Persistent Inequality · 2002-03-03 · Persistent Inequality 2 Economic Inequality .....is of...

Persistent Inequality 1

Persistent Inequality

Dilip Mookherjee

Boston University

Debraj Ray

New York University


Economic Inequality . . .

. . . is of interest, not only at some intrinsic level, but also for itspossible connections to diverse variables.

Natural to study the evolution of inequality.

Dominant View. Inequality as constant battle between conver-gence and “luck” (e.g., Becker and Tomes (1979), Loury (1981)).

In contrast, I want to emphasize a view of inequality as an inevitableconsequence of the market mechanism.

Basic Assertion. There are fundamental market forces responsiblefor the emergence of persistent inequality among ex ante identicalagents, even in a world of perfect certainty.

This is only one of several issues we address. Let’s first take aninformal look at the structure of the model.


An Informal Description

Continuum of dynasties, one person per generation.

Each person bequeaths a profession to her child.

“Profession”: occupations, skills, class-distinctions such as entrepreneuror worker, and even levels of capital holdings.

Cost is paid upfront by the parent.

The set of professions may be finite or infinite.

The cost for acquiring a profession may be endogenous. E.g., teacherwages.

No uncertainty.


The Questions

We have already discussed the question:

[1] Under what conditions might every steady state of this model ne-cessitate (utility) inequality?

Large literature argues that steady states with inequality are possible.Necessity is a different matter.

[Explain using a standard growth model with nonconvexities.]

Turns out that our conditions are simple but not necessarily mild.Will return to this.

Under what conditions are there multiple steady states with varyinglevels of inequality?

The policy implications of multiplicity are well-known.

Answer to multiplicity question will depend fundamentally on therichness of the set of professions.


Questions, contd.

We also ask

[3] Are steady states efficient?

Missing capital market, so no reason for efficiency.

We provide an almost-complete characterization of efficiency, andshow as corollaries that

• With a “small number” of occupations (finitely many) there isa continuum of efficient steady states and a continuum of ineffi-cient steady states.

• With two occupations, inefficiency is related to inequality.

• With a “large number” of occupations (a continuum), a steadystate is typically efficient.

And we ask

[4] Do competitive equilibria converge to some steady state?

In general, this is an open question. We answer it in the affirmativefor a two-occupation model.


Nonconvexities: Technological vs. Pecuniary

Richness of professional structure makes a difference to the results.

But apart from this, there is an intrinsic reason for studying richoccupational structure.

A small number of professions builds in nonconvexities by assump-tion. “Technological Nonconvexities.”

As the set of professions becomes finer, these vanish. Those thatremain (if any) are driven by relative prices, a GE phenomenon.“Pecuniary Nonconvexities.”

"Input" x

"Out

put"

y

"Production Function" f(x)

Figure 1: A “Production Function.”


Model

Agents and Preferences

Continuum of agents: index i on [0, 1]. Nonpaternalistic altruism.

Single consumption good c. Each agent has within-period utility udefined on c. Increasing, smooth, strictly concave.

Over time: generation t’s payoff from a sequence of bundles cs isthe “tail sum” ∞∑

s=tδs−tu(cs). (1)

where δ ∈ (0, 1) is a common discount factor.

Professions

H: set of professions. Labels, don’t need topology for now. Later wewill, so think of as compact subset of real line.

λt, a measure on H at date t, is the population distribution overprofessions at date t.


Technology

CRS production sector (makes consumption good) + CRS educa-tional sector (trains generation t+ 1).

Write as closed convex cone T : contains triples of the form:

(λ, c,λ′),

where

λ is input (population distribution over professions)

c is consumption output, and

λ′ is supply vector of professionals “in the next generation”.

Assume at least one profession needs no training.

Prices

Normalize the price of c to unity. Two price vectors remain:

• returns to professions, w(h) or more compactly, w.

• training costs, x(h) or more compactly, x.

Let q ≡ (w,x).


