Income inequality may influence macro-economic variables by affecting the money multiplier and the trade-off between inflation and output. In an AD-AS model with imperfect foresight income inequality intensifies the volatility of output and inflation rate by increasing the likelihood of oscillations as well as their magnitude. Volatility is, however, moderated when income inequality prolongs the business cycles.Rational eating, cycles

Levy, A.

English

Research Papers in Economics

 
 
 
 
 
 
 
 
 
A Note on Income Inequality and Macro-economic 
Volatility 
 
 
 
 
 
 
Amnon Levy 
 
 
 
 
 
 
 
 
 
 
WP 00-08 
 
University of Wollongong 
Department of Economics 
Working Paper Series 2000  
A NOTE ON INCOME INEQUALITY AND MACRO-ECONOMIC VOLATILITY 
 
Amnon Levy, University of Wollongong* 
 
 
Abstract 
Income inequality may influence macro-economic variables by affecting the 
money multiplier and the trade-off between inflation and output. In an AD-AS 
model with imperfect foresight income inequality intensifies the volatility of 
output and inflation rate by increasing the likelihood of oscillations as well as 
their magnitude. Volatility is, however, moderated when income inequality 
prolongs the business cycles. 
 
 (JEL D31, E32) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
* Department of Economics, University of Wollongong, Wollongong, NSW 2522 
E-mail: amnon_levy@uow.edu.au 
 
   1
A NOTE ON INCOME INEQUALITY AND MACRO-ECONOMIC VOLATILITY 
 
 
1.   Introduction  
During the last quarter of the twentieth century many countries suffering from low and 
fluctuating rate of growth and high and largely fluctuating inflation rates were characterized 
by a high degree of income inequality. At the same period, in contrast, many countries 
experiencing a steady rate of growth and low and moderately oscillating rate of inflation also 
enjoyed low levels of income inequality. A significant number of Latin American countries 
including Argentina, Brazil, Mexico and Uruguay are notable examples of the first group of 
countries. The technologically advanced European countries such as Germany, Norway, 
Sweden and the United Kingdom represent the second.  
 
Is there a causal relationship underlying the high correlation between macro-economic 
volatility and income inequality?  
 
One may argue that the high correlation between macro-economic volatility and income 
inequality can be attributed to the effect of the former factor on the latter: namely, severe 
fluctuations in major aggregate economic variables raise the levels of uncertainty, confusion, 
and employment instability, which, in turn, widen the income gap between those endowed 
with large stocks of human and physical capitals and those possessing small stocks of these 
capitals, especially when human and physical capital stocks are highly correlated. 
 
Is it possible that causality also flows in the opposite direction and income inequality 
intensifies macroeconomic volatility?  
 
Recent studies suggest that it is possible. Alesina and Perotti (1996) argue that income 
inequality has an indirect effect on macroeconomic volatility via increased political instability. 
Aghion, Banerjee and Piketty (1997) propose that inequality also means unequal access to 
investment opportunities and combined with a high level of capital market imperfection may 
generate persistent credit cycles. In this context, Aghion, Caroli and Garcia-Penalosa (1999) 
claim further that inequality of access to high-yield investment opportunities and the 
consequent separation of investors and savers generates macro-economic volatility.  
   2
Are there other channels through which inequality, and income inequality in particular, may 
cause macroeconomic volatility?  
 
This note suggests that income inequality reduces: 1. the aggregate propensity to consume and 
thereby the money multiplier, and 2. the trade-off (on the supply side) between inflation and 
output. The implications of these possible effects on the business cycles are  theoretically 
illustrated within a standard AD-AS macro-economic model with imperfect inflationary 
expectations. Section 2 uses this model as a benchmark for generating business cycles. 
Section 3 provides a rationale for the possible moderating effect of income inequality on the 
money multiplier and the inflation-output trade off. Section 4 shows that income inequality 
might intensify macro-economic volatility by increasing both the likelihood of oscillations in 
the output and inflation rate trajectories and by enlarging the magnitude of these oscillations. 
As the likelihood of business cycles oscillations and their magnitude are not the sole aspects 
of instability, the analysis also introduces the possible effects of income inequality on the 
length of the business cycles. Section 5 suggests that macro-economic volatility may be 
moderated by the possible effect of income inequality to prolong the business cycles. Section 
6 concludes. 
 
