EatCake Problem 1

[erineidschun]

I am confused about steps 3 and 4 of problem 1. Why do we have two matrices of two different sizes? It does not make sense to me how to iterate the value function here.

Is this along the right lines, where X is the original Nx Nm matrix made in question 2?
for i in range(N-1):
for j in range(N-1):
Vt[i][j] =X[i][j] + beta*Vt[i+1][j+1]*w[j]

[jmbejara]

There is a state variable (the initial W_t) and a choice variable (W_{t+1}). The choice that you make for the choice variable becomes the state variable next period. We're solving the problem of what choice to make for W_{t+1} given W_t for each possible state W_t. The N x N matrix gives us a matrix of possible values of consumption C_t = W_{t+1} - W_t for each possible W_t that we may encounter. If each row represents a different choice of W_{t+1}, then each column is a potential W_t we may encounter. Then, for each column, we must choose the optimal row. This choice of optimal row defines the policy function.

In your code, you need some way of choosing the best W_{t+1} given W_t.

Hope this helps!

[erineidschun]

What is the N X T+2 matrix for? Is it supposed to hold the V_T values? If not, how are we supposed to fill it out?

And for the second NxN matrix (question 4), do we just use values from the NxN matrix in question 2 and the NX T+2 matrix in question 3?

[jmbejara]

V_t takes values for t=0,1,2,...,T. Thus, we need T+1 columns. The extra column at t=T+2 is supposed to be a column of zeros representing V_{T+1}. This is a trick for convenience. The value function V_{T+1} is by definition zero, because no utility is gained after time T. This allows us to write the consumption-savings decision at time T the same as we would at time T-1. It's just that there is no benefit to saving cake at time T, because no consumption can occur at time T+1.

Yeah, I think you can use the same NxN matrix.