Behavior

Profit maximization. Given prices qt at any date, the economy gen-erates

• input demands for professions (λt “in equilibrium”)

• supplies of final goods at date t (ct)

• professional capacities for the next generation (λt+1 “in equilib-rium”)

That is, these choices must solve

max c+ xt.λ′ − wt.λ (2)

subject to (λ, c,λ′) ∈ T .

Utility maximization. qs is given. Generation t household i has“starting profession” h(i). Chooses hs, css≥t to solve

max∞∑s=t

δsu(cs) (3)

subject toht = h(i) (4)

andws(hs) = cs + xs(hs+1) (5)

for all s ≥ t. [Note: no time consistency problems.]


Equilibrium

Fix initial λ. An equilibrium is qt,λt, ct (with λ0 = λ) such that

[1] At each t, (λt, ct,λt+1) ∈ T maximizes profits under qt.

[2] There exist ht(i), ct(i) (for all i and t) such that for each i,ht(i), ct(i)∞

t=0 maximizes dynastic utility starting from h0(i), andsuch that

ct =∫[0,1]

ct(i)di (6)

andλt(B) = Measurei : ht(i) ∈ B (7)

for every Borel subset of H.

(q,λ, c) is a steady state if there exists an equilibrium qt,λt, ct with(qt,λt, ct) = (q,λ, c) for all t.


Examples

Special Cases of Independent Interest.

[1] Models without Interaction I. Convex Technologies.

Ramsey-Cass-Loury. Compact set of capital stocks (or educationalinvestments, as in Loury). Each “profession” now corresponds to aparticular level of investment.

[2] Models without Interaction II. Nonconvex Technolo-

gies. Single final good, two professions: 1 (unskilled) and 2 (skilled),so that H = 1, 2. Exogenous cost x to becoming skilled. of ac-quiring the skill. Exogenous wage rates w(1) and w(2) to unskilledand skilled labor. Generates history-dependent outcomes. See Galor

and Zeira [1993].

[3] Models with Interaction I. Skill Acquisition Can’t beexpressed as linear Markov processes. E.g. two skills, with g =f(µ(1), µ(2)). Then the returns w(h) are obtained as the value of thepartial derivatives of this function. See extended model in Galor and Zeira

[1993], Banerjee and Newman [1993], Ray and Streufert [1993], Ljungqvist [1993], ...

[4] Models with Interaction II. Entrepreneurship. “Pro-fessions” need not represent different grades of labor. E.g. thinkof “worker” and “entrepreneur”. Fixed investment x needed for en-trepreneurship (business setup costs). To see the interaction, assumethat (conditional on making the fixed investment I), an entrepreneurproduces final output g through the use of a production function Fthat depends only on labor (L). Then each entrepreneur chooses Lto

maxF (L) − w(1)L,

where w(1) is the wage rate for labor.


Examples, Continued

[4, contd.] In equilibrium, L is just the employment per entrepreneur,which is λ(2)/λ(1). So w(1) is given by

F ′λ(2)λ(1)

= w(1),

while w(2) — the return to entrepreneurship — is just profit:

w(2) = F

λ(2)λ(1)

− F ′

λ(2)λ(1)

λ(2)λ(1)

.

See Banerjee and Newman [1993] or Freeman [1996].

[5] Teaching. In general, setup or educational costs aren’t exoge-nous: they may depend on the return to physical and human capital.E.g., H = 1, 2, where 1 stands for unskilled worker and 2 standsfor skilled worker. As in Examples 1 and 2, there is a productionfunction F defined on unskilled and skilled labor. (a(1), a(2)). Buta(h) is now the number of production workers of type h (and is to bedistinguished from λ(h)).

x(h) now endogenous. Equals 0 if h = 1, and αw(2) if h = 2. [Theidea is that to acquire a single unit of type 2 skills, you must betrained by α units of type 2 people.]

