202: Dynamic Macroeconomic Theory

202: Dynamic Macroeconomic Theory

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

Mausumi Das

Lecture Notes, DSE

Aug 6-7, 2015

Inequality, Credit Market Imperfection & Development:

Moav (Econ Letters, 2002)

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

Credit market imperfection

Indivisibility (non-convexity) in the human capital formation technology Also recall that In the Galor-Zeira paper, savings/bequest behaviour of all households were symmetric: everybody left the same

proportion (1 α) of their income as bequest.

But non-symmetric savings behaviour could be another source of divergence between the rich and the poor.

If the savings propensities of the rich is higher than that of the poor andthe economy is credit constrained, then it can again get stuck in a long run poverty trap - even when all technologies (including human capital formation technology) are convex.

In such a scenario, initial inequality would again have a negative impact 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 was

…rst shown in a simple dynamic model by Moav (2002).

Recall however that earlier we had argued that a convex savings function could imply a positive e¤ect of inequality on output - due to the positive e¤ect of inequality on aggregate saving, as shown by Bourguignon (1981).

But Bourguignon did not allow for credit market imperfection.

Moav shows that when credit markets are imperfect, while inequality indeed increases the aggregate savings in the economy, it could still lead to lower output.

This is because in an imperfect capital market environment, capital is not allocated e¢ ciently to its most productive uses. Thus greater savings do not necessarily translate into higher output.

Moav (2002): The Model

Consider a small open economy with perfect capital mobility.

A single …nal commodity is produced - using a single technology that requires physicalcapital and skill (human capital) as inputs.

The …nal goods production technology is associated with a

neoclassical production function which is CRS and exhibits positive but diminishing marginal product of each factor:

Y_t = F(K_t,H_t) F_K,F_H > 0; F_KK,F_HH <0

Since F(K_t,H_t)obeys all neoclassical properties, including CRS:

Yt

H_t =f Kt

H_t ; f(0) =0;f⁰ >0;f⁰⁰ <0.

Production Side Story:

The …nal goods production tecnology is operated by pro…t-maximizing

…rms, operating under perfect competition. Thus the wage rate per unit of skilland the market interest rate are given respectively by:

w_t = f K_t Ht

f⁰ K_t Ht

K_t Ht

;

rt = f⁰ K_t Ht

.

A small open economy with perfect capital mobility implies:

r_t =r )^{f0 K}_H^t

t

=r . This …xes the domestic capital to skill ratio at:

Kt

H_t =f^{0 1}(r ) =k.^¯

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

¯

w =f (k^¯) f⁰(k^¯) (k^¯).

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 positive function of the amount of resources (et) spent in skill formation:

h_t+1 = ¹+γet for et 5^e;¯ h¯ 1+γe¯ for e_t >e¯,

whereh¯ is the maximum level of skills that one can acquire.

We assume that

¯

wγ>(1+r ) (Assumption 1) This assumption implies that for e_t 5^e,¯ the marginal return to human capital, , is larger than the marginal return to physical capital.

This ensures then until opportunities to invest in skill formation is exhausted (i.e., until h¯ is reached) people will spend their resources in skill formation rather physical capital formation.

Credit Market Imperfection:

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

Agents cannot borrowin order to …nance investment in human capital, although they can lend at the given interest rate r .

Alternatively, one can retain the Galor-Zeira assumption that agents can potentially borrow at a higher interest ratei, but assume that (1+i)>w¯γ.

Under the latter assumption, the opportunity cost of borrowing would be greater then the marginal return to skill formation - hence nobody would actually borrow to …nance investment in human capital (even though potentially they could).

Household Side Story:

A two-period overlapping generations economy with constant population.

There are L¯ households - each consisting of a young member and an old member at any point of timet.

An agent lives exactly for two periods - youth and maturity, and has an o¤spring at the beginning of the second period of his life.

He dies at the end of the 2nd period but the dynastic link is carried forward 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 …nal goods as bequest from parent.

Agents di¤er 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 generationt is as follows:

In the …rst period of his life:

He is endowed with one unit of time and some inherited wealth (x_t).

