The Wealth Effect and Nominal Interest Rates
Avner Bar-Ilan*
December 2000
Abstract
This short paper derives the IS curve with a wealth effect. The existence of nominal
bonds allows the central bank to use the nominal interest rate in order to affect wealth,
consumption and aggregate demand. This might provide an explanation to the
unresolved issue of interest rate smoothing.
* Department of economics, University of Haifa, 31905 Haifa, Israel.
E-mail: bar-ilan@econ.haifa.ac.il.
1. Introduction and Summary
The goal of this short paper is to use an explicit optimization in order to derive the
consumption function and the IS relationship with a wealth effect. This derivation is then
used as a possible explanation of the puzzle of interest rate smoothing. Another natural
application is a simplification of macroeconomic models suggested recently as models
appropriate to be used in classrooms.
We start by extending next section the setup of McCallum and Nelson (1999) by
introducing nominal bonds and overlapping generations. By changing the nominal
interest rate, the central bank changes bond prices. The valuation of bonds affects
public’s wealth and therefore its consumption. The resulting IS curve demonstrates that,
in addition to the standard depressing effect that real interest rate has on output, output
is negatively related to the nominal interest rate as well.
Its ability to affect wealth allows the central bank to use another channel of monetary
policy. This additional leverage implies that the movements in interest rates required to
achieve output and inflation targets are lower than without the wealth effect. This might
explain why interest rate movements are typically smaller than theory predicts, the oftendiscussed issue of interest rate smoothing. We demonstrate this point in section 3 by
using the widely used model of Clarida et al. (1999); the variance of nominal interest rate
is negatively related to the coefficient of this interest rate in the IS equation.
We suggest in section 4 that macroeconomic models presented in principles and
intermediate macro courses might include nominal instead of real interest rates. Basic
features of the models of Romer (2000) and Taylor (2000) are preserved with this
change. However, this frees the model from the assumption that the central bank is able
to control the real interest rate, an assumption that makes the model Keynesian. The
ability of the monetary policy to manipulate consumption in the short run using the
nominal interest rate, plus the use of the nominal rate as the instrument of monetary
policy, is all what is needed to present the aggregate demand.
The model presented here captures, in a very straightforward fashion, some main
features of the so-called stock market channel of monetary policy. That is, how the
impact that central banks have on valuation of financial assets could change aggregate
1
demand. For simplicity we use just bonds and ignore equities.1 Another simplifying
assumption used here is that all bonds are consols, perpetual bonds. It is true that central
banks control short-term interest rates much better than long rates. However, enriching
the range of financial assets will not change the qualitative results.2
2. Derivation of the IS Equation
We start by deriving the IS equation with a wealth effect. The derivation is based
on explicit optimization by economic agents, as in Fane (1985) and McCallum and
Nelson (1999). The model is an overlapping generations model where the economy
consists of many individuals who live for two periods. Each individual works, consumes,
and saves in the first period and consumes her saving in the second period. Unlike the
citizens, the government lives forever while spending, taxing, and borrowing using
perpetual bonds (consols). Each bond pays $1 at the start of every period, indefinitely.
Allowing the government to issue additional financial assets like short-term bonds and
money will not make any substantial difference.
The timing is as follows. During period t the young inelastically supplies one unit
of labor, consumes, and makes contingent consumption plan for period t+1. To finance
future consumption she purchases bonds and physical capital. Consumption of the old is
thus financed out of selling bonds and physical capital to the young and out of profit.
The government might then intervene in the financial market and change the nominal
interest rate and bond prices. This could potentially make actual consumption while old
different from planned consumption.
The individual maximizes lifetime utility u (cty )  u (cto1 ) where   (0,1) is the
discount factor, ct is period-t consumption, and the superscript y (o) denotes young
(old). The budget constraint is
Poterba (2000) surveys recent estimates of the response of consumption to wealth
changes. He quotes studies that show that consumption, especially in the long run,
responds much stronger to innovations in nonequity wealth than to equity wealth. He
also shows that as of December 1999 U.S. financial assets excluding equities are 60%
higher than the market value of equities.
2 Romer and Romer (2000) note that long-term interest rates typically rise in response to
tighter monetary policy. They attribute this to information advantage of the Fed. See also
Campbell (1995) on this point.
1
2
(1)
cty 

cto1
B
B
1 
 k t 1  t  wt nt  (1  rt ) 1  f (nt 1 , k t 1 )  txt 1  wt 1nt 1  (1   )k t 1  t (1 
).
1  rt
Pt I t
Pt 1
I t 1 

