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Academic wages, Singularities, Phase Transitions and Pyramid Schemes

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff

University of Toronto www.math.toronto.edu/mccann click on ‘Talk’ International Congress of Mathematicians at Seoul

15 August 2014

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 1 / 26

Outline

1

Mathematical challenges in economic theory Steady-state matching coupling the education and labor markets

2

A mathematical model A variational approach to competitive equilibria

3

Results Existence of equilibrium wages and matchings Specialization, uniqueness, and structural properties Description of singularities

4

Conclusions

5

References

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 2 / 26

Background, challenge, universality

• despite some celebrated successes, economic theory presents a largely

untapped source of interesting mathematical problems

• e.g. in a heterogeneous population of N collaborator/competitors, is

lim

N→∞

top wage average wage < +∞? i.e. lim

firm size→∞

CEO salary average salary = +∞?

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 3 / 26

Background, challenge, universality

• despite some celebrated successes, economic theory presents a largely

untapped source of interesting mathematical problems

• e.g. in a heterogeneous population of N collaborator/competitors, is

lim

N→∞

top wage average wage < +∞? i.e. lim

firm size→∞

CEO salary average salary = +∞? i.e. does (total economy) ∈ L1 imply (individual payoffs) ∈ L∞ ?

• some flavor of questions in statistical physics;
• do parallels exist that can be developed?

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 3 / 26

Matching in the education and labor markets

EDUCATION MARKET

• different students willing to pay teachers to enhance their skills
• different teachers seek students to pay their salaries

LABOR MARKET

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 4 / 26

Matching in the education and labor markets

EDUCATION MARKET

• different students willing to pay teachers to enhance their skills
• different teachers seek students to pay their salaries

LABOR MARKET

• adults choose a profession (worker, manager, teacher) based on earnings

potential given their skills (innate or acquired)

• workers seek managers to produce output (commensurate with skills)
• managers seek workers...
• fruits of output divided competitively (according to what each will bear)
• teachers seek students to educate (depending on the skills of each...)

Interrelation between these markets has unexpected potential for feedback!

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 4 / 26

Steady-state competitive equilibrium

PROFIT MOTIVE: individuals driven to maximize share of wealth (generated by labor production bL plus external value bE of education) LARGE MARKET HYPOTHESIS: no individual or small group has market power (i.e. can affect outcomes for a positive fraction of population) EQUILIBRIA are STABLE: no individual or small group should prefer to abandon their partners in favor of collaboration with each other STEADY-STATE: educational matching should reproduce the same endogenous distribution of adult skills α at each generation, given an exogenously specified distribution κ of student skills at each generation

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 5 / 26

A mathematical model

Student skills: k ∈ K = [0, ¯ k[ distributed according to dκ ≥ 0 on ¯ K ⊂ R Adult skill level a ∈ ¯ A has value cbE(a) outside the labor market, where 0 < bE ∈ C 1(¯ A) is strongly convex increasing, c ≥ 0, and w.l.o.g. A = K EDUCATION MARKET: parameterized by 0 < θ < 1 ≤ N and bE(·)

• a teacher can teach N students, each inheriting a fraction θ of their skill

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 6 / 26

A mathematical model

Student skills: k ∈ K = [0, ¯ k[ distributed according to dκ ≥ 0 on ¯ K ⊂ R Adult skill level a ∈ ¯ A has value cbE(a) outside the labor market, where 0 < bE ∈ C 1(¯ A) is strongly convex increasing, c ≥ 0, and w.l.o.g. A = K EDUCATION MARKET: parameterized by 0 < θ < 1 ≤ N and bE(·)

• a teacher can teach N students, each inheriting a fraction θ of their skill

i.e., if k ∈ K studies with a ∈ A they acquire skill zθ(k, a) = (1 − θ)k + θa. LABOR MARKET: parameterized by 0 < θ′ < 1 ≤ N′ and bL(·) like bE(·)

• worker a ∈ A and manager a′ ∈ A produce output bL((1 − θ′)a + θ′a′)
• each manager can manage up to N′ workers

