[PPT] - Econometrics of Matching Wednesday 1st April 11.00 12.30 Venue: PowerPoint Presentation

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ROYAL ECONOMIC SOCIETY CONFERENCE 2015 SPECIAL SESSION

Econometrics of Matching

Wednesday 1st April 11.00 — 12.30 Venue: University Place Lecture Theatre A

(RES 2015) 1 / 1

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ESTIMATING TRANSFER FRICTIONS IN THE MARRIAGE MARKET

Alfred Galichon Based on joint works with Arnaud Dupuy, Yu-Wei Hsieh, Scott Kominers, Bernard Salani´ e, and Simon Weber. The Econometric Journal special session RES conference, Manchester, April 1, 2015

TRANSFER FRICTIONS IN THE MARRIAGE MARKET MANCHESTER, APRIL 1, 2015 SLIDE 1/ 32

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Section 1 INTRODUCTION

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UTILITY TRANSFERS IN MATCHING MODELS

◮ Transfers of utility (under the form of money or other exchanges):

◮ Are sometimes clearly forbidden (e.g. school choice problems):

Nontransferable Utility (NTU)

◮ Are sometimes clearly allowed (e.g. wages in the market for CEOs):

Transferable Utility (TU).

◮ However, it is sometimes unclear (e.g. the marriage market).

Indeed, there is a tradition to model the marriage market with transfers (Shapley and Shubik; Becker; Choo and Siow) and without transfers (Gale and Shapley; Dagsvik; Hitsch, Hortacsu and Ariely; Menzel).

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MATCHING WITH IMPERFECT TRANSFERS

◮ It makes sense to assume that transfers whithin the couple are possible,

but not efficient in the sense that the utility transfered by i to j maybe more costly to i that it is beneficial to j: Imperfectly Transferable Utility (ITU).

◮ Also the case when matching is in the presence of nonquasilinear

utilities (Kelso and Crawford, Hatfield and Milgrom); of taxes (Jaffe and Kominers); of risk aversion (Legros and Newman, Chade and Eckehout); of investments (Samuelson and Noeldeke).

◮ However, in the case of the marriage matching market, this is an

empirical question

◮ Does the answer matter? I will argue that it does; both for econometric

analysis and for policy implications. For this we will need a framework for ITU matching.

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OUTLINE This talk:

1. Introduction
2. Theoretical framework
3. Empirical framework
4. Application

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Section 2 THE THEORETICAL FRAMEWORK

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MATCHING

◮ Let µij ∈ {0, 1} be a dummy variable that is equal to 1 if man i and

woman j are matched, 0 else.

◮ Hence, µij satisfies

∑

j

µij ≤ 1

∑

i

µij ≤ 1.

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UTILITIES

◮ Assume that if man i and woman j match, then i has utility αij and j

has utility γij, pre-transfer.

◮ Utilities of single individuals are normalized to zero. ◮ Assume that transfers ti←j and tj←i are decided, so that if matched, i

and j enjoy respectively ui = αij + ti←j vj = γij + tj←i.

◮ The link between ti←j (what i receives) and −tj←i (what j gives out) is

subject to a feasibility constraint, which will specify whether matching is TU, NTU, or ITU.

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FEASIBILITY

◮ If µij = 1, then ti←j and tj←i must satisfy a feasibility constraint:

◮ In the TU case ti←j + tj←i ≤ 0 ◮ In the NTU case, max

ti←j, tj←i

≤ 0

◮ More generally, in the ITU case, Ψij

ti←j, tj←i

≤ 0, where Ψij is continuous and nondecreasing in its two variables.

◮ We can rewrite the feasibility constraint as

µij = 1 = ⇒ Ψij

ui − αij, vj − γij

≤ 0.

◮ Of particular interest is the Exponentially Transferable Utility (ETU)

case, when Ψ (a, b) = τ log exp (a/τ) + exp (b/τ) 2

where τ > 0 is a transferability parameter. The ETU model interpolates

between TU and NTU; indeed, τ → 0 is NTU, while τ → +∞ is TU.

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STABILITY

◮ For any pair i and j, we need to rule out the case that i and j form a

blocking pair, i.e. each achieve higher payoffs by rebargaining. This implies ∀i, j Ψij

ui − αij, vj − γij

≥ 0.

◮ In the TU case, this is the well-known stability conditions

ui + vj ≥ αij + γij,

◮ while in the NTU case, this reads max

ui − αij, vj − γij

≥ 0.

