Consumption Theory

The Envelope Condition, Perfect Foresight CRRA, Random Walk, and the Consumption Function

Alan Lujan

Johns Hopkins University

February 10, 2025

The Envelope Condition

The Multiperiod Problem

The consumer maximizes discounted lifetime utility:

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

subject to the dynamic budget constraint:

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

where \(m_t\) denotes market resources (cash-on-hand) at the beginning of period \(t\).

Bellman Equation

The problem in recursive form:

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

First order condition:

\[u'(c_t) = R\beta \, v'_{t+1}(m_{t+1})\]

This relates marginal utility today to the marginal value of resources tomorrow, but how do we characterize \(v'\)?

The Lower-Bound Function

Define a lower-bound function:

\[\underline{v}_t(m_t, c_t) = u(c_t) + \beta \, v_{t+1}((m_t - c_t)R + y_{t+1})\]

Its partial derivatives are:

\[\underline{v}_t^{\,c} = u'(c_t) - R\beta \, v'_{t+1}(m_{t+1})\]

\[\underline{v}_t^{\,m} = R\beta \, v'_{t+1}(m_{t+1})\]

Note that \(\underline{v}_t^{\,c} = 0\) at the optimum (this is the FOC).

Applying the Envelope Theorem

By the chain rule:

\[v'_t(m_t) = \underline{v}_t^{\,m} + c'_t(m_t) \cdot \underline{v}_t^{\,c}\]

Since the FOC implies \(\underline{v}_t^{\,c} = 0\), the second term vanishes:

\[v'_t(m_t) = \underline{v}_t^{\,m} = R\beta \, v'_{t+1}(m_{t+1})\]

The marginal value of \(m\) equals the return on saving, regardless of how the consumption function responds to \(m\).

Key Result: \(v'(m) = u'(c)\)

The FOC says \(u'(c_t) = R\beta \, v'_{t+1}(m_{t+1})\).

The envelope result says \(v'_t(m_t) = R\beta \, v'_{t+1}(m_{t+1})\).

Equating the right-hand sides:

\[\boxed{v'_t(m_t) = u'(c_t)}\]

Marginal value of resources = marginal utility of consumption.

The Euler Equation

The envelope result \(v'(m) = u'(c)\) lets us eliminate the value function from the FOC entirely. Substituting into \(u'(c_t) = R\beta \, v'_{t+1}(m_{t+1})\):

\[u'(c_t) = R\beta \, u'(c_{t+1})\]

This is the Euler equation for general \(T\)-period problems: it relates marginal utility today to marginal utility tomorrow without referencing the value function.

The Mechanical Shortcut

The envelope theorem implies a practical rule: when differentiating the value function with respect to \(m\), treat \(c\) as constant (set \(c'_t(m) = 0\)).

Formally, from Bellman’s equation:

\[v_t(m) = u(c^*(m)) + \beta \, v_{t+1}((m - c^*(m))R + y_{t+1})\]

Differentiating and treating \(c^*\) as constant:

\[v'_t(m) = \beta R \, v'_{t+1}(m_{t+1})\]

This works because the FOC zeros out \(\underline{v}^c\), so any term multiplied by it vanishes.

Graphical Intuition

When \(m\) increases by \(dm\), the consumer can either:

1. Consume it: gain \(u'(c) \, dm\) in utility
2. Save it: gain \(R\beta \, v'(m_{t+1}) \, dm\) in continuation value

At the optimum, the FOC equates these two margins. The consumer is indifferent at the margin between consuming and saving the extra resources.

The total gain in value is the same regardless of which margin we evaluate. This is why the envelope works.

Code: Value Function and Envelope

Code: Saving Rate and the IES

Perfect Foresight CRRA Model

The Multiperiod Problem

A consumer with CRRA utility \(u(c) = c^{1-\rho}/(1-\rho)\) maximizes

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

subject to the dynamic budget constraint

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

where \(m_t\) is market resources (cash-on-hand), \(p_t\) is permanent labor income growing at factor \(G\):

\[p_{t+1} = G \, p_t\]

Consumption Growth

From the envelope condition, the Euler equation is \(u'(c_t) = R\beta \, u'(c_{t+1})\). With CRRA utility \(u'(c) = c^{-\rho}\):

\[c_t^{-\rho} = R\beta \, c_{t+1}^{-\rho} \quad \Longrightarrow \quad \frac{c_{t+1}}{c_t} = (R\beta)^{1/\rho} \equiv \Phi\]

The absolute patience factor \(\Phi\) governs consumption growth every period.

