purchase cash goods and credit to buy
credit goods, while the other party works.
• At the end of the period earnings are received by households and credit is paid. Furthermore, a central
bank makes helicopter drops Pt τt (could be negative). The total nominal asset holdings at the end of
period are At+1 .
18
Firms
• Producers hire labor and produce cash y1t and credit goods y2t . Both goods use the same technology,
thus the producer problem is
P1t y1t + P2t y2t − Wt nt
max
(y1t ,y2t ,nt )
subject to
y1t + y2t = zt nt
Since the production function exhibits constant returns to scale in equilibrium profits are zero.
• Since the technology for producing both goods is the same, their production costs are the same, thus
in competitive environment we have
P1t = P2t = Pt
(3.1)
If it were not true the producers would produce only the more expensive good.
• Therefore the producer problem becomes
Pt y1t + Pt y2t − Wt nt
max
(y1t ,y2t ,nt )
subject to
(3.2)
y1t + y2t = zt nt
Consumers
• Here we will have a representative agent who lives forever. In the beginning of each period consumers
hold nominal assets at , then they decide how to allocate this assets. They can buy cash Mt or bonds
Bt that offer the nominal interest rate Rt . Thus
M t + B t = At
(3.3)
Since cash goods can be purchased only with cash, we also have a cash-in-advance constraint
Pt c1t = Mt
(3.4)
This constraint introduces cash to the model, here consumers hold cash because they need it to carry
out transactions.
• Furthermore we assume that there is no disutility from labor (thus the consumer works one unit of
time, nt = 1) and for each unit of time consumers are paid the nominal wage rate Wt . Labor income
and savings can be spend on consumption of credit goods or can be saved for the next period in the
form of nominal assets At+1 . Hence the consumer budget constraint has the following form
At+1 + Pt c1t + Pt c2t = Wt nt + Bt Rt + Mt + Pt τt
(3.5)
• Instantaneous utility function is given by the following function u (c1t , c2t ) and the lifetime utility is
given by thus the consumer solves the following problem
∞
X
max
{c1t ,c2t ,nt ,Bt ,Mt ,At+1 }∞
t=0
β t u (c1t , c2t )
t=0
subject to
Mt + Bt
= At
Pt c1t
= Mt
At+1 + Pt c1t + Pt c2t
= Wt nt + Bt Rt + Mt + Pt τt
nt ∈ [0, 1]
19
(3.6)
Central bank
• Helicopter drops at the gross rate θt
s
Pt τt = Mts − Mt−1
s
where Mts = θt Mt−1
.
• If a central bank decides to increase the money supply it transfers it to the consumers τt > 0, and if
it decides to decrease the money supply is makes a negative transfer and takes money from consumers
τt < 0.
• Or, in real terms
τt =
where mst =
Mts
Pt
=
s
θt Mt−1
Pt−1
Pt
Pt−1
=
M s Pt−1
ms
Mts
− t−1
= mst − t−1
Pt
Pt Pt−1
πt
s
θt Mt−1
Pt−1
Pt−1
Pt
θt
s
πt mt−1
=
Market clearing
• Markets clear when demand and supply are equal. In the money market demand for money mt must
be equal to supply of money (decided by central bank) Mts
Mt = Mts
(3.7)
• Since government does not issue debt, in a closed economy the total private debt is equal to zero, thus
the aggregate bond holdings are equal to zero
Bt = 0
Since there are only two types of assets we have
At = Bt + Mt = Mts
q
0
• Furthermore, since the labor supply is always 1, the labor market clearing condition is
nt = 1
and the cash goods and credit goods market clearing conditions have the following form
c1t
=
y1t
c2t
=
y2t
Competitive equilibrium
Definition.
∞
A competitive equilibrium for this economy is an allocation {c1t , c2t , nt , y1t , y2t , Bt , At+1 , Mt }t=0 and prices
∞
{Wt , Rt , Pt }t=0 such that (given exogenous A0 )
∞
• {c1t , c2t , nt , Bt , Mt , At+1 }t=0 solves the consumer problem given prices and transfers.
W =
∞
X
max
{c1t ,c2t ,nt ,Bt ,Mt ,At+1 }∞
t=0
β t u (c1t , c2t )
t=0
subject to
Mt + Bt
At+1 + Pt c1t + Pt c2t
Pt c1t
20
= At
= Wt nt + Bt Rt + Mt + Pt τt
= Mt , nt ∈ [0, 1]
Definition 2. cont.
