202: Dynamic Macroeconomic Theoryecondse.org/wp-content/uploads/2015/08/C202-LectureNotes-Moav... · 202: Dynamic Macroeconomic Theory Inequality, Credit Market Imperfection & Development:

202: Dynamic Macroeconomic TheoryInequality, Credit Market Imperfection & Development: Moav (2002)

Mausumi Das

Lecture Notes, DSE

Aug 6-7, 2015

Das (Lecture Notes, DSE) Dynamic Macro Aug 6-7, 2015 1 / 26

Inequality, Credit Market Imperfection & Development:Moav (Econ Letters, 2002)

Recall that in the Galor-Zeira paper, there were two crucialassumptions:

Credit market imperfectionIndivisibility (non-convexity) in the human capital formation technology

Also recall that In the Galor-Zeira paper, savings/bequest behaviourof all households were symmetric: everybody left the sameproportion (1− α) of their income as bequest.But non-symmetric savings behaviour could be another source ofdivergence between the rich and the poor.If the savings propensities of the rich is higher than that of the poorand the economy is credit constrained, then it can again get stuck ina long run poverty trap - even when all technologies (including humancapital formation technology) are convex.In such a scenario, initial inequality would again have a negativeimpact on the long run economic performance - just as in Galor-Zeira.


Role of Convex Savings/Bequest Function:

That convex savings function can generate long run poverty traps wasfirst shown in a simple dynamic model by Moav (2002).

Recall however that earlier we had argued that a convex savingsfunction could imply a positive effect of inequality on output - due tothe positive effect of inequality on aggregate saving, as shown byBourguignon (1981).

But Bourguignon did not allow for credit market imperfection.

Moav shows that when credit markets are imperfect, while inequalityindeed increases the aggregate savings in the economy, it could stilllead to lower output.

This is because in an imperfect capital market environment, capital isnot allocated effi ciently to its most productive uses. Thus greatersavings do not necessarily translate into higher output.


Moav (2002): The Model

Consider a small open economy with perfect capital mobility.

A single final commodity is produced - using a single technology thatrequires physical capital and skill (human capital) as inputs.The final goods production technology is associated with aneoclassical production function which is CRS and exhibits positivebut diminishing marginal product of each factor:

Yt = F (Kt ,Ht )

FK ,FH > 0; FKK ,FHH < 0

Since F (Kt ,Ht ) obeys all neoclassical properties, including CRS:

YtHt= f

(KtHt

); f (0) = 0; f ′ > 0; f ′′ < 0.


Production Side Story:

The final goods production tecnology is operated by profit-maximizingfirms, operating under perfect competition. Thus the wage rate perunit of skill and the market interest rate are given respectively by:

wt = f(KtHt

)− f ′

(KtHt

)(KtHt

);

rt = f ′(KtHt

).

A small open economy with perfect capital mobility implies:

rt = r ∗ ⇒ f ′(KtHt

)= r ∗.

This fixes the domestic capital to skill ratio at:KtHt= f ′

−1(r ∗) = k.

This in turn implies that the wage rate per unit of skill is fixed at:

w = f (k)− f ′ (k) (k) .Das (Lecture Notes, DSE) Dynamic Macro Aug 6-7, 2015 5 / 26

Skill Formation Technology:

Unlike Galor-Zeira, skill levels are now divisible.Skill formation still requires spending one full time period in school,but the amount of skill level acquired in the next period is a positivefunction of the amount of resources (et) spent in skill formation:

ht+1 ={

1+ γet for et 5 e;h ≡ 1+ γe for et > e,

where h is the maximum level of skills that one can acquire.We assume that

wγ > (1+ r ∗) (Assumption 1)

This assumption implies that for et 5 e, the marginal return tohuman capital, , is larger than the marginal return to physical capital.This ensures then until opportunities to invest in skill formation isexhausted (i.e., until h is reached) people will spend their resources inskill formation rather physical capital formation.


Credit Market Imperfection:

Unlike Galor-Zeira, Moav assumes that imperfection in the creditmarket results in a complete absence of borrowing.

Agents cannot borrow in order to finance investment in humancapital, although they can lend at the given interest rate r ∗.

Alternatively, one can retain the Galor-Zeira assumption that agentscan potentially borrow at a higher interest rate i , but assume that(1+ i) > wγ.

Under the latter assumption, the opportunity cost of borrowing wouldbe greater then the marginal return to skill formation - hence nobodywould actually borrow to finance investment in human capital (eventhough potentially they could).


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 time.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 time and some inherited wealth (xt ).In the first period, he consumes nothing and only makes choices aboutskill formation and/or capital formation decisions.The crucial decision that he has to take here is: how much to invest inskill formation and how much to save (which would constitute hisphysical capital in the next period)

In the second period of his life:

Depending on his first period decision about the level of skill formation(ht+1) , he earns a wage income wht+1.If he had saved any wealth in the form of physical capital, then he alsoearns an additional interest income.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+ θ

); 0 < α < 1; θ > 0.

