Recursive Preferences in a Multiperiod Economy
Introduction
The Epstein-Zin preferences notebook derived the SDF in a two-date setting by guessing a homothetic consumption rule and solving a single maximization problem. That approach is clean but sidesteps the recursive structure that makes Epstein-Zin preferences genuinely useful in multiperiod environments. This notebook derives the same SDF directly from the multiperiod recursive utility aggregator, relying on three tools: implicit differentiation of the recursive equation, a perturbation argument to extract the pricing kernel, and Euler’s homogeneous function theorem to translate utility into observable quantities.
The payoff is the standard Epstein-Zin stochastic discount factor m_{t+1} = \left[\beta\!\left(\frac{c_{t+1}}{c_t}\right)^{\!-1/\psi}\right]^{\!\theta} \left[\frac{1}{R_{t+1}^w}\right]^{\!1-\theta}, \qquad \theta = \frac{1-\gamma}{1-1/\psi}, expressed entirely in terms of consumption growth and the return on the wealth portfolio. The parameter \theta controls the weight of each factor: when \psi = 1/\gamma, \theta = 1 and the formula collapses to the standard CRRA stochastic discount factor; otherwise the wealth return enters as a separate pricing factor, capturing the agent’s concern for future investment opportunities — the source of Epstein-Zin’s intertemporal hedging demand. The multiperiod derivation confirms that the two-date formula is exact in every period and not merely an approximation valid near a steady state.
General Recursive Utility
Kreps and Porteus (1978) develop a framework for recursive utility that separates attitudes toward risk from the willingness to substitute consumption over time. A general Kreps–Porteus aggregator takes the form V_t = U(c_t, \mu_t), where U(c, \mu) is an increasing aggregator and \mu_t is the certainty equivalent of next-period utility under a von Neumann–Morgenstern risk aggregator: \mu_t = \phi^{-1}\!\left(\operatorname{E}_t\!\left[\phi(V_{t+1})\right]\right), for some increasing, concave \phi. The concavity of \phi makes the agent averse to uncertainty about future utility levels, while the curvature of U in its first argument separately controls the willingness to substitute consumption across time. This is the key departure from standard expected utility, where a single function governs both simultaneously. Epstein-Zin preferences (Epstein and Zin 1989) are the leading special case, with U(c,\mu) = [(1-\beta)c^\rho + \beta\mu^\rho]^{1/\rho} and \phi(v) = v^{1-\gamma}.
To derive the SDF, consider a small perturbation in the consumption plan: give up \xi\,p_t units of consumption today in exchange for \xi\,x_{t+1} units tomorrow. At the optimum the agent is indifferent at the margin, so the derivative of V_t with respect to \xi vanishes at \xi = 0. Differentiating U(c_t - \xi p_t,\, \mu_t(\xi)) at \xi = 0 and using \partial V_{t+1}/\partial c_{t+1} = U_c(c_{t+1}, \mu_{t+1}) recursively gives the first-order condition p_t\,U_c(c_t,\mu_t) = U_\mu(c_t,\mu_t)\, \frac{\operatorname{E}_t\!\left[\phi'(V_{t+1})\,U_c(c_{t+1},\mu_{t+1})\,x_{t+1}\right]}{\phi'(\mu_t)}, so the SDF is m_{t+1} = \underbrace{\frac{U_\mu(c_t,\mu_t)}{U_c(c_t,\mu_t)}}_{\text{aggregator ratio}} \cdot \underbrace{\frac{\phi'(V_{t+1})}{\phi'(\mu_t)}}_{\text{risk adjustment}} \cdot \underbrace{U_c(c_{t+1},\mu_{t+1})}_{\text{marginal utility}}. \tag{1} The middle factor \phi'(V_{t+1})/\phi'(\mu_t) compares realized utility to the certainty equivalent through the lens of \phi. For power \phi(v) = v^{1-\gamma} this is (V_{t+1}/\mu_t)^{-\gamma}; for entropic \phi(v) = -e^{-\alpha v} it becomes e^{-\alpha(V_{t+1}-\mu_t)}.
