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Monetary Business cycles –
Lesson 1
Introducing money in the neoclassical model
1- The money-in-the-utility model (MIU)
2- The cash-in-advance model (CIA)
Introduction - Objective
We incorporate money in a standard RBC
model
What is the effect of money on the economy?
Can we generate a procyclical behavior for labor?
Introduction - Objective
But first, we need to introduce a demand and
supply for money
Supply of money
Budget constraint of the public sector
In nominal terms:
Where does this equation come from?
How does the Central Bank issue money?
What are the underlying assumptions?
Monetary aggregates
Supply of money: How is money created?
How to increase the supply of money
Increase the direct monetary transfers to
households
Open market operations: sell government
bonds
?
Supply of money
A short digression on monetary aggregates to
understand the underlying assumption(s)
Monetary aggregates
Notes and coins in circulation
Currency
Notes and coins in bank vaults
Central bank credit
Demand deposits
Savings and time deposits
Monetary
base
(MB)
Monetary aggregates – from narrow to broad
Notes and coins in circulation
Notes and coins in bank vaults
Central bank credit
Demand deposits
Savings and time deposits
M0
Bank reserves R
(minimum reserves
+ excess reserves)
MB
Monetary aggregates – from narrow to broad
More Liquid
Notes and coins in circulation
Notes and coins in bank vaults
Central bank credit
Demand deposits
Savings and time deposits
M0
M1
M2
Monetary aggregates – from outside to
inside money
Some elements of MB (bank reserves) are not in
M0, M1, M2
Why?
M0, M1, M2: liabilities of the banking sector visà-vis the private economy (households, firms…)
Exclusion of liabilities of the banking sector
(central bank+commercial banks) to itself
Avoid double counting
Monetary aggregates – from outside to
inside money
Notes and coins in circulation
Central bank money
Notes and coins in bank vaults
(« outside money »)
Central bank credit (minimum
=MB
reserves+excess reserves)
Demand deposits
Savings and time deposits
Commercial bank money
(« inside money »)
=M1-M0, M2-M0
Monetary aggregates – from outside to
inside money
Outside money (MB): liabilities of the central bank,
controlled by the central bank
=policy instrument.
Outside + inside money (M1 or M2 ): liabilities of the
banking system (Central Bank+commercial banks) vis-à-vis
the private economy
= actual supply of money to the private economy
How does the Central Bank control the actual money
supply through its policy instrument?
Monetary aggregates
Monetary aggregates - US
9000.0
8000.0
6000.0
M0
M1
M2
MB
5000.0
4000.0
3000.0
2000.0
1000.0
Source: Federal Reseve Bank of Saint-Louis
2009
2007
2005
2003
2001
1999
1997
1995
1993
1991
1989
1987
1985
1983
1981
1979
1977
1975
1973
1971
1969
1967
1965
1963
1961
0.0
1959
Billions of US$
7000.0
Monetary aggregates – from outside to
inside money: the money multiplier
Monetary multiplier m=M1/MB>1: Money is
created endogeneously by the banking system
« Fractional-reserve banking »: one unit of deposits
is backed by less than one unit of reserves: the
banks lend part of the deposits
Supply of money
Underlying assumption(s)?
Supply of money
Budget constraint of the public sector
In real terms:
Demand for money
Why is there a demand for money? Motive to
hold money?
Demand for money
Two main approaches:
Money in the utility function: money provides
transaction services holding money affects
directly the utility of agents (Sidrauski 1967)
Cash-in-advance constraint: money is held to
finance purchases affects indirectly the utility of
agents (through the utility of goods it helps
purchasing) (Lucas 1982, Svensson 1985, Cooley
and Hansen 1989)
Money-in-the-utility (MIU)
We first ignore uncertainty and labor-leisure
choice
Three types of assets (stores of value):
capital k:
, f increasing and
strictly concave
nominal bonds B: return it, nominal interest rate
and money M, no nominal return but flow of
services
MIU - Utility
Representative household with utility function:
where
is the stock of real money balances at
the end of the period
u is increasing in c and m, strictly concave and
continuously differentiable
U is separable in c and m: same utility out of
money whatever the level of consumption
Intertemporal utility:
MIU – Household’s budget constraint
Nominal budget constraint:
In real terms:
Inflation tax
Inflation Distortion on the demand for
money
MIU – FOC
Lagrangian:
FOC:
(1)
(2)
(3)
(4)
MIU – FOC
Intertemporal arbitrage:
Since resources must be divided between
consumption, capital, bonds and money balances,
each use must yield the same marginal benefit
MIU – Intertemporal arbitrage
These equivalences imply two relations that are
relevant for money and inflation:
No-arbitrage between nominal bonds and capital
(between nominal and real assets in general):
Notation for real return on capital:
Fisher relationship:
Nominal interest rate
Return on real asset
Approximation:
Inflation rate
MIU – Intertemporal arbitrage
No-arbitrage between money and consumption:
Demand for money:
User cost of money
(price of money)
Marginal rate of
substitution
between money
and consumption
=inflation tax/(1+rt)
Inflation affects the demand for money through the inflation
tax
MIU – Closing the model
Money supply: the government sets the
growth rate of money
Zero supply of bonds:
Aggregate budget constraint of the
economy:
MIU – Dynamic system
Euler equation
Budget constraint
Fisher relationship
Growth of real
money balances
Demand for money
Endogenous variables: m, k, c, i, п
MIU – Neutrality of money
Now that we have the dynamic system, we ask:
Is money neutral? real variables (capital, output
and consumption) are independent of the level of
money supply M?
