Neil Arnwine

English

Research Papers in Economics

 
 
 
 
The Fisher Equation and Output Growth 
 
 
 
A B S T R A C T 
 
 
 
Although the Fisher equation applies for the case of no output growth, 
I show that it requires an adjustment to account for non-zero output 
growth.  I demonstrate this using a standard model with constant 
relative risk aversion utility containing money, nominal bonds and 
output growth in a risk-free setting.   
 
Failing to incorporate a growth term in regressions will lead to biased 
estimation results. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
30 November 2004 
 
 
Neil Arnwine 
Lecturer 
Bilkent University 
 
I would like to thank Ivan Pastine and Kevin Hasker for their comments.  All remaining errors are 
my own.  
 
 
 
I.  Introduction 
The Fisher equation is a well-accepted theoretical construct in macroeconomics and 
finance.  It can be expressed as: 
() r E i + π =  (1) 
where i is the nominal interest rate,   is the expected inflation rate and r is the real 
interest rate.  In this paper I show that when there is growth in the level of consumption over 
time the Fisher equation becomes: 
() π E
() () y E E i & ⋅ α + ρ + π =  (2) 
where ρ represents the consumer’s real rate of time preference,   represents the expected 
rate of output (or consumption) growth, and α is the degree of relative risk aversion.  The 
logic is simple:  output growth reduces the marginal utility of real income over time 
increasing the real interest rate: 
() y E &
() y E r & ⋅ α + ρ = . (3) 
In equation 2 expected output growth increases the nominal interest rate as the consumer tries 
to smooth consumption by saving less.  Conversely, a decrease in expected output reduces 
the nominal interest rate.  When no output growth is expected equation 2 is identical to 
equation 1. 
This modification of the Fisher equation to account for growth does not alter Fisher’s 
original view that real and monetary factors are separable in their effect upon the economy.  
Expected inflation still effects output on a one-to-one basis, ceteris paribus, and has no effect 
upon the real rate of time preference of the consumer.   
Fisher (1896) introduces the Fisher equation, see Dimand (1999), and Fisher (1930) 
contained an extensive discussion of the relation.  After more than 100, years Fisher’s 
  1 relation is still a work horse.  In the two years 2003-2004 alone there were more than a dozen 
studies on the Fisher equation, mostly featured in empirical journals.  Fisher’s contribution 
here was to observe that real and nominal factors can be separated when analyzing their 
effects upon the economy.  This is a precursor to the theory of the neutrality of money.  His 
discussion of the equation demonstrated an understanding of rational expectations decades 
before the term was coined. 
The Fisher equation has various interpretations.  At one extreme we can view r as a fixed 
constant expressing the consumer’s rate of time preference so that expected inflation passes 
through on a one-to-one basis into nominal interest rates.  Even if this view is correct, since 
the researcher cannot observe the consumer’s ex-ante inflation expectation there will still be 
a stochastic element to empirical studies.  Alternatively, we can view the real rate as 
potentially time-varying.  Then, while there will be a one-to-one long run relationship 
between inflation and nominal interest rates it may not be exact in the short run.   
Mundell (1963) and Tobin (1965) provide a theoretical explanation for a time-varying 
real interest rate.  For Mundell (1963) higher inflation reduces wealth by reducing the desired 
level of real money balances.  This stimulates saving, which reduces the real interest rate.  
Tobin (1965) sees inflation as driving saving away from nominal assets to real assets, 
reducing the real interest rate by lowering the marginal product of capital.  Evans and Lewis 
(1995) considers the case in which inflation follows a Markov-switching process so that ex-
post observed the real interest rate features a non-zero expectation and is serially correlated.  
In empirical studies Mishkin (1984) and Pelaez (1995) reject the hypothesis that the real 
interest rate is constant for the OECD and U.S., respectively.  Fama (1981) concluded that 
variation in the real interest rate could be explained by real, rather than by nominal, factors.  
None of these studies necessarily invalidates the Fisher equation as a long run relation.  More 
  2 recently Rapach (2003) does reject the long run Fisher equation, concluding that interest rates 
adjust less than one-to-one with inflation in the long run, based upon a VAR study. 
One empirical observation that appears in the literature is that the inclusion of real 
variables in an estimation of the Fisher equation yields a significant coefficient; see Levi and 
Makin (1978), Evans and Makin (1979), VanderHoff (1984), and Dotsey, Lantz and Scholl 
2003).  This is in seeming contradiction to the Fisher equation’s prediction that only inflation 
explains the nominal interest rate.  The argument put forward by Levi and Makin (1978), 
Evans and Makin (1979), VanderHoff (1984), and others is that the Phillips curve introduces 
a short run bias.  They viewed this bias as disappearing in the long run because the economy 
reverts to full employment level of output (Levi and Makin 1979, p. 43).   
 
