The Consumption Euler Equation: Microeconomic Foundations of Macroeconomic Stabilization

This blog aims to help mathematicians understand Robert E. Hall and Michael Woodford’s contributions to the microeconomic foundations of macroeconomic stabilization, recognized by the 2026 Nobel Prize in Economic Sciences. I am not an expert in this field; as a mathematician learning about it, I wish to share the understanding I have developed with assistance from ChatGPT.

The central mathematical structure is familiar: a household chooses a consumption path subject to constraints, and an interior optimum must be stationary under every admissible variation. The consumption Euler equation expresses this condition for transfers of resources between dates. The equation predates both laureates. Hall developed and empirically tested its implications for consumption, while Woodford integrated forward-looking household behavior with firms’ pricing decisions and economic equilibrium to analyze monetary policy.

Begin with a deterministic economy at dates $t=0,\ldots,T$, where $T\geq1$. Let $c_t>0$ denote consumption, $y_t$ income, and $a_t$ assets entering period $t$. Negative assets represent borrowing. Let $R_t>0$ be the gross real return on assets carried over from the preceding period, so $R_t=1+r_t$ when $r_t$ is the net real interest rate. Consumption, income, and assets are measured in units of the consumption good; returns are dimensionless.

The budget constraint is $c_t+a_{t+1}=y_t+R_ta_t$. In period $t$, the household receives income $y_t$ and a gross payoff $R_ta_t$ from assets carried over from the preceding period. It allocates these resources between current consumption $c_t$ and saving $a_{t+1}$ for the next period.

The period utility function $u:(0,\infty)\to\mathbb R$ describes how the household values consumption during a single period. Assume $u\in C^2((0,\infty))$, $u'(c)>0$, and $u''(c)<0$. The condition $u'(c)>0$ means that more consumption is preferred, while $u''(c)<0$ means that marginal utility decreases as consumption rises. For example, $u(c)=\log c$ satisfies both conditions. Utility represents preferences rather than a directly measurable quantity of happiness.

The household evaluates a consumption path using $J=\sum_{t=0}^{T}\beta^tu(c_t)$, where $\beta\in(0,1)$ is the discount factor. A larger $\beta$ gives greater relative weight to future utility. This formulation assumes additive separability across time and the same period utility function at every date.

Fix initial assets $a_0$, terminal assets $a_{T+1}=\bar a$, incomes, and returns. Eliminating consumption leaves the intermediate asset vector $a=(a_1,\ldots,a_T)\in\mathbb R^T$ as the choice variable. Define $c_t(a)=y_t+R_ta_t-a_{t+1}$ and the admissible set $\mathcal A={a\in\mathbb R^T:c_t(a)>0\text{ for every }t=0,\ldots,T}$. We impose no additional borrowing constraints. For each fixed $t$, consumption $c_t(a)$ is a continuous affine function of the asset vector $a$. Since the consumption inequalities are strict and there are finitely many dates, $\mathcal A$ is open. Thus, at any admissible asset vector, sufficiently small variations in any direction remain admissible.

The optimization problem is $\max_{a\in\mathcal A}J(a)=\sum_{t=0}^{T}\beta^tu(c_t(a))$. Suppose $a^*\in\mathcal A$ is a local maximaizer, and write $c_t^*=c_t(a^*)$.  Let $h=(h_1,\ldots,h_T)\in\mathbb R^T$ be arbitrary, extended by $h_0=h_{T+1}=0$. The perturbed asset path is $a_t^\epsilon=a_t^*+\epsilon h_t$. Openness ensures that sufficiently small variations of either sign remain admissible. The induced consumption is exactly $c_t(a^*+\epsilon h)=c_t^*+\epsilon(R_th_t-h_{t+1})$.