Is money superneutral? real variables are
independent of the inflation rate п and the rate
of growth of nominal money θ?
MIU – Superneutrality of money
Take the Euler equation and the budget
constraint separately:
Ramsey model
in discrete time
Superneutrality?
MIU – Steady-state equilibrium – Real
variables
Steady state defined by:
Money grows at rate θ
c, m and k are constant
Long-run superneutrality: steady-state real
variables (capital and consumption) are
independent of the rate of inflation θ:
MIU – Steady-state equilibrium – Nominal
variables
m is constant Prices grow at the same rate as
money
Nominal interest rate defined by the Fisher
relationship:
Approximation:
MIU – Steady-state equilibrium – Effects
of inflation
The fact that inflation has no real effect does
not mean that it doesn’t affect households!
Then what is the effect of inflation on the
economy?
It affects the demand for money
It affects utility
MIU – Steady-state equilibrium – Demand
for money
Steady-state real money holdings are
determined by the steady-state user cost of
money:
Fisher relationship: inflation increases the user cost
of money
How does inflation affect real money balances m?
MIU – Steady-state equilibrium – Utility
How can the government increase welfare?
MIU – Superneutrality – More on intuition
Why doesn’t money growth θ have any
effect on the real economy?
Increase in money growth rate θ leads to an
increase in the cost of money (inflation tax)
Analysis in terms of wealth and intertemporal
substitution effects
MIU – Superneutrality – More on intuition
Wealth effect: the inflation tax weighs on
households’ budget
This should impair HH consumption (negative
wealth effect)
Is it the case? Why?
MIU – Superneutrality – More on intuition
Intertemporal substitution effect:
Intertemporal substitution of consumption:
Intertemporal substitution of money:
They are independent
Classical « dichotomy »
MIU – Taking stock
So far:
Important concepts:
Inflation tax
Fisher relationship: i=r+ п
Effect of money on the economy
Money has an effect on utility and on the demand for
money
However, money is both neutral and superneutral:
absolutely no effect on real variables!
Is that realistic? Which assumption should we relax?
MIU – Superneutrality and the separability
of utility function
Superneutrality relies on separability between
consumption and money:
It means that having more money does not affect the
marginal utility of consumption
However, the utility of money (eg. facilitating
transactions) is not independent of the amount of
goods we would like to buy (non-separability):
MIU – Non-separability – Dynamic
system
General case (non-separability):
Euler equation
Budget constraint
Fisher relationship
Growth of real
money balances
Demand for money
MIU – Non-separability – Superneutrality?
Importantly, the Euler equation is affected by
money holdings:
Money affects intertemporal substitution of cons.
Example: money and consumption are Edgeworth
complements:
How does money affect consumption?
What about wealth effects?
MIU – Non-separability – Superneutrality?
Real effect of money in the short run
What happens in the long run?
MIU – Introducing shocks
Analysis of the impact of monetary shocks
Uncertain environment:
Monetary shocks:
Productivity shocks:
Expected intertemporal utility:
MIU – Shocks – FOC
Lagrangian:
FOC:
(1)
(2)
(3)
(4)
MIU – Shocks – Intertemporal arbitrage
Intertemporal choices are affected by
uncertainty
Euler equation:
MIU – Shocks – Intertemporal arbitrage
Fisher relationship:
Approximation:
with
Any substantial change?
MIU – Shocks – Dynamic system
Euler equation
Budget constraint
Fisher
relationship
Growth of real
money balances
Demand for money
Productivity shocks
Monetary shocks
MIU – Shocks – Specifications
Functional forms:
The effect of money on real variables
depends on Ucm
Ucm has the same sign as b-Φ
What are b and Φ?
