202: Dynamic Macroeconomic Theory

202: Dynamic Macroeconomic TheoryInequality, Credit Market Imperfection & Growth: Galor-Zeira (1993)

Mausumi Das

Lecture Notes, DSE

July 24, 2015

Das (Lecture Notes, DSE) Dynamic Macro July 24, 2015 1 / 29

Inequality, Credit Market Imperfection & Growth:Galor-Zeira (RES, 1993)

This paper explores the negative link between inequality andgrowth/development that operates through the credit marketinperfection channel.

They argue that when credit markets are either absent or do not workproperly, this leads to ineffi cient allocation of resources across variousproductive activities.

So it is clear that imperfect credit market would lower growth.

But how does inequality get into the picture?

Inequality would get into the picture if for some reason less wealthypeople are also ‘potentially’more productive at the margin.


Inequality, Credit Market Imperfection & Growth: AnExample

Consider an economy where technology is non-convex (implyingthat the production function is not concave):

In particular, suppose the technology entails a threshold effect:

yh ={y for kh < k;y for kh = k. ; where y > y .

Note that marginal return to investment around k is infinitely high; andis zero before and after.If there was a market for capital where households/firms could lend orborrow from one another, then there would be a flow of funds from thecapital-rich households (with kh > k) to the capital-poor households(with kh < k) so that all households would be operating withkh = k;consequently the economy would be at its most effi cient level.But this will not happen if the capital market is imperfect.

Notice that here a redistribution of wealth from rich to poor iseffi cient (increases total output.)


Galor & Zeira (1993): The Model

Galor-Zeira explore similar mechanisms to show that initial incomedistribution matters for the macroeconomic outcomes in the long run.Their model is of course dynamic and much more complex than thestatic framework we have just discussed.Consider a small open economy which takes the world interest rate(r ∗) as given.A single final commodity is produced - using two possibletechnologies: a traditional cottage technology (associated with lowproductivity); a modern technology (associated with highproductivity).The traditional technology uses only unskilled labour and isassoacited with a constant marginal as well as average product:

Y Nt = ALNt .

The modern technology uses capital and skilled labour and isassocaited with a neoclassical production function exhibiting positivebut diminishing marginal as product of each factor:

Y St = F (KSt ,HSt )

FK ,FH > 0; FKK ,FHH < 0


Production Side Story: (Contd.)

Since F (KSt ,HSt ) obeys all neoclassical properties, including CRS:

Y StHSt

= f(KStHSt

); f (0) = 0; f ′ > 0; f ′′ < 0.

The modern tecnology is operated by profit-maximizing firms,operating under perfect competition. Thus the skilled wage rate andthe market interest rate are given respectively by:

w st = f(KStHSt

)− f ′

(KStHSt

)(KStHSt

);

r st = f ′(KStHSt

).

A small open economy with perfect capital mobility implies:

r st = r∗ ⇒ f ′

(KStHSt

)= r ∗.


Production Side Story: (Contd.)

This fixes the domestic capital to skilled labour ratio at:

KStHSt

= f ′−1(r ∗) = k .

This in turn implies that the wage rate for skilled labour is fixed at:

wS = f (k)− f ′ (k) (k) .

The traditional technology is operated by family firms angaged in ownproduction.

Wage earnings of an unskilled labour engaged in traditional sector:

wN = A.

If skill formation is costly, then of course, wS > wN . We shall make amore precise assumption about these two wage rates later.


Skill Formation Technology:

One unit of unskilled labour can be converted into one unit of skilledlabour by investing

(i) one full time period (in acquiring skills);

(ii) a fixed investment of h units of the final commodity.

The fixed investment h has to be paid upfront - before skill formationbegins.

Notice that since skill formation also require a time investment of onefull period, those who engage in skill formation cannot work for onetime period.


Household Side Story:

A two-period overlapping generations economy with constantpopulation.

There are L households - each consisting of a young member and anold member at any point of time t.

An agent lives exactly for two periods - youth and maturity, and hasan offspring at the beginning of the second period of his life.

He dies at the end of the 2nd period but the dynastic link is carriedforward over time by his progeny.

Each agent is born with an endowment of one unit of unskilledlabour.The young agent also receives an endowment of final goods asbequest from parent.

Agents differ in terms of the bequest received.


Household Side Story: (Contd.)