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 6 / 26

Payoffs and matchings

Recall: a map z : Rm − → Rn pushes a measure µ ≥ 0 on Rm forward to a measure z#µ on Rn assigning mass µ[z−1(V )] to each V ⊂ Rn (all Borel) Seek real functions u, v on K = A and measures ǫ, λ ≥ 0 on ¯ K × ¯ A where u(k) = lifetime net income of student of skill k (minus tuition invested) v(a) = salary (i.e. wage) of an adult of skill a

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 7 / 26

Payoffs and matchings

Recall: a map z : Rm − → Rn pushes a measure µ ≥ 0 on Rm forward to a measure z#µ on Rn assigning mass µ[z−1(V )] to each V ⊂ Rn (all Borel) Seek real functions u, v on K = A and measures ǫ, λ ≥ 0 on ¯ K × ¯ A where u(k) = lifetime net income of student of skill k (minus tuition invested) v(a) = salary (i.e. wage) of an adult of skill a dǫ(k, a) = fraction of skill k students who study with skill a teachers dλ(a, a′) = number of skill a workers who match with skill a′ managers whose marginals ǫi = πi

#ǫ under π1(k, a) = k and π2(k, a) = a

and push-forward zθ

#ǫ through zθ(k, a) := (1 − θ)k + θa satisfy...

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 7 / 26

MNEMONIC TABLE

Generation Skill range Skill distribution Distribution type Kids K = [0, ¯ k[ dκ(k) ≥ 0 exogenous Adults A = K dα(a) ≥ 0 endogenous: α = zθ

#ǫ

zθ(k, a) := (1 − θ)k + θa Direct Indirect Sector Exogenous Endogenous (exogenous) (endogenous) parameters matching payoff payoff Education (N, θ) dǫ(k, a) ≥ 0 cbE(z) u(k) Labor (N′, θ′) dλ(a, a′) ≥ 0 bL(z) v(a) MOTIVATING EXAMPLE: N = N′, θ = 1

2 = θ′ and bL(a) = ea = bE(a),

c ≥ 0, with c = 0 being a case of primary interest

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 8 / 26

Competitive equilbrium

STEADY-STATE ǫ1 = κ and (1a) λ1 + 1

N′ λ2 + 1 N ǫ2 = zθ #ǫ,

(1b) i.e. worker + manager + teacher skills = output of educational match STABLE u(k) + 1

N v(a) ≥ cbE(zθ(k, a)) + v(zθ(k, a))

and (2a)

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 9 / 26

Competitive equilbrium

STEADY-STATE ǫ1 = κ and (1a) λ1 + 1

N′ λ2 + 1 N ǫ2 = zθ #ǫ,

(1b) i.e. worker + manager + teacher skills = output of educational match STABLE u(k) + 1

N v(a) ≥ cbE(zθ(k, a)) + v(zθ(k, a))

and (2a) v(a) + 1

N′ v(a′) ≥ bL((1 − θ′)a + θ′a′)

• n ¯

K × ¯ A, (2b) BUDGET FEASIBLE equality holds ǫ-a.e. in (2a) and λ-a.e. in (2b) (3)

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 9 / 26

A variational approach...

But how can we find and analyze such equilibria? Recall a simpler matching problem: the STABLE MARRIAGE PROBLEM

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 10 / 26

A variational approach...

But how can we find and analyze such equilibria? Recall a simpler matching problem: the STABLE MARRIAGE PROBLEM Assume a marriage of man k to woman a generates surplus s(k, a), to be divided between them as they see fit. Given probability measures dκ(k) and dα(a) representing the frequency of different types of men and women in a given population, can we pair each man to a woman STABLY, meaning that, when the pairing is done, no man and woman would both prefer to leave their assigned partners and marry each other? e.g. M men and M women: κ = 1 M

M

• i=1

δki and α = 1 M

M

• j=1

δaj, payoff matrix (sij) = s(ki, aj)

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 10 / 26

Shapley and Shubik’s (1972) solution:

The solutions are precisely those pairings dǫ(a, k) of men to women which attain the maximum max

{ǫ≥0|ǫ1=κ, ǫ2=α}

• ¯

Kׯ A

s(k, a)dǫ(k, a). This is a LINEAR PROGRAM; in fact an optimal transportation problem.

