8. Exploring Recursive Utility#
Authors: Jaroslav Borovicka (NYU), Lars Peter Hansen (University of Chicago) and Thomas J. Sargent (NYU)
8.1. Introduction#
We explore the recursive utility preference specification of [Kreps and Porteus, 1978] and [Epstein and Zin, 1989] including versions of them proposed by [Hansen and Sargent, 2001] and [Anderson et al., 2003] that capture concerns about model misspecification as in. We deploy two distinct approximation approaches. One approach builds on a characterization by [Duffie and Epstein, 1992] and uses a continuous-time limiting approximation to a discrete-time specification in which underlying shocks are normally distributed, though our detailed derivations differ from those of [Duffie and Epstein, 1992]. We represent the limiting approximation with a Brownian motion information structure. Our second approximation m allows macroeconomic uncertainty to have first-order implications. We show how to explore model implications for (nonstandard) first- and second-order approximations to equilibria of dynamic stochastic models. We modify first- and second-order approximations routinely used in the macroeconomics literature in ways designed to focus on macroeconomic uncertainty. Our approximations apply to production-based macro-finance models with opportunities to invest in different kinds of capital. Both of our approximation approaches capture adjustments for uncertainty with a change in probability measure, one that differs from one that leads the so-called risk neutral distribution widely used to price derivative claims. Such a change of measure plays a big role in specifications of preferences like those in [Hansen and Sargent, 2001] and [Anderson et al., 2003] that build on a robust control literature initiated by [Jacobson, 1973] and [Whittle, 1981].
Our approximations allow macroeconomic uncertainty to have first-order implications. We assume risk averse economic decision makers with recursive utility preferences or closely related preferences that also express concerns about model misspecification. We show how to explore model implications for (nonstandard) first and second-order approximations to equilibria of dynamic stochastic models. We modify first and second order approximations routinely used in the macroeconomics literature in ways that focus on macroeconomic uncertainty. Our uncertainty adjustments are captured by a change in probability measure. Furthermore, our approximations apply to production-based macro-finance models with opportunities to invest in diverse forms of capital. We use this framework to advance our understanding of alternative preference specifications and their implications for production and asset pricing.
We extend work by [Schmitt-Grohé and Uribe, 2004] and [Lombardo and Uhlig, 2018] in ways that highlight consequences of uncertainty. We design approximations to make implied stochastic discount factors reside within an exponential linear quadratic class, a class that gives rise tractable formulas for asset valuation over alternative investment horizons. See, for instance, [Ang and Piazzesi, 2003] and [Borovička and Hansen, 2014]. The class is also useful for studying production-based macro-finance models with opportunities to invest in different forms of capital.
Note
We of course recognize that nonlinearities in some models are more accurately captured by global solution methods.
8.2. Recursive utility valuation process#
We construct continuation value and stochastic discount processes, important constituents of many dynamic stochastic models in macroeconomics and finance.