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

c + b = y .

This utility function is different from the one specified by Galor-Zeira.In particular, this utility function is nonhomothetic in c and b.The non-homotheticity property will imply that savings/bequestpropensities will depend on income.


Optimal Solution:

From the FONCs:α(b+ θ

)(1− α)c

= 1.

Putting this in the household’s budget constraint:

c +(1− α

α

)c = y + θ

which impliesc = α

(y + θ

);

b = (1− α) y − αθ.

Notice however that bequest cannot be negative. (Parents cannotpass on their own debt to their children).Thus the optimal bequest level is given by:

b =

0 for y ≤ y ≡ αθ

1− α;

(1− α) y − αθ for y > y .(I)


Skill Formation Decision of an Agent:

Given the skill formation technology and Assumption 1, the skillformation decision is almost trivial:

Agents with inherited wealth level xt 5 e spend their entire wealth inskill formation;Agents with inherited wealth level xt > e spend e amount of theirwealth in skill formation and saves the rest (which constitute theirphysical capital in the next period)

The corresponding second period income:

y ={

w(1+ γxt ) for xt 5 e;w(1+ γe) + (1+ r ∗)(xt − e) for xt > e.

(II)

Recall that out of this second period income the agents leaves abequest b as deyermined by equation (I).Equation (I) and (II) allow us to completely characterize the optiomalskill formation and bequest decisions of agents on the basis of thewealth distribution.


Wealth Cut-off for Zero Bequest:

Recall from equation (I) that b = 0 for any y ≤ y ≡ αθ

1− α.

On the other hand, from equation (II), y depends on the inheritedwealth level xt .Let us assume that y is attained at a wealth level < e.Then the corresponding wealth cut-off level for zero bequest is givenbelow:

y ≤ y ≡ αθ

1− α

⇒ w(1+ γxt ) ≤ y ≡αθ

1− α

⇒ xt ≤1wγ

[αθ

1− α− w

]We assume that this wealth cut-off is postive:

αθ

1− α> w (Assumption 2)


Wealth Distribution, Income and Corresponding BequestLevel:

1. For any xt such that 0 5 xt 5 1wγ

[αθ

1− α− w

]≡ f :

Income: w(1+ γxt )Bequest left to his child: bt = 0

2. For any xt such that f < xt 5 e :

Income: w(1+ γxt )Bequest left to his child: bt = (1− α) w(1+ γxt )− αθ

3. For any xt such that xt > e :

Income: w(1+ γe) + (1+ r ∗)(xt − e)Bequest left to his child:bt = (1− α) [w(1+ γe) + (1+ r ∗)(xt − e)]− αθ


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 =

0 for 0 5 xt 5 f ;

(1− α) w(1+ γxt )− αθ for f < xt 5 e;(1− α) [{w(1+ γ)− (1+ r ∗)} e + (1+ r ∗)xt ]− αθ

for xt >e.


Intergenerational Wealth Dynamics: Phase Diagram

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


Steady States, Stability & Existence of Poverty Trap:

Just as in Galor-Zeira, we now have three steady states: 0, x∗∗ andx∗∗∗: Out of these, the middle one is (locally) unstable, the other twoare (locally) stable.Once again the long run wealth position of any dyanstic householdi will be determined by its initial level of inherited wealth (x i0):

If x i0 > x∗∗ then the wealth level of the dynasty in the long run

approches x∗∗;If x i0 < x

∗∗ then the wealth level of the dynasty in the long runapproches 0.

Thus the middle steady state (x∗∗) can be interpreted as a povertytrap.Notice once again that Assumptions 1 & 2 do not gaurantee thatthere would always be three steady states; we need some additionalassumptions to ensure that there are indeed three intersection pointsin the phase diagram.(Derive a set of suffi cient conditions in the terms of theparameters such that this is indeed the case.)


Wealth Dynamics to Income Dynamics:

Since wealth and skill formation decisions are correlated, and theacquired skill level in turn determines an agent’s income, instead oflooking ar the wealth dynamics, we can equivalently look at theincome dynamics.Exercise: derive the difference equation in terms of yt and yt+1(instead of xt and xt+1 ) and draw the corresponding phasediagram.It is easy to show that, just as in Galor-Zeira, the inital distribution ofwealth (in particular the proportion of people in the initial distributionwith wealth level below x∗∗) matters for the long run average wealthas well as long run average income in the this economy.


Non-Homothetic Preferences: Why & How?

Moav shows that credit market imperfection along withnon-homothetic preferences can replicate the Galor-Zeira poverty trapresult - even when all technologies are convex.