Epstein-Zin Preferences
The agent has preferences over consumption streams (c_t, c_{t+1}, c_{t+2},\ldots) of the Kreps–Porteus form V_t = U(c_t, \mu_t), with aggregator U(c,\mu) = [(1-\beta)c^\rho + \beta\mu^\rho]^{1/\rho} and power risk aggregator \phi(v) = v^{1-\gamma}, so the certainty equivalent is \mu_t = \left[\operatorname{E}_t\!\left(V_{t+1}^{1-\gamma}\right)\right]^{\!1/(1-\gamma)}. The parameters \beta\in(0,1), \gamma>0, and \psi>0 are the time-discount factor, coefficient of relative risk aversion, and elasticity of intertemporal substitution. Defining \rho = 1-1/\psi, the aggregator takes the explicit form V_t = \left[(1-\beta)\,c_t^\rho + \beta\,\mu_t^\rho\right]^{1/\rho}, \tag{2} or equivalently, V_t^\rho = (1-\beta)\,c_t^\rho + \beta\,\mu_t^\rho. \tag{3} When \psi = 1/\gamma the aggregator collapses to standard CRRA utility with coefficient \gamma.
Homotheticity and the Wealth Identity
The aggregator (2) is linearly homogeneous in consumption: scaling every current and future consumption level by \lambda > 0 scales V_t by \lambda. Euler’s theorem therefore gives V_t = \sum_{\tau=0}^{\infty} \frac{\partial V_t}{\partial c_{t+\tau}}\,c_{t+\tau} = \operatorname{E}_t\!\left[ \sum_{\tau=0}^{\infty} \frac{\partial V_t}{\partial c_{t+\tau}}\,c_{t+\tau} \right], where the second equality holds because the sum equals V_t state by state, hence in conditional expectation as well. Dividing by U_c(c_t,\mu_t), the right-hand side is the present value of the entire consumption stream — total wealth W_t: \frac{V_t}{U_c(c_t,\mu_t)} = \operatorname{E}_t\!\left[ \sum_{\tau=0}^{\infty} \frac{U_c(c_{t+\tau},\mu_{t+\tau})}{U_c(c_t,\mu_t)}\,c_{t+\tau} \right] = W_t. \tag{4} Substituting U_c(c_t,\mu_t) = (1-\beta)\,c_t^{-1/\psi}\,V_t^{1/\psi} into V_t = U_c(c_t,\mu_t)\,W_t and simplifying: V_t = (1-\beta)\,c_t^{-1/\psi}\,V_t^{1/\psi}\,W_t \implies V_t^\rho = (1-\beta)\,c_t^{-1/\psi}\,W_t, \tag{5} so that V_t = \left[(1-\beta)\,c_t^{-1/\psi}\,W_t\right]^{1/\rho}. \tag{6} The entire infinite-dimensional consumption stream is thus compressed into two quantities: current consumption c_t and total wealth W_t.
