Assignment 5

Introduction and Setup

In this assignment you will implement a choice probability estimator for a toy model of dynamic labor supply.

Choices

Agents in the model make one of four choices (\(j \in \{0,1,2,3\}\)) that correspond to a binary work decision (\(H_{j}\)) and a binary program participation decision (\(P_{j}\)):

\[ \{H_0,\ H_1,\ H_2,\ H_3\} = \{0,1,0,1\},\qquad \{P_0,\ P_1,\ P_2,\ P_3\} = \{0,0,1,1\} \]

Income and Transfers

Potential earnings are given by:

\[ \log(W) = \phi_0 + \phi_1 \kappa \]

where \(\kappa\) is accumulated labor market experience that evolves according to:

\[ \kappa_{t+1} = \kappa_{t} + H_{jt} \]

Weekly net income is a function of this work decision and the participation decision:

\[ Y_{j}(W,\omega) = H_{j}W + P_{j}\mathbf{1}\{\omega<\Omega\}(200 - 0.5\times H_{j}W) \]

where:

  • \(W\) is the individual’s potential wage
  • \(\Omega\) is the lifetime limit on program participation
  • \(\omega\) is the individual’s cumulative prior program use.

Utility

Individuals discount the future at an exponential rate of \(1-\beta\). Per-period payoffs are given by:

\[ u_{j}(W) = \log(\gamma + Y_{j}(W,\omega)) - \alpha P_{j} - \varphi H_{j} \]

The per-period payoff for any individual \(n\) at time \(t\) is given by

\[ U_{ntj} = u_{j}(W_{nt}) + \epsilon_{ntj} \]

where \(\epsilon_{nt}\) is a preference shock that has a type 1 extreme value distribution with scale parameter \(\sigma\). It is iid across individuals and time periods and us only observed by individuals when period \(t\) arrives.

Life-Cycle

All individuals begin in \(t=1\) with \(\omega_{1}=0\) and \(\kappa_{1}=0\) and live for \(T\) periods.

Measurement

Assume that log-wages are observed with measurement error:

\[ \log(W^o) = \log(W) + \xi,\qquad \xi\sim\mathcal{N}(0,\sigma^2_\xi) \]

Model Solution

Here is code to solve the model for a default set of parameters.

T = 10 #<- length of life-cycle
ϕ = [2., 0.05]
Ω = 4 #<- time limit
wagefunc(κ,ϕ) = exp(ϕ[1] + ϕ[2]*κ)
net_income(W,H,P,eligible) = W*H + P*eligible*(200 - 0.5*W*H) 
j_idx(H,P) = P*2 + H + 1
utility(W,H,P,eligible,pars) = log(pars.γ + net_income(W,H,P,eligible)) - pars.α*P - pars.φ*H   
pars = (;α = 1., φ = 1., σ = 1., γ = 1., Ω, T,ϕ,β = 0.9)

function iterate!(P,v,V,pars,ω_idx,κ_idx,t)
    (;β,ϕ,Ω,σ) = pars
    ω = ω_idx - 1
    κ = κ_idx - 1
    W = wagefunc(κ,ϕ)
    eligible = ω<Ω
    for P in 0:1, H in 0:1
        j = j_idx(H,P)
        κ_next = min(κ_idx + H,T)
        ω_next = min(ω_idx + P,Ω+1)
        v[j]  = utility(W,H,P,eligible,pars) + β*V[ω_next,κ_next,t+1]
    end
    norm = sum(exp.(v ./ σ))
    P[:,ω_idx,κ_idx,t] = exp.(v ./ σ) ./ norm
    V[ω_idx,κ_idx,t] = σ * log(norm)
end

function solve!(P,v,V,pars)
    for t in reverse(axes(P,4))
        for κ_idx in axes(P,3), ω_idx in axes(P,2)
            iterate!(P,v,V,pars,ω_idx,κ_idx,t)
        end
    end
end

V = zeros(Ω+1,T,T+1)
P = zeros(4,Ω+1,T,T)
v = zeros(4)

solve!(P,v,V,pars)

Part 1

Write a function that simulates a panel dataset consisting of choices and wages for \(N\) individuals in every period \(t\) of their life-cycle.

Part 2

For a simulated sample of size \(N=1000\), write a function to estimate the array \(P\) of choice probabilities using a frequency estimator. Note that in period \(t\), only experience levels up to \(t-1\) can be observed so you do not need to estimate this array in its entirety.

Part 3

Write an estimation routine for the structural parameters \((\phi,\alpha,\varphi,\sigma,\gamma,\beta)\) that utilizes the first stage estimates \(\hat{P}\) from part 2. You can either work backwards from \(T\) or exploit shorter finite dependence horizons.

One small hint: you can estimate \(\phi\) directly using OLS.

Part 4

If you can, write a monte-carlo simulation in which you repeatedly draw a sample of size \(N=1000\) and produce an estimate. Use this sample to look at how the estimates are distributed around the true parameters and therefore check that your estimator is working properly.