I have to discuss a macroeconomics paper in which the author maximizes utility subject to a dynamic budget constraint and derives first order conditions by forming an HJB equation. This is my first time working with stochastic optimization problem in continuous time and I been struggling with it for quite a while. Being able to solve this will help me and others who are working with similar problems. Here is the problem set-up:
- A continuum of agents with utility $$ U(c,m) = \mathbb{E}\left[ \int_{0}^{\infty}e^{-\rho t}((1-\beta)\log c_{t} + \beta\log m_{t})\,dt \right], \hspace{1cm} m \equiv M/p $$ where $c_{t}$ is consumption, $m_{t}$ is real money, $\rho$ is discount factor, $\beta$ is a constant, $M$ is nominal money and $p$ is nominal price.
- Capital is used to produce consumption according to $y_{t} = ak_{t}$ where $y_{t}$ is consumption, $a$ is a constant and $k_{t}$ is capital.
- Capital is subject to idiosyncratic Brownian quality of capital shocks $$ d\Delta_{i,t}^{k} = k_{i,t}\sigma d W_{i,t} $$ which wash away in the aggregate.
- Aggregate capital stock evolves as $dk_{t} = (x_{t} - \delta k_{t})dt$ where $x_{t}$ is investment and $\delta$ is depreciation.
- Aggregate resource constraint is $c_{t} + x_{t} = ak_{t}$
- Total wealth in the economy is $w_{t} = k_{t} + m_{t} + h_{t}$ where $h_{t}$ is capitalized real value of future money transfers $$ h_{t} = \int_{0}^{\infty}e^{-\int_{t}^{s}r_{u}d_{u}}\frac{dM_{s}}{p_{s}} $$
- Dynamic budget constraint for an agent $$ dw_{t} = (r_{t}w_{t} + k_{t}\alpha_{t} - c_{t} - m_{t}i_{t})dt + k_{t}\sigma dW_{t} $$ where $w_{t}$ is wealth, $r_{t}$ is real interest rate, $\alpha_{t} \equiv a - \delta - r_{t}$ is excess return on capital, $i_{t} = r_{t} - \pi$ is nominal interest rate, $\pi$ is inflation rate, $\sigma$ is a constant and $W_{t}$ is Brownian motion.
- The first order conditions with respect to $c_{t}$, $k_{t}$ and $m_{t}$ are written in the said paper as follows. There is no time subscript $t$ since all variables are growing at the same rate on the balanced growth path and they have been normalized by $k_{t}$; e.g. $\hat{m}_{t} = \dfrac{m_{t}}{k_{t}}$ $$ r = \rho + (\hat{x} - \delta) - \sigma_{c}^{2} $$ $$ r = \alpha - \delta - \sigma_{c}\sigma $$ $$ \hat{m} = \frac{\beta}{1 - \beta}\frac{a - \hat{x}}{r + \pi} $$
My problem is I have been trying to derive the HJB for this problem and find the above three FOCs myself but I am not getting there. I formed the following HJB using Ito's Lemma $$ 0 = \max_{c,m,k}\{ ((1-\beta)\log c + \beta\log m) - \rho J + (x - \delta k)J^{\prime} + \frac{1}{2}k^{2}\sigma^{2}J^{\prime\prime} \} $$
I then take FOCs with respect to $c$, $m$ and $k$: $$ \frac{1-\beta}{c} = J^{\prime} \quad \text{where I made use of}\quad x_{t} =ak_{t} - c_{t} $$ $$ \frac{\beta}{m} = 0 $$ and $$ \delta J^{\prime} = k\sigma^{2}J^{\prime\prime} $$ I have trouble getting past this point. I spent a lot of time on this but cannot figure out what to do next even though I know the answers as written above. Please help!