ec9aa term 3: lectures on economic inequality

EC9AA Term 3: Lectures on Economic Inequality

Debraj Ray, University of Warwick, Summer 2020

Slides 1: Introduction

Slides 2: Personal Inequality: Occupational Choice

Slides 3: Functional Inequality: The Falling Labor Share

Slides 4: Inequality and Conflict

Slides 4s: Supplement: More on Cross-Group Violence

Postscript [if there is time]: Publishing without Perishing

From Personal to Functional

We now downplay personal endowments and accumulation

Though still very much in the background

Our focus: the functional distribution across capital and labor

Death of a Kaldor Fact

The falling labor share

Harrison (2002), Rodrıguez and Jayadev (2010), Karabarbounis and Neiman (2014)

Explanations

China Shock Autor, Dorn and Hansen (2016)

globalization + cheap labor

but happening everywhere (e.g., jobless growth in India)

Concentration Autor et al (2017), Azar and Vives (2019)

increasing product di�erentiation, gig economy, decaying unions

explanation or outcome?

Technical progress Acemoglu (1998, 2002), Acemoglu and Restrepo (2019)

robotics (hardware), machine learning (so�ware)

Explanations

These are add-ons to the standard growth model:

Ramsey-Solow-Uzawa-Lucas . . .

Balanced growth, steady states

But . . . have we got the standard model right?

Is balanced growth the “natural” baseline?

Re-Thinking Balanced Growth

Solow

Concavity of f ⇒ steady state

Balanced growth at the rate of technical progress

Endogenous growth theory Uzawa, Lucas, Romer

Still hard to let balanced growth and dynamic steady states go . . .

k(t) = sky(t) + (n + δ)k(t)

h(t) = shy(t) + (n + δ)h(t)

Mankiw-Romer-Weil 1992

Five Pillars

Five Pillars for a Theory

Human-physical asymmetry

Machines and robots

Preference neutrality

Economic growth

Singularity

I. The Human-Physical Asymmetry

k(t) = sky(t) + (n + δ)k(t)

h(t)= shy(t) + (n + δ)h(t)

What does the second equation mean?

Physical capital can be indefinitely replicated:

And so can individual claims to them.

But human capital cannot be replicated in the same way.

That equation is really there to generate balanced growth.

We drop it.

II. Machines and Robots

Capital: both complements (machines) and substitutes (robots).

Many sectors: final goods + machines + robots + education:

yj = f j(k j, `j) [CRS]

k: machines, complementary to labor

`: human or robot labor:

`j = hj + νjrj

νj: a replacement threshold: sector specific, but never 0.

Automation if the price is right:

in principle, the technology already exists.

Already There?

Automation if the price is right:

“nothing humans do as a job is uniquely safe anymore. From hamburgers to

healthcare, machines can be created to successfully perform such tasks with no

need or less need for humans, and at lower costs than humans. . . ” Scott Santens, The

Boston Globe, 2016

III. Preferences and Neutrality

People have (possibly di�erent) utility functions and discount factors.

Start with financial wealth + wage income in sector j

consume

get educated [evolution of human capital]

invest [evolution of physical capital]

End with new wealth, new sector, restart.

Neutrality

No bias in consumption preferences:

towards or against sectors that are robot-friendly

Price System

Competitive Pricing

numeraire: rental rate on machine capital

(p, pr, pk, pe): prices

(w, wr, wk, we): wages

Price System

Some Properties of Prices

price = unit cost, or zero profits:

pj = cj(1, λj), where λj ≡ min{wj, ν−1j pr} is e�ective price of labor.

profit-maximization:

e.g., λj = pj∂ f j(kj ,`j)

∂`j.

whether to automate:

e.g., if wj > pr/νj, then j is fully automated.

utility maximization:

consumption-savings choices are optimal at these prices.

IV. Growth

Summary of Ingredients

Production sectors

Physical capital can replicate, human capital expands across sectors Pillar I

machines and robots Pillar II

Preferences

neutral across human- and robot-friendly sectors Pillar III

Price system

standard competitive

Assume: Per-capita growth occurs in equilibrium Pillar IV

Need conditions on preferences relative to technology

An Example

Three Sectors

one final good:

y1 = k1/21 `1/2

1 c1(1, λ) =√

λ

capital:

yk = k1/2k `1/2

k ck(1, λ) =√

λ

robots:

yr =

[12

k−1r +

12`−1

r

]−1cr(1, λ) =

12

[1 +√

λ]2

free mobility of labor: no education sector.

common value of robot productivity ν.

An Example

robot price = unit cost:

pr(t) = cr(1, λ(t)) =12

[1 +

√λ(t)

]2.

Case 1: No automation if ν ≤ 1/2

Suppose not, then

pr(t) =12

[1 +

√ν−1 pr(t)

]2> pr(t), Contradiction

If economy grows forever, labor share constant at 50% in the long run.

An Example

robot price = unit cost:

pr(t) = cr(1, λ(t)) =12

[1 +

√λ(t)

]2.

Case 2: Long-run automation for all growing sectors if ν > 1/2.

Suppose not, then for large t:

pr(t) =12

[1 +

√w(t)

]2=

12+√

w(t) +w(t)

2< νw(t) Contradiction

because w(t)→ ∞ with growth.

If economy grows forever, labor income share falls to zero.