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 11 / 26

Shapley and Shubik’s (1972) solution:

The solutions are precisely those pairings dǫ(a, k) of men to women which attain the maximum max

{ǫ≥0|ǫ1=κ, ǫ2=α}

• ¯

Kׯ A

s(k, a)dǫ(k, a). This is a LINEAR PROGRAM; in fact an optimal transportation problem. The solution (u, v) ∈ C(¯ A)2 to its DUAL PROGRAM, inf

u(k)+v(a)≥s(k,a)

• ¯

K

u(k)dκ(k) +

• ¯

A

v(a)dα(a) shows how the surplus s(k, a) will be split between the husband and wife in each couple at equilibrium, provided the infimum is attained; it satisfies u(k) + v(a) ≥ s(k, a) on ¯ K × ¯ A, with equality ǫ-a.e.

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 11 / 26

The analogous linear programs for our steady-state match

PLANNER’S PROBLEM: a maximization over steady-state matches (ǫ, λ) LP∗ := max

{ǫ,λ≥0|(1a)−(1b)} c

• ¯

Kׯ A

bE ◦ zθdǫ +

• ¯

Aׯ A

bL ◦ zθ′dλ (recall zθ = (1 − θ)k + θa)

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 12 / 26

The analogous linear programs for our steady-state match

PLANNER’S PROBLEM: a maximization over steady-state matches (ǫ, λ) LP∗ := max

{ǫ,λ≥0|(1a)−(1b)} c

• ¯

Kׯ A

bE ◦ zθdǫ +

• ¯

Aׯ A

bL ◦ zθ′dλ (recall zθ = (1 − θ)k + θa) DUAL LINEAR PROGRAM: a minimization over stable payoffs (u, v) LP∗ := inf

{u,v∈F|(2a)−(2b)}

• ¯

K

u(k)dκ(k) F = {u0 + u1 = u ∈ L1(dκ) | u0 ∈ C( ¯ K) and u1 > 0 non-decreasing}

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 12 / 26

Proof of duality (LP∗ = LP∗): ≥ ‘easy’; ≤ standard

Rockafellar-Fenchel duality in (C(K), · ∞)2 implies LP∗ ≤ LP∗.

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 13 / 26

Proof of duality (LP∗ = LP∗): ≥ ‘easy’; ≤ standard

Rockafellar-Fenchel duality in (C(K), · ∞)2 implies LP∗ ≤ LP∗. The reverse inequality is formally clear: integrating educational stability u(k) − cbE(zθ(k, a)) ≥ v(zθ(k, a)) − 1 N v(a) (2a) against dǫ(k, a) yields

• ¯

K

udκ − c

• ¯

A

bEd(zθ

#ǫ)

≥

• ¯

A

vd(zθ

#ǫ) − 1

N

• ¯

A

vdǫ2

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 13 / 26

Proof of duality (LP∗ = LP∗): ≥ ‘easy’; ≤ standard

Rockafellar-Fenchel duality in (C(K), · ∞)2 implies LP∗ ≤ LP∗. The reverse inequality is formally clear: integrating educational stability u(k) − cbE(zθ(k, a)) ≥ v(zθ(k, a)) − 1 N v(a) (2a) against dǫ(k, a) yields

• ¯

K

udκ − c

• ¯

A

bEd(zθ

#ǫ)

≥

• ¯

A

vd(zθ

#ǫ) − 1

N

• ¯

A

vdǫ2 =

• ¯

A

vd(λ1 + 1 N′ λ2)

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 13 / 26

Proof of duality (LP∗ = LP∗): ≥ ‘easy’; ≤ standard

Rockafellar-Fenchel duality in (C(K), · ∞)2 implies LP∗ ≤ LP∗. The reverse inequality is formally clear: integrating educational stability u(k) − cbE(zθ(k, a)) ≥ v(zθ(k, a)) − 1 N v(a) (2a) against dǫ(k, a) yields

• ¯

K

udκ − c

• ¯

A

bEd(zθ

#ǫ)

≥

• ¯

A

vd(zθ

#ǫ) − 1

N

• ¯

A

vdǫ2 =

• ¯

A

vd(λ1 + 1 N′ λ2) ≥

• ¯

Aׯ A

bL((1 − θ′)a + θ′a′)dλ(a, a′) for all (ǫ, λ) & u, v ∈ F ⊂ L1(dκ) satisfying (1)-(2) provided the integrals converge. Strict inequalities would violate the budget constraint.