In the …rst period, he consumes nothing and only makes choices about skill formation and/or capital formation decisions.

The crucial decision that he has to take here is: how muchto invest in skill formation andhow muchto save (which would constitute his physical capital in the next period)

In the second period of his life:

Depending on his …rst period decision about the level of skill formation (h_t+1) , he earns a wage income w h¯ _t+1.

If he had saved any wealth in the form of physical capital, then he also earns an additional interest income.

He spends his entire second period income in own consumption and in leaving 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 bequest for his child (b).

His preference is represented by the following utility function (identical for all agents):

U(c,b) =αlogc+ (1 α)log b+θ^¯ ; 0<α<1;θ¯ >0.

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

c+b =y.

This utility function isdi¤erentfrom the one speci…ed by Galor-Zeira.

In particular, this utility function is nonhomotheticin c andb.

The non-homotheticity property will imply that savings/bequest propensities 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 implies

c =α y+θ^¯ ; b = (1 α)y αθ.¯

Notice however that bequest cannot be negative. (Parents cannot pass on their own debt to their children).

Thus the optimal bequest level is given by:

b = 8<

:

0 for y yˆ αθ¯ 1 α; (1 α)y αθ¯ for y >y.ˆ

(I)

Skill Formation Decision of an Agent:

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

Agents with inherited wealth levelxt 5^e¯ spend their entire wealth in skill formation;

Agents with inherited wealth levelxt >e¯ spende¯ amount of their wealth in skill formation and saves the rest (which constitute their physical capital in the next period)

The corresponding second period income:

y = ^w¯(1+γx_t)for x_t 5^e;¯

¯

w(1+γ¯e) + (1+r )(xt e¯) forxt >e.¯ (II) Recall that out of this second period income the agents leaves a bequestb as deyermined by equation (I).

Equation (I) and (II) allow us to completely characterize the optiomal skill formation and bequest decisions of agents on the basis of the wealth distribution.

Wealth Cut-o¤ for Zero Bequest:

Recall from equation (I) that b=0 for any y yˆ αθ¯ 1 α. On the other hand, from equation (II), y depends on the inherited wealth level x_t.

Let us assume that yˆ is attained at a wealth level<e.¯

Then the corresponding wealth cut-o¤ level for zero bequest is given below:

y yˆ αθ¯

1 α

) ^w¯(1+γx_t) yˆ αθ¯

1 α

) ^{xt 1}

¯ wγ

αθ¯

1 α w¯

We assume that this wealth cut-o¤ is postive:

αθ¯

1 α > w¯ (Assumption 2)

Wealth Distribution, Income and Corresponding Bequest Level:

1. For any x_t such that 05^xt 5 w_¯¹γ

αθ¯

1 α w¯ f :

Income: w¯(1+γx_t)

Bequest left to his child: b_t =0 2. For any xt such that f <xt 5 ^{e¯ :}

Income: w¯(1+γx_t)

Bequest left to his child: b_t = (1 α)w¯(1+γx_t) αθ¯ 3. For any x_t such that x_t > e¯ :

Income: w¯(1+γe¯) + (1+r )(x_t e¯) Bequest left to his child:

b_t = (1 α) [w¯(1+γe¯) + (1+r )(x_t e¯)] αθ¯

Intergenerational Wealth Dynamics:

Notice that for any dynasty, b_t (bequest left by the agent of

generation t for his child) is nothing byx_t+1(the inherited wealth of the next generation in the same dynasty).

Thus we can represent the bequest/wealth dynamics for any dynasty by the following di¤erence equation:

xt+1 = 8>

><

>>

:

0 for 05^xt 5^f;

(1 α)w¯(1+γx_t) αθ¯ for f <x_t 5 ^e¯; (1 α) [f^w¯(1+γ) (1+r )g^e¯+ (1+r )xt] αθ¯

for x_t >e.¯

Intergenerational Wealth Dynamics: Phase Diagram

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

Steady States, Stability & Existence of Poverty Trap:

Just as in Galor-Zeira, we now have three steady states: 0,x and x : Out of these, the middle one is (locally) unstable, the other two are (locally) stable.