8.2.1. Basic recursion#
A homogeneous of degree one representation of recursive utility is
where
Value
In equation (8.1),
Continuation values are determined only up to an increasing transformation. For computational and conceptual reasons, it is useful to work with the transformation
where
The right side of recursion (8.3) is the logarithm of a constant elasticity of substitution (CES) function of
Remark 8.1
The limit of
We shall construct small noise expansions of
8.2.2. Preference for robustness#
When
Let the random variable
where
The inequality follows from Jensen’s inequality because the function
Relative entropy measures a discrepancy between two probability distributions. Think of
Remark 8.2
To solve minimization problem (8.5), we attach a Lagrange multiplier
where
which implies the minimizer
and the minimized objective
To determine
whose first-order necessary condition is
The maximizing
and the minimized objective function is
The minimizing
The minimizer (8.6) of problem (8.5) evidently shifts probabilities toward low continuation values. [Bucklew, 2004] called this a stochastic version of Murphy’s law. Notice that the minimized objective satisfies
where earlier we described
It follows from (8.6) that
The random variable
8.2.3. Stochastic discount factor process#
A stochastic discount factor (SDF) process
To deduce an SDF process, we positing that a date zero value of a risky date
To compute the ratio
We think of
For recursive utility, a one-period increment in the stochastic discount factor process is
where
and
The recursive structure of preferences makes the time
Remark 8.3
To verify formula (8.8), we compute a one-period intertemporal marginal rate of substitution. Given the valuation recursions (8.3) and (8.4), we construct two marginal utilities familiar from CES and exponential utility:
From the certainty equivalent formula, we construct the marginal utility of the next-period logarithm of the continuation value:
where the
Putting these four formulas together using the chain rule for differentiation gives a marginal rate of substitution:
Now let
8.3. Continuous-time limit#
We now explore a continuous-time limit that approximates a discrete-time specification. Since we continue to work with normal shocks, the continuous-time counterpart to these shocks are Brownian increments. The continuation value in continuous time will evolve as:
for some drift (local mean),
where
8.3.1. Discrete-time approximation#
To study the utility recursion, start with a discrete-time specification:
where
by local log normality. We turn now to the first recursion and compute time derivatives in three steps. First, we
evaluate the term inside the logarithm as
This term is in the denominator as implied by the derivative with respect to a logarithm. Second, we
differentiate the term inside the logarithm with respect to
Third, we differentiate the term inside the logarithm with respect to
Putting together the derivative components together gives:
Notice that this relation imposes a restriction across the local mean,
8.3.2. Robustness to misspecification#
To investigate an aversion to model misspecification in continuous time,
we now treat the distribution of
We start by considering positive martingales
The martingales have local evolutions:
and we initialize them at
We use these martingales as relative densities or likelihood ratios.
Write the discrete-time counterpart as
Let
The altered conditional density has mean
which measures the statistical divergence or relative entropy between original and altered conditional probabilities.
For the continuous-time limit, under the
where
We may justify focusing on drift distortions for Brownian increments because of our imposition of absolute continuity of the alternative probabilities with respect to the baseline specification of a multivariate standard Brownian motion. This is an implication of the Girsanov Theorem.
We are now in a position to deduce a robustness adjustment in continuous time.
Consider formula (8.10) when
Modify this equation to include minimization over
The minimizing solution is
with a minimized objective
Notice that this agrees with formula (8.10) for
8.3.3. Uncertainty pricing#
For the purposes of valuation, we compound the equilibrium version of
provided that the constructed process is a martingale.[1] With this construction we interpret
8.4. Small noise expansion of dynamic stochastic equilibria#
We next consider a different type of characterization that sometimes gives good approximations for dynamic stochastic equilibrium models. While the approximations build from derivations in [Schmitt-Grohé and Uribe, 2004] and [Lombardo and Uhlig, 2018], and we extend them in a way that features the uncertainty contributions more prominently and are reflected even in first-order contributions. By design, the implied approximations of stochastic discount factors used to represent market or shadow values reside within the exponential linear quadratic class. This class is known to give tractable formulas for asset valuation over alternative investment horizons. See, for instance, [Ang and Piazzesi, 2003] and [Borovička and Hansen, 2014]. Moreover, they are applicable to production-based macro-finance models with investment opportunities in alternative forms of capital.
While these approximations are tractable for the reasons described, researchers may be concerned about some more fundamental aspects of uncertainty that are disguised by these methods. Indeed, for some models nonlinearities are more accurately captured by global solution methods. Nevertheless, the approximations still provide further understanding of the preferences and their implications for asset pricing in endowment and production economies.
8.4.1. Approximate state dynamics#
We follow [Lombardo and Uhlig, 2018] by considering the following class of stochastic processes indexed by a scalar perturbation parameter
Here
We denote a zero-order expansion
and assume that there exists a second-order expansion of
where
In the remainder of this chapter, we shall construct instances of the second-order expansion (8.13) in which the generic random variable
Processes
In this chapter, we use a prime
The first-derivative process obeys a recursion
that we can write compactly as the following a first-order vector autoregression:
We assume that the matrix
It is natural for us to denote second derivative processes with double subscripts. For instance, for the double script used in conjunction with the second derivative matrix of
Recursions (8.14) and (8.15) have a linear structure with some notable properties. The law of motion for
Remark 8.4
Perturbation methods have been applied to many rational expectations models in which partial derivatives of
Let
Approximate this process by:
where
In models with endogenous investment and savings, the consumption dynamics and some of the state dynamics will emerge as the solution to a dynamic stochastic equilibrium model. We use the approximating processes (8.13) and (8.16) as inputs into the construction of an approximating continuation value process and its risk-adjusted counterpart for recursive utility preferences.