Absolute Impatience

The value of \(\Phi\) relative to 1 determines the absolute impatience condition (AIC):

• \(\Phi < 1\): the consumer is absolutely impatient (consumption falls over time)
• \(\Phi > 1\): the consumer is absolutely patient (consumption rises over time)
• \(\Phi = 1\): the consumer is absolutely poised (constant consumption)

The Intertemporal Budget Constraint

The present discounted value of consumption must equal total wealth:

\[\text{PDV}(c) = b_t + \text{PDV}(p) \equiv o_t\]

Human wealth \(h_t\) is the PDV of future labor income:

\[h_t = \sum_{n=0}^{T-t} R^{-n} p_{t+n} = p_t \sum_{n=0}^{T-t} \left(\frac{G}{R}\right)^n = p_t \left(\frac{1 - (G/R)^{T-t+1}}{1 - G/R}\right)\]

Overall wealth \(o_t = b_t + h_t\) is the sum of nonhuman and human wealth.

Finite Human Wealth Condition (FHWC)

For the infinite-horizon case (\(T \to \infty\)), human wealth must be finite:

\[\boxed{G < R} \qquad \text{(FHWC)}\]

If income grows faster than the interest rate forever, the PDV of future income is infinite, and the problem has no well-defined solution.

When the FHWC holds:

\[h_t = \frac{p_t}{1 - G/R} = \frac{p_t \, R}{R - G}\]

Return Impatience Condition (RIC)

Since consumption grows at rate \(\Phi\) each period, its PDV is finite only if

\[\boxed{\Phi_R \equiv \frac{\Phi}{R} = \frac{(R\beta)^{1/\rho}}{R} < 1} \qquad \text{(RIC)}\]

The return patience factor \(\Phi_R\) measures whether the desired growth rate of consumption exceeds the interest rate.

• If \(\Phi_R \geq 1\), the consumer wants consumption to grow at least as fast as \(R\), and the PDV of consumption is infinite.
• The RIC imposes a minimum level of impatience for the model to have a well-defined solution.

The Consumption Function (Finite Horizon)

Combining the IBC with consumption growth at rate \(\Phi\):

\[c_t = \underbrace{\left(\frac{1 - \Phi_R}{1 - \Phi_R^{\,T-t+1}}\right)}_{\equiv \, \kappa_t} \cdot \, o_t\]

where \(\kappa_t\) is the marginal propensity to consume out of overall wealth.

Key properties of \(\kappa_t\):

• In the last period, \(\kappa_T = 1\) (consume everything)
• As the horizon grows, \(\kappa_t\) falls (spread wealth over more periods)
• The MPC satisfies the recursion \(\kappa_t^{-1} = 1 + \Phi_R \, \kappa_{t+1}^{-1}\)

The Infinite Horizon Solution

Taking \(T \to \infty\) under the FHWC and RIC:

\[c_t = \underbrace{(1 - \Phi_R)}_{\equiv \, \kappa} \cdot \, o_t = \left(\frac{R - (R\beta)^{1/\rho}}{R}\right) o_t\]

The infinite-horizon MPC \(\kappa = 1 - \Phi_R\) is:

• Positive (by the RIC)
• Constant over time
• Independent of the level of wealth

This is a linear consumption function: spending is proportional to total wealth.

Code: Finite vs Infinite Horizon MPC

Sustainable Consumption

What spending rate leaves total wealth intact forever?

\[o_{t+1} = (o_t - c_t)R \quad \Longrightarrow \quad \bar{c} = \frac{r}{R} \, o_t\]

The sustainable (wealth-preserving) spending rate is \(r/R\), the interest earnings on total wealth divided by the return factor.

Whether the consumer spends above or below sustainability depends on impatience:

• \(\kappa > r/R\): absolutely impatient (\(\Phi < 1\)), spending exceeds interest income
• \(\kappa = r/R\): absolutely poised (\(R\beta = 1\)), spending equals interest income
• \(\kappa < r/R\): absolutely patient (\(\Phi > 1\)), wealth accumulates over time

Normalizing by Permanent Income

Divide all variables by permanent income \(p_t\): let \(\hat{c}_t = c_t/p_t\), \(\hat{m}_t = m_t/p_t\), \(\hat{b}_t = b_t/p_t\).

The budget constraint becomes

\[\hat{b}_{t+1} = (\hat{m}_t - \hat{c}_t)(R/G), \qquad \hat{m}_{t+1} = \hat{b}_{t+1} + 1\]

and the normalized consumption function is

\[\hat{c}_t = (1 - \Phi_R) \, \hat{o}_t\]

The Growth Impatience Condition

Whether the wealth-to-income ratio is rising or falling depends on the growth impatience condition (GIC):

\[\boxed{\Phi_G \equiv \frac{\Phi}{G} = \frac{(R\beta)^{1/\rho}}{G} < 1} \qquad \text{(GIC)}\]

A “growth impatient” consumer (\(\Phi_G < 1\)) spends more than income, drawing down \(\hat{o}\).