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EQUILIBRIUM

◮ We are now ready to define an equilibrium in this market. ◮ An outcome (µ, u, v) is an equilibrium if

◮ (i) µij ∈ {0, 1}, ∑j µij ≤ 1 and ∑i µij ≤ 1 ◮ (ii) Ψij

ui − αij, vj − γij

≥ 0

◮ (iii) µij = 1 implies Ψij

ui − αij, vj − γij

= 0.

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Section 3 THE EMPIRICAL FRAMEWORK

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MATCHING

◮ Assume that there are groups, or clusters of men and women who share

similar observable characteristics, called types. There are nx men of type x, and my women of type y.

◮ Let µxy ≥ 0 be the number of men of type x matched to women of

type y. This quantity satisfies

∑

y

µxy ≤ nx

∑

x

µxy ≤ my

◮ We shall denote µx0 and µ0y the number of single men of type x and

single women of type y.

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UTILITIES

◮ Assumption 1: Assume that if man i of type x and woman j of type y

match, then αij = αxy + εiy γij = γxy + ηjx Utilities of single man i and woman j are respectively εi0 and ηj0. Recall that transfers ti←j and tj←i are decided, so that if matched, i and j enjoy respectively ui = αxy + εiy + ti←j vj = γxy + ηjx + tj←i

◮ Assumption 2: there are a large number of invididuals per group and

the ε and the η’s are i.i.d. Gumbel.

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FEASIBILITY AND STABILITY

◮ Assumption 3: If i is of type x and j is of type y, then

Ψij (a, b) = Ψxy (a, b) (i.e. we assume that Ψij only depends on i and j through their types).

◮ Thus, we can rewrite the feasibility constraint as (for i in x and j in y)

µij = 1 = ⇒ Ψxy

ui − αxy − εiy, vj − γxy + ηjx

≤ 0. and stability ∀i ∈ x, j ∈ y, Ψxy

ui − αxy − εiy, vj − γxy + ηjx

≥ 0.

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EQUILIBRIUM TRANSFERS Theorem 1 (Galichon, Kominers and Weber). Under Assumptions 1, 2 and 3 above, equilibrium transfers ti←j and tj←i only depend on x and y, the

bservable types of i and j. Hence, let us denote these quantitites by tx←y

and ty←x. This theorem extends to the general ITU case a result which was known in the TU case (Choo and Siow, Chiappori, Salani´ e and Weiss, Galichon and Salani´ e).

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MATCHING AND DISCRETE CHOICE Implication of this theorem: the matching problem now embeds two sets of discrete choice problems. Indeed, man i and woman j (of types x and y) solve respectively max

y

αxy + tx←y + εiy, εi0
max

x

γxy + ty←x + ηjx, ηj0
which are standard discrete choice problems; thus the log-odds ratio formula

applies, and ln µxy µx0 = αxy + tx←y ln µxy µ0y = γxy + ty←x But remember that Ψxy (tx←y, ty←x) = 0, thus Ψxy

ln µxy

µx0 − αxy, ln µxy µ0y − γxy

= 0.

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EQUILIBRIUM CHARACTERIZATION: RESULT Theorem 2 (GKW). Equilibrium in the ITU problem with logit heterogeneities is fully characterized by the set of nonlinear equations in µxy, µx0 and µ0y Ψxy

ln µxy

µx0 − αxy, ln µxy µ0y − γxy

= 0

∑

y

µxy + µx0 = nx

∑

x

µxy + µ0y = my Under very mild conditions on Ψ it exists; under mild conditions on Ψ it is also unique. Galichon and Hsieh extend this result to the NTU case with general stochastic utilities.

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EQUILIBRIUM CHARACTERIZATION, BIS Note that first equation defines implicitely µxy as a function of µx0 and µ0y, which can be written as a matching function µxy = Mxy

µx0, µ0y
hence we can restate the previous result as:

Theorem 2’ (GKW). Equilibrium in the ITU problem with logit heterogeneities is fully characterized by the set of nonlinear equations in µx0 and µ0y

∑

y

Mxy

µx0, µ0y

+ µx0 = nx

∑

x

Mxy

µx0, µ0y

+ µ0y = my.

