Money in the Competitive
Equilibrium Model
Part 2
Explicit Money Demand
Cash-in-Advance Model
Optimal Monetary Policy
Money and Real Ecomomic
Variables
• Neutrality of Money  a one-time change in the
level of the nominal money supply has no effect
on real economic variables (only nominal).
• Superneutrality of Money  a change in the
growth rate of the money supply has no effect on
real economic variables.
– Sometimes “superneutrality” definition exclued the
real money supply as a “real” economic variable.
• CE model with Ad-Hoc money demand (e.g.
Cagan model)  money is neutral and
superneutral.
• An increase in the money growth rate 
    and   e  MD  R and  (M / P)
• Classical dichotomy  No change in CE values
of y*,N*,c*, and r*.
• This result may not be true in CE model with
explicit money demand.
• Reminder: Nominal versus Real Interest Rates:
1  Rt
(1  rt ) 
1  t
or
r=R–
where R = nominal rate
r = real rate
inflation rate =  t 
(exact)
(approx)
Pt 1  Pt Pt 1

1
Pt
Pt
Explicit Money Demand
• Incorporate use of money as a decision of the
representative household.
• Assumptions:
(A1)
Income yt is exogenous
(A2)
Households make an asset allocation
decision between nominal money (M)
and bonds (B).
(A3)
TO BE ADDED
(A4)
Government directly sets nominal Ms
(A6)
No uncertainty
• Money Supply:
M ts1  M ts  X t  (1   )M ts
where Xt = transfer of money to public
(“helicopter drop”) and  = money growth rate
• Reminder: Real vs Nominal Interest Rates:
(1+r) = (1+R)/(1+) or
r = R –  (approx)
where
Pt 1  Pt Pt 1
t 

1
Pt
Pt
• Timeline
• Budget Constraint (nominal terms)
Pt yt  Bt (1  Rt )  M t  X t  Pt ct  M t 1  Bt 1
Total Sources of Income = Total Uses
• Optimization: Choose {ct, Mt, Bt} to

max  t 1u (ct )
t 1
subject to (BC)
(BC)
• State Variables: M t , Bt
Control Variables: c , B
t
t 1
• Bellman Equation:
V ( M t , Bt )  max u (ct )  V ( M t 1 , Bt 1 )
ct , Bt 1
subject to
M t 1  Pt yt  Bt (1  Rt )  M t  X t  Pt ct  Bt 1
(transition equation)
• FOC and Envelope conditions contradict unless
R = 0.
• If R > 0 then M = 0. Money is an inferior asset
to bonds and valueless.
• Need another constraint to give money value.
Cash-in-Advance Model
(A3): Consumption must be purchased with cash
carried in advance from previous period.
• New Timeline
• Cash-In-Advance Constraint
M t  Pt ct
(CIA)
• State Variables: M t , Bt
Control Variables: c , B
t
t 1
• Bellman Equation:
V ( M t , Bt )  max u (ct )  V ( M t 1 , Bt 1 )  t [ M t  Pt ct ]
ct , Bt 1
CIA Constraint
subject to
M t 1  Pt yt  Bt (1  Rt )  M t  X t  Pt ct  Bt 1
• FOC & Envelope
MRSct ,ct1
 1  Rt
u ' (ct )

 
u ' (ct 1 )  1   t

  (1  rt )

(1)
• Market-Clearing (MC):
Goods:
ct = yt = y*
Money:
Mt = Mts
Bonds:
Bt = 0
(Note from BC if two of the three markets clear, the third
one will also clear)
• The CE are values for {ct, yt, Bt, rt, m=(M/P), R,
} solving (1), (2) and (MC) conditions.
• CE Values:
c* = y*
(exogenous)
*  
r* = (1/ – 1) = r
(M/P)* = c*
(Neutrality)
(1+R) = (1+r*)(1+*) (Fisher Effect)
• One time changes in the level of Ms are neutral.
• Increases in the growth rate of money () leads
to an increase in * and R* while leaving c*, y*,
r* unchanged. (Superneutrality)
• This result comes from exogenous income and
is not general when model is modified.
• Consider adding labor market and firms to the
model.
Figure 15.4 Scatter Plot of the Inflation Rate vs.
the Growth Rate in M0 for the United States,
1960–2003
CIA Model with Production
• Cooley and Hansen (1989 – AER)
• Modify to Include Labor and Production
(1) yt = f(Nt)
(2) Utility in each period: U(ct,lt) = u(ct) + u(lt)
(3) Firms demand labor to max P = f(N) - wN
(4) Modify (BC)
Ptwt Nts  Bt (1  Rt )  M t  X t  Pt Pt  Pt ct  M t 1  Bt 1
(5) (CIA) is the same
(BC)
• Household FOCs
MRSct ,ct1
 1  Rt
u ' (ct )

