Dynamic programming with housing consumption and labor

Question

I try to solve the following maximization problem of a representative household with dynamic programming. However, my last result is not similar to the solution. Could any one help me?

$$\max\limits_{C_{t},H_t,N_t} E_0 \sum_{t=0}^{\infty} \beta^t\Bigg[logC_t+jlogH_t-\dfrac{(N_t)^\eta}{\eta}\Bigg]$$ $C_t,H_t,N_t$ represent consumption at time t, housing stock and working hours, respectively. Household budget constraint:

$C_t + b_t +q_t(H_t-H_{t-1})=\dfrac{R_{t-1} b_{t-1}}{\pi_t}+w_t N_t+F_t$

$b_t$ denotes bank deposits, $R_t$ is the return from deposit, $q_t$ is the price of housing, $\pi_t$ is the inflation rate and $w_t$ is the real wage rate. $F_t$ denotes profits received from firms. To sole this problem, I use dynamic programming with these steps: Step 1: Bellman equation:

I identify $b_{t-1}$ as the state variable, since the household must know the deposit in the previous period. The budget constraint can be rewrite as: $b_t=f(C_t,b_{t-1},H_t,H_{t-1},N_t,F_t,q_t,R_{t-1},\pi_t)$

Step 1: Setting up the Bellman equation:

$V(b_{t-1})=\max\limits_{C_{t},H_t,N_t} \Bigg\{logC_t+jlogH_t-\dfrac{(N_t)^\eta}{\eta} +\beta E_tV(b_t) \Bigg\}$

Evolution of the costate variable $b_t$, using envelope theorem:

$$\dfrac{\partial V(b_{t-1})}{\partial b_{t-1}} =\beta E_t \dfrac{\partial V(b_{t})}{\partial b_{t}} \dfrac{\partial b_t}{\partial b_{t-1}} = \beta E_t V'(b_t) \dfrac{R_{t-1}}{\pi_t}$$ (1)

Step 2: FOC

First order condition (FOC1) $$\dfrac{\partial V(b_{t-1})}{\partial C_t}=\frac{1}{C_t}+\beta E_t V'(b_t)\dfrac{\partial b_t}{\partial C_t}=0$$ $\dfrac{\partial b_t}{\partial C_t}=-1$

$\dfrac{1}{C_t}=\beta E_t V'(b_t)$ (2)

Inserting (2) into (1) twice to get the Euler equation, we have:

$$\dfrac{1}{C_t}=\beta E_t \Bigg(\dfrac{R_t}{\pi_{t+1 C_{t+1}}}\Bigg)$$

This result is similar to the solution

FOC2: $$\dfrac{\partial V(b_{t-1})}{\partial N_t}=-(N_t)^{\eta-1}+\beta E_t V'(b_t)w_t=0$$ Inserting (2), we have: $\dfrac{1}{C_t} w_t = -(N_t)^{\eta-1}$, This result is similar to the solution

FOC3: The third FOC involves taking the partial derivative of $V(b_{t-1})$ with $H_t$, but I cannot get the same result with the solution: The solution is: $$\dfrac{j}{H_t}=\dfrac{1}{C_t}q_t-\beta E_t \dfrac{1}{C_{t+1}}q_{t+1}$$

I only end up with:

$$\dfrac{\partial V(b_{t-1})}{\partial H_t}=\dfrac{j}{H_t}+\dfrac{1}{C_t}(-q_t)=0$$ Did I miss something? Did I correctly identify the state variable? I suspect that it has something to do with $H_t$ and $H_{t-1}$, but I cannot figure it out

Answer 1

$$V(b_{t-1})=\max\limits_{C_{t},H_t,N_t} \Bigg\{\ln C_t+j\ln H_t-\dfrac{(N_t)^\eta}{\eta} +\beta E_tV(b_t) \Bigg\}$$

so

$$\beta E_tV(b_{t})=\beta E_t\left[\max\limits_{C_{t+1},H_{t+1},N_{t+1}} \Bigg\{\ln C_{t+1}+j\ln H_{t+1}-\dfrac{(N_{t+1})^\eta}{\eta} +\beta E_{t+1}V(b_{t+1}) \Bigg\}\right]$$

But forwarding the budget constraint

$$C_t + b_t +q_t(H_t-H_{t-1})=\dfrac{R_{t-1} b_{t-1}}{\pi_t}+w_t N_t+F_t$$

we get after re-arranging,

$$C_{t+1} =- b_{t+1} -q_{t+1}(H_{t+1}-H_{t})+\dfrac{R_{t} b_{t}}{\pi_{t+1}}+w_{t+1} N_{t+1}+F_{t+1}$$

So $\beta E_tV(b_{t})$ is also a function of $H_t$ through $C_{t+1}$.