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CONSEQUENCE 1: IDENTIFICATION

◮ In the TU case, Mxy

µx0, µ0y

= √µx0µ0y exp αxy +γxy

2

, thus

µxy = Mxy

µx0, µ0y
identifies

αxy + γxy = ln µ2

xy

√µx0µ0y which is the Choo and Siow identification formula, extended to general stochastic utilities by Galichon and Salani´

e. Note that α and γ are not

individually identified, but α + γ is.

◮ In the NTU case, Mxy

µx0, µ0y

= min

µx0 exp αxy, µ0y exp γxy
,

thus α and γ are partially identified by min

αxy + ln µx0, γxy + ln µ0y

= ln µxy.

◮ In the ETU case, α and γ are partially identified by

µxy =

e−αxy /τµ−1/τ

x0

+ e−γxy /τµ−1/τ

0y

2 −τ .

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CONSEQUENCE 2: COMPUTATION

◮ Computation of

∑

y

Mxy

µx0, µ0y

+ µx0 = nx

∑

x

Mxy

µx0, µ0y

+ µ0y = my. is very easy (and computationally very efficient) by iterative fitting. Convergence is guaranteed, and in practice, very fast.

◮ This is a very convenient alternative to the Kelso-Crawford algorithm to

get approximate equilibrium solutions when the populations are large.

◮ This will allows us to efficiently compute the likelihood for the purpose

f maximum likelihood estimation.

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CONSEQUENCE 3: COMPARATIVE STATICS

◮ Comparative statics are very easy to compute in this model, thanks to

the implicit function theorem.

◮ Of particular interest are the impact of a change in α and γ

(endowments) and of n and m (number of men and women of each types) on U and V ( equilibrium welfare) and µ (equilibrium matching patterns).

◮ Some results from the TU case extend: an increase in the number of

men of type x decreases the welfare of men of type x.

◮ Some results don’t extend from the TU case:

◮ In the TU case, the Becker-Coase theorem predicts that the welfare only

depends on α + γ: thus, if αxy is changed into αxy − cxy and γxy is changed into γxy + cxy, then the welfare and matching patterns are unchanged.

◮ In the ITU case, this is no longer the case. Worse, an increase in γ may

result in a decrease in V (unintended/averse consequence).

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Section 4 APPLICATION: WHAT IS THE COST OF

MARITAL CONCESSIONS?

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THE US MARRIAGE MARKET

◮ Data. We use 2010 ACS data to construct the population vectors. ◮ Age selection. We select couples in which one of the partner is head of

the household. The head must be between 20 and 30 years old. Every single individual in this age interval is selected.

◮ Markets. We have enough data to compute the observed matchings µk

xy

for several markets k, a market being defined as a state in this

application. Thus k ∈ S ={1, ..., 51}.

◮ Types. A person belongs to one of the following type

◮ below high school, high school, college. TRANSFER FRICTIONS IN THE MARRIAGE MARKET MANCHESTER, APRIL 1, 2015 SLIDE 24/ 32

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MARRIAGE GAINS AND CONSERVATIVENESS

◮ The utilities Uxyand Vxy are computed by the standard logit inversion

formula (log-odds ratio) Uxy = log µxy µx0

Vxy = log

µxy µ0y

.

◮ We plot the estimated U versus the estimated V .

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MARRIAGE GAINS AND CONSERVATIVENESS (CONT)

●
●
●
●
−2

−1 1 2 −3 −2 −1 1 2 3

Marriage Gains by Conservativeness (Quartiles %Republican)

U V

●
●
●
●
●
●●
● ●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
Q1

Q2 Q3 Q4

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MARRIAGE GAINS IN THE US

Utility (Men) [−0.559,0.438] (0.438,0.645] (0.645,0.698] (0.698,1.03] Utility (Wom) [−0.699,0.229] (0.229,0.364] (0.364,0.482] (0.482,0.833]

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ESTIMATION

◮ The previous figure shows too much heterogeneity between markets. ◮ Let k ∈ S be the index of a market (states in this application). Assume

that pre-transfer value of marriage in market k is given by αk

xy = α0 xy + ηk

for men, and γk

xy = γ0 xy + νk

◮ Equilibrium implies Ψ(Uxy − αxy, Vxy − γxy) = 0.Then estimation

consists in finding τ, α0

xy, ηk, γ0 xy and νk that minimizes

min∑

k ∑ xy

τ log
exp
Uxy − αk

xy

τ

+ exp
Vxy − γk

xy

τ

2

in the ETU case.