• for all t, (y1t , y2t , nt ) solves the producer problem given prices
Pt y1t + Pt y2t − Wt nt
max
(y1t ,y2t ,nt )
subject to
y1t + y2t = zt nt
• central bank monetary policy satisfies
s
Pt τt = Mts − Mt−1
s
where Mts = θt Mt−1
Definition 2. cont.
• markets clear
3.2
Mt
= Mts
Bt
=
At
= Mts
nt
=
c1t
= y1t ; c2t = yt
0
1
Solution
Solving Firm Problem
• Lagrangian
L = Pt y1t + Pt y2t − Wt nt − $t (y1t + y2t − zt nt )
• FOCs
y1t
: Pt = $t
y2t
: Pt = $t
nt
: Wt = $t zt
• Eliminating λs we get
Wt
= wt = zt
Pt
(3.8)
y1t + y2t = zt nt
(3.9)
and the production function
Solving HH Problem
• First we simplyfing the consumer problem. Dividing (3.4) by Pt and using mt =
Mt
Pt
c1t = mt
we get
(3.10)
• Take (3.5) and divide by Pt
At+1 Pt+1
+ c1t + c2t
Pt Pt+1
at+1 πt+1 + c1t + c2t
Wt
Mt
Bt
nt +
+
Rt + τt
Pt
Pt
Pt
= wt nt + mt + bt Rt + τt
=
Next, we divide by Pt (3.3) to get
mt + bt = at
multiplying both sides by πt and substituting for at from the budget constraint
(mt + bt )πt = wt−1 nt−1 + mt−1 + bt−1 Rt−1 + τt−1 − c1t−1 − c21t−1
21
(3.11)
• Constructing Lagrangian
n
o
L = β 0 u(c10 , c20 ) + ... + β t u( c1t , c2t ) + β t+1 u(c1t+1 , c2t+1 ) + ...
− {... + µt [ c1t − mt ] + λt [ bt πt + mt πt
− wt−1 nt−1 − Rt−1 bt−1 − mt−1 − τt−1 + c1t−1 + c2t−1 ]
+ µt+1 [c1t+1 − mt+1 ] + λt+1 [bt+1 πt+1 + mt+1 πt+1
− wt nt − Rt bt − mt − τt + c1t + c2t ] + ...}
or
L =
∞
X
β t u(c1t , c2t ) −
t=0
∞ h
X
µt (c1t − mt )
t=0
+λt (bt πt + mt πt − wt−1 nt−1 − Rt−1 bt−1
i
−mt−1 − τt−1 + c1t−1 + c2t−1 )
• FOCs
c1t
: β t u1 (t) − µt − λt+1 = 0
c2t
:
β t u2 (t) − λt+1 = 0
bt
: −λt πt + λt+1 Rt = 0
mt
: µt − λt πt + λt+1 = 0
and CIA
c1t = mt
Simplifying
c1t
: β t u1 (t) = λt πt
c2t
: β t u2 (t) = λt+1
bt
: λt πt = λt+1 Rt
mt
: µt + λt+1 = λt πt
and (note we do not need the last equation anymore)
β t u1 (t)
= λ t πt
λ t πt
β t u2 (t) =
Rt
λt
Rt
=
λt+1
πt
(3.12)
(3.13)
(3.14)
Eliminating λs we get:
– cash-credit good choice
u1 (t)
= Rt
u2 (t)
(3.15)
– Euler
λt
=
λt+1
β t u1 (t)
πt
β t+1 u1 (t+1)
πt+1
=
Rt
πt
u1 (t)
βu1 (t + 1)
=
Rt
πt+1
(3.16)
– and CIA
c1t = mt
22
(3.17)
Equilibrium
• From firms problem we have (3.8) - (3.9)
wt
= zt
(3.18)
yt
= zt nt
(3.19)
where yt = y1t + y2t
• From consumers problem we have (3.15) - (3.17)
u1 (t)
u2 (t)
u1 (t)
βu1 (t + 1)
c1t
= Rt
(3.20)
Rt
πt+1
= mt
(3.22)
=
(3.21)
• Market clearing conditions
mt
=
mst
(3.23)
bt
=
0
(3.24)
at
=
mt
(3.25)
nt
=
1
(3.26)
c1t + c2t
=
yt
(3.27)
• A competitive equilibrium is fully described by the following equations: from the producer problem
(3.18) - (3.19), from the consumer problem (3.20) - (3.22), and market clearing conditions (3.23) (3.27).
