Advanced Consumption Models

Habit Formation, Durable Goods, and Quasi-Hyperbolic Discounting

Alan Lujan

Johns Hopkins University

February 17, 2025

Habit Formation

Utility with Habits

Consider a consumer whose past consumption affects current utility (Carroll, 2000). The goal is to solve

\[\max \sum_{n=0}^{T-t} \beta^n \, u(c_{t+n}, h_{t+n})\]

where the habit stock $h_t$ enters the utility function with $u^h < 0$: higher past consumption raises the reference point, making any given level of current consumption less satisfying.

The dynamic budget constraint is the usual

\[m_{t+1} = (m_t - c_t)R + y_{t+1}\]

How Habits Evolve

Simple version: last period’s consumption becomes this period’s habit stock:

\[h_{t+1} = c_t\]

General version: habits adjust partially toward consumption:

\[h_{t+1} = h_t + \lambda(c_t - h_t)\]

When $\lambda = 1$, the general version reduces to the simple one.

In either case, a consumer who has been spending lavishly finds it painful to cut back, because today’s utility is judged against yesterday’s standard.

The Bellman Equation

With two state variables $m_t$ and $h_t$, Bellman’s equation becomes

\[v_t(m_t, h_t) = \max_{c_t} \left\{ u(c_t, h_t) + \beta \, v_{t+1}((m_t - c_t)R + y_{t+1}, \, c_t) \right\}\]

Note the second argument of $v_{t+1}$: tomorrow’s habit stock is $h_{t+1} = c_t$, so the choice of $c_t$ affects the future through two channels: it reduces resources via $m_{t+1}$ and raises the habit stock via $h_{t+1}$.

First Order Condition

Differentiating with respect to $c_t$ yields

\[u^c_t = \beta\left(R \, v^m_{t+1} - v^h_{t+1}\right)\]

Without habits, $v^h_{t+1} = 0$ and we recover the standard FOC. With habits, the right-hand side is larger (since $v^h_{t+1} < 0$), so a higher marginal utility is needed. The consumer who accounts for habits chooses a lower level of consumption.

Intuition: spending a dollar today not only depletes resources but also raises the bar against which future consumption is judged.

Envelope Condition for $m_t$

Applying the envelope theorem, treating $c_t$ as constant when differentiating with respect to $m_t$:

\[v^m_t = \beta R \, v^m_{t+1}\]

This is identical to the standard envelope result. An extra dollar of wealth still raises value by the return on saving, regardless of habits.

Envelope Condition for $h_t$

Differentiating Bellman’s equation with respect to the habit state, again treating $c_t$ as constant:

\[v^h_t = u^h_t\]

The marginal value of a higher habit stock equals the direct marginal utility effect. Habits enter only through today’s utility, not through the budget constraint.

Combining the Optimality Conditions

From the FOC and the envelope results:

\[v^m_t = u^c_t + \beta \, v^h_{t+1} = u^c_t + \beta \, u^h_{t+1}\]

Rolling forward one period and substituting into $v^m_t = \beta R \, v^m_{t+1}$:

\[\boxed{u^c_t + \beta \, u^h_{t+1} = R\beta \left[u^c_{t+1} + \beta \, u^h_{t+2}\right]}\]

When $u^h = 0$, this reduces to the standard Euler equation $u'(c_t) = R\beta \, u'(c_{t+1})$.

A Specific Utility Form

Suppose utility depends on consumption relative to habits:

\[u(c, h) = f(c - \alpha h)\]

where $\alpha \in (0, 1)$ is the habit strength. Define $z_t = c_t - \alpha h_t$. The derivatives are $u^c = f'$ and $u^h = -\alpha f'$.

With $h_{t+1} = c_t$, the surplus evolves as

\[z_t = c_t - \alpha c_{t-1}\]

Serial Correlation in Consumption Growth

Assuming $f(z) = z^{1-\rho}/(1-\rho)$ and a perfect-foresight solution where $f'$ grows at a constant rate, one can show (after log-linearization):

\[\Delta \log c_{t+1} \approx \frac{1-\alpha}{\rho}\log(R\beta) + \alpha \, \Delta \log c_t\]

Habit formation predicts that consumption growth is positively serially correlated: a period of rising consumption tends to be followed by further rises, because the habit stock adjusts gradually.

When $\alpha = 0$, we recover the standard result with no serial correlation.

Code: Habit Formation and Consumption Dynamics

Durable Goods

What Makes a Good “Durable”?

A durable good provides utility over multiple periods rather than being consumed immediately. The consumer maximizes

\[\max \sum_{s=t}^{T} \beta^{s-t} u(c_s, d_s)\]

where $c_s$ is nondurable consumption and $d_s$ is the stock of the durable good.

Examples: housing, cars, appliances. The key distinction is that spending on durables (a flow) differs from the stock that yields utility.