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 13 / 26

Numerics: Equilibrium wage v(a) as a function of adult skill a ∈ [0, ¯ a[

0.5 1 1.5 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

κ = L1 uniform, c = 0, bL(a) = ea, and (N, θ) = (N′, θ′) = (10, 1

2),

Note segregation: workers=yellow, managers=brown, and teachers=beige

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 14 / 26

Doubling condition

To guarantee this convergence, we henceforth assume a doubling condition

• n the student skill distribution at its upper endpoint: for some C < ∞

and all D > 0:

• [¯

k−2D,¯ k]

dκ ≤ C

• [¯

k−D,¯ k]

dκ. (D.C.)

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 15 / 26

Doubling condition

To guarantee this convergence, we henceforth assume a doubling condition

• n the student skill distribution at its upper endpoint: for some C < ∞

and all D > 0:

• [¯

k−2D,¯ k]

dκ ≤ C

• [¯

k−D,¯ k]

dκ. (D.C.)

Proposition (Variational characterization of competitive equilibria)

(D.C.) implies LP∗ = LP∗. If LP∗ is attained, then for (u, v) and (ǫ, λ) to

• ptimize LP∗ and LP∗ is equivalent to forming a competitive equilibrium.

Thus it is crucial to know the infimum is attained — if not in C(¯ A)2 — then at least in the larger class u, v ∈ F. Moreover, we want to analyze the optimal (ǫ, λ) and (u, v).

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 15 / 26

Theorem (Existence of equilibrium wages)

Fix c ≥ 0 and positive constants ¯ k = ¯ a and max{θ, θ′} < 1 < min{N, N′}. If bE/L ∈ C 1(¯ A) are positive, uniformly convex and increasing and κ satisfies (D.C.) on K = [0, ¯ k[, then LP∗ is attained by convex non-decreasing functions u, v ∈ F, uniformly convex and increasing if either c > 0 or Nθ2 ≥ 1.

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 16 / 26

Theorem (Existence of equilibrium wages)

Fix c ≥ 0 and positive constants ¯ k = ¯ a and max{θ, θ′} < 1 < min{N, N′}. If bE/L ∈ C 1(¯ A) are positive, uniformly convex and increasing and κ satisfies (D.C.) on K = [0, ¯ k[, then LP∗ is attained by convex non-decreasing functions u, v ∈ F, uniformly convex and increasing if either c > 0 or Nθ2 ≥ 1. Moreover, v = max{vw, vm, vt} and u(k) = sup

a∈¯ A

cbE(zθ(k, a)) + v(zθ(k, a)) − 1 N v(a) where the worker / manager / teacher wages for an adult of skill a ∈ K are vw(a) := sup

a′∈¯ A

bL((1 − θ′)a + θ′a′) − 1 N′ v(a′) vm(a′) := N′ sup

a∈¯ A

bL(zθ′(a, a′)) − vw(a) vt(a) := N sup

k∈ ¯ K

cbE(zθ(k, a)) + v(zθ(k, a)) − u(k).