Code: Impatience Conditions

The Approximate Consumption Function

When \(\beta = 1/(1+\delta)\) with time preference rate \(\delta\), an approximation gives

\[c_t \approx \left(\underbrace{r}_{\text{income}} - \underbrace{\rho^{-1}(r - \delta)}_{\text{substitution}}\right) \cdot o_t\]

where \(o_t\) includes human wealth \(h_t = p_t \cdot R/(r - g)\) (the human wealth effect).

Three effects of a change in \(r\):

• Income effect: higher \(r\) raises the payout rate on wealth (\(c \uparrow\))
• Substitution effect: higher \(r\) makes future consumption cheaper (\(c \downarrow\))
• Human wealth effect: higher \(r\) reduces the PDV of future labor income (\(c \downarrow\))

Applications

How Large Is the Human Wealth Effect?

With \(b_t = 0\) (no financial wealth), approximate consumption is

\[c_t \approx \left(r - \rho^{-1}(r-\delta)\right) \frac{p_t}{r - g}\]

Numerical example with \((r, \delta, g, \rho) = (0.04, 0.04, 0.02, 2)\):

\[c_t \approx 0.04 \cdot \frac{p_t}{0.02} = 2 \, p_t\]

The Human Wealth Effect Dominates

Now suppose the interest rate \(r\) falls from \(0.04\) to \(0.03\):

\[c_t \approx 0.035 \cdot \frac{p_t}{0.01} = 3.5 \, p_t\]

A one percentage point drop in \(r\) raises consumption by 75%. The human wealth effect is far stronger than the income and substitution effects combined (Summers, 1981).

Code: Human Wealth Effect

The Saving Rate and Interest Rates

For a wealthy consumer (\(a_{t-1} \to \infty\)), the saving rate asymptotes to

\[\varsigma \approx \frac{\rho^{-1}(r - \delta)}{r}\]

and the response of the saving rate to the interest rate \(r\) is

\[\frac{d\varsigma}{dr} = \rho^{-1} \delta \, r^{-2}\]

Saving Rate: A Numerical Example

With \(r = \delta = 0.05\) and \(\rho = 2\):

\[\frac{d\varsigma}{dr} = \frac{1}{2} \cdot \frac{0.05}{0.0025} = 10\]

A one percentage point rise in \(r\) raises the saving rate by 10 percentage points. The model predicts an enormous sensitivity of saving to interest rates.

Code: Saving Rate Response

The Random Walk Model

From Euler Equation to Random Walk

When future consumption is uncertain, the Euler equation becomes

\[u'(c_t) = R\beta \, \mathbb{E}_t[u'(c_{t+1})]\]

Suppose the utility function is quadratic: \(u(c) = -(1/2)(\bar{c} - c)^2\), so \(u'(c) = \bar{c} - c\), where \(\bar{c}\) is the bliss point (assumed unreachable).

The Random Walk Result

With \(R\beta = 1\), the Euler equation under quadratic utility reduces to

\[\bar{c} - c_t = \mathbb{E}_t[\bar{c} - c_{t+1}] \quad \Longrightarrow \quad \boxed{\mathbb{E}_t[c_{t+1}] = c_t}\]

Consumption follows a martingale: the best forecast of next period’s consumption is today’s consumption.

Hall’s Random Walk Proposition

Defining the consumption innovation \(\epsilon_{t+1} = c_{t+1} - c_t\):

\[\Delta c_{t+1} = \epsilon_{t+1}, \qquad \mathbb{E}_t[\epsilon_{t+1}] = 0\]

Hall (1978): No information known to the consumer at time \(t\) can predict how consumption will change between \(t\) and \(t+1\).

Testable implication: lagged income, lagged stock returns, or any other variable in the consumer’s information set should have zero predictive power for \(\Delta c_{t+1}\).

This shifted macroeconomics from estimating consumption functions to testing Euler equations.

Code: Simulating a Consumption Random Walk

Implications

The random walk result is powerful because it provides a model-free test: we do not need to know the consumer’s income process, preferences, or wealth.