◮ The problem has dimension |X | + |Y| + 2 × |S|, and we have

|X | × |Y| × |S| observations.

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ESTIMATED TRANSFERS (PRELIMINARY)

◮ In this setting, we estimate

ˆ τ ≃ 8

●
●
●
●
●
●
●
●●
●
●
●
●
●
●
● ●
●
●
−4

−2 2 4 −4 −2 2 4

Transfers

t t'

Transfer function for estimated tau = 8.63103517578035

t t' −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 TRANSFER FRICTIONS IN THE MARRIAGE MARKET MANCHESTER, APRIL 1, 2015 SLIDE 29/ 32

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ESTIMATED FIXED EFFECTS

●
●
●
10

20 30 40 50 60 −1.5 −0.5 0.0

Fixed Effects (Men)

Conservativeness (% Republican) Fixed effects (Men)

●
●
10

20 30 40 50 60 −1.0 0.0 0.5

Fixed Effects (Women)

Conservativeness (% Republican) Fixed effects (Women)

●
10

20 30 40 50 60 −2 −1 1

Fixed Effects (sum)

Conservativeness (% Republican) Fixed effects (sum)

●
−1.5

−1.0 −0.5 0.0 −1.0 0.0 0.5

Scatter plot of Fixed Effects

Fixed effects (Men) Fixed effects (Women) TRANSFER FRICTIONS IN THE MARRIAGE MARKET MANCHESTER, APRIL 1, 2015 SLIDE 30/ 32

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A MAP OF THE US MARRIAGE MARKET

State FE (Men) [−1.51,−0.169] (−0.169,−0.0353] (−0.0353,0.0954] (0.0954,0.415] TRANSFER FRICTIONS IN THE MARRIAGE MARKET MANCHESTER, APRIL 1, 2015 SLIDE 31/ 32

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EXPLAINING THE FIXED EFFECTS β t β t (Intercept) 0.12 1.75

0.27
1.14

regionNortheast Region

0.25
2.46

0.00 0.02 regionSouth Region

0.29
3.27
0.23
2.99

regionWest Region

0.22
2.38
0.16
2.11

abortionrate

0.02
2.82

pct rep 0.01 3.19

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ROYAL ECONOMIC SOCIETY CONFERENCE 2015 SPECIAL SESSION

Econometrics of Matching

Wednesday 1st April 11.00 — 12.30 Venue: University Place Lecture Theatre A

(RES 2015) 1 / 1

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▼✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❙❦✐❧❧s✱ ❙♦rt✐♥❣✱ ❛♥❞ ❍✉♠❛♥ ❈❛♣✐t❛❧ ❆❝❝✉♠✉❧❛t✐♦♥

Jeremy Lise Fabien Postel-Vinay

UCL, IFS UCL, IFS, Sciences Po

Royal Economics Society 2015 Meeting April 1, 2015

1 / 19

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■♥tr♦❞✉❝t✐♦♥

◮ Questions:

◮ How well matched are worker skills to job requirements? ◮ How easily can workers acquire the skills necessary for a given job? ◮ How costly is early career skill mismatch?

◮ We generalize the sequential auction model of Postel-Vinay and Robin (2002)

to allow for multi-dimensional skills (cognitive, manual, interpersonal. . . ) and

n-the-job learning.

◮ We estimate the model using occupation-level measures of skill requirements

based on O*NET data, combined with a worker-level panel (NLSY).

2 / 19

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❙♦♠❡ ❘❡❧❛t❡❞ ▲✐t❡r❛t✉r❡

◮ Search with two-sided heterogeneity:

Postel-Vinay and Robin (2002); Lise, Meghir and Robin (2014); Lamadon, Lise, Meghir and Robin (2014); Bagger and Lentz (2014); Hagedorn and Manovskii (2013); Bagger, Fontaine, Postel-Vinay and Robin (2014); Lise and Robin (2014)

◮ Multidimensional frictionless assignment:

Lindenlaub (2014)

◮ Returns to tenure/experience, task-specific human capital:

Surveyed in Sanders and Taber (2012)

◮ Roy models of occupation choice with task-specific human capital:

Keane and Wolpin (1997); Lee and Wolpin (2006); Yamaguchi (2012); Sanders (2012)

3 / 19

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▼♦❞❡❧ ❙✉♠♠❛r②

◮ The economy is populated by heterogeneous workers and firms:

◮ Workers are characterized by their skill bundles x ∈ X ⊂ RK (ie, cognitive,

manual, interpersonal, general ability ...)