Finally, we determine the wage function in equilibrium. To this end,let λt and λt+1 be two “adjacent” population distributions over pro-fessions. Then notice that

at(1) = λt(1),

whileat(2) = λt(2) − αλt+1(2),

so that the wage function at date t is given by

wt(h) = Fh (λt(1), λt(2) − αλt+1(2))

for h = 1, 2. See, e.g., Ljungqvist [1993].


Persistent Inequality

Distinct Professions.

Say that two professions h and h′ are indistinguishable (relative tosome steady state (q,λ)) if w(h) = w(h′) and x(h) = x(h′). Other-wise, they are distinct.

Remarks.

• If two professions have distinct training technologies, they willgenerically be distinct (though we don’t pursue the exact conditions to make

this precise).

• “Vertical comparisons”: if training for h requires more of everymaterial good and every kind of teacher than training someonefor occupation h′, then h and h′ must be distinct.


Zero Mobility.

Proposition 1 Let (q,λ, c) be a steady state. Then no positivemeasure of individuals will switch across distinct professions.

Result is in deliberate contrast to “ergodic theories” of inequalitybased on exogenous random shocks. In such models (e.g., Loury[1981]), the entire steady-state distribution of economic characteris-tics will be experienced at different dates by every dynasty.

Proof relies on applying a supermodularity argument.

"wages"

today tomorrow

"crossing" inevitableif proposition is false


Inequality.

No uncertainty and zero mobility. Yet in every steady state, inequal-ity must emerge:

Proposition 2 Suppose that two dynasties inhabit two distinct pro-fessions in some steady state. Then they must enjoy different levelsof consumption (and utility).

No dynamics yet, but this suggests that even starting from perfectequality (and with perfect certainty), inequality must emerge.

When is the hypothesis satisfied? One example: say that a professionis necessary if without it, the final good cannot be produced (maketechnical adjustments if there is a continuum of professions).

Corollary to Proposition 2. Suppose that there are at least twovertically ordered professions that are necessary. Then every steadystate with positive consumption must involve (utility) inequality.

But the distinctness assumption is actually subtle. Leads to anotherpaper . . .


Multiple Steady States and History-Dependence

Same fundamentals generally consistent with numerous steady stateoutcomes.

Such history-dependence permits varying degrees of inequality, out-put, unemployment, productive efficiency . . .

[Note: usual notion of multiple equilibrium quite unrelated to thisdiscussion.]

Role for judicious policy intervention (see Hoff and Stiglitz [2001] for exam-

ples).

By changing initial conditions, the policy intervention may changethe steady state that forms the attractor for the process and therebygenerate permanent effects.

There is no need to change the set of steady states themselves.

This view to be contrasted with a more classical notion in whichconvergence to some unique limit obtains. In that case, only policiesthat are permanently in place can have persistent effects: in theabsence of the policy reversion to the unique limit distribution wouldoccur.

This is why an exploration of the multiplicity of steady states isimportant.


Two Notions of History-Dependence

Useful to distinguish between two notions of history-dependence.

Individual History Dependence or “Micro-Multiplicity”.

In which initial endowments or perturbations at the level of a house-hold shape the long run outcome of that household.

Societal History Dependence or “Macro-Multiplicity”.

In which which initial society-wide conditions determine the finaldestiny of the economy as a whole.

Individual History Dependence

So

ciet

al H

isto

ry D

epen

den

ce

YesNo

Yes

No Convergence literature,Loury [1981]

Piketty [1997]

Most recent literature: e.g.,Banerjee-Newman [1993],

Galor-Zeira [1993],Ljungqvist [1993], Ray-

Streufert [1993]

This paper


Exploring Macro-Multiplicity

Steady state has full support if every available profession is occupied.

Not a big deal if every profession is (directly or indirectly) necessaryin producing the final good.

Fix a (full-support) steady state (q,λ, c). Characterization:

[1] (λ, c) related to q via profit maximization:

(λ, c,λ) ∈ arg max c+ xλ′ − wλ, subject to (λ, c,λ′) ∈ T . (8)

[2] No individual contemplates a “one-shot deviation” to anotherprofession. By zero-mobility and full-support, the new professionwill prevail thereafter. So, for every h and alternative h′:

u (w(h) − x(h)) ≥ (1 − δ)u (w(h) − x(h′)) + δu (w(h′) − x(h′)).(9)

By Blackwell unimprovability, these conditions are sufficient as well.(8) and (9) completely characterize the set of steady states.


