Labor Economics I

FDPE Spring 2015
Marko Terviö (Aalto University)

Ben-Porath Model

Why human capital?

What explains differences in earnings potential?

HC is like physical capital because ...

Investment = give up something today to increase future output

Depreciation?

HC is unlike physical capital because ...

Inalienability

Lumped with preferences, limited liability

Innate + acquired heterogeneity

Life cycle

The Ben-Porath Model

Yoram Ben-Porath: “The Production of Human Capital and the Life Cycle of Earnings” (JPE 1967).

Model of an individual lifetime decision of how to divide time between work and learning. 
What is the optimal time profile of human capital investment and earnings?

Key features

Finite career with a fixed retirement time: t ∈ [0,T]

Earning potential at time t depends on the level of human capital H(t)

Building up human capital requires time and money

Fixed leisure/labor divide: one unit of non-leisure time per “period”

Choices at t ∈ [0,T]

What fraction of productive time to devote to human capital accumulation 
	s(t) ∈[0,1]

How much money to spend on human capital
	M(t) ≥ 0.

The choice of M(t) is less interesting and could be ignored. (Footnote 5 in the paper is odd!)

Model setup

Here with modernized notation and terminology

Maximizing PV of lifetime utility = maximizing PV of lifetime consumption, because

Perfect capital market: no credit constraints

Fixed “time budget constraint” with no interactions with leisure in the utility

Exponential discounting

Present value of lifetime consumption

(Subscript[W, 0]) = ((∫_0)^T)(e^(-rt))(wH(t)(1 - s(t)) - M(t))dt

Evolution of human capital

H'(t)=A(((s(t)H(t)))^α)((M(t))^β)-δH(t)

Choice variables

s	share of time devoted to investment in human capital
M	monetary spending on human capital

State variable

H	human capital 	
H(0)	endowment

Parameters

r	discount rate
w	wage rate per units of human capital 
δ	depreciation rate of human capital
T	length of career
A,α,β	parameters of a human capital production function, α+β ≤ 1

The economic problem

Maximize (Subscript[W, 0]), choosing s(t) ∈ [0,1]  and M(t) ≥ 0, t ∈ [0,T].

Solution method

Set up the present value Hamiltonian

ℋ=(e^(-rt)) (wH(t) (1-s (t))-M (t)) + λ(t)(((A(s(t) H(t)))^α)((M(t))^β)-δH(t))

Costate variable λ is the shadow value of investment

Recall the Maximum Principle (aka Pontryagin’s maximum principle).
Maximization requires satisfying these first-order conditions:

((∂ℋ)/(∂s)) = 0   ⟹

(e^(-rt))w H(t)= λ(t) Aα ((s(t))^(α-1))((H(t))^α)((M(t))^β)

((∂ℋ)/(∂M)) = 0  ⟹

(e^(-rt))= λ(t) Aβ ((s(t))^α)((H(t))^α)((M(t))^(β-1))

((∂ℋ)/(∂H)) = -λ'(t)  ⟹

(e^(-rt))w(1-s(t))+λ(t)[α A((s(t))^α)((H(t))^(α-1))((M(t))^β)-δ]  =-λ'(t)

Terminal condition

Human capital cannot be sold and has no “scrap value”

It is not in general optimal to let H run down to zero in the end. Endogenous H(T) > 0.

Shadow value of H declines to zero when time runs out: λ(T) = 0.

Investment in the last instant of the career is useless: H’(T) = 0 is optimal, so s(T) = 0, M(T) = 0.

Solution steps

Relation of money and time spending

What is the cheapest way to combine time and money to produce units of H?

Optimal money investment M(t) is mechanically related to the optimal “human capital investment” s(t)H(t).

Production function is Cobb-Douglas, so expenditure shares are proportional to coefficients α and β.
“Prices” are w for human capital and 1 for money.

(( w s(t)H(t) )/(M(t))) = (α/β) ⟹

M(t)=((w β)/α) s(t)H(t)

This holds if interior solution s(t) ∈ (0,1).

Equivalently, solve both (4) and (5) for λ(t), then solve for M(t).

Reduce the choice variable FOC

With just one choice variable the problem gets simpler

((∂ℋ)/(∂s)) = 0

⟹   (e^(-rt))w H(t)= λ(t) Aα ((s(t))^(α-1))((((H(t))^α)[((w β)/α) s(t)H(t)])^β)

⟺   (e^(-rt))s(t) =   λ(t)A(Subscript[, 1])s(((t))^(α+β))((H(t))^(α+β-1))

where   A(Subscript[, 1]):=    ((A((α/w)))^(1-β)) (β^β)

Reduce the state variable FOC

((∂ℋ)/(∂H)) = -λ'(t)

⟹    (e^(-rt))w(1-s(t))+λ(t)[α A((s(t))^α)((((H(t))^(α-1))[((w β)/α) s(t)H(t)])^β)-δ]  =-λ'(t)

⟺    (e^(-rt))w-(e^(-rt))s(t)w+λ(t)[w A(Subscript[, 1])((s(t))^(α+β))((H(t))^(α+β-1))-δ]  =-λ'(t)

⟺    (e^(-rt))w -δ λ(t) =-λ'(t)

where (7) was used to eliminate s(t) and H(t) from ((∂ℋ)/(∂H)) in (8)

Eliminating the costate variable

Evolution of the shadow value of investment follows a simple ODE

λ'(t)+ δ λ(t)  - (e^(-rt))w = 0

Solution is

λ(t)= ((e^(-rt))/(r+δ))w+(e^(δt))(Subscript[C, 1])

Terminal condition λ(T) = 0 pins down the constant (Subscript[C, 1])

λ(t)= ((e^(-rt))/(r+δ))w+(e^(δt))(-((e^(rT-δT))/(r+δ))w)=(w/(r+δ))((e^(-rt))-(e^(δt-(δ+r)T)))

⟹ λ(t)= (w/(r+δ))(e^(-rt))(1-(e^(-(δ+r)(T-t))))

Optimal policy

Insert λ(t) from (11) to ((∂ℋ)/(∂s)) = 0  from (7) and then solve for s(t)

s(t) =   λ(t)(e^rt)A(Subscript[, 1])s(((t))^(α+β))((H(t))^(α+β-1))                                          ⟹
    s(t) =   (w/(r+δ))(1-(e^(-(δ+r)(T-t))))A(Subscript[, 1])s(((t))^(α+β))((H(t))^(α+β-1))      ⟺ 
s(t)=(1/(H(t)))(((A(Subscript[, 1]) (w/(r+δ))(1-(e^(-(δ+r)(t-T))))))^(1/(1-α-β)))

This solution is only valid when s(t) ∈ (0,1).

But s(t) = 1 can be optimal for some t ∈[0,(SuperStar[t])], “specialization phase.”

Empirical earnings profiles

Pekkarinen, Tuomas and Roope Uusitalo. “Productivity-wage gaps and aging", IZA working paper (2012).

Numerical example with interior solution

r=0.05;δ=0.03;A=1;w=1;α=1/2;β=1/8;T=10;H0=20;

A1=A ((((α/w)))^(1-β))(β^β);
s[t_]=(1/(H[t]))(((A1 (w/(r+δ))(1-(E^(-(δ+r)(T-t))))))^(1/(1-α-β)));
M[t_]=((w β)/α)s[t]H[t]; 

Hsol=NDSolve[{H'[t]==A (((s[t] H[t]))^α)((M[t])^β)-δ H[t],H[0]==H0},H,{t,0,T}][[1]];
GraphicsRow[Plot[#[t]/.Hsol,{t,0,T}]&/@{H,s},ImageSize->Full]

Earnings

Earnings[t_]=w H[t](1-s[t])/.Hsol;
Spending[t_]=M[t]/.Hsol;
NetEarnings[t_]=Earnings[t]-Spending[t];

Plot[{Earnings[t],NetEarnings[t],Spending[t]},{t,0,T},PlotRange->All,
PlotLegends->"Expressions",AxesLabel->{"t","€"}]

NIntegrate[NetEarnings[t],{t,0,T}]

Note about the evolution of investment

Outside the possible specialization phase the MC of new human capital is equated with its shadow value. 

The present value of a unit of H keeps declining as the remaining career gets shorter
MC is increasing in the quantity Q of human capital produced at that moment. Convex adjustment costs!

Q(t)=((A[s(t)H(t)])^α)((M(t))^β)
          =((A[s(t)H(t)])^α)(((((w β)/α) s(t)H(t)))^β)

= ((w )/α)A(Subscript[, 1])(([s(t)H(t)])^(α+β))

Investment cost I of using the optimal combination of inputs is

I=w s H +M 
       =w s H (1+ (β/α) )                                   using  (6)
       = w (((Q (α/(w A(Subscript[, 1])))))^(1/(α+β))) (1+ (β/α) )          using  (12)

Marginal cost is

((∂I)/(∂Q)) = w(1+ (β/α) )  ((((α/(w A(Subscript[, 1])))))^(1/(α+β)))((Q^((1/(α+β))-1))/(α+β))=A(Subscript[, 2])(Q^((1-α-β)/(α+β)))

where (Subscript[A, 2]):= ugly and positive.