Define $\Phi_h(\epsilon)=J(a^*+\epsilon h)$. Since $a^*$ is an interior local maximizer of $J$, $\Phi_h$ has a local maximum at $\epsilon=0$. Therefore, $0=\Phi_h'(0)=\sum_{t=0}^{T}\beta^tu'(c_t^*)(R_th_t-h_{t+1})$. Reindexing the second term and using $h_{T+1}=0$ gives $\sum_{t=0}^{T}\beta^tu'(c_t^*)h_{t+1}=\sum_{t=1}^{T}\beta^{t-1}u'(c_{t-1}^*)h_t$. Combining the two sums and using $h_0=0$, we obtain $0=\Phi_h'(0)=\sum_{t=1}^{T}\left[\beta^tR_tu'(c_t^*)-\beta^{t-1}u'(c_{t-1}^*)\right]h_t$. Since this equality holds for arbitrary $h=(h_1,\ldots,h_T)\in\mathbb R^T$, each coefficient must vanish. Thus $\beta^tR_tu'(c_t^*)=\beta^{t-1}u'(c_{t-1}^*)$ for $t=1,\ldots,T$. Equivalently, $u'(c_t^*)=\beta R_{t+1}u'(c_{t+1}^*)$ for $t=0,\ldots,T-1$. This is the consumption Euler equation. 

The Euler equation balances the marginal utility sacrificed today against the discounted marginal utility gained tomorrow. Under concavity, this stationarity condition is sufficient for global optimality; strict concavity gives uniqueness.

Under uncertainty, let $\mathbb E_t[X]=\mathbb E[X\mid\mathcal F_t]$, where $\mathcal F_t$ represents information available at date $t$. At an interior optimum, variations in saving based on that information yield $u'(c_t)=\beta\mathbb E_t[R_{t+1}u'(c_{t+1})]$. If $R$ is constant and $\beta R=1$, marginal utility is a martingale: $\mathbb E_t[u'(c_{t+1})]=u'(c_t)$. Thus, given today’s information, its expected change is zero. Consumption itself need not be a martingale; this follows when marginal utility is affine, as with quadratic utility. Under these additional assumptions, Hall’s result implies that available information is already reflected in current consumption, so consumption changes have zero conditional mean.

Woodford’s framework connects this behavior to monetary policy. With nominal interest rate $i_t$, gross inflation $\Pi_{t+1}=P_{t+1}/P_t$, and marginal utility $u'(c)=c^{-\sigma}$, the stochastic consumption Euler equation becomes $\underbrace{c_t^{-\sigma}}_{u'(c_t)}=\beta\mathbb E_t [\underbrace{\frac{1+i_t}{\Pi{t+1}}}_{R_{t+1}}\underbrace{c_{t+1}^{-\sigma}}_{u'(c{t+1})} ].$ Since $i_t$ and $c_t$ are known at date $t$, dividing by $c_t^{-\sigma}$ and taking $1+i_t$ outside the conditional expectation gives $1=\beta(1+i_t)\mathbb E_t[\Pi_{t+1}^{-1}(c_{t+1}/c_t)^{-\sigma}]$.  A first-order approximation shows that a higher expected real interest rate encourages households to postpone consumption. This can weaken current demand and output. Expectations of future interest rates therefore matter today, explaining why credible central-bank announcements can influence spending before rates change. In the deterministic continuous-time model, the Euler–Lagrange equation expresses the same trade-off: households balance the benefit of consuming now against the return from saving for later. With the utility function above, optimal consumption rises over time when the real interest rate exceeds the household’s preference discount rate.

P.S. The sharp decline in semiconductor stocks between October 1 and 15, 2026, offers a possible illustration of the expectations channel emphasized in Woodford’s framework. If investors expect interest rates to remain higher for longer, higher discount rates can reduce the present value of future earnings, even when current profits remain strong. Higher borrowing costs may also delay investment in data centers and other equipment, weakening expected semiconductor demand. Share prices can therefore fall before earnings decline. This illustrates how monetary policy expectations can affect markets and investment today, without establishing the cause of this particular sell-off.