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Chapter 2: Applications of discrete and dynamic choice
models
Joan Llull
Structural Empirical Methods for Labor Economics
(and Industrial Organization)
IDEA PhD Program
Introduction
Chapter 2: Applications of discrete and dynamic choice models 2
Introduction
This chapter:
Spatial equilibrium: Diamond (2016)
Married woman's labor force participation.
Human capital accumulation: Heckman, Lochner, and Taber (1998).
Chapter 2: Applications of discrete and dynamic choice models 3
Spatial equilibrium: Diamond (2016)
Chapter 2: Applications of discrete and dynamic choice models 4
Spatial Equilibrium ModelsDate back from Rosen (1979) and Roback (1982).
The Rosen-Roback model is static (similar to the random utility models described in Chapter I,one time migration decisions).
These models typically feature (some of) the following:
Perfectly mobile labor market (with moving costs?).
Fixed land and endogenous housing markets.
Local amenities.
Productivity di�erentials across cities.
Local price di�erences.
They are at the intersection of urban economics and labor economics (cities vs individuals asthe subject of interest).
Chapter 2: Applications of discrete and dynamic choice models 5
Diamond (2016)
I am going to present a stylized version of Diamond (2016) as a canonical example.
Spatial equilibrium model to determine causes and welfare consequences of increasedskill sorting.
The model features:
Perfectly mobile labor market (one time settlement).
Fixed land and endogenous housing markets.
Endogenous amenities (main novelty).
Productivity di�erentials across cities.
Chapter 2: Applications of discrete and dynamic choice models 6
Motivation
Chapter 2: Applications of discrete and dynamic choice models 7
ModelAggregate �rm in each city:
Yj = K1−αj (θUjU
ρj + θSjS
ρj )
αρ .
Wage rates: workers' marginal product Wkj for k ∈ {S,U}.
Workers of skill k choose consumption of housing services and goods, as well as
location to maximize utility:
max{j,c,h}
ζ ln c+ (1− ζ) lnh+G(Aj(Uj , Sj),Xj , z) + εj
s.t. Pc+Rjh ≤Wkj .
Housing supply:
Rj = F (Cj , Lj).
Chapter 2: Applications of discrete and dynamic choice models 8
EstimationExcept for amenity parameters associated to z, all relevant variation in the model is
at the city-education group level.
Two-stage estimation:
1. MLE/Logit estimation of parameters for z, collapsing city-education group-
speci�c parts in city-education dummies.
2. Joint estimation using two-step GMM (in di�s) of all other parameters, using
city level variation.
Instruments: exogenous variables plus Bartik shocks and their interactions with
variables that a�ect the housing supply elasticity.
Chapter 2: Applications of discrete and dynamic choice models 9
GMM equations
Amenities: construct an amenity index Aj as ∆Aj = γ∆ lnSjUj
+ εj .
Production function: standard expressions from FOCs.
Housing: equilibrium expression linking changes in rents with regulation and land
availability indexes, interest rates, and endogenous housing demand variables.
Location choice: once the combination of all city-education group-speci�c elements
of the utility are estimated as δjk, a linear expression relates them to primitives.
Chapter 2: Applications of discrete and dynamic choice models 10
Results
Chapter 2: Applications of discrete and dynamic choice models 11
Married woman's labor force participation
Chapter 2: Applications of discrete and dynamic choice models 12
Utility
Unitary household model to describe female labor force participation decisions.
Single decision unit that takes into account the utilities of the two members of
the couple in the decision process.
The couple's utility is: U(c, d, n,x, ε(1− d)), with:
∂U/∂c > 0.
∂2U/∂c2 < 0.
U(c, 1, n,x, ε) > U(c, 0, n,x, 0) for some values of ε.
Typically, U(c, 1, n,x, ε) > U(c, 1, n′,x, ε) for n > n′ as well.
Chapter 2: Applications of discrete and dynamic choice models 13
Budget constrain and distr. of unobservables
The husband is assumed to work, generating income y.
The wife receives a wage o�er ω(x, υ) and decides whether to work or not accordingly.
If the wife works, the household incurs in child care cost of π per child.
The budget constraint is:
c = y + [ω(x, υ)− πn]d.
Unobservables ε and υ are serially uncorrelated and are jointly distributed as F (ε, υ|y,x, n).The probability that the wife participates is:
Pr(d = 1|x, n, y)
=
∫1{U(y + ω(x, υ)− πn, 1, n,x, ε)− U(y, 1, n,x, 0) > 0}dF (ε, υ|y,x, n)
≡ G(y,x, n).
Chapter 2: Applications of discrete and dynamic choice models 14
Estimation approaches
Primitives to recover: U(·), ω(·), and F (·).
Four estimation methods: structural vs non-structural + parametric vs non-
parametric.
We evaluate the convenience of these approaches considering three goals, to what
extent the following elements a�ect participation:
1. wages.
2. husband's earnings.
3. childcare costs.
Chapter 2: Applications of discrete and dynamic choice models 15
Non-structural approaches
In a non-structural non-parametric approach, we do not need to make further assump-tions: estimate G(·) non-parametrically.
First goal requires further assumptions: exclusion restriction.
Let x1 denote the partition of the vector x that a�ects wages but does not enter the utilityfunction directly.
The e�ect of wage changes on participation can be inferred from ∂G/∂z′.
The second goal is clearly feasible (within sample) without further assumptions, since∂G/∂y is identi�ed.
The third goal, on the contrary, is unfeasible without further assumptions, because G andπ cannot be separately identi�ed.
Parametric speci�cation of G(·) (e.g. probit or logit): similar results but ∂G/∂y is alsoidenti�ed out of sample.
Chapter 2: Applications of discrete and dynamic choice models 16
Non-parametric structural
The non-parametric structural approach requires identifying U(·), ω(·), and F (·)separately without imposing additional assumptions about functional forms.
This is infeasible provided wages are only observed for the individuals who work.
With further non-parametric assumptions and data on wages for the women
who work, one could go a bit further.
For example, if ω(·) is assumed to be additively separable⇒ deterministic part of
the wage function is identi�ed (y, n, and potentially some elements in x not included
in x1, denoted by x2 are exclusion restrictions).
Further assumptions on F (·) could also lead to partial identi�cation of U(·).
Chapter 2: Applications of discrete and dynamic choice models 17
Parametric structuralConsider the following very standard parametric assumptions:
U(c, d, n,x, n, ε(1− d)) ≡ c+ (1− d)[x′2β + γn+ ε],
ω(x, υ) = x′δ + υ,
and:
(ε, υ)′|y,x, n ∼ N (0,Σ).
Given this parameterization, the di�erence in utilities is:
U(y + ω(x, υ)− πn, 1, n,x, ε)− U(y, 1, n,x, 0) > 0}dF (ε, υ|y,x, n)
= x′δ − [π + γ]n− x′2β + υ − ε.
Chapter 2: Applications of discrete and dynamic choice models 18
Parametric structuralData on choices⇒ π+γ, δ1, and δ2−β, where δ1 and δ2 are the partitions of δ associated,respectively, to x1 and x2.
Further data on wages for women who work ⇒ β and δ are separately identi�ed (Heck-man selection approach).
Wage data and exclusion restrictions⇒ Σ could also be identi�ed (σνε from Heckman,σ2ν from variance in wages, and σ2
ε from coe�cient of x1δ1 the probit).
Goals:
1. Elasticity of labor supply with respect to wages is only identi�ed if there are exclusionrestrictions.
2. E�ect of husband's income is assumed to be zero in this case; other utility functionswould lead to di�erent e�ects.
3. E�ect of changing the cost of child care can also be identi�ed, even though only γ+πis identi�ed.
Chapter 2: Applications of discrete and dynamic choice models 19
Human capital accumulation: Heckman, Lochner,and Taber (1998)
Chapter 2: Applications of discrete and dynamic choice models 20
Equilibrium model for wage inequality
Literature on inequality: partial equilibrium.