 
u ' (ct 1 )  1   t

  (1  rt )

u ' (lt )
wt
MRSl ,c 

u ' (ct ) 1  Rt
(FOC1)
(FOC2)
• Firm FOC:
wt  f ' ( Nt )  MPNt
• Market-Clearing (MC):
Goods:
ct = y t
Money:
Mt = Mts
Bonds:
Bt = 0
Labor:
Nts = Ntd = Nt
• Utility: Assume u(c,l) = ln(c) + ln(l)
(FOC3)
• A steady state equilibrium occurs where N, c, y,
(M/P) are constants (to be determined, NOT
exogenous):
N t  N t 1  N *
ct  ct 1  c *
yt  yt 1  y *
M t M t 1

 m*
Pt
Pt 1
• Steady State CE Values:
(s1)
*  
(s2)
r* = (1/ – 1) = r
(s3)
(1+R) = (1+r*)(1+*)
(s4)
(s5)
(Fisher Effect)
f ( N *) f ' ( N *)

1 N *
1 
c* = y* = f(N*) = (M/P)=m*
• Notice (s4)  N* and there will be an inverse
relationship between N* and .
• In CIA model with production money is neutral
but not superneutral.
• Money growth and inflation negatively affects
employment, consumption, output, real
balances.
• Inverse Phillips Curve - relation between
inflation and “unemployment” is upward sloping.
• Inflation “taxes” work and households substitute
towards leisure.
Inflation & Employment: Cross Country
Study [Cooley & Hansen (1989)]
Xass
1976-1985
Austria, Belgium
Demark, Finland
France, Germany
Greece, Ireland, Italy
Netherlands, Norway
Portgual, Spain
Switzerland, UK
Canada, US, Australia
New Zealand, Japan
Chile, Venezuela
Vertical Axis =
employment
Costs of Inflation and Optimal
Monetary Policy
• Recall relation between nominal and real
interest rates:
r*  R  
(approx)
1 R
(1  r*) 
1 
(actual)
• CEM (in steady state)  r* = r constant.
• increase   increases   increases R
• High inflation leads to higher costs of conducting
transactions with currency (“shoe-leather” costs).
• Welfare costs of inflation: Lucas (2000,
Econometrica) estimates that reducing US
steady inflation from 10% to 0% is equivalent to
1% gain in real GDP.
• What is the optimal money growth rate * in the
CE/CIA model with production?
• What’s the “optimal” inflation rate in the longrun?
• What value of  maximizes utility of the
representative household?
• The best (welfare maximizing) allocation is
the Pareto Optimal allocation:
MRSl,c = w
MRSct,ct+1 = (1+r*)
• Money distorts the optimal decisions of
individuals away from social planner.
• The “Friedman Rule” says that the optimal
monetary policy is to deflate the money supply
and prices at a rate which drives R = 0:
(i) If R = r* +  ,
R = 0  *    r* < 0
(ii) If (1+R)
= (1+r)(1+) = (1+)/
R = 0
*     - 1 < 0
• The Friedman Rule requires deflation at the real
interest rate or rate of time preference.
(M. Friedman – The Optimum Quantity of
Money, 1969)
• Practical Considerations
* Drive the nominal rate on riskless assets
(government bonds) to zero.
* Nominal variables (wages) are downward
rigid.
* There are always temptations to inflate the
money supply (funding G, business cycles).
* Assumes certainty about money/prices.
* Most economists agree that low inflation
(rather than deflation) is more practical.
M1 Money Supply, 2000-2010
Levels
M1 Money Supply, 2000-2010
Growth Rate
• Current monetary policy and the Friedman rule:
– High money growth rate
– Historically Low Nominal Interest Rates
– Moderate/Low Inflation
• Model provides good description of long-run or steady
inflation but lacks “liquidity effects” important for
business cycle analysis.
• Solution? Modify Model or abandon market-clearing
(stick prices, IS-LM?)
• Readings:
Williamson, Ch 10, p 363-368, 377-388, 395-399
Williamson, Ch 15, p 559-575