Online Appendix to “The Allocation of Talent
and U.S. Economic Growth”
(Not for publication)
Chang-Tai Hsieh, Erik Hurst, Charles I. Jones, Peter J. Klenow
February 22, 2013
A
Derivations and Proofs
The propositions in the paper summarize the key results from the model. This appendix shows how to derive the results.
Proof of Proposition 1. Occupational Choice
As given in equation (5), the individual’s utility from choosing a particular occupaβ
tion, U (τi , wi , ǫi ), is proportional to (w̄ig ǫi ) 1−η , where w̄ig ≡ wi h̄ig sφi i (1 − si )
1−η
β
/τig . The
solution to the individual’s problem, then, involves picking the occupation with the
largest value of w̄ig ǫi . To keep the notation simple, we will suppress the g subscript in
what follows.
Without loss of generality, consider the probability that the individual chooses occupation 1, and denote this by p1 . Then
p1 = Pr [w̄1 ǫ1 > w̄s ǫs ] ∀s 6= 1
= Pr [ǫs < w̄1 ǫ1 /w̄s ] ∀s 6= 1
Z
= F1 (ǫ, α2 ǫ, . . . , αN ǫ)dǫ,
(30)
where F1 (·) is the derivative of the cdf with respect to its first argument and αi ≡ w̄1 /w̄i .
Recall that
 "
#1−ρ 
N


X
−θ
F (ǫ1 , . . . , ǫN ) = exp −
Ts ǫs
.


s=1
Taking the derivative with respect to ǫ1 and evaluating at the appropriate arguments
gives
F1 (ǫ, α2 ǫ, . . . , αN ǫ) = θ(1 − ρ)T1 T̄
−ρ −θ(1−ρ)−1
1
ǫ
h
i1−ρ −θ
· exp − T̄ ǫ
(31)
2
where T̄ ≡
HSIEH, HURST, JONES, AND KLENOW
P
−θ
s Ts αs .
Evaluating the integral in (30) then gives
p1 =
=
=
=
=
Z
F1 (ǫ, α2 ǫ, . . . , αN ǫ)dǫ
h
Z
i1−ρ T1
1−ρ
−θ(1−ρ)−1
−θ
T̄
θ(1 − ρ)ǫ
· exp − T̄ ǫ
dǫ
T̄
Z
T1
· dF (ǫ)
T̄
T1
T̄
T1
P
−θ
s Ts αs
T1 w̄θ
=P 1 θ
s Ts w̄s
A similar expression applies for any occupation i, so we have
1/θ
where w̃i ≡ Ti
w̃θ
pi = P i θ
s w̃s
w̄i .
Proof of Proposition 2. Average Quality of Workers
Total efficiency units of labor supplied to occupation i by group g are
Hig = qg pig · E [hi ǫi | Person chooses i] .
Recall that h(e, s) = h̄i sφi eη . Using the results from the individual’s optimization problem, it is straightforward to show that
hi ǫi = h̃i
wi (1 − τiw )
1 + τih
η
1−η
1
ǫi1−η ,
1
where h̃i ≡ (η η h̄i sφi i ) 1−η . Therefore,
Hig = qg pig h̃i
wi (1 − τiw )
1 + τih
η
1−η
1
1−η
· E ǫi | Person chooses i .
(32)
3
THE ALLOCATION OF TALENT
To calculate this last conditional expectation, we use the extreme value magic of the
Fréchet distribution. Let yi ≡ w̄i ǫi denote the key occupational choice term. Then
y ∗ ≡ max{yi } = max{w̄i ǫi } = w̄∗ ǫ∗ .
i
i
Since yi is the thing we are maximizing, it inherits the extreme value distribution:
Pr [y ∗ < z] = Pr [yi < z] ∀i
= Pr [ǫi < z/w̄i ] ∀i
z
z
=F
,...,
w̄1
w̄N
 "
#1−ρ 


X
= exp −
Ts w̄sθ z −θ


s
= exp{−[T̄ z −θ ]1−ρ }.
(33)
That is, the extreme value also has a Fréchet distribution, where T̄ ≡
P
θ
s Ts w̄s .
Straightforward algebra then reveals that the distribution of ǫ∗ , the ability of people
in their chosen occupation, is also Fréchet:
G(x) ≡ Pr [ǫ∗ < x] = exp{−[T ∗ x−θ ]1−ρ }
where T ∗ ≡
PN
s=1 Ts (w̄s /w̄
∗ )θ .
(34)
This result is useful later in the paper in that it implies
that the wage distribution across people within an occupation will also be Fréchet, with
a parameter that depends on θ(1 − ρ). Therefore, to recover an estimate of θ from the
wage distribution, we will need to adjust for the correlation parameter ρ.
Finally, one can then calculate the statistic we needed above back in equation (32):
the expected value of the chosen occupation’s ability raised to some power. In particular, let i denote the occupation that the individual chooses, and let λ be some positive
exponent. Then,
E[ǫλi ]
=
=
Z
Z
∞
0
0
∞
ǫλ dG(ǫ)
θ(1 − ρ)T ∗(1−ρ) ǫ−θ(1−ρ)−1+λ e−[T
∗ ǫ−θ ]1−ρ
dǫ
(35)
4
HSIEH, HURST, JONES, AND KLENOW
Recall that the “Gamma function” is Γ(α) ≡
x = [T ∗ ǫ−θ ]1−ρ , one can show that
E[ǫλi ] = T ∗λ/θ
Z
R∞
0
∞
x
xα−1 e−x dx. Using the change-of-variable
λ
− θ(1−ρ)
−x
e
dx
0
= T ∗λ/θ Γ(1 −
λ
).
θ(1 − ρ)
(36)
Applying this result to our model, we have
1
1−η
E ǫi
1
∗ θ1 · 1−η
Γ 1−
| Person chooses i = T
=
Ti
pig
1
θ 1−η
1·
1
1
·
θ(1 − ρ) 1 − η
1
1
Γ 1−
·
.
θ(1 − ρ) 1 − η
(37)
Substituting this expression into (32) and rearranging leads to the last result of the
proposition.
Proof of Proposition 3. Occupational Wage Gaps
The proof of this proposition is straightforward given the results of Proposition 1.
Note that η̄ ≡ η η/(1−η) .
B Data Appendix
Table B.1: Sample Statistics By Census Year
THE ALLOCATION OF TALENT
Table B.2: Occupation Categories for our Base Occupational Specification
5
6
HSIEH, HURST, JONES, AND KLENOW
Table B.3: Examples of Occupations within Our Base Occupational Categories
THE ALLOCATION OF TALENT
Table B.4: Occupation Categories for our Broad Occupation Classification
7