All agents born the beginning of period t will be called ‘generation t’.The life cycle of an agent belonging to generation t is as follows:In the first period of his life:

He is endowed with one unit of unskilled labour and some inheritedwealth (xt ).In the first period, he consumes nothing in the first period and onlytakes the decision as to whether to acquire skill or not.If he decides not to acquire skill then he works as unskilled labour toearn wN .If he decides to to acquire skill then he spends the first period of hislife in schooling.In the latter case he also spends a fixed amount h is school fees.

In the second period of his life:Depending on his first period decision to go for skill formation on not,he works either as a skilled labour, (earning wS ) or as an unskilledlabour (again earning wN ).He spends his entire second period income in own consumption and inleaving a bequest for his child.


Preferences of an Agent:

Consider an agent with a second period income y .The agent spends this income in own consumption (c) and in bequestfor his child (b).His preference is represented by the following utility function(identical for all agents):

U(c , b) = α log c + (1− α) log b; 0 < α < 1.

The agent maximises the above utility function subject to his secondperiod budget constraint:

c + b = y .

Optimal solution:

c = αy ;

b = (1− α)y .

Corresponding indirect utility:

U = α log (αy) + (1− α) log ((1− α)y)Das (Lecture Notes, DSE) Dynamic Macro July 24, 2015 10 / 29

Occupational/Skill Formation Decision of an Agent:

Notice that the indirect utility of an agent depends on his secondperiod income:

U = [α log α+ (1− α) log(1− α)] + log y

= M + log y .

The second period earning on the other hand depends on the skillformation decidion undertaken by the agent in the first period of hislife.

Let us now analyse the optimal occupational/skill formation choice ofan agent.

Given his inherited wealth level xt , a forward-looking agent endowedwith perfect foresight will choose his skill and therefore occupation soas to maximise his second period utility U.


Skill Formation Decision of an Agent: (Contd.)

In deciding whether to go for skill formation or not, there are twopossibilities:

Case I: xt = hIn this case the agent can invest in skill formation directly from hisinherited wealth. The rest of the wealth can then be invested in thecapital market to earn an additional interest income in the next period(over and above the skilled wage rate that he would be earning).Case II: xt < hIn this case the agent can invest in skill formation only if he borrowsfrom the capital market an amount (h− xt ). If he does so, then he hasto repay the loan with interest in the next period from the skilled wagerate that he would be earning.

We shall now bring in credit market impefection into the story,which will affect the skill formation decision of the rich agents (withxt = h ) vis-a-vis the poor agents (with xt < h ).


Credit Market Imperfection:

Credit market imperfection can show up in an economy in threepossible ways:

1 Complete absence of credit (missing credit market): In this casethere is no borrowing possible. Thus only rich agents can invest in skillformation; the poor agents are completely excluded form acquiring skilland therefore working in the modern sector.

2 Collateral Requirement & Credit Rationing: In this case theamount of credit obtained depends on the wealth level of an agent(which can be pledged as collateral). Once again agents who are toopoor (such the inherited wealth + limited credit fall short of h) will beexcluded form acquiring skill and therefore working in the modernsector.

3 A Wedge Between Borrower’s and Lender’s Rate: In this case theagents who are who are borrowing from the credit market face a higherinterest rate (i) than the interest rate recieved by the lenders (r). Ifthe borrower’s rate is too high, this again will deter the poorer agentsto avail the credit and go for skill formation, since they won’t be ableto pay back that high an interest.


Reason for Imperfect Credit Market:

Typically the credit market does not work perfectly because ofassocaited moral hazard problem.

One can take the loan and run away. In order to prevent this, thefinancial institution will have to incur some cost either in monitoringor in tracing the borrower who has ran away.

Such costly activities would either lead to credit rationing or wouldcreate a wedge between borrower’s and lender’s rate.

Galor & Zeira adopts a credit market imperfection story that entailsthe latter.


Credit Market Impefection Story a la Galor-Zeira:

Imagine a world where borrowing and lending are done by a numberof competing financial institutions/banks - each taking deposits fromthe lenders and lending it out to the borrowers.

Since the lender has the option of directly investing in theinternational capital market (small open economy with perfect capitalmobility), he has to be given the world interest rate r ∗.

The borrower on the other hand can run away with the loan amount.

The bank can spend some amount z is checking his whereabouts; butthe borrower can still evade by spending a higher amount βz (β > 1).

Suppose a borrower has asked for a loan amount d .

The bank now has to decide what interest to charge the borrower (i)and how much to spend in checking him (z).