Optimality requires (e^(-rt))MC(t) = λ(t) ⟹

A(Subscript[, 2])Q(((t))^((1-α-β)/(α+β)))=(w/(r+δ))(1-(e^(-(δ+r)(T-t))))
Q(t) = A(Subscript[, 3])(((1-(e^(-(δ+r)(T-t)))))^((α+β)/(1-α-β)))

where (Subscript[A, 3]):= uglier and positive.

Investment declines during the working career, reaching zero at T.   What about implied earnings?

Example with specialization phase

If the solution imples s(0) > 1  then s(t) = 1 for some t ∈ [0,(SuperStar[t])].  
During the specialization phase H keeps increasing the shadow value of investment equals MC

End time for specialization is a choice variable t*

A unit of human capital at time t has the present value of earnings potential (e^rt)λ(t) 
This exceeds the MC human capital until (SuperStar[t])

For this case Ben-Porath considers a CRS version with 
 β = 0 		no “purchases”
 α = 1 		CRS
 δ = 0

Now MC of producing human capital is always (w/A). Set it equal to (e^rt)λ(t):

(w/r)(1-(e^(-r(T-t)))) = (w/A)  ⟹
(SuperStar[t]) = T+(1/r)log(1-(r/A))

A note about notation

JPE 1967 ⟷ This lecture note

(Subscript[K, t])   ⟷   H(t)
(Subscript[(OverDot[K]), t])   ⟷   H'(t)
(Subscript[s, t])   ⟷   s(t)
(Subscript[D, t])  ⟷  M(t)
(Subscript[α, 0  ])⟷ w
(Subscript[β, 0  ])⟷ A
(Subscript[β, 1 ])⟷ α
(Subscript[β, 2  ])⟷ β
q  ⟷ λ 
ψ  ⟷ λ (e^(r t))

Normalization

(Subscript[P, d])   := 1

Unchanged

r, T, (Subscript[Q, t]),(Subscript[W, t]), (SuperStar[t])

Indirect inference of earnings dynamics

Altonji, Smith, and Vidangos: “Modeling Earnings Dynamics.” (Econmetrica 2013)

What explains the age-profile of earnings?  What explains cross-sectional earnings inequality?

Key features

tenure and experience effects

transitions between jobs

extensive margin transitions to and from unemployment

intensive margin may have job-specific hours constraints

tenure effects

observed and unobserved heterogeneity of individuals and jobs

Reduced form model motivated by theory (theories)

equations are “approximations” of what arise in structural models

Main outputs: decompositions, impulse-response analysis (, proof-of-concept)

Earnings model

Career outcomes by year: participation, hours, wages

Exogenous variables

education

race

age

Unobserved heterogeneity

productivity

switching propensity

Latent match value variables

current job

outside job

unemployment

Shock variance, persistence, measurement error, state-dependent variance, etc

See model schema on blackboard

Inference

PSID data on male household heads 1975-1997

Model too complicated too jumpy (latent variables) for MLE

too many moving parts for a structural life-cycle model (?)

too complicated too jumpy (latent variables) for MLE

very unbalanced panel

Method: Indirect inference

Pick (long list of) summary statistics θ that describe the complex outcome dynamics

1. Assume a value for model parameters β
2. Simulate careers based on β and observed exogenous variables
3. Calculate (OverTilde[θ])(β), the average θ from simulated careers

Pick β that minimizes distance between (OverTilde[θ])(β) and the (OverHat[θ]) calculated from actual career data

Some results /sanity checks

Low labor supply elasticity

Persistent wage effect of unemployment shocks decomposition (lost tenure, job-specific

Very negative tenure effect, esp. for low-educated

Wage offer shocks persistent across jobs

Wage growth decomposition (job shopping, tenure, experience)

Cross-section variance decomposition of hours, wages, earnings

Decomposition of experience profile

Log wage and potential experience. Altonji, Smith, and Vidangos (Ecma 2013)

(Log) wage rate determination in Altonji, Smith, and Vidangos (Ecma 2013)