Economics of Inequality

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INFLATION, GROWTH, AND INCOME INEQUALITY 141

stock and skill level, respectively. The capital share ki has a distribution with mean1. Denote its variance by 2k . Similarly, the distribution of a i has unit mean. Denoteits variance by 2a . For simplicity we assume that ki and a i are uncorrelated; that

is, their covariance k,a = 0.At every point in time, the household works h i hours, rents out capital K i to

rms, receives real lump-sum transfer i from the government, and invests inphysical capital and real balance. Thus the budget constraint is

c i + I i + m i = rK i + (wK)(A i h i ) + i m i , (5)

where is the ination rate.Let be the timediscount rate and let li be the leisure time enjoyed by household

i . Thus h i 1 li . The household solves the problem

max

0

1

ci li

e t dt, > 0, < 0, (6)

subject to (3)(5).Let i and i be the Lagrangian multipliers associated with (4) and (5) in the

current-value Hamiltonian. The rst-order conditions are

li : m i l

1i = i wKA i , (7)

m i : i i = m 1i l i i (1 + ) , (8)

K i : i i = i r, (9)

I i : i = i . (10 )

Transversality conditions are lim t i K i e t = 0 and lim t i m i e t = 0.Dene l A i li as the aggregate effective leisure, and L A i h i as theaggregate effective labor. Then labor market clearing requires L j = L = A l .The capital and goods market clearing conditions are

K j = K and

(c i + I i ) =

Q Q j , respectively. Finally, the real money supply followsmm

= , (11 )

where is the growth rate of the nominal money stock. For simplicity, we assumethat the rate of real transfer is proportional to individual real balance; i.e., i = m i ,i [0 , 1].

3. THE BALANCED-GROWTH EQUILIBRIUMEquations (7)(10) yield

( 1)m im i

+ lili

= + r (L) L

1 + + r(L)+ r(L), (12 )


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m im i

+ ( 1)lili

= r(L) +w (L)L

w(L)LL

+KK

, (13 )

implying that the growth rate of real balance holdings (and hence consumption)does not depend on i, nor does the growth rate of leisure:

cici

=cc

=m im i

=mm

,lili

=ll

. (14 )

Dividing both sides of the household budget constraint (5) by K i and using i = m i , we obtain the growth rate for the individual capital stock,

i K i

K i= r (L ) + w (L )

K

K iA i h i

1 h i

1

1 + + r (L). (15 )

Aggregating over all households yields

KK

= i i K iK

= r (L ) + w (L ) L A L

1

1 + + r (L). (16 )

Using (16), we can rewrite (12) and (13) as

L = ( ) w L A L

11+ + r

[( 1) / (A L) /L ], (17 )

= [( 1) ( ) + r ] (1 + + r) r + 1 + + r

A LL. (18 )

LEMMA 1. There exists a unique balanced growth path , along which L = 0, = 0, and ki = 0, for all i.

Proof. Some algebraic work shows that, evaluated at the steady state, the Ja-cobian matrix associated with (17) and (18) has positive determinant and positive

trace; hence both eigenvalues are positive. Therefore there exists a unique perfectforesight equilibrium path leading to the steady state. The steady state L is givenby g(L ) = 0, where

g(L) r(L)

1 w(L) L

A L

= 0

and

1 + + r (L) = 1 + + r (L)

1 . (19 )

L is unique because lim L 0 g(L) = + , lim L A g(L) < 0, and g (L) < 0. Tounderstand the evolution of ki , note that

ki =K iK i

KK

ki = w (L) A i h i 1 h i

L

A L

ki , (20 )


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which is linear in ki . The aggregate transversality condition, lim t Ke t = 0,requires that / + K/K + [d(e t )/dt ]/e t < 0, or r(L) > (the rate of return on capital must exceed the growth rate in the equilibrium). This implies that

L 0,

and < 1 when the growth rate is positive.

Proof. The expression of is derived using (22) and (23). Obviously > 0.Positive growth rate implies that < 1.5

5. INFLATION, GROWTH, AND INCOME INEQUALITY

Now we examine the relationship among ination, growth, and income inequalitywhen the economy is subject to changes in long-run money growth.

PROPOSITION 1. An increase in increases the ination rate and reduces the

aggregate growth rate unambiguously.

Proof. Totally differentiating g(L) = 0 gives

dLd

=(A L)/

(1 )1

AL

(1 ) + ( 1 )1 L (A L)/

< 0.

From equation (21) and = , we obtain d/d = [( 1 )/(1 ) ]L dL/d < 0 and d/d = 1 d/d > 0.

Some discussion is in order. The higher ination caused by an increase in leadshouseholds to economize on real balances and substitute away from consumptiontoward leisure time, which is a credit good. As a result, aggregate labor supplygoes down. The lowered labor supply reduces the rate of return to capital, whichin turn reduces growth. The reduced growth then causes the ination rate to riseby more than the money growth rate.

