# Habit forming Model & State Variables

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.

1. 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.

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.