Two Professions

Suppose only two professions: “unskilled” (h = 1) and “unskilled”(h = 2), with x(1) = 0 and x(2) = x.

Let λ = proportion of skilled people. Production function f(λ, 1−λ).

Let w be the wage to skilled and w the wage to unskilled. Then

w(λ) ≡ f1(λ, 1 − λ),

while the unskilled wage will be given by

w(λ) ≡ f2(λ, 1 − λ).

Yields, along with earlier conditions, the following characterization:a fraction λ is compatible with a steady state if and only if

u (w(λ)) − u (w(λ) − x) ≤ δ1−δ [u (w(λ) − x) − u (w(λ))]

≤ u (w(λ)) − u (w(λ) − x).(10)

Left-most term is the “utility education cost” (call it κs(λ)) for askilled parent. The last term is the cost (κu(λ)) for an unskilledparent. The middle term is the present value benefit to being skilled:call it b(λ).

If f satisfies the “usual conditions”, κs(λ) is increasing in λ, whileremaining functions are decreasing in λ.


Education Costs and Benefits in Two-Profession Model

λ6

λ

κs(λ)

κu(λ)

b(λ)

λ5 λ4 λ3 λ2 λ1

Proposition 3 There is a continuum of steady states in the two-profession model, and total output net of training costs unambigu-ously rises as the skill proportion in steady state rises.


A “Rich Space” of Professions

One way to conceptualize “richness”: introduce continuity in the costof creating professional slots.

To this end, assume first that H = [0, 1]. [Can approach the contin-uum by taking a nested grid, with similar results. Earlier version.]

Next, define unit cost functions for each output. To do this, assume:

[T.1] T generated from a collection of individual production func-tions, one for the consumption good, and one each for the training ofa professional in every h.

So for each category h, there is a production function g(µh, yh, h),where

• µh is a measure on [0, 1], and

• yh is an amount of the final good

that enter into the training of h-professionals.

For the final good, can simply write production function as f(µ).


Unit Costs

[T.1] implies the existence of a unit cost function for each h

ψ(w, h) ≡ infµ,y′y

′ +∫Hw(h′)dµ(h′), subject to g(µ, y′, h) ≥ 1, (11)

In a steady state, ψ(w, h) must equal training cost x(h) for every h.

Now we are ready for the richness assumption.

[T.2] The unit cost function ψ(w, h) is continuous in h for everymeasurable w.

[T.2] typically satisfied when the required human inputs to train anh-professional can be represented by a density, which varies continu-ously in h.

Can replace [T.2] by weaker requirement that the range of possibletraining costs is an interval in each steady state: the set of investmentoptions is “perfectly divisible”.


Unique Steady State

Proposition 4 Assume [T.1] and [T.2] and full-support. Then, ifsome steady state exists with a strictly positive wage function, there isno other steady state wage function. If, in addition, every productionfunction is strictly quasiconcave, then there is no other steady state.

One aspect is very intuitive, provide we suppose (provisionally) thatx(h) is given.

Test the steady state condition by moving a tiny amount “up” or“down” in “profession space”. Curvature of the utility function canbe (almost) neglected. So for such tiny changes ∆(h),

w(h+ ∆h) − w(h) 1δx(h+ ∆h) − x(h).

Recall that one profession (say 0) needs no training. So using aboveequation we may conclude that

w(h) =1δx(h) + w(0),

where w(0) is just the wage for occupation 0.

Thus, given x(h), there is a one-parameter family of w(h)’s whichare potential candidates for a steady state.

But only one of these can be compatible with profit maximization.


Discussion, contd.

The less intuitive part of the proposition is that there is only onew-function even when x is endogenously determined.

