VIII. Counterfactual Lifetime Skill Premiums
Are surgeons and other specialists really more highly skilled than their generalist colleagues, or just better suited for the specialized branch of medicine
they practice? The key counterfactual question is: how much would a surgeon earn as an internal medicine doctor relative to what internal medicine doctors make? In this
section, I answer this question using the model of Section III to calculate the lifetime opportunity costs of picking a specialty.
A. Counterfactual wage paths
Estimating opportunity costs requires calculating alternate wage streams in each spe- cialty for each doctor starting from medical school graduation, through residency
training, and up to retirement. Let T be the length of working life starting from med- ical school graduation.
30
Let these streams be denoted by ,
, t d
s
~ where s denotes
each of the alternate specialties, d denotes the actually chosen specialty, and t
denotes the number of years after graduation from medical school. Thus, ,
t d
d
~ gives the wage path of a doctor in his chosen specialty, d
that is,
s
~ with s d
= .
Let L
s
be the length of the residency training period in each specialty. Using information from the AAMC, I set
, d
s
~ to the real average resident
wage in specialty s in the year when the doctor graduated medical school.
31
During residency, wages rise by about 5 percent every year, so:
, ,
. t d
d for t
L and d 15
1 05 1
1 5
s s
t s
f f
= =
= ~
~ The model predicts the alternate wage paths after residency through to retirement:
, ;
exp exp
t d E w
specialty d
X g t
E v
v specialty
d 16
2
s st
s s
s s
s s
s 2
1 1
1 2
1 2
= =
= +
+ +
= ~
b c
v d
d J
L K
K N
P O
O
8
9
B
C for t
L T
1
s
f =
+ . Applying Bayes’ Rule to the last term in Equation 16 yields:
1 1
1 1
, ,
冱 冱
冱 冱
exp exp
E v
v specialty
d a
a
specialty d v
a v
a v
a v
a specialty
d v a
v a
v a
v a
P P
P P
P P
17
, ,
, ,
, ,
, ,
, ,
s s
s k
s m
r q
r q
k m
k m
1 1
1 2
1 2
2 2
1 1
1 2
1 2
2 2
1 1
2 2
1 1
2 2
1 1
2 2
1 1
2 2
+ =
= +
= =
= =
= =
= =
= =
d d
d d
= =
= =
q r
m m
`
a a
j
k k
9 C
where Prob v
a
, k
1 1
= and Prob
v a
, m
2 2
= are defined in Equation 5. Specialty selec-
tion probabilities are multinomial-logit, so for , ,
k 1 2
= and
, m
1 2 =
: Bhattacharya
133
30. I assume that each doctor retires at age 65. 31. Using the average wage should not introduce too much error because the distribution of resident wages
each year is tight—the difference between the 75th and 25th percentile resident wage is less than 15 percent of the median wage for every year.
1
,
冱
exp exp
specialty d v
a v
a Z
a a
Z a
a P
18
, ,
, ,
, ,
k m
r r
r k
r m
d d
d k
d m
1 1
2 2
5 1
2 1
2 2
2 1
2 1
2 2
2
= =
= =
+ +
+ +
a d
d a
d d
= r
a k
It is difficult to compare the counterfactual wage paths across specialties because they vary across a large number of dimensions—one dimension for each year in practice.
Let r be the discount rate and H a fixed supply of hours. The usual convention in
empirical human capital applications, which this paper follows, is to consider the net present value of income in the alternate specialties—
NPV d
s
—as a summary index of the desirability of each occupation:
32
, ,
冱 冱
NPV d r
t d H
H r
t d 19
1 1
s T
t s
T t
s
= +
= +
~ ~
= =
t t
B. Lifetime Skill Premiums
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