This then begs the question: why are preferences non-homothetic?

Moav does not provide any explanation for this.

But one can construct economically meaningful stories that generatesuch non-homothetic preferences.

We now turn to one such story.


An Economic Rationale for Non-Homothetic Preferences:Das (JDE, 2007)

Decision to educate the child is typically undertaken by the parent -not the child himself.

Therefore, the degree of parental altruism plays an important role indetermining the educational investment bestowed on a child - whichin turn determines his earning capability upon adulthood.

But parental perceptions about the utility of children’s educationdiffer across rich and poor households.

Typically in a poor family, which is close to subsistence, consumptionof the family assumes more importance than the level of education ofthe children.

Crucial assumption of the paper: degree of parental altruism isendogenously determined and it varies with the earning ability of theparent (‘Limited’Parental Altruism)


‘Limited’Parental Altruism: Supporting Arguments

Poverty creates stress: Various socio-medical studies link parentalcare for children to parents’socio-economic status.

“poverty creates a heightened parental stress, straining or limiting thecapacity of parents to provide warmth, understanding and guidance fortheir children”.

Poverty increases agent’s rate of time preferences: Irving Fisher(1930) argued that poverty compels people to evaluate current andfuture differently: current consumption needs gets priority over futureconsumption needs.

“Poverty bears down heavily on all portions of a man’s expected life,both that which is immediate and that which is remote. But itenhances the utility of immediate income even more than the futureincome.”


‘Limited’Parental Altruism & Agents’Preferences:

Consider a representative agent with an income y .

The agent derives utility from own consumption (c) as well as fromthe investment made on children’s education (b) (where investmentincurred on children’s education can be thought of as a form ofbequest).

Let c be some subsistence level of consumption which each householdmust maintain in order to survive.

Any consumption above this subsistence level gives them positiveutility, as does children’s education.

Assume however that for all agents y > c , so that the subsistenceconsumption does not play any significant role in the optimalbehaviour of the agent.

Let c ≡ c − c denote the consumption over and above thesubsistence level. By the above assumption, it is always positive.


‘Limited’Parental Altruism & Agents’Preferences:(Contd.)

Utility function of an agent:

W (c , b) = u(c) + β(c)u(b), (1)

where β(c) is the weight attached to utility derived from children’seducation, which represents the degree of parental altruism.Assumptions:1. u(0) = 0, and for all c , b = 0, u′(.) > 0; u′′(.) < 0. Further, thefunction u(.) exhibits constant elasticity with respect to its argument,

such that σu =xu′(x)u(x)

; x = c , b is a constant ∈ (0, 1).

2. β(0) = 0, and for all c , b = 0, β′(c) > 0; β′′(c) < 0. Further, thefunction β(c) exhibits constant elasticity with respect to its argument,

such that σβ =cβ′(c)β(c)

is a constant ∈ (0, 1).

Each adult agent maximises (1) subject to his budget constraint:

c + b = yDas (Lecture Notes, DSE) Dynamic Macro Aug 6-7, 2015 23 / 26

Implication of ‘Limited’Parental Altruism: ConvexBequest Function

Under Assumptions 1 and 2, the marginal rate of substitutionbetween consumption (above subsistence) and educationalexpenditure decreases along a ray from the origin. (Proof?)

Implication: The resulting income-consumption curve will be convexwith respect to c , implying that the optimal bequest function b(y)will be convex in income.


Convex Bequest Function & Long Run Poverty Trap

If the bequest function is convex, it is easy to replicate theGalor-Zeira/Moav type poverty trap result.

Assume that adulthood income is a linear function of the educationalbequest received during childhood (as in Moav):

yt = Abt−1.

Then the convex b(y) function will immediately generate a convexphase curve representing the intergenerational bequest (wealth)dynamics, which under suitable parametric restrictions will result inmultiple steady states and long run poverty trap.


Long Run Poverty Trap: Will Re-Distribution Help?

Since unequal wealth distribution lies at the heart of the problem, willa redistribution of wealth at the inital point of time help?

Notice that a direct redistributive policy that taxes the wealth of therich and gives it to the poor would make the economy better off inthe long run (in an average sense) if and only if post- redistribution,the proportion of people with wealth level > x∗∗ goes up.

Any other redistributive measure will either leave the long runoutcome unchanged, or might even worsen it! (How?)Moral of the story: Redistribution is good for growth (development,actually) but only if it is carefully designed.

Alternative Policies:

Provide easy credit to the poor?Public education?


Documents

202: Dynamic Macroeconomic Theoryecondse.org/wp-content/uploads/2015/08/C202-LectureNotes-Moav... · 202: Dynamic Macroeconomic Theory Inequality, Credit Market Imperfection & Development: