Lecture 1: The Human Capital Model - University of Minnesotausers.econ.umn.edu/~guvenen/HANDOUT1_BENPORATH.pdf · 2012-09-04 · Key observations 1 Ben-Porath gets several things

Lecture 1: The Human Capital Model

Fatih Guvenen

University of Minnesota

September 4, 2012

Fatih Guvenen (2012) Lecture 1: Ben Porath September 4, 2012 1 / 18

Why Study Wage Income?

Labor income is 2/3 of GDP.

Most individuals outside of top 10% have very little capital income.Labor income is all they have.

So it’s crucial to understand:

I Why do wages grow over the life cycle?

I Why does labor income varies so much across individuals?

I Why has wage inequality increased so much since 1970s in the US?

I What is the nature of income risk faced by workers?


The Human Capital Model

Developed with seminar contributions by Becker, Schultz, Ben-Porath,Mincer, and Rosen.

Key Assumptions:

Skills are general (No firm-specificity. Can you have industry- oroccupation-specific skills? Will discuss this later.)

No slavery (i.e, no commitment by worker).

Labor markets are competitive.

Lack of commitment can be solved by making workers pay for theirown training (Becker).


Ben-Porath (1967) Model

Individuals can borrow and lend at a constant interest rate (denotedby r), which implies that markets are complete. Let � = 1/ (1 + r) .

Individuals solve

max{it}Tt=1

"

TX

t=1

�t�1Rht(1 � it)

#

s.t

ht+1

= ht + A (ht it)↵ , with h

0

> 0 givenit 2 [0, 1]

This formulation does not rest on the assumption of risk-neutrality,but instead requires markets to be complete. [Homework: Prove thisstatement.]


Ben-Porath (cont’d)

↵ : measures the returns to scale in investment. When it is small(large), you want to spread (bunch) investment over time.

Technology could be generalized: Ah�s i↵s :

1 Estimates of ↵ and � are typically close.

2 Present technology gives closed form solutions.

3 Can also introduce pecuniary costs of investment.


Analyzing the Individual’s ProblemDefine:

C (Qs) ⌘ hs is =

✓

Qs

A

◆

1/↵

.

With this transformation, the problem of an individual can be writtenas

max{Qs}Ss=1

"

SX

s=1

�s�1R (hs � C (Qs))

#

subject tohs+1

= hs + Qs , with h0

given

Now write it as a Bellman equation:

Vs (hs) = maxQs2[0,hs ]

[R (hs � C (Qs)) + �Vs+1

(hs + Qs)]


Assuming interior solution, the FOC:

RC 0(Qs) = �V 0s+1

(hs + Qs)

Envelope:V 0s (hs) = R + �V 0

s+1

(hs + Qs)

Lead by one period:

V 0s+1

(hs+1

) = R + �⇥

R + �V 0s+2

(hs+2

)⇤

..repeatedly substitute for the RHS to get

RC 0(Qs) = �n

R + �R + ...+ �S�s�1Ro

(1)

C 0(Qs) =1r

"

1 �✓

11 + r

◆S�s#

(2)

! Q⇤s = A

1

1�↵ [↵MB (r , S � s)]↵

1�↵ (3)


Solve the model:RHS does not depend on Q.

LHS is proportional to Q1�↵↵

s . If ↵ > 0.5, this is a concave function.Alternatively convex.

How curvature depends on ↵ :


Notice that Q⇤s does NOT depend on current human capital level.

This is a consequence of the neutrality assumption.

Is this realistic? (is still depends on hs of course).

It would imply for example that somebody who has high initial humancapital will spend less time investing in human capital.. seems a bitunrealistic.

It also implies a convergence in the wage distribution (var(log(w)) ifthe only heterogeneity is in initial human capital.


Key observations

1 Ben-Porath gets several things right:

1 Investment declines over the lifecycle in a convex fashion. Forexample, interpreting i = 1 as college enrollment, we have collegeattendance early on.

2 Wages grow in a concave fashion.

3 Can generate a hump shaped wage profile by assuming positivedepreciation.

4 Investment and wage growth are higher for high ability people.

5 Can generate a rise in wage inequality over the lifecycle without shocksif A differs across people. More on this later.


1 Shortcoming: No well-defined concept of return to skill as mentionedbefore (eq. (3)).

2 Why do wages grow over the lifecycle?1 ws = h⇤s � C (Q⇤

s ) and ws+1 = h⇤s+1 � C�

Q⇤s+1

�

ws+1 � ws = hs+1 � hs + C (Qs)� C (Qs+1)

= Q⇤s + C (Q⇤

s )� C�

Q⇤s+1

�

So it is partly because (i) the stock of human capital grows, and also(ii) simply because individuals reduce investment.

2 Kuruscu (2006, AER): if ↵ is large the second effect is the main reasonfor the rise in wages. He estimates ↵ to be about 0.95.

3 If true, the benefit of training is rather low, less than 1%.4 Some are skeptical. One reason is that an ↵ that high implies:

Qs = A1

1�↵ [↵MB (r , S � s)]↵

1�↵

1 extreme sensitivity of investment to ability differences. (Does this makesense?)

2 extreme sensitivity to changes in MB (interest rates, returns, etc.)


Open Questions1 If you can say something useful about the value of ↵ you can grab

many people’s attention.I Move beyond the average wage growth over the lifecycle and look at

other moments to identify ↵.

2 One difficulty of human capital model is that OJT is very difficult tomeasure.

Homework:I Take the Ben-Porath model. Assume two periods.I Introduce endogenous leisure: u (c , l) = log c + log l . Allow for savings

between periods 1 and 2.I Solve for the value function in period 2 and plug into period 1 problem.

Can you solve for (h1, c1, l1) using the first order conditions? Why orwhy not?

I Modify the problem to make it well-behaved and easily solved throughthe FOCs. Play with preferences if necessary.


Discussion

Missing from Ben-Porath model:

1 No Returns to skill: one cannot easily study what happens to humancapital accumulation, etc, when the return to education, etc goes up.We will return to this point.

2 No Jobs, Occupations, etc: A bit too simple in its treatment of jobs,occupations.. none of them exist here. They do in real life. So how tocombine the two?

1 How about start with specific human capital and the notion ofoccupation? Consider a model with two types of human capital wherethe wage of each is determined by a CES production technology. Whatcan you say about the relative employment, wages, education in eachoccupation depending on movements in relative productivity.. whatkinds of transitions would we observe? KEY: is to model what happenswhen you transit from one occupation to another?


Open Questions

1 How about learning about your ability? Suppose there is learningand two human capital types. Non-trivial two-armed bandit problem.

1 Policy questions: labor market frictions could have substantial welfarelosses by discouraging experimentation and learning about one’s bestskill.

2 What if learning ability is match-specific and you learn about itafter the match. Like Jovanovic (1979) but with human capitalaccumulation.

Question: These models are tools for studying economic questions.What other type of question can you fit into this framework?

I Irreversible investment problems with physical capital.


Leisure in Ben-Porath Model

Second period problem

V1

(k1

, h1

) = maxc1

,n1

[log (c1

) + log ((1 � n1

))]

c1

= (1 + r) k1

+ h1

n1

FOC:h1

(1 + r) k1

+ h1

n1

=1

1 � n1

! Solve 1 � n1

and then c1

= (1 � n1

) h1

= h1

+(1+r)k1

2

. !

V1

(k1

, h1

) = log✓

h1

+ (1 + r) k1

2

◆

+ log✓

h1

+ (1 + r) k1

2h1

◆

= 2 log (h1

+ (1 + r) k1

)� log (h1

)� 2 log (2)


Is V1

increasing and concave in h1

?

@V1

@h1

=h1

� (1 + r) k1

h1

(h1

+ (1 + r) k1

)> 0 if h

1

> (1 + r) k1

@2V1

@h2

1

=2 (1 + r) k

1

h1

� h2

1

+ (1 + r)2 k2

1

#

You can show that if k1

is large compared to h1

(still satisfyingh1

> (1 + r) k1

) one can get a convex value function in h1

.

So in period 1, one cannot solve this using Euler equation is notsufficient.


Workaround: Modify preferences:

U = log (c1

) + log ((1 � n1

) h1

)

= log (c1

) + log ((1 � n1

)) + log (h1

)

Now V1

will be:

eV1

= [2 log (h1

+ (1 + r) k1

)� log (h1

)� 2 log (2)]| {z }

V1

+ log (h1

)

= 2 log (h1

+ (1 + r) k1

)� 2 log (2)


Next Lecture

We will have somebody present the Rosen version of Ben-Porath.


Documents

Lecture 1: The Human Capital Model - University of Minnesotausers.econ.umn.edu/~guvenen/HANDOUT1_BENPORATH.pdf · 2012-09-04 · Key observations 1 Ben-Porath gets several things