2.  Business cycles in an imperfect foresight augmented AD-AS model 
Although the AD-AS model is criticised for lacking microeconomic foundations, for 
excluding perfect nominal adjustment, and for focusing on the quantity of money rather than 
the interest rate as the central banks’ policy instrument (Romer, 2000), its simplicity rendered 
it a comprehensible, wide, baseline framework for analysing short-run fluctuations of output 
and prices. In a standard AD-AS model (Dornbusch and Fischer, 1978) the aggregate supply 
schedule is given by 
 
p p d = + - * ( ) Y Yp      
            (1) 
and the aggregate demand schedule by  
 
Y Y f m = + + - -1 g f p ( )              (2) 
   3
where Y is the output level, Yp is the potential output level, p  is the actual inflation rate, 
p* is the expected inflation rate,  f  is the increase in autonomous spending,  m is the 
growth rate of nominal money stock, g  is the fiscal multiplier, f is the money multiplier and 
d  is a positive parameter reflecting, on the supply side, the short run trade-off between 
inflation and GNP. 
 
Imperfect inflationary expectations are essential for generating business cycles in the AD-AS 
model. As in Cagan (1956), the public inflationary expectations are assumed here to be 
adaptive: namely, adjusted to the last period unanticipated rate of inflation 
 
p p b p p
*
-
*
- -
* - = - 1 1 1 ( )                (3) 
 
where 0 1 < < b . This adaptive inflationary expectations can be equivalently rendered by 
p
b
b
p *
- =
- - 1 1
1 ( )L
                (4) 
where L denotes the lag operator. 
 
By substituting equation (4) into equation (1) the aggregate supply scheduled can be now 
expressed as  
 
p p d b d db - - + - = - - - 1 1 1 Y Y Yp ( ) .          (5) 
By pre-multiplying both sides of the system of equations (5) and (2) by the inverse of the 
matrix of the coefficients associated with the current output level and inflation rate, the 
adaptive expectation augmented AD-AS model can be expressed as a system of two first-
order difference equations whose solution is  
p l l t
t t m a a = + + 11 1 12 2               (6) 
and  
Y Y a a t p
t t = + + 21 1 22 2 l l               (7)   4
where,   
l
df
df b df b dfb 1
2 1
2 1
2 1 1 4 =
+
+ - + - -
( )
{ ( ) [ ( )] }    (8) 
 
l
df
df b df b dfb 2
2 1
2 1
2 1 1 4 =
+
+ - - - -
( )
{ ( ) [ ( )] }    (9) 
and  a￿1 and  a￿2 are the eigenvectors associated with the eigenvalues  l1 and  l2, 
respectively. 
 
The trajectories of inflation rate and output display oscillations when the discriminant in 
equations (8) and (9) is negative, in which casel1 and l2 are complex conjugate pair. This 
is the case where 
4
1 2
b
b
df
( ) -
> .                  (10) 
 
Condition  (10) is satisfied, for example, in the limiting case of naïve inflationary 
expectations (b =1). When this condition is fulfilled, the stationary point of inflation rate 
m and output Yp is a spiral and the deviations of the current inflation rate and output from 
their stationary levels are given by  
 
p
df
y q t
t
m a t - =
+
Ø
º
Œ
ø
ß
œ + 1
05
1
1
~ .
cos( )          (11) 
 
p
df
y q t
t
m a t - =
+
Ø
º
Œ
ø
ß
œ + 2
05
1
1
~ .
cos( )           (12) 
 
where  the complex roots’ amplitude, q, satisfies  
   5
tgq
df
df b
=
+
+ -
-
￿
￿
￿
￿
￿
￿
4 1
2 1
1 2
05
( )
[ ( )]
.
            (13) 
and the parameters a
~
1 and a
~
2 and y  are chosen so as to satisfy the initial conditions.  
 
3. Possible effects of income inequality on the money multiplier and the inflation-output 
trade off 
In recalling Engel’s law of  negative relationship between the marginal propensity and  
income, one may argue that the less equal the distribution of income is the lower the fraction 
of the aggregate income spent on consumption. In other words, the higher the degree of  
income inequality the lower the private sector’s propensity to consume. In recalling further 
that the AD-AS’ money multiplier increases with the private sector’s propensity to consume, 
one may continue and argue that the higher the degree of income inequality (s ) the lower the 
money multiplier:  
 
¶f
¶s
< 0.                    (14) 
 
One may also argue that the less equal the distribution of income is the larger the number of 
`people willing to work for relatively low wages. In this case, a reduction of the output gap 
causes a smaller rise in the price level than in the case of more equal distribution of income. It 
is therefore suggested, in terms of the AD-AS model, that the higher the degree of income 
inequality the smaller the trade-off (on the supply side) between inflation and output: 
 
¶d
¶s
< 0.                    (15) 
 
The implications of these p ossible negative effects of income inequality on the money 
multiplier and the trade off between inflation and output for business cycles is analysed in the 
next two sections.  
 
4. Possible effects of income inequality on the oscillations of the business cycles 
The possible effects of income inequality on the oscillations of the business cycles are 
summarised in the following claims.   6
Claim 1: The greater the degree of income inequality, the higher the likelihood that the 
trajectories of inflation rate and aggregate income oscillate.  
Proof: In recalling condition (10)  and that  df decreases with s , a rise in s , ceteris 
paribus, increases the likelihood that l1 and l2 are a complex conjugate pair. QED 
 
Comment: Recalling that the parameters d  and f are positive, the modulus of the complex 
roots, 1 1 / ( ) +df , is smaller than 1 and hence the joint oscillations of income and inflation 
rate are damped.  
 
Claim 2: The higher the degree of income inequality the larger the oscillations of the inflation 
rate and output. 
Proof: The greater the complex roots’ modulus  the larger the oscillations of p  and Y. The 
modulus, 1 1 / ( ) +df , decreases with df which, by inequalities 14 and 15,  decreases with 
s . QED 
 
5. Possible effects of income inequality on the length of the business cycles 
The possible effects of income inequality on the length of the business cycles is summarised 
in following claim. 
 
Claim 3: If dfb <1, the higher the degree of income inequality the longer the economic 
cycle. However, if  dfb >1, the higher the degree of income inequality the shorter the 
economic cycle.  
Proof: The length of the economic cycle is  l = 2P /q where  0 05 £ £ q . P. By 
differentiating equation (13) with respect to s   
 
¶ q
¶s
df
df b
dfb
df b
¶ df
¶s
tg
=
+
+ -
-
￿
￿
￿
￿
￿
￿
-
+ -
￿
￿
￿
￿
￿
￿
-
2
4 1
2 1
1
1
2 1 2
05
3
( )
[ ( )]
( )
[ ( )]
( )
.
 
   7
where 
¶ df
¶s
( )
< 0 by inequalities (14) and (15). Hence, 
¶ q
¶s
tg
> 0 if  dfb >1, 
¶ q
¶s
tg
= 0 if  dfb =1 and 
¶ q
¶s
tg
< 0 if  dfb <1. Recalling that tgq   rises with q 
and that  l decreases with q, then 
¶
¶s
l
> 0 if  dfb >1, 
¶
¶s
l
= 0 if  dfb =1 and 
¶
¶s
l
< 0 if  dfb <1. QED 
Comment: The inequalities (14) and (15) also suggest that the probability of  dfb <1 
increases with the degree of income inequality. Moreover, in the extreme case of  b =1 
(naïve inflationary expectations)  tgq df = - 1 1 /  which, combined with the 
inequalities (14) and (15), implies that 
¶ q
¶s
tg
< 0 and consequently income inequality 
prolongs the economic cycles. 
 
6. Conclusion 
This note was concerned with the possible effects of income inequality on the business cycles 
within the framework of a standard AD-AS macroeconomic model with imperfect foresight. It 
was argued that income inequality reduces the private sector’s propensity to consume as well 
as the trade off (on the supply side) between inflation and output. Consequently, the effect of 
income inequality is to increase both the likelihood of short-run oscillations of output and 
inflation rate and their magnitude on the one hand, but not necessarily to shorten the economic 
cycles. It may prolong the economic cycles when the product of the inflation-output trade off 
coefficient, money multiplier and the expectation correction coefficient is smaller than one. 
 
 
References 
Alesina, Alberto and Roberto Perotti, 1996, “Income Distribution, Political Instability, and 
investment”, European Economic Review 40:6, pp.1203-1228. 
 
Aghion, Philippe, Abhijit Banergee, and Thomas Piketty, 1997, “Dualism and 
Macroeconomic Volatility” mimeo, University College, London. 
   8
Aghion, Philippe, Eve Caroli, and Cecilia Garcia-Penalosa, 1999, “Inequality and Economic 
Growth: The Perspective of the New Growth Theories”,  Journal of Economic Literature 
XXXVII:4, pp. 1615-1660. 
 
Cagan, Phillips, 1956, “The Monetary Dynamics of Hyper-Inflation”, In Milton Friedman 
(ed.) Studies in the Quantity Theory of Money, Chicago, University of Chicago Press, pp. 25-
117. 
 
Dornbusch, Rudiger and Stanley Fischer, Macroeconomics, McGraw-Hill, Inc. (1978) 
 
Romer, David, 2000, “Keynesian Macroeconomic without the LM Curve”, Journal of 
Economic Perspectives 14:2, pp. 149-169. 
 
 
 

Dualism and Macroeconomic Volatility” mimeo,

Income Distribution, Political Instability, and investment”,

Inequality and Economic Growth: The Perspective of the New Growth Theories”,

The Monetary Dynamics of Hyper-Inflation”, In

A Note on Income Inequality and Macro-economic Volatility

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A Note on Income Inequality and Macro-economic Volatility

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