Coursework 2025 - Cake Eating Problem

Name:

Surname:

After completing the following questions, send the edited notebook, to pablo.winant@ensae.fr

You are allowed to use any online available resource, even to install Julia packages, but not to copy/paste any code.

Also, don’t forget to comment your code and take any initiative you find relevant.

Part Ι: Useful Skills

  1. Create a random 10x10 matrix R using rand(10,10). Normalize each row in R, in order to create a stochastic matrix (the sum of each row is equal to 1, and all coefficients are positive). Compute the only vector v with positive coefficients summing to 1, such that v' R == v'
# your code...
# R = ...
# v = ...
# check
@assert maximum(abs, v' R - v')<1e-8
  1. Find \(u\) such that \(tanh(u)=0.5\) by writing your own solver (no solver library but any algorithm you want).
# your code...
# u = 
# check
@assert maximum(tanh(u) - 0.5)<1e-8
@assert maximum(u-tan(0.5))<1e-8

Part II: Cake-Eating Problem with Regeneration

We consider the following variant of the cake-eating problem. It can be interpreted as a model of sustainable consumption for a renewable resource.

An agent has a finite amount of cake \(k_t\) and must decide in each period how much of it to consume: \(c_t\in[0,k_t]\).

The part of the cake that is not consumed degenerates at rate \(\delta\) but new cake is also regenerated at rate \(\rho\) so that the law of motion for the cake is:

\[k_{t+1}=\delta (k_{t}-c_{t}) + \rho\]

with \(\delta\in]0,1]\) and \(\rho>0\)..

The goal is to maximize the present discounted value of utility over an infinite horizon:

\[\sum_{t\geq0} \beta^t u(c_t)\]

with \(u(x) = \frac{x^{1-\sigma}}{1-\sigma}\)

We can use the parameters \(\beta=0.95\), \(\sigma=2\), \(\delta=0.9\).

  1. What is the maximal value \(\overline{K}\) for the cake that can be obtained using natural regeneration? Choose \(N>0\) and discretize the values for the cake into a vector \(K=(K_1, ... ,K_N)\) of linearly spaced values.
# your code
  1. Create a named tuple (or any other Julia construct) to store all parameters of the (discretized) model. In what follows, all Julia functions will take an additional parameter m carrying all these informations.
# your code

Since the transition for the cake is deterministic, we can identify the choice of consumption with the choice of next period cake. Here we will make the assumption that next period level must stay on the discretized grid.

This yields the following, discretized, Bellman equation \[V(K_i) = \max_{j \in A(i)} U(K_i,K_j) + \beta V(K_j)\] where \(A(i)\) is the set of all possible cake indices that can be chosen for tomorrow given today’s index and where \(U(K_i,K_j)\) is utility obtained from consumption today.

  1. What is \(U(K_i,K_j)\)? Write a function U(a,b,m)::Float64 to represent it in Julia.
# your code
  1. Write a function A(i,m)::Tuple{Int, Int} which maps the index \(i\) of today’s cake \(K_i\) with a tuple containing the lowest and highest possible indices for the cake tomorrow.
# your code
  1. Make an initial guess v0 for the value function represented (a vector of Float64 of length N) and an admissible guess x0 for the optimal choices (a vector of Int64 of length N).
# your code
  1. For the initial guess, plot the values and the consumptions against the size of the cake
# your code
  1. Write a function bellman_step(v::Vector{Float64}, m)::Tuple{Vector{Float64}, Vector{Int64}} which associates \(\tilde{V}(K_1), ..., \tilde{V}(K_N)\) to \(V(K_1), ... V(K_N)\) in \[V(K_i) = \max_{j \in A(i)} U(K_i,K_j) + \beta \tilde{V}(K_j)\] This function returns the new value and the controls resulting from the optimization.
# your code
  1. Implement the value function algorithm to find the optimal consumption policy by iterating on Bellman step until convergence.
# your code
  1. Plot the optimal consumption rule.
# your code
  1. Implement the policy iteration algorithm.
# your code
Note

The policy iteration algorithm can be obtained from the value function algorithm: after each bellman step which yields a new policy rule \(p(i)\), compute the value of this policy, i.e. the value \(V\) such that:

\[V(K_i) = \left. \left\{U(K_i,K_j) + \beta V(K_j)\right\} \right|_{j=p(i)}\]

  1. Comment on the results (interpret, vary the parameters, …)
# your code