In equation (1) rt denotes the one-period real interest rate between period t and
period t+1 while I t is the long-term nominal interest rate at period t. Bt is period-t
quantity of bonds whose nominal price at that period, right after the coupon payment, is
1/ I t . Nominal value of the bond one period later is 1+1/ I t 1 . Pt is the price level at t.
We denote as k t the capital stock at the start of period t. The depreciation rate is  per
period, and the real price of capital is assumed to be constant equal to one. The period-t
real wage and employment are wt and nt , respectively. The production function is
f (,) and txt denotes period-t lump sum tax on the old. Imposing the lump sum tax
on the young will not make any difference. All quantities are expressed in per-capita
terms. The population growth, for simplicity, is fixed at level zero.
Taking derivatives of the Lagrangean with respect to ct , ct 1 , nt 1 , Bt , and k t 1
yields, after some simplification, the following first-order conditions,
(2) u ' (cty )   (1  rt )u ' (cto1 ),
(3)
f1 (nt 1 , k t 1 )  wt 1 ,
(4) (1  rt )(1   t ) 
(5)
It
 It ,
I t 1
f 2 (nt 1 , k t 1 )  rt   .
In equations (2)-(5) f i denotes partial derivative with respect to the ith argument
while  t is the inflation rate between periods t and t+1. Equations (2), (3), and (5) are
3
standard. Equation (4) determines the short-term real interest rate in response to paths of
long-term nominal rates and inflation.3
The terminal condition of no-bankruptcy is
(6) cto1  f (nt 1 , k t 1 )  txt 1  wt 1 nt 1  (1   )k t 1 
Bt
1
(1 
)  prt 1  as t 1 .
Pt 1
I t 1
The last term of the middle equation in (6) defines the value of financial assets
and is denoted by as. The sum of the other terms in (6) represents return on capital and
denoted by pr for ‘profit’. Equations (1)-(6) determine cty , cto1 , Bt , k t 1 , nt 1 , and rt in
response to paths for I t , txt , wt ,  t , and Pt . The last three variables are determined by
the following equations,
 B  Bt 1 
Bt 1
,
 txt   t
Pt
I
P
t t