II.  Model 
I examine a standard cash-in-advance consumption model with constant relative risk 
aversion utility function and output growth.  The model contains no risk. This assumption is 
made to simplify notation and so that it is clear that my findings do not depend upon Phillips 
curve explanations, Mundell-Tobin arguments or on a deviation from the quantity theory of 
money.  There are no frictions in the model. 
Production is exogenously determined so we can interpret this as a Lucas-type tree 
economy.  The representative household can be thought of as a worker-shopper pair.  The 
worker sells the exogenously arriving output in a competitive market and the shopper collects 
the couple’s income, as cash balances, at the end of the period and proceeds to shop in the 
next period.  I assume a constant output growth rate.  Output evolves according to: 
1 − ⋅ γ = t t y y .   (4) 
Throughout I will assume a risk-free steady output and monetary growth so these growth rates 
have no time subscript.   
  3 Government in this model introduces money at a constant growth rate, sells one-period 
bonds on a discounted basis, and provides a lump sum transfer payment.  The money supply 
evolves according to  t t M M ⋅ ω = +1 , where a bar over the variable indicates it is the supply of 
the variable.  The government’s finance constraint insures that the revenue from seigniorage 
plus the net revenue from bond sales and redemptions is equal to the lump sum transfer 
payment: 
() () t t t t t B B i M T − ⋅ + + ⋅ − ω = +
−
1
1 1 1 .   (5) 
I assume that the government does not indefinitely let the size of debt grow faster than output. 
The representative consumer has a constant relative risk aversion utility function: 
()
() 


 


