Value function iteration with habit

I would like to know how I could write a value function when there are habits in preferences. I have the following equations:

$$u\left(C, t, H_{t}, L_{t}\right)=\frac{\left(C_{t} / H_{t}^{\kappa}\right)^{1-\gamma}}{1-\gamma}-\theta \frac{L_{t}^{1+1 / \chi}}{1+1 / \chi}$$ Habits are external in the sense that they depend on past aggregate consumption, so from the perspective of atomistic households, they are exogenous. $$H_{t}=C_{t-1}$$

Resource constraint gives: $$Y_t=C_t+ I_t=C_t +K_t-(1-\delta) K_{t-1}$$. There is technology $$Z_t$$ following a standart AR(1) process.

How should I write the policy functions in Matlab's code? Should I write them as 2 dimensional, that is to say, they would depend only on the grid of capital and the grid of technology? Or should they be 3 dimensional, namely they would depend on the grid of capital, the grid of technology, and the grid of $$H_t$$ ?

Thank you

• You seem to have three law of motions: (i) $K_{t+1} = Y_t - C_t + (1-\delta) K_t$, (ii) $Z_t = \rho Z_{t-1} + \varepsilon_t$ and (iii) $H_{t+1} = C_t$, so I guess your state space will indeed be 3 dimensional.
– tdm
May 11, 2021 at 10:21