• Note that if we simplify the equations above we get the equations that characterize the competitive
allocation
u1 (t)
βu1 (t + 1)
c1t
u1 (t)
u2 (t)
c1t + c2t
with mt =
=
Rt
πt+1
mt
=
Rt
(3.30)
=
yt = zt nt = zt
(3.31)
=
(3.28)
(3.29)
θt
πt mt−1 .
• Consider u(c1t , c2t ) = log c1t + φ log c2t then (3.28) - (3.31) become
c1t+1
βc1t
c1t
c2t
c1t
c1t + c2t
=
Rt
πt+1
mt
=
Rt
(3.34)
=
yt = zt nt = zt
(3.35)
=
Substituting from (3.33) into (3.32) we get and using mt =
mt+1
Rt
=
βmt
πt+1
θt+1
πt+1 mt
βmt
23
=
Rt
πt+1
(3.32)
(3.33)
θt
πt mt−1 .
Simplifying
Rt = β −1 θt+1
(3.36)
Next, substituting from (3.34) into (3.35)
c1t + c2t = zt nt = zt
c1t + Rt c1t = zt
c1t =
Substituting from (3.36)
c1t =
zt
1 + Rt
zt
1 + Rt
and using (3.34)
c2t = Rt c1t =
Rt zt
1 + Rt
and it is straightforward to find the rest.
3.3
Steady state
Non-superneutrality of Money
• In the steady state from (3.28) - (3.31) we get
R
π
= m
u1 (c1 , c2 )
βu1 (c1 , c2 )
c1
u1 (c1 , c2 )
u2 (c1 , c2 )
c1 + c2
=
= R
= z
with π = θ. Simplifying
R
u1 (c1 , c2 )
u2 (c1 , c2 )
c1 + c2
=
β −1 θ
=
β −1 θ
=
z
• Therefore money is not superneutral since it drives a wedge between cash and credit goods. For
example since separable functions we have u12 = u21 = 0 the an increase in growth rate of money
lowers consumption of cash goods and increases consumption of credit goods.
• Endogenous labor does not change the result.
3.4
Optimal Monetary Policy.
Optimal Monetary Policy
Definition.
An efficient allocation is an allocation {c1t , c2t , yt , lt }∞
t=0 such that it solves the following problem
max
{c1t ,c2t ,yt ,nt }∞
t=0
subject to
∞
X
β t u (c1t , c2t )
t=0
c1t + c2t = yt = zt nt
nt ∈ [0, 1]
24
• To find optimal allocation we simplify
∞
X
max ∞
• Lagrangian
L=
β t u (c1t , c2t )
{c1t ,c2t }t=0
t=0
subject to
c1t + c2t = zt
∞
X
β t u (c1t , c2t ) −
t=0
∞
X
λt (c1t + c2t − zt )
t=0
• FOCs
c1t
: β t u1 (t) − λt = 0
c2t
: β t u2 (t) − λt = 0
and
: c1t + c2t = zt
Eliminating λs
u1 (t)
c1t + c2t
= u2 (t)
(3.37)
= zt
(3.38)
• Hence, an optimal allocation is characterized by equations (3.37) - (3.38). In order to find optimal
monetary policy we compare competitive allocation (characterized by equations from (3.28) - (3.31))
with optimal allocation (characterized by (3.37) - (3.38)). Notice, that equations (3.28) - (3.31) and
(3.37) - (3.38) are the same for Rt = 1. Which, after we define a new term, allows us to state the
following Proposition.
Definition.
We say that monetary policy satisfies the Friedman rule if it leads to the zero (net) nominal interest rate,
i.e. Rt = 1.
Theorem
Monetary policy is optimal if and only if it satisfies the Friedman rule.
Proof.
(1) Suppose that the monetary policy satisfies the Friedman rule, then by definition Rt = 1. Which, since
equations (3.37) - (3.38) and (3.30) - (3.31) are the same, implies that a competitive allocation is the same
as an efficient allocation. (2) If monetary policy does not satisfy the Friedman rule (which means Rt > 1),
then a competitive allocation is not efficient, since equations (3.30) - (3.31) and (3.37) - (3.38) are not the
same.