= α =
≠ α ⋅
α −
= α −
1 , ln
1 ,
1
1 1
c c u
c c u
   (6) 
The consumer’s budget constraint is: 
() t t t t t t t t t t T y P B M B i M c P + ⋅ + + ≤ ⋅ + + + ⋅ +
−
+ 1
1
1 1  (7) 
where the right hand side represents the consumer’s wealth.  This consists of money carried 
into this period from the previous period, the maturing one-period nominal bond, output sold 
in a competitive market, and the lump sum government transfer payment.  The left hand side 
of equation 6 shows how the consumer allocates his wealth.  There is a single consumption 
good.  Saving takes the form of nominal money balances and discounted government bonds. 
Money demand is motivated by a cash-in-advance constraint placed upon the 
consumption good.  This creates a demand for money, even though money is dominated in 
return by bonds and equities.   
t t t M c P ≤ ⋅    (8) 
Money is selected a period in advance of its use. 
  4 We are only interested in variables in two different time periods, time t and t+1.  
Therefore I drop the time subscript and use a prime to denote variables evaluated at time t+1.  
It is necessary to re-pose the problem in a way that is time stationary.  I divide equations 6 
and 7 by next period's money stock, M ′, to define these new variables:   
M
T
tr
M
B
b
M
P
p
M
M
m
′
=
′
′
= ′
′
=
′
′
= ′ , , ,  .  (9) 
Here m is the representative consumer’s share of money supply.  The other variables, p, b, and 
tr, are the real goods price, bond value, and transfer payment.  Basically, the share of the 
money stock held by the consumer is selected as a numerairé. 
After making the change of variables, we get stationary representations of the consumer's 
budget constraints.   
() tr y p
b m
b i m c p + ⋅ +
ω
+
= ′ ⋅ + + ′ + ⋅
−1 1    (10) 
ω
= ⋅
m
c p    (11) 
Each constraint is satisfied as equality since consumption is valued and there is an opportunity 
cost to holding money. 
Let   define the current state of the world.  The level of money supply  { ω′ γ = , , ,b y S } M  
has no effect upon the consumer’s decision or on the equilibrium, other than determining the 
absolute value of P. 
In equilibrium the representative consumer’s share of money stock is one, m , and 
all output is consumed c .   
1 = = ′ m
y =
The consumer’s problem can be stated as a value function problem: 
() () ( ) { S v c u S v
b m c
′ ⋅ β + =
′ ′
max
, ,
}    (12) 
subject to equations 10 and 11. 
  5 Finally, I assume that output does not grow too fast,  .  The consumer’s first order 
conditions are as follows: 
1
1 < γ ⋅ β α −
() ( ) 0 = µ + λ ⋅ − = α − p c S vc    (13) 
() ( ) [] 0 = ′ ⋅ β + λ − = ′ ′ S v E S v m m   () ( µ + λ ⋅
ω
=
1
S vm )  (14) 
() ( ) ( ) [] 0 1
1 = ′ ⋅ β + λ ⋅ + − = ′
−
′ S v E i S v b b   ()
ω
λ
= S vb  (15) 
The second term in equations 14 and 15, vm and vb, makes use of the envelope theorem.  
Combining equations 13-15 together with the constraints, equations 10 and 11, and the two 
market clearing conditions yields a stationary (in growth rates) equilibrium.  In evaluating the 
envelope condition in equations 14 and 15 it is not necessary to take expectations because 
there is no uncertainty, however it is important to observe the time superscript of the 
variables.  A priori we know that   and  .    ω = ω′ γ = γ′
The price level is found from the cash-in-advance constraint, equation 11: 
y
p
⋅ ω
=
1
   (16) 
Combining equations 13 and 16 I obtain: 
α − ⋅ ω = µ + λ
1 y    (17) 
Equation 14 together with equation 17 yields the Lagrange multiplier on wealth: 
()
()
α − ⋅ γ ⋅ β =
µ′ + λ′ ⋅
ω′
β
= λ
1 y
   (18) 
Finally combining equations 15 and 18 yields the modified Fisher equation: 
1
1
1 −
γ ⋅ β
ω
=
−
λ′
λ
⋅
β
ω′
=
α −
i
   (19) 
  6 Using equation 9 with the price level in equation 16 yields the inflation rate: 
γ
ω
=
′ ⋅ ω′
=
′
p
p
P
P
   (20) 
The modified Fisher equation presented in equation 19 is perhaps more recognizable if I 
make the following change of variables: 
() ( ) 1 ln ln , ln , ln ln , 1 ln ~
− − = β − = ρ γ − ω = π + = t t y y y i i &  (21) 
Then the equation becomes: 
y i & ⋅ α + ρ + π = ~ .   (22) 
There is a one-to-one relationship between inflation and the nominal interest rate if we control 
for output growth.   
Finally, I would like to make a couple of observations concerning the solutions presented 
in equations 16-20.  Firstly, I examine the derivative of the value function with respect to 
output: 
() ()
α −
−
⋅ γ ⋅
ω
γ
⋅ β = λ ⋅ = y p S vy
1
.   (23) 
This is clearing declining over time if output is growing.  This is the reason for an output 
growth term belonging in the Fisher equation.  Next, observe that   is constant, 
confirming that the transformed budget constraint is indeed time stationary.   
y p⋅
 
III.  Conclusion 
The Fisher equation is only correct when there is no output growth.  In practice this 
condition is not likely to hold.  I have derived a modification of the Fisher equation that 
allows us to directly estimate the consumer’s degree of relative risk aversion.   
This observation provides theoretical support to the practice of including real variables in 
the estimation of the Fisher equation.  For example Dotsey, Lantz and Scholl’s (2003) finding 
  7 that the ex-ante real interest rate is positively correlated with output, and high rates are 
correlated with high output in the next quarter is completely consistent with the modified 
Fisher equation. 
 
References 
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Dotsey, Michael, Lantz, Carl, and Scholl, Brian, “The Behavior of the Real Interest Rate.” 
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  8 

Anticipated Inflation and Interest Rates: Further

Appreciation and

Are Real Interest Rates Equal Across Countries? An Empirical

Do Expected Shifts in Inflation Affect

Evidence on the Varying Effect of Expected Inflation on Interest Rates.”

Expected Inflation, and Inflation Risk Premia.”

Fisher and the Fisher Relation: Setting the Record Straight.”

Friedman and the Measured Impact of Inflation on

Inflation and Real Interest.”

International Evidence on the Long-Run Empact of Inflation.”

Stock Returns, Real Activity, Inflation,

The Behavior of the Real Interest Rate.”

The Fisher Effect:

The Fisher Effect: Reprise.”

The Long-Run Relationship between Nominal and Inflation: The Fisher Equation Revisited.”

The Theory of Interest.

Fisher Equation and Output Growth

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Fisher Equation and Output Growth

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Research Papers in Economics