Heckman, Lochner, and Taber (1998):
General equilibrium.
Overlapping generations.
Human capital accumulation (education and on the job).
New methods for estimating these models.
Using their estimated model, evaluate the mechanisms behind increasing wageinequality.
Chapter 2: Applications of discrete and dynamic choice models 21
ModelConsider the following life-cycle maximization problem:
V (ha, ba, e,it, ret) ≡ maxc,g
{c1−γ
1− γ + βV (ha+1, ba+1, e, it+1, ret+1)
},
s.t. ba+1 ≤ ba[1 + (1− τ)it] + (1− τ)retha(1− g)− c.
On-the-job human capital accumulates as:
ha+t(ω, e) = ωgηeha(ω, e)ψe + (1− δ)ha(ω, e),
with 0 < ηe < 1 and 0 < ψe < 1 for e ∈ {S,U}.
Discrete distribution for ω, with eight points of support (four observable types, denoted by k,and based on quartiles of AFQT test, two education groups).
Individuals are assumed to have perfect foresight of future prices and interest rates in equilibrium(no aggregate shocks in the economy).
Individuals work until age aR, when are forced to retire, and afterwards live until age a withoutperceiving labor income.
Chapter 2: Applications of discrete and dynamic choice models 22
ModelEducation decision is taken at the front end:
maxe
[V E(ω, e, t)− πe + εe
].
Output Yt is determined by the following nested CES technology:
Yt ={αKφ
t + (1− α)[θtLρSt + (1− θt)LρUt]
φρ
} 1φ
.
Skill-biased technical change, determined by the evolution of θt is given by:
ln
(θt
1− θt
)= ln
(θ0
1− θ0
)+ ϕt.
Equilibrium is given by the sequence of interest rates {it}∞t=0 and skill prices {rUt, rSt}∞t=0
that clear the market subject to aggregate �rm pro�t maximization, and workers' lifetimeutility maximization.
Chapter 2: Applications of discrete and dynamic choice models 23
EstimationThey assume values for β, γ, δ(= 0), and τ .
The tuition costs πe are estimated from the data.
The estimation of the remaining parameters is carried with a step-wise procedure.
First step: production function.
At old ages, say a > a∗ for some a∗, individuals no longer invest in human capital (that is,g ≈ 0). Therefore:
w(a∗ + 1, t+ 1, ha∗+1) ≡ ret+1ha∗+1 = ret+1ha∗(1− δ),
which implies:
w(a∗ + `, t+ `, ha∗+`)
w(a∗, t, ha∗)=ret+`(1− δ)`
ret.
Normalizing re0 = 1, skill prices are identi�ed up to a scale (1− δ)t.Chapter 2: Applications of discrete and dynamic choice models 24
EstimationGiven these skill prices, the aggregate stocks of skill units can be recovered from the skill prices:
wage billetret(1− δ)t
=Let
(1− δ)t .
Relative demands of the two labor inputs give:
lnrStrUt
=θt
1− θt+ (φ− 1) ln
LStLUt
= ln
(θ0
1− θ0
)+ ϕt+ (φ− 1) ln
LStLUt
.
⇒ θ0, ϕ, and φ can be recovered by OLS.
The remaining aggregate PF parameters estimated analogously.
Second step: lifetime maximization problem.
NLS for wages (g is unobserved, replaced by the solution of the dynamic problem). This solutionis computed by backwards induction.
To estimate h0(k, e), parameterize haR(k, e) and recover backwards.
Chapter 2: Applications of discrete and dynamic choice models 25
Estimation
Third step: education decision.
First estimate an auxiliary probit model to recover a non-parametric estimate of
{(1− τ)[V E(ω, S, t)− V E(ω,U, t)] + µk}/σ and a coe�cient associated to estimated
tuition costs πe.
Then, they recover the structural parameters from these estimates:
σ is recovered as the coe�cient associated to πe.
µk is recovered comparing the non-parametric estimates to the values predicted
by the model (given the parameters estimated in steps one and two).
Chapter 2: Applications of discrete and dynamic choice models 26
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