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 Mathematical challenges in economic theory 1 Steady-state matching coupling the education and labor markets A mathematical model 2 A variational approach to competitive equilibria Results 3 Existence of equilibrium wages and matchings Specialization, uniqueness, and structural properties Description of singularities Conclusions 4 References 5 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 top wage lim average wage < + ∞ ? N →∞ i.e. CEO salary lim average salary = + ∞ ? firm size →∞ 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 top wage lim average wage < + ∞ ? N →∞ i.e. CEO salary lim average salary = + ∞ ? firm size →∞ i.e. does ( total economy ) ∈ L 1 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 b L plus external value b E 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 d κ ≥ 0 on ¯ K ⊂ R Adult skill level a ∈ ¯ A has value cb E ( a ) outside the labor market, where 0 < b E ∈ 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 b E ( · ) • 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 d κ ≥ 0 on ¯ K ⊂ R Adult skill level a ∈ ¯ A has value cb E ( a ) outside the labor market, where 0 < b E ∈ 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 b E ( · ) • 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 z θ ( k , a ) = (1 − θ ) k + θ a . LABOR MARKET: parameterized by 0 < θ ′ < 1 ≤ N ′ and b L ( · ) like b E ( · ) • worker a ∈ A and manager a ′ ∈ A produce output b L ((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 : R m − → R n pushes a measure µ ≥ 0 on R m forward to a measure z # µ on R n assigning mass µ [ z − 1 ( V )] to each V ⊂ R n (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 : R m − → R n pushes a measure µ ≥ 0 on R m forward to a measure z # µ on R n assigning mass µ [ z − 1 ( V )] to each V ⊂ R n (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 d ǫ ( k , a ) = fraction of skill k students who study with skill a teachers d λ ( a , a ′ ) = number of skill a workers who match with skill a ′ managers whose marginals ǫ i = π i # ǫ under π 1 ( k , a ) = k and π 2 ( k , a ) = a and push-forward z θ # ǫ through z θ ( k , a ) := (1 − θ ) k + θ a 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 K = [0 , ¯ Kids k [ d κ ( k ) ≥ 0 exogenous endogenous: α = z θ Adults A = K d α ( a ) ≥ 0 # ǫ z θ ( k , a ) := (1 − θ ) k + θ a Direct Indirect Sector Exogenous Endogenous (exogenous) (endogenous) parameters matching payoff payoff Education ( N , θ ) d ǫ ( k , a ) ≥ 0 cb E ( z ) u ( k ) ( N ′ , θ ′ ) d λ ( a , a ′ ) ≥ 0 Labor b L ( z ) v ( a ) 2 = θ ′ and b L ( a ) = e a = b E ( a ), MOTIVATING EXAMPLE: N = N ′ , θ = 1 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 ǫ 1 = κ (1a) and λ 1 + 1 N ′ λ 2 + 1 N ǫ 2 = z θ # ǫ, (1b) i.e. worker + manager + teacher skills = output of educational match STABLE u ( k ) + 1 N v ( a ) ≥ cb E ( z θ ( k , a )) + v ( z θ ( k , a )) (2a) and 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 ǫ 1 = κ (1a) and λ 1 + 1 N ′ λ 2 + 1 N ǫ 2 = z θ # ǫ, (1b) i.e. worker + manager + teacher skills = output of educational match STABLE u ( k ) + 1 N v ( a ) ≥ cb E ( z θ ( k , a )) + v ( z θ ( k , a )) (2a) and on ¯ K × ¯ v ( a ) + 1 N ′ v ( a ′ ) ≥ b L ((1 − θ ′ ) a + θ ′ a ′ ) 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 d κ ( k ) and d α ( 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: M M κ = 1 δ k i and α = 1 � � δ a j , payoff matrix ( s ij ) = s ( k i , a j ) M M i =1 j =1 RJ McCann + A Erlinger, X Shi, A Siow, and R Wolthoff (University of Toronto) Academic Wages and Pyramid Schemes 15 August 2014 10 / 26
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