Four Lessons and One Puzzle

1. Non-substitution theorem

robot-sector automation pins down robot prices completely:

p∗r =12

[1 +

√ν−1 p∗r

]2,


2. Full long-run automation in growing sector once robot prices are pinned

for if false, human wages must be unbounded in growing sector.

⇒ ultimately robots must be cheaper.


3. Usually, partial economy-wide automation at any finite date

otherwise, wages would crash and humans would undercut robots.

(uses free mobility)


4. Labor income share declines to zero once robot prices are pinned

the decline is gradual but

by (eventual) full automation, inexorable.

Note: statement about relative, not absolute income.


Puzzle

What, precisely, separated Case 2 from Case 1?

V. Singularity

Stan Ulam quotes von Neumann:

“The accelerating progress of technology, and changes in the mode of human life,

give the appearance of approaching some essential singularity in the history of the

race beyond which human a�airs, as we know them, can not continue.”

V. Singularity

pr450

-1pr = cr (1, !r pr)

V. Singularity

pr450

-1pr = cr (1, !r pr)

cr (1, !r pr)-1

V. Singularity

pr

cr (1, !r pr)

Singularity guarantees this cut.

450

pr*

-1

-1pr = cr (1, !r pr)

V. Singularity

pr450

cr (1, !r pr)-1

-1pr = cr (1, !r pr)

No singularity

V. Singularity

pr

cr (1, !r pr)

Singularity guarantees this cut.

450

pr*

-1

-1pr = cr (1, !r pr)

Formal Condition νr > limρ→0

cr(ρ, 1)

Always holds for Cobb-Douglas production

Or all CES production with elasticity of substitution no less than 1.

Our example: yr =[

12 k−1

r + 12 `−1r

]−1

CES: condition holds when ν > 1/2, fails when ν ≤ 1/2.

Automation and the Declining Labor Share

Theorem 1assume (a) long run growth, and (b) the singularity condition; then:

(i) every growing sector must become automated a�er some finite date;

(ii) though sectoral co-existence of robots and humans is possible, each growing

sector must become asymptotically fully automated in the long-run limit;

(iii) the share of human labor in national income must converge to zero.

Two Remarks

1. Machine capital scales, human capital does not.

2. Singularity in robot sector infects every other growing sector.

The Unbounded Accumulation of Human Capital

Give Humans a Chance!

Infinite number of final sectors

agents can move from sector to sector, accumulating human capital indefinitely

All utility functions are asymptotically homothetic.

(other technical restrictions on demand: continuity, positivity)

The Unbounded Accumulation of Human Capital

Theorem 2assume (a) long run growth, (b) the singularity condition, and (c)

asymptotically homothetic preferences; then:

(i) every sector, except possibly the machine capital and education sectors must

become automated a�er some finite date;

(ii) each such sector is asymptotically fully automated in the long run;

(iii) the share of human labor in national income must converge to zero.

A relative, not absolute crisis: If education costs are bounded and there is a

sequence of sectors such that νi → 0, then every human wage goes to infinity.

Link to Piketty

Escape Hatches

Four escape routes

No growth:

With a positive measure of highly patient agents, this cannot happen.

No singularity:

This is an empirical question. It could happen.

No homotheticity:

Again, an empirical question.

But homotheticity will need to fail in a particular way.

Technical progress:

next topic

Directed Technical Progress

Directed technical progress to the rescue?

That depends on what you choose to assume.

Extensive Margin:

Acemoglu and Restrepo (2018): new goods are un-automatable

N(t)N(t)-1 I(t)i

automatedcannot beautomated




Intensive Margin: write `i = [νiri] + [θihi]

i

Set of GoodsAutomated Non-Automated

!-1wii "-1pri




Intensive Margin: write `i = [νiri] + [µihi]

i


!-1pri"-1wii


Production Function

yj = f j(θjtk jt, µjthjt + νjtrjt)

Productivities θjt, µjt, νjt all a�ected by R&D

R&D

Inventor can advance productivity at rate ρ for a factor-sector pair:

Gets temporary patent protection, which she licenses to an active firm.

A�er one period the advance goes public.

Spillover fraction γ > 0 (public) for this factor in other sectors.

Cost κ(ρ) increasing, convex, prohibitive at ρ, same for every factor and sector.


Theorem 3 (The Extended Dismal Scenario)Assume the singularity condition, and mild technical restrictions (see paper).

Then in any equilibrium with capital growth, the income share of human labor

must converge to zero as t→ ∞.

i


!-1wii "-1pri





i


!-1pri"-1wii





i

wi


!-1pri"-1i

Summary

The Falling Labor Share

A natural consequence of any theory that rests on five pillars:

An asymmetry in physical and human capital accumulation

A recognition that physical capital can be machine-like or robot-like.

Preference neutrality with respect to human-friendly or robot-friendly goods.

Acceptance that economic growth will continue in the long run.

Singularity: The production of automata by means of automata.

Under these conditions, labor income share→ 0:

full automation in the long run . . .

. . . despite wages rising over time (slow automation).

Summary

An age-old anxiety: that “capital” will inherit the earth.

But the underlying worry is about the personal distribution of income.

that will depend on how much people save, and in what form they save.

Financial education is fundamentally important.

i’m pessimistic about the prospects of intelligent, informed savings in equity

but probably this is the only way to avoid a long-run crisis

Social Alternatives

universal basic income (e.g. Ideas for India special issue, Economic Survey)

social stock portfolios (e.g., Ghosh and Ray 2019 on the India Fund)

ec9aa term 3: lectures on economic inequality

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