This part makes use of constant returns to scale. Suppose (by contra-diction) that there is another steady state (w, x, λ, c) with a distinctwage function. Once again, we know that that

w(h) =1δx(h) + w(0).

Doing some division, we see that

x(h)x(h)

=w(h) − w(0)w(h) − w(0)

(12)

for all h such that both x(h) and x(h) are not simultaneously zero,interpreting this ratio to be ∞ in case x(h) = 0.

Use this expression to contradict constant returns to scale in theproduction of professions.


Two Quick Examples

[1] (Leontief): Workers proceed incrementally over successive traininglevels. To go from h − dh to h requires α(h) teachers with traininglevel h. This gives us

x(h) =∫ h

0α(h′)w(h′)dh′ (13)

Combining this with the description of the wage function given x, wesee that any steady state must belong to the family

w(h) = w(0) exp[∫ h

0

α(h′)δ

dh′]

Now profit maximization must uniquely pin down w(0), and we aredone.

[2] (Cobb-Douglas): Level-h training is described by

log s(h) =∫ h

0α(h′) log t(h′)dh′ (14)

where s(h) is the number of type h students turned out by a processthat uses t(h′) teachers of type h′ ∈ [0, h]. Scope for substitution. Bysolving minimization problem, we see that

x(h) = exp[∫ h

0α(h′) log

w(h′)α(h′)

dh′]

implying that a steady state wage function must satisfy the d.e.

w′(h) =1δα(h) log

w(h)α(h)

exp

∫ h

0α(h′) log

w(h′)α(h′)

dh′

Once again, profit maximization gets us uniqueness.


Pecuniary Nonconvexities

The uniqueness property in the continuum, combined with the factthat the steady state must display inequality, throws light on thenecessity of endogenous pecuniary nonconvexities.

At the steady state, the wage for each profession, viewed as a functionof the cost of acquisition, is affine. So there is no failure of convexityat the steady state.

However, the transitional dynamics involve endogenous functions suchas wt(h) and xt(h). If we believe that transitional dynamics converge,these functions cannot be concave throughout. Pecuniary nonconvex-ities must be the rule rather than the exception.

This finding is masked in a model with a finite number of occupations.


Pareto Optimality of Steady States

Universe of agents to whom the Pareto criterion should be applied:all pairs (i, t), where i indexes dynasty and t time.

Given some initial λ0 over occupations, say that ct(i),λt is efficientif, first, it is feasible:

(ct,λt,λt+1) ∈ Tfor all dates t, where ct ≡ ∫

[0,1] ct(i)di, and if there is no other feasibleallocation c′t(i),λ′

t (with λ′0 = λ0) such that

∞∑s=t

δs−tu(c′s(i)) ≥∞∑s=t

δs−tu(cs(i)),

with strict inequality holding over a set of agents of positive measure.

Proposition 5 Suppose that a steady state (q,λ, c) = (w,x,λ, c)has the property that

x(h) − x(h′) = a[w(h) − w(h′)] (15)

for some a ≥ δ, and for all occupations h and h′. Then such a steadystate is Pareto-efficient.

Apply to continuum economy discussed earlier. Shows that outcomesare (Pareto) efficient.


Pareto Optimality, Contd.

What if the condition of the proposition is violated; that is, what ifeither

x(h) − x(h′) < δ[w(h) − w(h′)] (16)

for some pair of professions with x(h) > x(h′), or if

w(h2) − w(h1)x(h2) − x(h1)

>w(h4) − w(h3)x(h4) − x(h3)

(17)

for four professions h1, h2, h3 and h4 (not necessarily all distinct)with x(h1) < x(h2) and x(h3) < x(h4)?

Proposition 6 Assume that there are a finite number of profes-sions, and that T has smooth boundary. Suppose that either (16) or(17) holds at some steady state with all professions occupied. Thenthe steady state cannot be Pareto-efficient.

In two-skill model, there is always a continuum of Pareto-optimalsteady states, and a continuum of Pareto-sub-optimal steady states.