◮ Jobs (“occupations”) are characterized by their skill requirements y ∈ Y ⊂ RL

◮ Workers can be matched to a firm (an “occupation”) or unemployed. ◮ Workers receive job offers both when unemployed and employed. ◮ The firm’s technology y is fixed. . . ◮ . . . but the worker’s skills evolve depending on the firm’s requirements:

˙ x = g(x, y), where g : RK × RL → RK is a continuous function

◮ Wages are determined by Bertrand competition, similar to Postel-Vinay and

Robin (2002)

4 / 19

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❱❛❧✉❡ ❋✉♥❝t✐♦♥s ❛♥❞ ❲❛❣❡ ❊q✉❛t✐♦♥

Value of Unemployment: (r + µ)U(x) = b(x) + g(x, 0) · ∇U(x) Match Value: (r + µ + δ)P(x, y) = f(x, y) − c(x, y) + δU(x) + g(x, y) · ∇xP(x, y) Worker Value: W(x, y, σ) = (1 − σ)U(x) + σP(x, y)

. . . where the share σ is determined by the worker’s second best offer.

Implied Wage Equation: w(x, y, σ) = σf(x, y) + (1 − σ)b(x) + (1 − σ)c(x, y) − λ1Ey′∼Υ max

0, min
P(x, y′) − P(x, y), 0
+ (1 − σ) (P(x, y) − U(x))
− (1 − σ) (g(x, y) − g(x, 0)) · ∇U(x)

5 / 19

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❚❤❡ ❉❛t❛

◮ Workers:

◮ Estimation is based on 1,773 males from the NLSY79 cohort, whom we follow

for 15 years from first entry into the labor market.

◮ We keep track of wages, transitions, occupations. ◮ Initial skill bundles x(0) are derived from education, ASVAB scores (details),

psychometric tests, and measures of criminal and antisocial behavior, using PCA.

Skill bundles x(t) at positive experience levels will depend on worker employment histories.

◮ Jobs:

◮ Skill requirements derived from the O*NET data set. ◮ O*NET describes 970 occupations in terms of of 277 descriptors of skill and

knowledge requirements, work practices, and work settings.

◮ We use a subset of about 200 descriptors, which we collapse to three dimensions

(interpreted as “cognitive”, “manual”, and “interpersonal” skill requirements) using PCA.

6 / 19

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❚❤❡ ❉❛t❛

◮ Examples of skill requirements:

Skill requirements: Occupation title Cognitive Manual Interpers. Physicists 1 0.207 0.741 Graders and Sorters, Agricultural Products 0.343 0.001 Elevator Installers and Repairers 0.497 1 0.485 Anthropologists 0.940 0.752 Preventive Medicine Physicians 0.999 0.095 1 Pressers, Textile, Garment, and Related Materials 0.005 0.578 Economists 0.891 0.009 0.697 Source: O*NET and authors’ calculations

7 / 19

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❊♠♣✐r✐❝❛❧ ▼♦❞❡❧ ❙♣❡❝✐❢✐❝❛t✐♦♥

◮ Production function:

f(x, y) = xT  αT +

k=C,M,I

αkyk −

k=C,M,I

κu

k min {xk − yk, 0}2

 

◮ Utility cost of being under-matched (over qualified):

c(x, y) = xT

k=C,M,I

κo

k max {xk − yk, 0}2 ◮ Unemployment income:

b(x) = b xT

◮ General worker efficiency:

xT (t) = exp

g t+ζS YRS OF SCHOOLING+ζC xC(0)+ζM xM(0)+ζI xI(0)+ε0
ε0: uncorrelated time-invariant unobserved heterogeneity

8 / 19

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❊♠♣✐r✐❝❛❧ ▼♦❞❡❧ ❙♣❡❝✐❢✐❝❛t✐♦♥

◮ Skill adjustment function:

g(x, y) =       ˙ xC ˙ xM ˙ xI ˙ xT       =       γu

C max {yC − xC, 0} + γo C min {yC − xC, 0}

γu

M max {yM − xM, 0} + γo M min {yM − xM, 0}

γu

I max {yI − xI, 0} + γo I min {yI − xI, 0}

gxT      

◮ Skill requirements:

yC = yξC

C

yM = yξM

M

yI = yξI

I

where yC, yM and yI are the empirical measures of skill requirements.