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 16 / 26

Idea of proof:

1) the conclusions becomes true if we restrict the infimum by replacing F with the compact set F0 = {v ∈ F | v convex, non-decreasing} 2) we then need to check that these artificially imposed constraints do not bind for the minimizing pair u, v ∈ F0 under this restriction; 3) this requires positive lower bounds for first two derivatives of vw/m/t 4) for c > 0 this follows from analogous bounds for bE/L & convexity of v

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 17 / 26

Idea of proof:

1) the conclusions becomes true if we restrict the infimum by replacing F with the compact set F0 = {v ∈ F | v convex, non-decreasing} 2) we then need to check that these artificially imposed constraints do not bind for the minimizing pair u, v ∈ F0 under this restriction; 3) this requires positive lower bounds for first two derivatives of vw/m/t 4) for c > 0 this follows from analogous bounds for bE/L & convexity of v A few technical issues: 5) get v = max{vw, vm, vt} for a.e. adult, but need it L1-a.e. in K 7) need to perturb the problem to ensure L1 < < κ and L1 < < α = zθ

#ǫ

8) finally, let this perturbation (and c > 0 if desired) tend to zero, using convexity and monotonicity of (uk, vk) to extract a convex monotone limit 9) more work shows uniform convexity/monotonicity survives if Nθ2 ≥ 1

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 17 / 26

Let b′

E/L

:= b′

E/L(0)

¯ b′

E/L

:= b′

E/L(¯

a).

Lemma (Specialization by type; the educational pyramid)

In any equilibrium: a) N′θ′ ≥ ¯ b′

L/b′ L =

⇒ least manager skill ≥ highest worker skill

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 18 / 26

Let b′

E/L

:= b′

E/L(0)

¯ b′

E/L

:= b′

E/L(¯

a).

Lemma (Specialization by type; the educational pyramid)

In any equilibrium: a) N′θ′ ≥ ¯ b′

L/b′ L =

⇒ least manager skill ≥ highest worker skill b) Nθ ≥ ¯ b′

L/b′ L =

⇒ highest worker and/or manager skill < ¯ a

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 18 / 26

Let b′

E/L

:= b′

E/L(0)

¯ b′

E/L

:= b′

E/L(¯

a).

Lemma (Specialization by type; the educational pyramid)

In any equilibrium: a) N′θ′ ≥ ¯ b′

L/b′ L =

⇒ least manager skill ≥ highest worker skill b) Nθ ≥ ¯ b′

L/b′ L =

⇒ highest worker and/or manager skill < ¯ a c) if Nθ > 1 education strictly improves everyone’s skill and d) in this case the academic descendents of a skill a ∈ A teacher display at most finitely many skill types unless differentiability of v fails at a. i.e. finitely many academic descendents, yet INFINITELY many ancestors

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 18 / 26

Corollary (Uniqueness; positive assortativity)

a) If (ǫ, λ) maximize the planner’s problem, spt λ ⊂ R2 is non-decreasing; i.e. managerial skill varies directly with worker skill. b) Moreover, there exist maximizers (ǫ, λ) with spt ǫ non-decreasing also, i.e. with teacher skill varying directly with student skill.

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Corollary (Uniqueness; positive assortativity)

a) If (ǫ, λ) maximize the planner’s problem, spt λ ⊂ R2 is non-decreasing; i.e. managerial skill varies directly with worker skill. b) Moreover, there exist maximizers (ǫ, λ) with spt ǫ non-decreasing also, i.e. with teacher skill varying directly with student skill. c) If the minimizing payoffs (u, v) are strictly convex, all maximizing matches have this monotonicity. d) If also κ is free from atoms, the equilibrium match (ǫ, λ) is unique.

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 19 / 26

Corollary (Uniqueness; positive assortativity)

a) If (ǫ, λ) maximize the planner’s problem, spt λ ⊂ R2 is non-decreasing; i.e. managerial skill varies directly with worker skill. b) Moreover, there exist maximizers (ǫ, λ) with spt ǫ non-decreasing also, i.e. with teacher skill varying directly with student skill. c) If the minimizing payoffs (u, v) are strictly convex, all maximizing matches have this monotonicity. d) If also κ is free from atoms, the equilibrium match (ǫ, λ) is unique. e) If also Nθ > 1, then u′(k) and v′(a) are uniquely determined for κ-a.e. student k and α = zθ

#ǫ-a.e. adult a ∈ K

f ) If also κ dominates some a.c. measure whose support fills ¯ K, then u is unique (among locally Lipschitz functions on K = [0, ¯ k[).

