Homework 2024

Name:

Surname:

After completing the following questions, send the edited notebook to pablo.winant@ensae.fr. You can work in (small) teams but are not allowed to copy/paste any code. Don’t forget to comment your code and take any initiative you find relevant.

General Problem

We seek to solve numerically the following consumption-saving problem using the time-iteration algorithm.

Income is an iid normally distributed process with standard deviation \(\sigma\). Interest rate is \(r\in [1,\frac{1}{\beta}[\) where \(\beta\in]0,1[\) is the discount factor.

Available income is \(w_t\) and follows the law of motion:

\[w_{t+1} = e^ {\epsilon_{t+1}} + (w_{t} - c_{t}) r\]

where \(c_t \in ]0,w_t]\) is consumption chosen at date \(t\) (which includes an implicit no borrowing constraint).

Calibration

We will use the following calibration:

\(\sigma = 0.01\)

\(\beta = 0.96\)

\(r = 1.03\)

Choose a structure model to represent the model parameters.


# model = ...

Representing the decision function

The solution of the model is a decision rule \(c(w)\).

In what follows, we will approximate it by a function defined over \([0,w_{max}[\), pinned down by the values it takes on \(N\) linearly spaced grid points.

In our solutions algorithm, we will use the initial guess \(\varphi_0(w) = min(w, p_0+p_1(w-p_0))\).

Create a structure approx to represent the approximation space. This structure should contain the parameters, as well as a points::Vector{Float64} array coontaining the (regularly spaced) list of grid points.

To start with you can take \(w_{max}=5\), \(N=20\), \(p_0=0.95, p_1=0.03\) but don’t hesitate to change these values later if needed.

# approx = ...

Define a method φ_0(w::Float64)::Float64 for the initial guess. Plot it against \(w\) in the approximation space.

Compute the values c_0::{Float64}of φ_0() on the grid points

Use the Interpolations.jl library to define a function φ(w::Float64) defined for any w using piecewise linear interpolation in such a way that it takes values c_0 on the grid points.

Create a single plot with:

  • the function φ_0()
  • the interpolated function \(φ\)
  • its values c_0 on the grid points

Bonus: add to the graph the function obtained using cubic spline interpolation. What could be the problem?

Discretizing \(\epsilon_{t+1}\)

Represent the shock \(\epsilon\) by two vectors \((e_1, ... e_q)\) and \((l_1, ..., l_q)\) with \(q=10\) such that for a suitable function \(g\) we can approximate \(E_{\epsilon} (g(\epsilon))\) by \(\sum_{i=1}^q l_i e_i\) .

(choose the method you want)

# e = ...
# w = ...

Test that it works by computing \(E_{\epsilon} \left[ \epsilon^2 \right]\).

Redefine approx so as to include the discretized shock

Euler equation

Write down the Euler equation, paying attention to the slackness condition.

It should be of the form \[0 \leq \underbrace{E_t \left[ f(w_t, c_t, w_{t+1}, c_{t+1}, \epsilon_{t+1}) \right]}_{\Phi_t} \perp c_t \leq w_t\]

where \(f\) is a function to be explicited.

Define the function Phi(w::Float64,c::Float64,φ::Fun, model, approx) which approximates the residuals of the euler equation given the available income today, the consumption choice today, the consumption function tomorrow and the model/approx structures.

Overload Phi function with another method Phi(w::Float64,c::Float64,φ::Fun, model, approx, slackness=true) which uses the Fisher-Burmeister transform to incorporate the credit constraint.

Plot the optization residuals (i.e. values of \(\Phi_t\)) corresponding to the initial guess function with and without the complementarity constraints.

Time Iteration

(prep): Assuming the initial guess \(φ_0()\) is the decision rule followed tomorrow, determine, for a given value w in the state-space, the optimal consumption choice made today. This can be achieved by feeding the appropriate function into the right nonlinear root-finder.

(prep): Find the vector of all the optimal consumption choices on the grid today, given the decision rule tomorrow.

Plot the optimization residuals (i.e. values of \(\Phi_t\)) for the initial guess function

Write down the time-iteration algorithm.

You can use the course as reference and/or repeat the recurrence steps below:

  • given an initial guess for the consumption vector c0
  • create a function defined φ on [0,wmax] which interpolates c0 on the grid
  • for each grid point w in the grid, solve the system u->Phi(w,u,φ, model, approx, slackness=true)
    • store the result as a vector c1
  • check whether c1 is close to c0
    • yes: check the system is indeed solved and return
    • no: start again with c1 as c0
# here is a placeholder implementation to help you structure your program.
#  feel free to modify or discard


"""
φ: Float64->Float64 Initial guess for the consumption function
model: parameters representing the model
approx: parameters defining the approximation and solution method
K: max number of iterations
... other parameters for you to choose
"""
function time_iteration(φ0, model, approx; K=200, ... )

    # ...

    # convert initial function φ0 into a consumption vector by evaluating φ on the grid
    # c0 = ... :: Vector{Float64}

    for k=1:K

        # convert consumption vector c0 into a function φ
        # φ = ...

        # solve for the optimal consumption for all grid points
        # ...
        # c1  :: Vector{Float64}

        # check distance between c0 and c1

        # η = ...

        # if distance is small return function `varphi`

    end




end

Check that the value returned by time_iteration is actually a solution to the system. How fast is the convergence? Graphical representation.

Bonuses

(easy): perform some sensitivity analysis on the model to explain the effect of the main parameters

(medium): Give the solution to the above problem, simulate the law of motion for the available income. Find a way to plot the distriution of that income over a long period of time.

(hard): Propose and implement some ideas to speed up the solution process.

One promising avenue consists in solving for all consumption values at all grid points as one single system of equation after recognizing that the jacobian of this particular system has a specific structure.