Pareto-Optimality with Two Skills

λ1λ2λ3λ4λ5

λ6

λ

κs(λ)

κu(λ)

b(λ)

λ

Pareto Sub-Optimal ParetoOptimal


Transition Dynamics

A satisfactory theory of long run inequality must account for thedynamic process by which initial conditions determine final outcomes.

If there are several steady states, only a few of them may be stableattractors.

Even if convergence can be established, need to know the map be-tween initial conditions and eventual steady state.

Even if there is a unique steady state (as in the thick professionsmodel), can we be sure that it is a good predictor?

The transitory process is of interest in its own right. When doesinequality tend to increase or decrease over time? How fast is theconvergence? What are the transitory and long term effects of one-shot redistributions?

To our knowledge, there is no general theorem that guarantees con-vergence in this class of models. Competitive versions of the turnpiketheorem are available (see Bewley [1982], Coles [1985] or Yano [1984]) but donot apply here, as those arguments rely on the equivalence betweencompetitive equilibria and full Pareto-optimality. Such equivalencedoes not obtain in our setting because the credit market is missing.

We report on a special case.


Dynamics in the Two-Skill Model

We study the two-skill model with exogenous training cost.

Adapt the definition of perfect foresight competitive equilibrium tothis context.

Imagine that an infinite sequence of wages is given, one for each cat-egory of labor. Then for each household, there is a sequence of valuesVt, V t describing the infinite-horizon payoffs to each generation ateach date, conditional on starting skilled or unskilled. That is, foreach t,

Vt = maxct,xt

[u(ct) + δVt+1(xt)]

subject to the conditions that

ct + xt = wt,

and

Vt+1(xt) = Vt+1 if xt ≥ x

= V t+1 if xt < x.

Similar conditions describe V t, starting from wt.

Given λ0, a competitive equilibrium is a sequence wt, wt, λt∞t=0 such

that

[i] λt is generated by the maximization problems just described,

[ii] For each t, wt = w(λt) and wt = w(λt).


Dynamics, Continued

The main observation: if in any competitive equilibrium the percent-age of skilled households rises over any adjacent dates, the unskilledat those dates must be indifferent between acquiring and not acquir-ing skills. [Also true if “unskilled” and “skilled” are switched.]

Proof. Single-crossing argument used again: if an unskilled householdwere to educate its children, then

u(wt) − u(wt − x) ≤ δ[Vt+1 − V t+1].

By strict concavity and the fact that wt < wt for

u(wt) − u(wt − x) < δ[Vt+1 − V t+1].

But this means that a skilled household has a strict incentive toeducate its children.

The observation implies right away that

Vt =∞∑s=t

δs−tu(ws − x) (18)

andV t =

∞∑s=t

δs−tu(ws). (19)

These equations yield dynamic analogues of the steady-state no-switch conditions. For a currently skilled household,

u(wt) − u(wt − x) ≡ cs(λt) ≤∞∑

s=t+1δs−t[u(ws − x) − u(ws)], (20)

with equality holding whenever a switch from “skilled” to “unskilled”occurs at date t. Likewise, for a currently unskilled household,

u(wt) − u(wt − x) ≡ cu(λt) ≥∞∑

s=t+1δs−t[u(ws − x) − u(ws)], (21)

with equality holding whenever a switch from “unskilled” to “skilled”occurs.


Dynamics, Continued

Consider two possible non-steady-state zones in which λ might lie.The first is when the steady state condition fails because the skilledhouseholds have insufficient incentives. See (λ3, λ1).

In the second subset (denoted by B), the steady state condition failsbecause unskilled families strictly prefer not to educate their children.B is the union of all skill ratios lower than λ3 that do not constitutesteady states.

It is clear that A and B are disjoint.