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 19 / 26

Theorem (Transition to unbounded wage gradients)

If κ given by an L∞ probability density, continuous and positive at ¯ k, and i) all sufficiently skilled adults become teachers (as when Nθ ≥ ¯ b′

L/b′ L)

ii) spt ǫ ⊂ Graph(at) for at : ¯ K − → ¯ A non-decreasing (as when Nθ2 > 1) iii) the student-to-teacher skill map a = at(k) is differentiable at ¯ k iv) v(a) is differentiable on ]¯ a − δ, ¯ a[ for some δ > 0, then for Nθ = 1

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 20 / 26

Theorem (Transition to unbounded wage gradients)

If κ given by an L∞ probability density, continuous and positive at ¯ k, and i) all sufficiently skilled adults become teachers (as when Nθ ≥ ¯ b′

L/b′ L)

ii) spt ǫ ⊂ Graph(at) for at : ¯ K − → ¯ A non-decreasing (as when Nθ2 > 1) iii) the student-to-teacher skill map a = at(k) is differentiable at ¯ k iv) v(a) is differentiable on ]¯ a − δ, ¯ a[ for some δ > 0, then for Nθ = 1 dv da (¯ a − ∆a) = const |∆a|

log Nθ log N

+ c¯ b′

E 1 Nθ − 1 + o(1)

as ∆a ↓ 0. Note divergence exponent independent of model details such as bE/L(·)

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 20 / 26

Theorem (Transition to unbounded wage gradients)

If κ given by an L∞ probability density, continuous and positive at ¯ k, and i) all sufficiently skilled adults become teachers (as when Nθ ≥ ¯ b′

L/b′ L)

ii) spt ǫ ⊂ Graph(at) for at : ¯ K − → ¯ A non-decreasing (as when Nθ2 > 1) iii) the student-to-teacher skill map a = at(k) is differentiable at ¯ k iv) v(a) is differentiable on ]¯ a − δ, ¯ a[ for some δ > 0, then for Nθ = 1 dv da (¯ a − ∆a) = const |∆a|

log Nθ log N

+ c¯ b′

E 1 Nθ − 1 + o(1)

as ∆a ↓ 0. Note divergence exponent independent of model details such as bE/L(·)

Corollary

If Nθ > 1 then (i)&(ii) = ⇒ a singularity must occur in u or in v near ¯ a. If also (iii)-(iv) hold, then lim

a→¯ a v(a) < +∞

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 20 / 26

Idea of proof (theorem and corollary):

A student-teacher match produces equality in the stability constraint u(k) + 1

N v(a) − [cbE + v]((1 − θ)k + θa) ≥ 0

(2a) Assuming differentiability, the first-order conditions for equality u′(k) 1 − θ = [cb′

E + v′](1−θ)k+θa =

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 21 / 26

Idea of proof (theorem and corollary):

A student-teacher match produces equality in the stability constraint u(k) + 1

N v(a) − [cbE + v]((1 − θ)k + θa) ≥ 0

(2a) Assuming differentiability, the first-order conditions for equality u′(k) 1 − θ = [cb′

E + v′](1−θ)k+θa = v′(a)

Nθ = ⇒ a = at(k) = 1 θ[cb′

E + v′]−1

u′(k) 1 − θ

• + (1 − θ−1)k

and

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 21 / 26

Idea of proof (theorem and corollary):

A student-teacher match produces equality in the stability constraint u(k) + 1

N v(a) − [cbE + v]((1 − θ)k + θa) ≥ 0

(2a) Assuming differentiability, the first-order conditions for equality u′(k) 1 − θ = [cb′

E + v′](1−θ)k+θa = v′(a)

Nθ = ⇒ a = at(k) = 1 θ[cb′

E + v′]−1

u′(k) 1 − θ

• + (1 − θ−1)k

and v′(a) = Nθ[cb′

E + v′](1−θ)k+θa

This last formula shows Nθ acts as a multiplier relating the marginal wage v′(z) of an adult with skill z = (1 − θ)k + θa to the marginal wage v′(a) of his or her teacher. If Nθ > 1 and we know to first-order how a and hence z relate to k, we can compute the rate at which v′(a) diverges as a → ¯ a. QED

