Habit forming Model & State Variables

Question

Taking the habit formation model of consumption, as a standard dynamic programming problem.

Bellman Value Function for Habit Model

Max$\sum_{t=1}^Tβ^tu(c_t - γc_{t-1})$ $\qquad \qquad (1)$

s.t. $w_{t+1} = (1+r)(w_t + y_t - c_t)$,

Here $c_t$ is consumption at time $t$, $w_t$ are financial assets yielding constant net interest $r$, $y_t$ is labour income.

1. How would I determine the state variables. They are stated as $w_t$ and $c_{t-1}$ I believe this is because these change endogenously within the model, and they both impact future consumption. However I'm not satisfied with this answer, and i'm not 100% certain why $y_t$ isn't a state variable.

2. Discount Factor:

In general these models always end up with a Bellman Equation looking something like:

$V(w_t, c_{t-1}, t) = max_{c_t} {u(c_t, c_{t-1}) + β V(w_{t+1}, c_t, t+1)}$ $\qquad \qquad (2)$

$s.t. w_{t+1} = (1+r)(w_t + y_t - c_t)$

I understand how to derive the Bellman equation when we don't have a discount factor, but how do we derive the equation with a discount factor, particularly so that the $t$ power on $β$  goes away.

3. Is my understanding of the model correct:

• $V(w_t, c_{t-1},t)$ is the value function, which represents the lifetime utility of an agent who starts with initial wealth from financial assets $w_t$ and previous consumption $c_{t-1}$ at time $t$ and makes optimal consumption decisions over their lifetime.
• $V(w_t, c_{t-1},t+1)$ is the value function for our $t+1$ up to time $T$ i.e. it captures the value from optimal decision making, working backwards from terminal time point $T$ .i.e. through backwards induction?
• Our problem is to maximise consumption decisions maximise value today as given by $u(c_t - γc_{t-1})$ and future value as given by $V(w_t, c_{t-1},t+1)$
• $u(c_t - γc_{t-1})$ is the utility function, which captures the instantaneous utility that an individual derives from consuming $c_t$ at time $t$, given their previous consumption $c_{t-1}$.
• $\beta$ is the discount factor, which determines the individual's time preference for consumption.

Answer 1

It seems that your transition equation is not standard. I took the liberty to change it like this: $$ \begin{align} &\max \sum^T_{t=0} \beta^t ( u(c_t) + \gamma u(c_{t-1}) ), \\ &\text{s.t. } c_t + w_{t+1} = (1+r)w_t + y_t. \end{align} $$ In your formulation, both income and consumption yield interest, which is not normal. So, I wrote in a form that the LHS shows the expenses, and the RHS shows the source of funding. And $y_t = f(w_t)$ where $f'>0,f''<0$. (By the way, $w$ seems like a weird choice of the variable for assets because it normally represents wage. I think it's better to use $k$.)

There will be 2 state variables $(c_{t-1}, w_t)$. Based on that, you can write the Bellman equation

$$ V(c_{t-1},w_t) = \max_{c_t, w_{t+1}} \left( u(c_t) + \gamma u(c_{t-1}) + \beta V(c_t, w_{t+1}) \right). $$ Using the constraint, we can reduce it to a problem of 1 choice variable $$ V(c_{t-1},w_t) = \max_{w_{t+1}} \left( u( \underbrace{(1+r)w_t + f(w_t) - w_{t+1}}_{c_t} ) + \gamma u(c_{t-1}) + \beta V( \underbrace{(1+r)w_t + f(w_t) - w_{t+1}}_{c_t}, w_{t+1}) \right). $$ Your choice variable is $w_{t+1}$, so you take FOC with respect to it $$ u'(c_t) + \beta V_1(c_t, w_{t+1}) = \beta V_2(c_t, w_{t+1}). $$ The envelope theorem conditions are $$ \begin{align} V_1(c_{t-1}, w_t) &= \frac{\partial V}{\partial c_{t-1}} = \gamma u'(c_{t-1}), \\ V_2(c_{t-1}, w_t) &= \frac{\partial V}{\partial w_t} = [(1+r) + f'(w_t)]u'(c_t) + \beta [(1+r) + f'(w_t)] V_1 (c_t, w_{t+1}). \end{align} $$ Forwarding one period, we have $$ \begin{align} V_1(c_t, w_{t+1}) &= \gamma u'(c_t), \\ V_2(c_t, w_{t+1}) &= [(1+r) + f'(w_{t+1})](u'(c_{t+1}) + \beta V_1(c_{t+1}, w_{t+2})]. \end{align} $$ Combining these 2 yields $$ V_2 (c_t, w_{t+1}) = [(1+r) + f'(w_{t+1})](1 + \beta \gamma) u'(c_{t+1}). $$ Replacing $V_1(c_t, w_{t+1})$ and $V_2(c_t, w_{t+1})$ back to the FOC to get $$ u'(c_t) + \beta\gamma u'(c_t) = \beta[(1+r) + f'(w_{t+1})](1 + \beta \gamma) u'(c_{t+1}), $$ which can be written as $$ (1+\beta\gamma) u'(c_t) = \beta [(1+r) + f'(w_{t+1})](1 + \beta \gamma) u'(c_{t+1}). $$ Cancelling common terms and rearranging give you the Euler equation $$ \frac{u'(c_t)}{u'(c_{t+1})} = \beta[(1+r) + f'(w_{t+1})], $$ which does not depend on the lagged consumption term. Sargent (1987) explains that the choice of $c$ involves comparing the loss in current marginal utility and the gain in future consumption so the state $c_{-1}$ does not influence the choice of $c$. It only has a scale effect on the utility function.

From here, you can use Guess and Verify to work out the Value function. Since we have 2 state variables, the usual form should be $$ V(c_{t-1}, w_t) = E + F \ln w_t + G \ln c_{t-1}. $$ Since the Euler equation is the same as a problem without lagged consumption, you can guess that the Value Function is identical to a problem without Habit Formation.