The bank’s participation constraint:

(1+ i)d = (1+ r ∗)d + z (1)


Credit Market Impefection Story a la Galor-Zeira: (Contd.)

Given it’s participation constraint, the bank will choose i and z sothat the borrower has no incentive to run away.

Notice that if the borrower does not renege, then he has to pay back:(1+ i)d

On the other hand if he evades, then his cost is: βz

Hence the incentive compatibility constraint for the borrower setby the bank:

βz = (1+ i)d (2)

When there are many banks competing with one another, all financialintermediaries will eventually earn zero profit, whch implies that theabove two conditions will hold with equality.

Solving, we get:

i =1+ βr ∗

β− 1 > r ∗



Recall that an agent with an inherited wealth level xt , will go for skillformatio if his second period utility U is higher under skill formationthan under no-skill.His indirect utility in turn is an increasing function of his secondperiod income:

U = M + log y .

So let us now calculate the second period income of an agent underskill formation and no-skill.Two possible cases:

Case I: xt = hsecond period income of the agent under skill formation:(

y skillt

)Case I

= (1+ r ∗)(xt − h) + wS

second period income of the agent under no skill:(y noskillt

)Case I

= (1+ r ∗)(xt + wN ) + wN



In Case I, agents will go for skill formation iff

(1+ r ∗)(xt − h) + wS = (1+ r ∗)(xt + wN ) + wN

i.e., wS − wN = (1+ r ∗)(wN + h

)Now let us consider the other case:

Case II: xt < hsecond period income of the agent under skill formation:(

y skillt

)Case II

= wS − (1+ i)(h− xt )

second period income of the agent under no skill:(y noskillt

)Case II

= (1+ r ∗)(xt + wN ) + wN

In Case II, agents will go for skill formation iff

wS − (1+ i)(h− xt ) > (1+ r ∗)(xt + wN ) + wN

i.e., xt >(1+ r ∗)

(2+ wN

)+ (1+ i)h− wS

i − r ∗ ≡ f



Assumption 1:

(1+ r ∗)(2+ wN

)+(1+ i)h > wS > (1+ r ∗)

(2+ wN

)+(1+ r ∗)h

The latter part of Assumption 1 implies that rich agents (withxt = h) would always prefer to go for skill formation.The first part of Assumption 1 ensures that there exists a positivecut-off wealth level f (< h) such that

agents with f < xt < h would prefer to go for skill formation byborrowing;but for agents with 0 5 xt 5 f , the cost of repayment is toohigh and they prefer to remain unskilled.

In any time period t , given the distribution of wealth (among thecurrent youth), we can break up the entire distribution in threedistinct intervals and analyse the corresponditing optimal occuptionaland other choices.


Wealth Distribution and Skill Formation:

1 For any xt such that 0 5 xt 5 fThe agent works as unskilled labour in the traditional sector:Income: yt = (1+ r ∗)(xt + wN ) + wN

Bequest left to his child:bt = (1− α)yt = (1− α)

[(1+ r ∗)(2+ wN ) + (1+ r ∗)xt

]2 For any xt such that f < xt < h

The agent acquires skills by borrowing and works as skilled labour inthe modern sector:Income: wS − (1+ i)(h− xt )Bequest left to his child:bt = (1− α)yt = (1− α)

[wS − (1+ i)h+ (1+ i)xt

]3 For any xt such that xt = h

The agent acquires skills from own inherited wealth and works asskilled labour in the modern sector:Income: (1+ r ∗)(xt − h) + wSBequest left to his child:bt = (1− α)yt = (1− α)

[wS − (1+ r ∗)h+ (1+ r ∗)xt

]Das (Lecture Notes, DSE) Dynamic Macro July 24, 2015 20 / 29

Intergenerational Wealth Dynamics:

Notice that for any dynasty, bt (bequest left by the agent ofgeneration t for his child) is nothing by xt+1(the inherited wealth ofthe next generation in the same dynasty).

Thus we can represent the bequest/wealth dynamics for any dynastyby the following difference equation:

xt+1 =

(1− α)

[(1+ r ∗)(2+ wN ) + (1+ r ∗)xt

]for 0 5 xt 5 f ;

(1− α)[wS − (1+ i)h+ (1+ i)xt

]for f < xt < h;

(1− α)[wS − (1+ r ∗)h+ (1+ r ∗)xt

]for xt = h.