• Intuition. If Rt = 1 then the nominal interest rate on bonds and money is the same (and equal to
0). So holding money or bonds does not make any difference. Since money is needed for purchases of
cash goods, and is not needed to purchases of credit goods positive nominal interest rate would disturb
the the cash-credit good choice. The efficiency is achieved when marginal rate of substitution between
both goods is equal to one, and the positive nominal interest rate works similarly to a tax on cash
goods. It imposes a wedge that distorts the choice.
• There is also another way of looking at this. From the CIA condition we can see that people need to
buy money (instead of bonds) to purchase cash goods. Thus they have to give up the interest that
they could have made, had they bought bonds. Thus the price of cash goods from the point of view
of consumers is increased by the foregone interest. As a result if R > 1 consumers buy to little cash
goods.
• If the nominal interest rate is zero, then this problem is solved and efficiency restored.
• Note, if in the steady state R = 1 then in the steady state, from R = β −1 θ we get θ = β and from
mt = πθtt mt−1 in the steady state we get π = θ = β. Which implies that if the monetary policy satisfies
the Freidman rule we get deflation.
25
Cost of inflation
• We want to see quantitative implications of costs of inflation. Monetary policy leads to a change in an
allocation that generates different values of utility. So far we were only able to establish that a deviation
from the Friedman Rule is costly. Now we will address the question how costly. The procedure is as
follows. Consider two model economies: one economy is the economy in which the growth rate of
money follows the Friedman Rule, θt = β, denote it by θF ; and the other one is the economy with
θt = θi > θF labeled by i. The equilibrium path associated with the economy in which the growth rate
of money θt = θF (follows the Friedman Rule) gives the following utility
UF =
∞
X
F
β t [log cF
1t + φ log c2t ]
t=0
• The equilibrium path associated with economy in which the growth rate of money θt = θi gives the
following utilityU i :
∞
X
Ui =
β t [log ci1t + φ log ci2t ]
t=0
• There is a problem with comparing utilities because (recall from Intermediate Microeconomics) the
values of utility allow only to compare if something is better not how much better. But we can do
something else. Consider the economy with the growth rate of money θi . Let xi be percentage increase
in consumption of cash good such that the following equation is satisfied
UF =
∞
X
β t [log 1 + xi ci1t + φ log(1 + xi )ci2t ]
(3.39)
t=0
Interpretation: decreasing the growth rate of money from θi to θF has the same effect on welfare as
an increase in consumption by xi %. In this case xi can be interpreted as the cost of inflation.
• Notice
U
F
=
∞
X
β t [log 1 + xi ci1t + φ log(1 + xi )ci2t ]
t=0
UF
=
∞
X
β t [log 1 + xi + φ log(1 + xi ) + log ci1t + φ log ci2t ]
t=0
UF
∞
X
U
∞
X
(1 + φ) log 1 + xi
=
1
+ Ui
(1 + φ) log(1 + x )
1−β
t=0
F
βt +
=
β t [log ci1t + φ log ci2t ]
t=0
i
Solving with respect to xi
log 1 + xi
i
elog(1+x )
=
=
i
=
xi
=
1+x
Cooley and Hansen (1989)
26
1−β
UF − Ui
1+φ
1−β
F
i
e 1+φ (U −U )
1−β
F
i
e 1+φ (U −U )
1−β
F
i
e 1+φ (U −U ) − 1
Table 1: Moment matching
US data (qtrly)
Model
St. Dev. Corr(·, y) St. Dev. Corr(·, y)
Series
Output
1.74
1.00
1.73
1.00
0.81
0.65
0.62
0.72
Consumption
Investment
8.45
0.91
5.69
0.97
0.38
0.28
0.48
0.06
Capital Stock
Hours
1.41
0.86
1.33
0.98
0.89
0.59
0.5
0.87
Productivity
Price Level
1.59
-0.48
1.7
-0.27
Welfare cost of inflation (consumption equiv): 0% : 0.144 and 10% : 0.52
3.5
Summary
Summary
• Model behaves in a similar fashion as MIU model.
• Money is not superneutral.
• The mechanism for a negative impact of inflation on the economy because it disturbs the ability of the
price mechanism to convey the proper information about costs from producers to consumers (see the
results of the first and the second welfare theorems). Inflation sticks a wedge between cash and credit
goods, similarly like a tax.
• Optimal monetary policy satisfies the Friedman rule - efficiency is restored.
• We can use this model to quantitatively compute the costs of inflation (later).
27