Strengths:

• Testable without specifying the income process
• Applies to any consumer satisfying quadratic utility and \(R\beta = 1\)
• Spawned a large empirical literature testing for excess sensitivity and excess smoothness

Limitations:

• Requires quadratic utility (no precautionary saving motive)
• Requires \(R\beta = 1\) (knife-edge condition)
• With CRRA utility, consumption growth depends on \(R\), \(\beta\), and \(\rho\), and the random walk breaks down

The Consumption Function

The CEQ Consumer

A consumer with quadratic utility \(u(c) = -(1/2)(\bar{c} - c)^2\) and \(R\beta = 1\) faces the budget constraint

\[b_{t+1} = (b_t + y_t - c_t)R\]

The Income Process

The income process has permanent and transitory components:

\[p_{t+1} = p_t + \psi_{t+1} \qquad \text{(permanent income)}\]

\[y_{t+1} = p_{t+1} + \theta_{t+1} \qquad \text{(observed income)}\]

where \(\psi_{t+1}\) and \(\theta_{t+1}\) are mean-zero white noise shocks.

Solving via the IBC

The intertemporal budget constraint in expectation:

\[\mathbb{E}_t\left[\sum_{n=0}^{\infty} R^{-n} c_{t+n}\right] = b_t + \mathbb{E}_t\left[\sum_{n=0}^{\infty} R^{-n} y_{t+n}\right]\]

Since \(\mathbb{E}_t[c_{t+1}] = c_t\) (random walk), the left side simplifies to \(c_t \cdot R/(R-1)\).

The expected PDV of income is \(b_t + \theta_t + p_t \cdot R/(R-1)\), because the transitory shock \(\theta_t\) appears only in the current period, while \(p_t\) persists forever.

Key Result: The Consumption Function

Equating the two sides and solving for \(c_t\):

\[\boxed{c_t = \frac{r}{R}(b_t + \theta_t) + p_t}\]

The MPC depends on the nature of the income shock:

Shock type	MPC	Typical value
Transitory (\(\theta_t\))	\(r/R\)	\(\approx 0.05\)
Permanent (\(\psi_t\))	\(1\)	\(1.00\)

A dollar of transitory income raises consumption by only 5 cents; a dollar of permanent income raises it by a full dollar.

Code: MPC Comparison

Why the Keynesian Consumption Function Fails

The “Keynesian” consumption function \(c_t = \alpha_0 + \alpha_1 y_t\) assumes a single MPC \(\alpha_1\) relating consumption to income. But the true MPC depends on the nature of the shock:

• If income changes are mostly transitory: \(\alpha_1 \approx r/R \approx 0.05\)
• If income changes are mostly permanent: \(\alpha_1 \approx 1\)

There is no “true” value of \(\alpha_1\). The consumption function is meaningless without specifying the income process.

Hall’s (1978) innovation: test the theory by examining whether lagged variables predict \(\Delta c_{t+1}\), bypassing the need to specify what consumers believe about income. This avoids arbitrary and difficult-to-test assumptions about the structure of the income process.

Code: Transitory vs Permanent Shocks

Summary

Key Takeaways (1/2)

• Envelope theorem: \(v'(m) = u'(c)\); marginal value equals marginal utility, extending the Euler equation to general \(T\)-period problems
• Perfect foresight CRRA: consumption grows at factor \(\Phi = (R\beta)^{1/\rho}\); the consumption function is \(c = \kappa \cdot o\) where \(\kappa = 1 - \Phi/R\)
• Three conditions for the infinite-horizon solution:
  • AIC (\(\Phi < 1\)): consumption falls over time
  • FHWC (\(G < R\)): human wealth is finite
  • RIC (\(\Phi/R < 1\)): PDV of consumption is finite

Key Takeaways (2/2)

• Human wealth effect dominates income and substitution effects: consumption is very sensitive to interest rate changes (Summers, 1981)
• Random walk (Hall, 1978): with quadratic utility and \(R\beta = 1\), no lagged variable predicts consumption changes; a powerful, model-free test
• The consumption function: MPC depends on whether a shock is transitory (\(r/R \approx 0.05\)) or permanent (\(\approx 1\))
• Muth’s insight: the “Keynesian” consumption function \(c = \alpha_0 + \alpha_1 y\) is meaningless without specifying the income process

How the Topics Connect

The four topics trace a logical arc:

• Envelope condition: generalizes the Euler equation to \(T\) periods via \(v'(m) = u'(c)\)
• Perfect foresight CRRA: solves the multiperiod problem analytically; introduces impatience conditions that discipline the model
• Random walk: specializes the Euler equation under quadratic utility; shows how to test the theory without knowing the income process
• Consumption function: derives the optimal response to different income shocks; explains why naive regressions of \(c\) on \(y\) are misleading