The proposition characterizes dynamics:

Proposition 7 If λ0 ∈ A, then there exists a unique competitiveequilibrium from λ which goes to the steady state in one period: λ =λ0 > λ1 = λt for all t ≥ 1.If λ0 ∈ B, then there exists a unique competitive equilibrium in whichthe proportion of skilled people increases strictly in every period, andconverges to some steady state: λt < λt+1 for all t ≥ 0.If λ is a steady state, there is a unique competitive equilibrium fromλ0 = λ, given by λt = λ for all t.


Competitive Equilibria from Zones A and B

λ0λ3λ4

λ

κs(λ)

κu(λ)

b(λ)

l(λ0)λ0'

b'

B A

Note that

the map from initial to long-run skill distributions is nonmonotonic.

local policies when in A have perverse long-run effects.

local policies when in the steady state set have persistent effects.

local policies in B have no long-run effects (unless they cause animmediate jump over some interval of steady states).


Unpacking the Distinctness Assumption

Several layers. Haven’t explored all the implications yet.

Main problem: in its most abstract form, a “profession” is a pair ofthe form (occupation, financial capital).

So if financial bequests are allowed, distinctness of occupations doesimply distinctness of professions.

In what follows, we explore this theme using a simpler variant.


Simplified Features

1. Occupations and Money. We unpack these two componentsof a profession. In particular, parent can supplement education withfinancial bequest at fixed return r. [Easy to modify.]

2. Occupations produce a single final good.

That is, λ is just a bundle of inputs, which produces output using aconvex CRS technology.

Wages endogenous as before. Wage function w = w(h), dependson λ.

Assumption. For each λ there is a unique “supporting” wage func-tion w. Conversely, every w admits some scaling kw such that kwis a supporting wage function with unique profit-maximizing inputchoices.

3. Exogenous Educational Costs.

Measured in units of final good, occupation h costs x(h) to acquire.We assume that there is h with x(h) = 0 and h′ with x(h′) > 0.

4. Utility from own consumption and child’s wealth.

Write as u(ct) + v(Wt+1).

Note. Wealth includes wages w of child and (interest-augmented)financial bequests b made by parent. So W = w + b(1 + r).

5. Equilibrium and Steady State: Just as before.


Artificial Scenario: No Occupational Choice

Suppose constant wage w at every date.

Only financial bequests permitted.

Given resources W , individual chooses b to maximize

u(W − b) + v(W ′),

where W ′ ≡ [1 + r]b+ w.

Let Ψ(W,w) be resulting choice of W ′. Ψ nondecreasing in W .

An Example.

v = δu for some discount factor δ and u(c) = c1−σ/(1 − σ) for someσ > 0. Then

Ψ(W,w) =(1 + r)ρ1 + ρ+ r

W +ρ

1 + ρ+ rw

if ρW ≥ w, and equals zero otherwise, where

ρ ≡ [δ(1 + r)]1/σ.


Benchmark, contd.

Imperfect Persistence Assumption. Increase in next period’sassets following a unit change in assets today is bounded below unity.

See Becker and Tomes (1979)), Mulligan (1997), Bowles and Gintis (2001).

Under imperfect persistence, for any given w, Ψ(., w) intersects the450 line once and only once.

In constant-elasticity example, imperfect persistence satisfied iff

ρ ≡ [δ(1 + r)]1/σ < 1 +1r. (22)

Under imperfect persistence, policy function precipitates unique limitwealth Ω(w). In example with (22) satisfied,

Ω(w) = w if ρ ≤ 1,

=ρ

1 − r(ρ− 1)w otherwise.


Steady State Inequality

Define special wage function w∗ by

w∗(h) − w∗(h′) = (1 + r)[x(h) − x(h′)]

for every pair of occupations h and h′, and scaled suitably to serveas a supporting wage function.

By CRS, there is a unique wage function of this form. Let w∗ be thelowest wage in w∗, and w∗ be the highest wage.

Note. Easy to compute w∗ from parameters of system.

Proposition 8 As long as

w∗ > Ω(w∗), (23)

every steady state must involve (utility) inequality. If (23) fails, asteady state with perfect equality exists, and must display the wagefunction w∗.