Corresponding Phase Diagram:

Under suitable parametric conditions, one can draw the correspondingphase diagram as follows:


Identification of Steady States and Stability:

Given the above phase diagram, one can identify there steady statesfor this economy: x∗, x∗∗ and x∗∗∗.Out of these, the middle one is (locally) unstable, the other twoare (locally) stable.This implies that the long run wealth position of any dyanstichousehold h will be determined by its initial level of inherited wealth:xh0 :

If xh0 > x∗∗ then the wealth level of the dynasty in the long run

approches x∗∗;If xh0 < x

∗∗ then the wealth level of the dynasty in the long runapproches x∗.

This has implication for the occupational pattern of the dyansties inthe long run:

If xh0 > x∗∗ then the members of the dynasty in the long run are

skilled and engaged in modern production;If xh0 < x

∗∗ then the members of the dynasty in the long run areunskilled and engaged in cottage production.


Identification of Steady States and Stability: (Contd.)

Thus the middle steady state (x∗∗) can be interpreted as a povertytrap or a low-skill trap or an occupational trap.

Notice that Assumption 1 alone does not gaurantee that there wouldalways be three steady states.

There are other possible parametric conditions where there is no trap- there is uniform convergence to a single steady state irrespective ofthe initial wealth position of the households .

We need some additional assumptions to ensure that there are indeedthree intersection points in the phase diagram.(Derive a set of suffi cient conditions in the terms of theparameters such that this is the case).


Wealth Dynamics & The Macroeconomy in the Long Run:

Interestingly, the initial distribution of wealth and the correspondingwealth dynamics have important implications for the overall positionof the macroeconomy in the long run.Recall that total number of households in the economy is L, which isalso the number (measure) of young population at any point of timet.Let the D0(xh0 ) denote the initial distribution of wealth (inheritence)across all the L young agents, such that

∞∫0

dD0(xh0 ) = L

Out of these L young agents, let Lx∗∗0 denote that measure of young

agents at time 0 who have an inherited wealth level xh0 5 x∗∗ :x ∗∗∫0

dD0(xh0 ) = Lx ∗∗0


Initial Distribution, Wealth Dynamics & TheMacroeconomy:

From the intergenerational wealth dyanamics , we know that thechildren of all these Lx

∗∗0 agents will get trapped in the low skill-low

income cottage sector in the long run.Children of the rest (L− Lx ∗∗0 ) on the other hand will be skilled in thelong run and will get employed in the high skill-high income modernsector.Then the long run average wealth in this economy:

x =Lx∗∗0 x∗ +

(L− Lx ∗∗0

)x∗∗∗

L

= x∗ − Lx ∗∗0

L(x∗∗ − x∗)

It is easy to see that the long run average wealth of the entireeconomy depends on the initial distribution of wealth:

the higher is the proprtion of people below x∗∗, the lower in the longrun average wealth.


Initial Distribution, Wealth Dynamics & TheMacroeconomy: (Contd.)

Similar conclusion would hold even for the long run averageincome. (Verify)Notice that a direct redistributive policy that taxes the wealth of therich and gives it to the poor such that post- redistribution theproportion of people with wealth level > x∗∗ goes up, would make theeconomy better off in the long run (in an average sense).

Also notice that any random redistribution can actually make theeconomy worse off in the long run!!


Inequality & The Macroeconomy:

Now what about inequality?

Again consider two economies, A and B, which are identical in everyrespect except in terms of initial wealth distribution.

In particular suppose the average initial wealth holding is the same inboth economies, but wealth in economy A is more unequallydistributed than in economy B such that(

Lx∗∗0

L

)A>

(Lx∗∗0

L

)B.

Then economy A will experience a lower average income in the longrun than economy B.


Whither Growth?

But what about growth?The Galor-Zeira model exhibits no growth - in the long run all agentsreach their respective steady state levels of income and wealth and sodoes the economy.However one can derive similar conclusions in terms of the long rungrowth rate of the economy if we slightly modify the model in thefollowing way:

Assume that credit market in completely missing;Allow for a Harrod-Neutral technical progress in both sectors such thatlabour productivity in the cottage sector grows at the constant rate nwhile labour productivity in the modern sector grows at the constantrate m (m > n);Assume that fixed investment requirement for skill formation alsogrows at the same rate as modern sector productivity: mIn this modified structure one can now show that the higher inequalityimplies lower average growth in the long run.(Try this exercise)


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202: Dynamic Macroeconomic Theory