The Certainty Equivalent
Substituting (5) into the recursive equation (3) gives (1-\beta)\,c_t^{-1/\psi}\,W_t = (1-\beta)\,c_t^\rho + \beta\,\mu_t^\rho. Writing (1-\beta)c_t^\rho = (1-\beta)c_t^{-1/\psi}\cdot c_t and rearranging: (1-\beta)\,c_t^{-1/\psi}(W_t - c_t) = \beta\,\mu_t^\rho. Here W_t - c_t is the amount saved after current consumption. Those savings are invested and grow to next-period wealth: W_{t+1} = R_{t+1}^w\,(W_t - c_t), where R_{t+1}^w is the gross return on the wealth portfolio. Solving for the certainty equivalent: \mu_t = \left[\frac{(1-\beta)\,c_t^{-1/\psi}(W_t - c_t)}{\beta}\right]^{\!1/\rho}. \tag{7}
Deriving the Pricing Kernel
We apply (1) to the Epstein-Zin aggregator U(c,\mu) = [(1-\beta)c^\rho + \beta\mu^\rho]^{1/\rho} with risk aggregator \phi(v) = v^{1-\gamma}, for which \mu_t = [\operatorname{E}_t(V_{t+1}^{1-\gamma})]^{1/(1-\gamma)}. The partial derivatives of U are U_c = (1-\beta)\,c^{\rho-1}\,U^{1-\rho}, \qquad U_\mu = \beta\,\mu^{\rho-1}\,U^{1-\rho}, so their ratio is \frac{U_\mu}{U_c} = \frac{\beta\,\mu^{\rho-1}}{(1-\beta)\,c^{\rho-1}}. Since \phi'(v) = (1-\gamma)v^{-\gamma}, the risk-adjustment factor reduces to \frac{\phi'(V_{t+1})}{\phi'(\mu_t)} = \left(\frac{V_{t+1}}{\mu_t}\right)^{-\gamma}. Substituting into (1) and collecting terms using \rho - 1 = -1/\psi: m_{t+1} = \beta\!\left(\frac{c_{t+1}}{c_t}\right)^{\!-1/\psi} \mu_t^{\rho-1+\gamma}\,V_{t+1}^{1/\psi-\gamma} = \beta\!\left(\frac{c_{t+1}}{c_t}\right)^{\!-1/\psi} \left(\frac{V_{t+1}}{\mu_t}\right)^{\!1/\psi-\gamma}. \tag{8} This is an exact expression for the SDF in terms of the utility process; (6) and (7) now translate the ratio V_{t+1}/\mu_t into observables: \frac{V_{t+1}}{\mu_t} = \frac{\left[(1-\beta)\,c_{t+1}^{-1/\psi}\,W_{t+1}\right]^{1/\rho}} {\left[\dfrac{(1-\beta)\,c_t^{-1/\psi}(W_t-c_t)}{\beta}\right]^{1/\rho}} = \left[\beta\,\frac{W_{t+1}}{W_t-c_t}\,\left(\frac{c_{t+1}}{c_t}\right)^{\!-1/\psi}\right]^{1/\rho} = \left[\beta\,R_{t+1}^w\,\left(\frac{c_{t+1}}{c_t}\right)^{\!-1/\psi}\right]^{1/\rho}, where the last step uses W_{t+1}/(W_t - c_t) = R_{t+1}^w. Substituting into (8) and collecting terms: m_{t+1} = \beta^{(1-\gamma)/\rho} \left(R_{t+1}^w\right)^{(1/\psi-\gamma)/\rho} \left(\frac{c_{t+1}}{c_t}\right)^{-(1-\gamma)/(\rho\psi)}. Defining \theta = (1-\gamma)/\rho = (1-\gamma)/(1-1/\psi), so that 1 - \theta = (\gamma - 1/\psi)/\rho, the Epstein-Zin stochastic discount factor is m_{t+1} = \beta^\theta \left(\frac{c_{t+1}}{c_t}\right)^{\!-\theta/\psi} (R_{t+1}^w)^{\theta-1}, \tag{9} or equivalently, m_{t+1} = \left[\beta\!\left(\frac{c_{t+1}}{c_t}\right)^{\!-1/\psi}\right]^{\!\theta} \left[\frac{1}{R_{t+1}^w}\right]^{\!1-\theta}, \qquad \theta = \frac{1-\gamma}{1-1/\psi}. \tag{10} The parameter \theta measures the weight placed on the consumption-growth factor relative to the wealth-return factor. When \theta = 1 — which occurs exactly when \psi = 1/\gamma, the CRRA case — the wealth return drops out entirely. When \theta \neq 1, the wealth return enters the SDF as a separate pricing factor, reflecting the agent’s concern for news about future investment opportunities beyond what current consumption growth reveals. The further \theta departs from one, the more the wealth return matters as an independent pricing factor.