MIU – Shocks – Specifications
1/Φ: elasticity of intertemporal substitution: 0.5-3
b: elasticity of substitution between goods and money
Equation of demand for money:
1/b is the interest elasticity of money: 0.3-0.5 b=2-3.3
Benchmark: Φ=2 , b=2.56
Since b-Φ>0, then
(c and m are complements)
MIU – Shocks – Baseline calibration
MIU – Shocks
We consider two cases:
1)
No autocorrelation in the monetary shock:
γ=0
2)
Positive autocorrelation in the monetary
shock: γ>0
% deviation from SS
Responses to a shock in money growth
1
inflation
0.5
0
-2
nominal rate
0
2
4
Years after shock
6
Responses to a shock in money growth
1
0.5
0
investment
-0.5
-1
-2
0
2
4
Years after shock
6
8
Responses to a shock in money growth
1
0.5
0
real rate
-0.5
-1
-2
8
% deviation from SS
% deviation from SS
% deviation from SS
MIU: Responses to a money growth shock,
γ =0 (no autocorrelation)
0
2
4
Years after shock
6
8
Responses to a shock in money growth
1
0.5
0
output
-0.5
-1
-2
0
2
4
Years after shock
6
8
MIU: Responses to a money growth shock,
γ =0.5 (autocorrelation)
Responses to a shock in money growth
1
0.5
0
-2
0
2
4
Years after shock
6
8
0.04
0.02
0
-0.02
-2
0
2
4
Years after shock
Responses to a shock in money growth
5E-3
0E-3
investment
-5E-3
consumption
-10E-3
-2
0
2
4
6
Years after shock
8
% deviation from SS
% deviation from SS
1.5
nominal rate
% deviation from SS
0.06
inflation
% deviation from SS
% deviation from SS
Responses to a shock in money growth
2
6
Responses to a shock in money growth
0
-0.5E-5
-1E-5
-1.5E-5
8
-2E-5
-5
real rate
0
5
Years after shock
Responses to a shock in money growth
4E-4
output
3E-4
2E-4
1E-4
0
-2
0
2
4
Years after shock
6
8
10
MIU: Responses to a money growth shock
Only expected inflation has an impact
Is money neutral?
Is it superneutral?
What is the channel?
MIU – Introducing labor supply
Elastic labor supply l:
Equilibrium in the labor market:
How does money affect labor supply?
MIU – Introducing labor supply
Functional forms:
MIU: Responses to a money growth shock,
φ=0.5, with elastic labor supply (baseline)
1
0.5
0
-2
0
2
4
Years after shock
6
8
nominal rate
0.04
0.02
0
-0.02
-2
0
Responses to a shock in money growth
5E-3
0
investment
-5E-3
consumption
-10E-3
-2
0
2
4
6
Years after shock
8
2
4
Years after shock
% deviation from SS
% deviation from SS
1.5
% deviation from SS
% deviation from SS
inflation
0.06
% deviation from SS
Responses to a shock in money growth
Responses to a shock in money growth
2
6
8
Responses to a shock in money growth
0
-1E-5
-2E-5
-3E-5
-2
real rate
0
2
4
6
Years after shock
Responses to a shock in money growth
5E-4
0
-5E-4
output
employment
-10E-4
-2
0
2
4
6
Years after shock
8
8
MIU: Responses to a money growth shock –
conclusion
Real effect on output only if anticipated rise in money
growth: money is neutral but not superneutral
The effect on output is negative
Channel:
Positive effect on the nominal interest rate (Fisher equation),
which increases the cost of money (inflation tax)
Intertemporal substitution effects
Intertemporal substitution of consumption (effect on
capital)
Intertemporal substitution of leisure (effect on labor
supply)
Small quantitative effects
Non-neutrality comes from a supply effect ≠
keynesian view of fluctuations
Cash-in-advance constraint (CIA)
Same model as MIU
Only differences:
Utility does not depend directly on mt :
Cash-in-advance constraint (households must hold cash
in order to make transactions):
We also start by ignoring uncertainty and laborleisure decisions
CIA – Demand for money
Cash-in-advance constraint:
This constraint is binding (why?):
Budget constraint of the government:
CIA – Specifications
We add stochastic productivity and monetary
shocks
We add labor
Same functional forms as MIU, except:
Same calibration
% deviation from SS
Responses to a shock in money growth
1.5
inflation
1
0.5
nominal rate
0
-0.5
-2
0
2
4
Years after shock
6
8
Responses to a shock in money growth
1
investment
0.5
0
consumption
-0.5
-2
0
2
4
Years after shock
6
Responses to a shock in money growth
0
-0.5E-3
-1E-3
real rate
-1.5E-3
-2
% deviation from SS
% deviation from SS
% deviation from SS
CIA: Responses to a money growth shock
8
0
2
4
6
Years after shock
8
Responses to a shock in money growth
0.02
0
-0.02
-0.04
-2
output
employment
0
2
4
Years after shock
6
8
CIA: Responses to a money growth shock
Similar effects
But stronger responses
Why?
Empirical validity? None!
See next lesson