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 21 / 26

Returing to point (9) of our earlier proof: uniform convexity of vc for Nθ2 ≥ 1 survives c ↓ 0 is derived from the analogous second-order condition for a minimum of v(a) + 1

N′ v(a′) − bL((1 − θ′)a + θ′a′) ≥ 0 :

(2b) v′′

c (a) ≥

     (1 − θ′)2b′′

L|(1−θ′)a+θ′a′

≥ δ if a works N′(θ′)2b′′

L|(1−θ′)a′+θ′a

≥ δ if a manages Nθ2[cb′′

E + v′′ c ](1−θ)k+θa

≥ 0 if a teaches.

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 22 / 26

Returing to point (9) of our earlier proof: uniform convexity of vc for Nθ2 ≥ 1 survives c ↓ 0 is derived from the analogous second-order condition for a minimum of v(a) + 1

N′ v(a′) − bL((1 − θ′)a + θ′a′) ≥ 0 :

(2b) v′′

c (a) ≥

     (1 − θ′)2b′′

L|(1−θ′)a+θ′a′

≥ δ if a works N′(θ′)2b′′

L|(1−θ′)a′+θ′a

≥ δ if a manages Nθ2[cb′′

E + v′′ c ](1−θ)k+θa

≥ 0 if a teaches. Thus v′′

c := inf a∈A v′′ c (a) ≥

δ independent of c > 0 or Nθ2(cb′′

E + v′′ c)

Since v′′

c ≥ 0, we get a c > 0 independent bound ¯

v′′

c > δ > 0 if Nθ2 ≥ 1.

QED

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 22 / 26

Conclusions and open questions:

1) Hidden recursion in education market can generate wage singularities with universal exponents; 2) but only if a teacher’s impact Nθ ≥ 1 does not decrease from one generation of students to the next. 3) Such singularities lead to subtle questions, some remaining open. 4) In this model, they occur at the level of gradients rather than wages. 5) Do singularities persist with discounting? (ie. interest on student loans) 6) What about models with a countable number of management layers? 7) Does competition allow a tiny fraction of the population to extract a positive fraction of the total wealth? 8) Can one analyze the limiting behaviour of finite population models? 9) Parallels to statistical physics? 10) Economic theory remains largely ripe for mathematization...

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 23 / 26

References: www.math.toronto.edu/mccann/publications

R.J. McCann. Academic wages, singularities, phase transitions and pyramid schemes

• transport. Proceedings of the 2014 ICM at Seoul.

Alice Erlinger, R.J. McCann, X. Shi, A. Siow and R. Wolthoff. Academic wages and pyramid schemes. A mathematical model. Submitted. R.J. McCann, Xianwen Shi, Aloysius Siow and Ronald Wolthoff. Becker meets Ricardo: Multisector matching with communication and cognitive skills. Submitted. R.J. McCann and N. Guillen. Five lectures on Optimal Transport... In Analysis and Geometry of Metric Measure Spaces, G. Dafni et al,

• eds. Providence: Amer. Math. Soc. (2013) 145-180.
• Y. Brenier.

D´ ecomposition polaire et r´ earrangement monotone des champs de

• vecteurs. C.R. Acad. Sci. Paris S´
• er. I Math. 305 (1987) 805–808.

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 24 / 26

Related economics literature

L.S. Shapley and M. Shubik. The assignment game I: The core.

• Internat. J. Game Theory 1 (1972) 111–130.
• S. Rosen. The economics of superstars.
• Amer. Econom. Rev. 71 (1982) 845–858.

G.S. Becker and K. Murphy. The division of labor, coordination costs, and knowledge.

• Quart. J. Econom. 107 (1992) 1137–1160.
• L. Garicano and E. Rossi-Hansberg.

Organization and inequality in a knowledge economy. Quarterly J. Econom. 121 (2006) 1383–1435.

• X. Gabaix and A. Landier.

Why has CEO compensation increased so much?

• Quart. J. Econom. 123 (2008) 49–100.

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 25 / 26

Kamsa hamnida (Thank you)

RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 26 / 26