Intuition: Argue by contradiction.


Configurations that Favor w∗ > Ω(w∗).

Very broadly: the bequest motive cannot be so strong as to over-whelm all earning differences.

Discounting. For instance, if δ(1 + r) ≤ 1 in constant-elasticityexample, Ω(w) must equal w, and so the inequality condition musthold.

Occupational Variety. Inequality condition more likely to ap-ply when occupational structure exhibits large differences in trainingcosts. In example, condition reduces to

M

w∗ >ρ− 1

1 + r(1 − ρ),

M is the cost of the most expensive occupation.

Poverty. Inequality condition more likely to hold in poor economies.Limit wealth Ω(w) moves more than one-to-one with w.

Growth. Inequality condition more likely to hold in growing economies.Attenuates bequest motive. In constant-elasticity example, supposewages start at base w and grow at rate g. Then steady state wealthalso grows at g, but with “level coefficient”

Ω(w) = ρ/[1 − r(ρ− 1) + g(1 + ρ+ r)]w.

[Smaller than old value.]


What do Steady States Look Like?

Consider the special case of a “rich” set of occupations. Explain whyimportant.

Assume a continuum of occupations, with x ∈ [0,M ].

x

w(x)

M

w

x*x

_

_w

~

Ω(w)_

Figure 2: A Steady State Wage Function.


What do Steady States Look Like?, contd.

Wealth Clustering at Bottom. Across occupations with wagesbetween w and Ω(w), wages must be linear in costs with slope (1+r).All individuals in occupations in this zone will haev identical wealth.

Wealth Inequality Elsewhere. To encourage the settlementof “high-cost” occupations, the rate of return on occupational choicemust depart from the financial rate. And this departure must createinequality.

Inequality May “Accelerate” with Wealth. Typically, willneed higher and higher rates of return to support the more expensiveoccupations. True in the constant-elasticity example.

Financial Bequests... are made at the bottom of the occupa-tional ladder. But this result to be qualified for two reasons: (a)possible gaps in occupational structure; (b) interpretation of an oc-cupational category.

Unique Steady State. True with the continuum structure; nottrue with a “small” set of occupations. To see why, look at figure.


Equilibrium Dynamics

A distribution of wealth prevails at any date; this will map into adistribution of wealth for the next generation.

Several cases possible. We focus on one. Assume that initial wealthis perfectly equal at W0, and suppose that

w∗ > Ω(w∗).

Observation 1. Let tomorrow’s wage function be w1, with lowestwage w1. Let W1 ≡ Ψ(W0, w1). Then for every x with w1(x) ≤ W1,

w1(x) = w1 + (1 + r)x.

Observation 2. For any N , there exists a threshold such thatstarting from any equal initial wealth above this threshold, there isperfect equality for at least N generations.

Note. This “declining wealth” story can be replaced by a an ar-gument based on economy-wide growth + endogenous training costs.


But Equality Can’t Last Forever . . .

_w*

W0

x

w(x), W

M

w

W1

WN

WN-1

w at N

w = w* up to N-1

xNxN+1

WN+1

_ w at N+1

Figure 3: Symmetry-Breaking along Equilibrium Paths.


Summary

We study the general equilibrium of inequality with imperfect creditmarkets.

Persistent inequality may be the rule rather than the exception, evenin a world of ex-ante identical agents and perfect certainty.

Whether or not such outcomes are fundamentally history-dependentseems to depend on the richness of occupational structure.

Pareto-optimality may or may not obtain at steady states, but com-plete characterizations appear to be possible.

Transitional dynamics represent the big open question. We solve thisfor the case of two professions.

As for the distinctness condition that yields persistent inequality:a wide diversity of occupations (relative to bequest motive) seemscrucial.

Above condition more likely to hold when:

• there is low discounting.

• the range of training costs is large.

• the economy is poor; or when

• the economy grows quickly

Persistent Inequality · 2002-03-03 · Persistent Inequality 2 Economic Inequality .....is of...

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