Stock Accumulation

The durable stock evolves as

\[d_{t+1} = (1 - \delta)d_t + x_{t+1}\]

where $x_t$ is period-$t$ expenditure and $\delta$ is the depreciation rate. A lower $\delta$ means a more durable good.

The budget constraint subtracts both nondurable and durable expenditures:

\[m_{t+1} = (m_t - c_t - x_t)R + y_{t+1}\]

The Two-Control Bellman Equation

Treating $d_t$ (stock) as the control instead of $x_t$ (expenditure), with $x_t = d_t - (1-\delta)d_{t-1}$:

\[v_t(m_t, d_{t-1}) = \max_{c_t, d_t} \left\{ u(c_t, d_t) + \beta \, v_{t+1}(m_{t+1}, d_t) \right\}\]

where $m_{t+1} = (m_t - c_t - d_t + (1-\delta)d_{t-1})R + y_{t+1}$.

This problem has two control variables and two state variables.

First Order Conditions

With respect to $c_t$:

\[u^c_t = R\beta \, v^m_{t+1}\]

With respect to $d_t$:

\[u^d_t = R\beta \, v^m_{t+1} - \beta \, v^d_{t+1}\]

The nondurables FOC is standard. The durables FOC balances the marginal utility of the durable stock against both the opportunity cost of funds ($R\beta \, v^m_{t+1}$) and the continuation value of the stock ($\beta \, v^d_{t+1}$).

Envelope Conditions

For $m_t$ (treating both controls as constant):

\[v^m_t = R\beta \, v^m_{t+1}\]

For $d_{t-1}$ (the durable stock inherited from last period):

\[v^d_t = (1-\delta) R\beta \, v^m_{t+1} = (1-\delta) v^m_t\]

Interpreting $v^d = (1 - \delta) v^m$

When $\delta = 1$: the “durable” depreciates completely, so $v^d_t = 0$. Last period’s stock has no effect on current value, exactly as for a nondurable.

When $\delta = 0$: the good never depreciates, so $v^d_t = v^m_t$. An extra unit of the indestructible durable is worth exactly one unit of wealth, because it can be costlessly converted back and forth.

For intermediate values, $v^d_t$ is a fraction $(1-\delta)$ of the marginal value of wealth.

The Intratemporal Condition

Combining the two FOCs with the envelope results:

\[u^d_t = u^c_t - \frac{(1-\delta)}{R} u^c_t = \frac{r + \delta}{R} \, u^c_t\]

\[\boxed{u^d_t = \left(\frac{r + \delta}{R}\right) u^c_t}\]

The marginal utility of the durable in the current period is lower than the marginal utility of the nondurable. Why? Because the durable yields utility in future periods too. You do not buy a car because it is worth $20,000 today; you buy it because its discounted lifetime value exceeds $20,000.

Cobb-Douglas Utility

With $u(c, d) = \frac{(c^{1-\alpha} d^{\alpha})^{1-\rho}}{1-\rho}$, the intratemporal condition implies a constant ratio:

\[\frac{d}{c} = \frac{\alpha}{1-\alpha} \cdot \frac{R}{r + \delta} \equiv \gamma\]

Whenever nondurable consumption changes, the durable stock adjusts proportionally.

Why Durable Spending Is Volatile

Since expenditure is $x_t = d_t - (1-\delta)d_{t-1}$, a small change in $c$ triggers a large adjustment in $x$. If nondurable consumption grows by a fraction $\epsilon_t$ (so $c_t/c_{t-1} = 1 + \epsilon_t$) from a previously stable level:

\[\frac{x_t}{x_{t-1}} = \frac{\epsilon_t + \delta}{\delta}\]

For a good with low depreciation (say $\delta = 0.05$), a 5% consumption adjustment doubles durable spending. The stock-to-income ratio is much larger than the spending-to-income ratio, so even small income revisions cause large spending swings.

Code: Durables Spending Volatility

Frequency and Correlation

The degree of correlation between nondurables and durables spending depends on the time horizon. For a quarterly depreciation rate of 5%, the good almost fully depreciates over 10 years (40 quarters: $0.95^{40} \approx 0.12$).

Over intervals long enough for the stock to have depreciated, durables spending growth matches nondurables spending growth, because both are effectively nondurable at that frequency.

At high frequencies: durables spending is far more volatile and weakly correlated with nondurables. At low frequencies: the two converge.

Quasi-Hyperbolic Discounting

Present Bias

Standard exponential discounting treats all future periods symmetrically: the discount between periods $t+5$ and $t+6$ is the same $\beta$ as between $t$ and $t+1$.

Laibson (1997) proposes that something special happens “now.” Brain regions associated with emotional reward activate for immediate gratification but not for future prospects (McClure et al., 2004).