Next we study the effect on income inequality of changes in .

PROPOSITION 2.(i) If capital heterogeneity dominates skill heterogeneity in the sense that 2a

[(1 )/ ] 2k , then an increase in enlarges income inequality.(iii ) If the two types of heterogeneity cancel out in the sense that 2a = [(1 )/ ]

2k ,

then is neutral for income inequality.


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Proof. With k,a = 0, equation (24) implies

2y = (1 )2 2k +

2 2a . (25 )

And d 2y /d = 2(1 )(d/d) 2k + 2(d/d)

2a . Because d/d = 1

[ ( 1 )/( 1 ) ]L dL/d < 0,

dd

=A (1 )

(1 + )2 L

1L

(1 + ) + dd

> 0.

Given that 2k is xed along the balanced growth path, the result follows.

Proposition 2 says that when income inequality is decomposed into inequalitycaused by capital heterogeneity and inequality caused by skill heterogeneity [seeequation (25)], a higher money growth rate reduces the contribution of 2k , whereasit raises the contribution of 2a . Thus the overall effect depends on the relative sizeof the two types of heterogeneity. Alternatively, we can examine this effect byconsidering inequality of different income sources. Denote the variance of n i by 2n . Observe that from (23), we have

2y = 2 2k + (1 )

2 2n + 2 (1 ) k,n . (26 )

That is, the overall income inequality consists of capital income inequality, labor

income inequality, and the covariance between the two income components. Therelative labor supply function gives 6

2n = 1

1

2

2k +

1

2

2a ,

k,n = 1

1 2k < 0.

We have the following observations. (i) An increase in does not affect capital

income inequality, 2 2k . (ii) Because d/d > 0 and > 1 (see footnote5), a higher increases the contribution of both 2k and

2a to labor income

inequality, 2n . (iii) A higher makes capital and labor income more negativelyrelated. Thus, the total effect of on 2y is thereby determined by the two oppositeforces specied in (ii) and (iii). When there is only capital heterogeneity, laborincome becomes more unequally distributed as increases. But this is alwaysdominated by the enlarged negative covariance. As a result, a higher reducesincome inequality unambiguously. With the additional skill heterogeneity, laborincome will become even more unequally distributed. Therefore, as long as skill

heterogeneity is large enough, the increase in labor income inequality can offsetthe more negative income covariance, leading to a more unequally distributedincome as goes up.

We now present the relationship among ination, growth, and income inequalityin the following proposition.


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146 YI JIN

PROPOSITION 3.(i) Changes in the long-run money growth generate a negative inationgrowth rela-

tionship unambiguously.

(ii) When 2a > [(1 )/ ]

2k , we obtain a positive inationinequality relationship

and a negative growthinequality relationship.(iii ) When 2a < [(1 )/ ]

2k , we obtain a negative inationinequality relationship

and a positive growthinequality relationship.

Proof. This immediately follows from Proposition 1 and 2.

6. CONCLUSIONS

In this paper we develop a monetary endogenous growth model to analyze the

relationship among ination, growth, and income inequality. In the absence of skill heterogeneity, the model predicts that lower income inequality is associatedwith higher ination, which runs counter to the evidence. Introducing skill het-erogeneity has the potential to reverse the inationinequality relationship to bein line with the evidence.

Furthermore, the long-run growthinequality relationship traced out by varia-tions in money growth also depends on which type of heterogeneity dominates.This same insight carries over to other sorts of fundamental changes, such asvariations in total factor productivity, labor supply elasticity, and the rate of time

preference. The ambiguity concerning the empirical evidence on the growthinequality relationship is therefore not surprising in light of our analysis. Ourresults suggest that, to obtain a denitive relationship between growth and in-equality, one has to condition their relationship on an appropriate variable, such asthe relative importance of capital and skill heterogeneity emphasized in our study.

NOTES

1. Fischer (1991) reports that the slow-growth countries have an average ination rate slightlyabove 30%, whereas the fast-growth countries average only 12% ination.

2. This threshold level has been found through testing to be at 1% ination rate for industrializedcountries and at 11% for developing countries [Khan and Senhadji (2001)].

3. Note that conning attention to this type of heterogeneity causes no problem in Garc a-Penalosaand Turnovsky because their real model is not concerned with ination and its relationship with growthand inequality.

4. For a treatment of endogenous accumulation of skills, see Jin (2007). It is shown that the maininsights in the exogenous-skill setup are largely retained.

5. We can actually show that 1 < < ( 1 )(A/L)/( 1 + ) < 1.6. k,n is negative because households with higher capital shares tend to supply smaller amounts

of effective labor.

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Fischer, Stanley (1991) Growth, macroeconomics, and development. In Olivier Blanchard and StanleyFischer (eds.), NBER Macroeconomics Annual , pp. 329364. Chicago: University of Chicago Press.

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Economics of Inequality