◮ Sampling distribution:

Υ(y) : Gaussian copula

9 / 19

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❊st✐♠❛t✐♦♥

◮ We estimate the model by indirect inference ◮ Human capital accumulation and the cost of mismatch are identified from

comparisons of the sets of job types y that are acceptable to workers with equal initial skills x(0), but different employment histories.

◮ Model fit is good

10 / 19

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❊st✐♠❛t❡s

production function

disu. of work

g 0.004 0.004 0.039 0.039 5.6e − 7 5.6e − 7 2.6e − 3 (14.5) (14.0) (1.48) (1.48) (>100,000) (>100,000)

genl. efficiency

transition rates ζS ζC ζM ζI λ0 λ1 δ 0.045 0.690 −0.048 0.031 0.409 0.155 0.021

⋆: half-life in years in parentheses

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SLIDE 46

❊st✐♠❛t❡s

production function

disu. of work

g 0.004 0.004 0.039 0.039 5.6e − 7 5.6e − 7 2.6e − 3 (14.5) (14.0) (1.48) (1.48) (>100,000) (>100,000)

genl. efficiency

transition rates ζS ζC ζM ζI λ0 λ1 δ 0.045 0.690 −0.048 0.031 0.409 0.155 0.021

⋆: half-life in years in parentheses

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SLIDE 47

❊st✐♠❛t❡s

production function

disu. of work

g 0.004 0.004 0.039 0.039 5.6e − 7 5.6e − 7 2.6e − 3 (14.5) (14.0) (1.48) (1.48) (>100,000) (>100,000)

genl. efficiency

transition rates ζS ζC ζM ζI λ0 λ1 δ 0.045 0.690 −0.048 0.031 0.409 0.155 0.021

⋆: half-life in years in parentheses

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SLIDE 48

❊st✐♠❛t❡s

production function

disu. of work

αT αC αM αI κu

C

SLIDE 50

❙♦rt✐♥❣✿ ❈♦❣♥✐t✐✈❡ s❦✐❧❧s ✶✺ ②❡❛rs ♦❢ ❡①♣❡r✐❡♥❝❡

13 / 19

SLIDE 51

❙♦rt✐♥❣✿ ▼❛♥✉❛❧ s❦✐❧❧s ✶ ②❡❛r ♦❢ ❡①♣❡r✐❡♥❝❡

14 / 19

SLIDE 52

❙♦rt✐♥❣✿ ▼❛♥✉❛❧ s❦✐❧❧s ✶✺ ②❡❛rs ♦❢ ❡①♣❡r✐❡♥❝❡

15 / 19

SLIDE 53

❈♦❣♥✐t✐✈❡ ✈s ▼❛♥✉❛❧ s❦✐❧❧s ❛❢t❡r ✺ ②rs ❡①♣❡r✐❡♥❝❡

16 / 19

SLIDE 54

❈♦❣♥✐t✐✈❡ ✈s ▼❛♥✉❛❧ s❦✐❧❧s ❛❢t❡r ✶✺ ②rs ❡①♣❡r✐❡♥❝❡

17 / 19

SLIDE 55

❚❤❡ ❈♦st ♦❢ ▼✐s♠❛t❝❤✿ P❧❛♥♥❡r✬s ❭❖✉t♣✉t✧ ❣❛✐♥

20 40 60 80 100 120 140 160 180 1 2 3 4 5 6 7 8 9 10 ◮ Blue line internalizes human capital accumulation on other matches ◮ Green line eliminates mismatch

◮ initial gain from reallocation ◮ continued gain from optimal skill accumulation 18 / 19

SLIDE 56

❈♦♥❝❧✉s✐♦♥s

◮ The model sees cognitive, manual, and interpersonal skills as very different

productive attributes:

◮ Manual skills have low returns and adjust quickly ◮ Cognitive skills have much higher returns and adjust slowly ◮ Interpersonal skills have very modest returns and are essentially fixed over a

worker’s lifetime

◮ The cost of mismatch is very high for cognitive skills, substantial for manual

skills, and negligible for interpersonal skills

◮ The cost of mismatch is asymmetric: employing an under-qualified worker in

either cognitive or manual skills is several orders of magnitude more costly than employing an over-qualified worker

◮ It is doubtful whether those various dimensions of worker skills can be

usefully summarized by a scalar index.

19 / 19