Once again the long run wealth position of any dyanstic household i will be determined by its initial level of inherited wealth (x₀ⁱ):

Ifx₀ⁱ >x then the wealth level of the dynasty in the long run approchesx ;

Ifx₀ⁱ <x then the wealth level of the dynasty in the long run approches 0.

Thus the middle steady state (x ) can be interpreted as a poverty trap.

Notice once again that Assumptions 1 & 2 do not gaurantee that there would always be three steady states; we need some additional assumptions to ensure that there are indeed three intersection points in the phase diagram.

(Derive a set of su¢ cient conditions in the terms of the parameters such that this is indeed the case.)

Wealth Dynamics to Income Dynamics:

Since wealth and skill formation decisions are correlated, and the acquired skill level in turn determines an agent’s income, instead of looking ar the wealth dynamics, we can equivalently look at the income dynamics.

Exercise: derive the di¤erence equation in terms of y_t and y_t+1 (instead of xt and xt+1 ) and draw the corresponding phase diagram.

It is easy to show that, just as in Galor-Zeira, the inital distribution of wealth (in particular the proportion of people in the initial distribution with wealth level belowx ) matters for the long run average wealth as well as long run average income in the this economy.

Non-Homothetic Preferences: Why & How?

Moav shows that credit market imperfection along with

non-homothetic preferences can replicate the Galor-Zeira poverty trap result - 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 generate such 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 in determining the educational investment bestowed on a child - which in turn determines his earning capability upon adulthood.

But parental perceptions about the utility of children’s education di¤er across rich and poor households.

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

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

‘Limited’Parental Altruism: Supporting Arguments

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

“poverty creates a heightened parental stress, straining or limiting the capacity of parents to provide warmth, understanding and guidance for their children”.

Poverty increases agent’s rate of time preferences: Irving Fisher (1930) argued that poverty compels people to evaluate current and future di¤erently: current consumption needs gets priority over future consumption 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 it enhances the utility of immediate income even more than the future income.”

‘Limited’Parental Altruism & Agents’Preferences:

Consider a representative agent with an incomey.

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

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

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

Assume however that for all agents y >c, so that the subsistence¯ consumption does not play any signi…cant role in the optimal behaviour of the agent.

Let cˆ c c¯ denote the consumption over and above the subsistence level. By the above assumption, it is always positive.

‘Limited’Parental Altruism & Agents’Preferences:

(Contd.)

Utility function of an agent:

W(cˆ,b) =u(cˆ) +β(cˆ)u(b), (1) where β(cˆ)is the weight attached to utility derived from children’s education, which represents the degree of parental altruism.

Assumptions:

1. u(0) =0, and for allc,b=^0,u0(.)>0;u⁰⁰(.)<0. Further, the functionu(.)exhibits constant elasticity with respect to its argument, such thatσu = ^xu0(x)

u(x) ^;x =c,ˆ bis a constant2(0,1).

2. β(0) =0, and for allc,b=^0,β0(c_ˆ)>0;β⁰⁰(c_ˆ)<0. Further, the functionβ(c_ˆ)exhibits constant elasticity with respect to its argument, such thatσ_β= ^cˆβ

0(cˆ)

β(c_ˆ) is a constant2(0,1).

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

c+b =y

Implication of ‘Limited’Parental Altruism: Convex Bequest Function

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

Implication: The resulting income-consumption curve will be convex with respect toc, implying that the optimal bequest functionb(y) will be convex in income.

Convex Bequest Function & Long Run Poverty Trap

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

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

y_t =Ab_{t 1}.

Then the convex b(y)function will immediately generate a convex phase curve representing the intergenerational bequest (wealth) dynamics, which under suitable parametric restrictions will result in multiple 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, will a redistribution of wealth at the inital point of time help?

Notice that a direct redistributive policy that taxes the wealth of the rich and gives it to the poor would make the economy better o¤ in the 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 run outcome 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?