(7) g t 
(8) nt  1,
(9)  t 
Pt 1  Pt
.
Pt
In equation (7), the government budget constraint, g t , is (per capita) government
spending and the term Bt 1 / Pt denotes real coupon payment in period t. The system (1)(9) determines cty , cto1 , Bt , k t 1 , nt 1 , rt , wt ,  t , and Pt in response to monetary and fiscal
paths for I t , g t , and txt .
3
Since 1  rt  (1  it ) /(1   t ) , the one-period nominal rate, it , is
it  I t 
I t  I t 1
.
It
4
We follow McCallum and Nelson (1999) in converting a system similar to (1)-(9)
into an IS equation. The first step is observing that combining equations (1), (6), and (7)
yields the following equation
(10) y t  cto  cty  k t 1  (1   )k t  g t  cto  cty  in t  g t .
Equation (10) is the national income identity. Output is yt  f (nt , k t ) and in t is
period-t investment. Log-linearization of equation (10) gives,
(11) log y t  d1 log cty  d 2 log cto  d 3 log in t  d 4 log g t ,
where the coefficients d i , 0  d i  1 , are the steady-state output shares of the
corresponding components of output. Assume next, still as in McCallum and Nelson
(1999), that the utility function is of isoelastic form,
u (c)  (1 /  )c ,  (  1) /  ,   0 . Substitute this functional form in equation (2)
yields,
(12) log cty  log cto1   log( 1  rt )   log  .
Interpreting equation (13) as a choice of period-t consumption in response to rt and
expectations of future consumption and approximating log( 1  rt ) by rt gives,
(13) log cty  b0'  b1' rr  Et log cto1 ,
b1'  0 .
We denote expectations as of time t by Et. Log-linearization of equation (6) gives,
(14) log cto  d 5 log prt  d 6 log ast .
In (12) d 5 and d 6 are steady-state ratios of profit and assets to consumption of the old,
0  d 5 , d 6  1 . Denoting the steady-state ratio of profit to output by  we obtain
5
(15) log cto  d 5 log y t  d 6 log( 1 
1
)  d 6 log Pt  d 7 .
It
The parameter d7 is defined as a function of constant and predetermined variables as
d 7  log   d 6 log Bt 1 . We next assume Cobb-Douglas production function where we
denote by  ,0    1 , the power of the capital stock. Equation (5) yields now
(16) log k t  d 8  d 9 log( rt   ) ,
where d 9  1 /(1   )  0 . In order to simplify, and without loss of generality, we assume
full depreciation within a period. Inserting   1 in (16) gives
(17) log int d 8d 9 rt .
Substitution of equations (13), (15), and (17) into (11) gives
(18)
log y t  b0  b1 rt  b2 log( 1 
1
1
)  b3 log Pt  b4 log g t  b5 Et log y t 1  b6 Et log( 1 
)  b7 Et log Pt 1
It
I t 1
This is our IS curve. The coefficients bi are functions of the various d i and bi'
and satisfy bi >0, i  1,...,7. The negative effect of the real interest rate rt comes from
intertemporal substitution in consumption and from investment. Government
expenditures have the usual expansionary effect. The negative coefficient of the price
level Pt and its expected level is the real balances effect coming here from nominal
bonds instead of money, the standard source of this effect. The positive relationship
between expected future output and current output is the focus of McCallum and Nelson
(1999).
Our focus here is the negative impact of nominal interest rates on output. This is
the wealth effect. Higher nominal rates devalue nominal bonds and depress
consumption. This is obvious in equation (6) where consumption of the old is financed
6
in part by bonds purchased one period earlier and whose price fluctuates with the
nominal rate.4 Future rates have similar effect via expected capital losses.
3. Interest Rate Smoothing
Clarida et al. (1999) write on p. 1688 that “understanding why central banks
choose a smooth path of interest rates than theory would predict is an important
unresolved issue.” A few possible explanations have been offered to this observation,
referred to as “interest rate smoothing.” Goodfriend (1991) and Campbell (1995) suggest
that volatility of interest rates might disrupt financial markets and risk financial
institutions. Clarida et al. (1999) discuss the role of model uncertainty in introducing
some interest rate smoothing. Williams (1997) and Rudebusch and Svensson (1999)
constrain the volatility of interest rates while estimating optimal monetary rules.
Rotemberg and Woodford (1997) discuss the related issue of lagged dependence in the
interest rate.
We suggest here a simple mechanism that reduces interest rate volatility. The
ability of central bank to affect aggregate demand using the nominal interest rate, via the
wealth effect, implies that the interest rate change needed to achieve a certain goal is
smaller than the standard change when only real interest rate can affect real variables. It
is easy to demonstrate this point. Consider a straightforward extension of the model of
Clarida et al. (1999), the model used in many recent monetary policy studies (Taylor
1999, 2000). The economy is described by an IS-type equation and a Phillips curve. The
former takes the following form,
(19) xt  b1rt  b2 I t  Et xt 1  g t .
The variable xt is the deviation of (log) output yt from the (log of) natural output level.
Equation (19), a simplified version of equation (18), is identical to the IS equation in
Clarida et al. (1999), except for the additional term b2 I t here that captures the wealth
effect. The real rate rt is rt  I t  Et  t 1 , where  t 1 is the inflation rate. The fiscal
policy term is taken as random, satisfying an AR(1) process,
4
Introducing a bequest motive will weaken this effect.
7
(20) g t  g t 1  g t* .
The disturbance term g t* is i.i.d. with zero mean and variance  g2 and  is weakly
bounded between 0 and 1. The Phillips curve that relates inflation to output gap is given
by
(21)  t  xt  Et  t 1  ut .
The random term u t satisfies
(22) u t  u t 1  u t* ,
when u t* is i.i.d. with zero mean and variance  u2 and the AR(1) term  satisfies
0    1.
The model is closed with the objective function of the central bank. The bank
chooses a path for nominal interest rates in order to minimize a loss function given by
1 

(23) Min Et   i [xt2i   t2i ] .
2  i 0

The parameter  denotes the weight on output variability relative to price variability, and
 is the discount factor.
When the central bank cannot commit to certain future actions we obtain the socalled discretionary solution. As Clarida et al. (1999) demonstrate, the optimal output and
inflation levels are given by
(23) xt  

   (1  )
2
ut ,
8
(24)  t 

ut .
   (1  )
2
Plugging the values of xt and  t from (23) and (24) into the IS equation (19) yields the
solution for the nominal interest rate,
 b1
(1   ) 
1
 Et  t 1 
(25) I t  

gt ,
b1  b2
 b1  b2 (b1  b2 ) 
where the expected inflation is
(26) Et  t 1 

ut .
   (1  )
2
The variance of I t ,Var ( I t ) , is given by
 b1
(1   ) 


(27) Var ( I t )  
 b1  b2 (b1  b2 ) 
2
 g2

  u2

1
 2


.
2
    (1  )  1   (b1  b2 ) 1  
2
The essence of the wealth effect on interest rate smoothing is
dVar ( I t ) / db2  0 . The extra leverage that the central bank has on the economy allows
it to adjust the nominal interest rate less in face of external shocks and still achieve the
same output and inflation levels. Another way of observing this point is by noting that
the coefficient of Et  t 1 in equation (25) is not necessarily larger than 1 when b2  0 .
Result 3 of Clarida et al. (1999) states that this coefficient is larger than 1; that is, the real
interest rate must optimally rise in response to an increase in expected inflation in order
to contract demand and stabilize the economy. But this is not necessarily true unless
b2  0 . The presence of the wealth effect allows some contraction to come from hike of
the nominal rate such that the real rate does not necessarily rise as well. The issue of
whether it is optimal to adjust real rates up or down when inflation accelerates is thus an
empirical issue, depending on the characteristics of the economy; in particular, on the
ratio b2 / b1 that measures the effectiveness of the wealth effect relative to the standard
channel of real interest rate on aggregate demand.
9
4. A Classroom Model
Romer (2000) and Taylor (2000) have recently suggested to adjust the way
macroeconomics is taught at the principles and intermediate levels to recent
developments. Their model is a simple version of the model of Clarida et al. (1999)
presented in the previous section. It is a system of three equations in three unknowns:
output, inflation and the real interest rate. The first equation is an IS curve that negatively
relates output to the real rate. A second equation is a positive relationship between the
real interest rate and inflation. It aims at capturing the way monetary policy is conducted,
and replaces the LM equation in simple models. These two equations are combined to
give an inverse relationship between output and inflation that is termed aggregate
demand. This aggregate demand is combined with a Phillips curve equation that relates
inflation to output in order to solve for these two variables. Determination of the
interest rate completes the solution.
I suggest replacing the real interest rate by the nominal one in the basic model
suggested above for teaching macroeconomics at the principles or even intermediate
levels. This has most of the advantages discussed by Romer (2000), and more. As Romer
notes, replacing the LM equation, with its dependence on the nominal interest rate, with
the assumption that the central bank directly controls the real rate, makes the model
more coherent; both the IS and the monetary policy equation include the same type of
interest rate. Using my suggestion this coherence is achieved via the wealth effect of the
nominal rate on consumption and the IS curve.
As Romer (2000) emphasizes, the assumption that the central bank is able to
follow a real interest rate rule is based on price rigidity and therefore makes the model
Keynesian. It is also hard to explain this assumption to students; Romer suggests to
explain it in stages, starting with complete price rigidity and proceeding by relaxing this
assumption. The ability of the central bank to affect the nominal interest rate, on the
other hand, is not constrained by any assumption. There is a unanimous agreement on
the ability of central banks to control short-term nominal interest rates. Moreover, as
discussed in the previous section, assuming that the real rate is controlled requires that
the central bank adjust the nominal rate more than one-for-one with inflation. This is a
10
strong quantitative assumption that does not always hold in reality5 and not easily
understood by beginners. All this complication is avoided by assuming that the nominal
interest rate is the monetary policy instrument.
A more advanced model should include both the nominal and the real interest
rates affecting aggregate demand. Simplifying by omitting the real rate is not a big loss, at
least at the principles level, in terms of economic intuition and dynamics of the model.
The aggregate demand relationship of output and inflation is qualitatively identical
whether the model includes nominal or real interest rates. On the other hand, the
mechanism by which higher nominal interest rates act to reduce consumption is easily
grasped even by beginners. Generally speaking, the wealth channel of monetary policy
transmission mechanism is widely discussed in the popular press. The students are aware
of the ability of the central bank to determine nominal interest rates and the prompt
effect that interest rate changes have on both the debt and equities markets. The last link,
that between wealth and consumption, is also easy to understand.
References
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Journal of Macroeconomics 7, Fall 1985, 493-508.
Goodfriend, Marvin S., “Interest Rates and the Conduct of Monetary Policy,” CarnegieRochester Conference Series on Public Policy 34, 1991, 7-37.
5
See Goodfriend (1993) and Clarida et al. (2000).
11
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