The quasi-hyperbolic model captures this by adding an extra discount factor $\delta_h < 1$ that applies only to the step from “now” to “all of the future.”

Two Value Functions

Given a period-$t+1$ value function $v_{t+1}(m_{t+1})$ and any consumption rule $\chi_t$:

\[v_t(m_t; \chi_t) = u(\chi_t(m_t)) + \beta \, \mathbb{E}_t[v_{t+1}((m_t - \chi_t(m_t))R + y_{t+1})]\]

\[\mathfrak{v}_t(m_t; \chi_t) = u(\chi_t(m_t)) + \delta_h\beta \, \mathbb{E}_t[v_{t+1}((m_t - \chi_t(m_t))R + y_{t+1})]\]

The first function $v_t$ discounts the future by $\beta$ (the “long-run self”). The second $\mathfrak{v}_t$ discounts by $\delta_h\beta$ (the “present-biased self”).

Two Consumption Functions

The standard consumer and the Laibson consumer solve different problems:

\[\mathbf{c}_t(m_t) = \arg\max_c \left\{ u(c) + \beta \, \mathbb{E}_t[v_{t+1}] \right\}\]

\[\mathfrak{c}_t(m_t) = \arg\max_c \left\{ u(c) + \delta_h\beta \, \mathbb{E}_t[v_{t+1}] \right\}\]

Since $\delta_h < 1$, the Laibson consumer values the future less from today’s perspective and therefore consumes more. Laibson calibrates $\delta_h \approx 0.7$ at an annual frequency.

The Modified Euler Equation

The envelope theorem applied to $\mathfrak{v}_t$ gives $\mathfrak{v}^m_t = u'(c_t)$, and the FOC gives $u'(c_t) = \delta_h R\beta \, \mathbb{E}_t[v^m_{t+1}]$.

A useful identity links the two value functions:

\[\delta_h \, v_t = \mathfrak{v}_t - (1 - \delta_h) \, u(\mathfrak{c}_t(m_t))\]

Differentiating and substituting:

\[\boxed{u'(c_t) = \mathfrak{v}^m_t - (1 - \delta_h) \, u'(c_t) \, \mathfrak{c}^m_t}\]

Interpreting the Modified Euler Equation

When $\delta_h = 1$, we recover the standard Euler equation: $u'(c_t) = \mathfrak{v}^m_t$.

When $\delta_h < 1$, three things follow:

The Laibson term $(1-\delta_h) \, u'(c_t) \, \mathfrak{c}^m_t$ reduces the right-hand side, so a lower $u'(c_t)$ (higher $c_t$) satisfies the equation. The present-biased consumer spends more.
The magnitude of the bias depends on the MPC $\mathfrak{c}^m_t$. If the MPC is small, present bias barely matters.
The model captures the psychological tension: “eating dessert this one time won’t make me fat” (small one-period cost), “but if I always give in, the consequences are large” (perpetual deviation).

Code: Present Bias and Overconsumption

When Does Present Bias Matter?

The modified Euler equation reveals that present bias is most consequential when:

Condition	MPC is…	Present bias is…
Young, little wealth	High	Large
Wealthy, long horizon	Low	Small
Liquidity constrained	High	Large
Unconstrained buffer-stock	Moderate	Moderate

A retiree with a 30-year horizon and substantial savings has a low MPC, so quasi-hyperbolic discounting barely distorts behavior. A young worker living paycheck to paycheck has a high MPC, and the temptation to overspend is strong.

Summary

Key Takeaways (1/2)

Habit formation: utility depends on consumption relative to a habit stock; the modified Euler equation is $u^c_t + \beta u^h_{t+1} = R\beta[u^c_{t+1} + \beta u^h_{t+2}]$, and the calibrated model predicts positively autocorrelated consumption growth
Durable goods: the intratemporal condition $u^d = [(r+\delta)/R] \, u^c$ equates the marginal utility of durables (adjusted for future service) to that of nondurables; with Cobb-Douglas utility the stock-to-nondurables ratio is constant

Key Takeaways (2/2)

Durable spending volatility: because the stock-to-income ratio exceeds the flow-to-income ratio, even small permanent income shocks produce large swings in durable expenditure
Quasi-hyperbolic discounting: the extra discount factor $\delta_h \approx 0.7$ captures present bias; the modified Euler equation shows that the distortion scales with the MPC, so present bias matters most for liquidity-constrained or low-wealth consumers

How the Topics Connect

All three models extend the baseline consumption-Euler framework from Module 4:

Habits add a second state variable that makes the value of consumption depend on the past, generating persistence in consumption growth
Durables add a second control variable that separates the stock yielding utility from the flow of spending, explaining why durable expenditure is so volatile
Laibson modifies the discount structure itself, showing that time inconsistency matters most precisely when the MPC is high, that is, when the consumer is most constrained