8.5. Incorporating preferences with enhanced uncertainty concerns#
To approximate the recursive utility process, we deviate from common practice in macroeconomics by letting the risk aversion or robust parameter in preferences depend on
The aversion to model misspecification or the aversion to risk moves inversely with the parameter
8.5.1. Order-zero#
Write the order-zero expansion of (8.3) as
where the second equation follows from noting that randomness vanishes in the limit as
For order zero, write the consumption growth-rate process as
The order-zero approximation of (8.3) is:
We guess that
This equation implies
Equation (8.17)
determines
In the limiting
8.5.2. Order-one#
We temporarily take
where
Notice how the parameter
When
To facilitate computing some useful limits we construct:
which we assume remain well defined as
Taking limits as
Subtracting
Substituting formula (8.22) into the right side of (8.19) gives the recursion for the first-order continuation value:
Remark 8.5
We produce a solution by “guess and verify.” Suppose that
It follows from (8.23) that
Deduce the second equation by observing that
is distributed as a log normal. The solutions to equations (8.25) are:
The continuation value has two components. The first is:
and the second component is a constant long-run risk adjustment given by:
This second term is the variance of
conditioned on
Remark 8.6
The formula for
converges to the variance of the martingale increment of
Remark 8.7
Consider the logarithm of the uncertainty-adjusted continuation value approximated to the first order. Note that from (8.24),
Substitute this expression into formula (8.22) and use the formula for the mean of random variable distributed as a log normal to show that
Associated with the first-order approximation, we construct:
Equation (8.22) is a standard risk-sensitive recursion applied to log-linear dynamics. For instance, see [Tallarini, 2000]’s paper on risk-sensitive business cycles and [Hansen et al., 2008]’s paper on measurement and inference challenges created by the presence of long-term risk.[4] Both of those papers assumed a logarithmic one-period utility function, so that for them
Remark 8.8
The calculation reported in Remark 8.7 implies that
As a consequence, under the change in probability measure induced by
and with the same covariance matrix given by the identity. This is an approximation to robustness adjustment expressed as an altered distribution of the underlying shocks. It depends on
As we noted in Remark Remark 8.6,
8.5.3. Order two#
Differentiating equation (8.3) a second time gives:
Equivalently,
Rewrite transformations (8.20) and (8.21) as
Differentiating twice with respect to
Differentiating (8.21) with respect to
and thus
where subtracting
Substituting this formula into (8.27) gives:
Even if the second-order contribution to the consumption process is zero, there will be nontrivial adjustment to the approximation of
8.6. Stochastic discount factor approximation#
We approximate
where
We now consider two different approaches to approximating
8.6.1. Approach 1#
Write
Form the ``first-order’’ approximation:
We combine a first-order approximation of
which preserves the quadratic approximation of
8.6.2. Approach 2#
Next consider an alternative modification of Approach 1 whereby:
and
By design, this approximation of
To understand better this choice of approximation, consider the family of random variables (indexed by
The corresponding family of exponentials has conditional expectation one and the
Thus this family of random variables has the same first-order approximation in
As a change of probability measure, this approximation will induce state dependence in the conditional mean and will alter the covariance matrix of the shock vector. We find this approach interesting because it links back directly to the outcome of the robustness formulation we described in Section 3.1.
8.7. Solving a planner’s problem with recursive utility#
The [Bansal and Yaron, 2004] example along with many others building connections between the macro economy and asset value take aggregate consumption as pre-specified. As we open the door to a richer collection of macroeconomic models, it becomes important to entertain more endogeneity, including investment and other variables familiar to macroeconomics.
Write a triangular system with stochastic growth as:
where
where the first equation is a vector of static constraints and the second constructs the measure of consumption that enters preferences.
We extend the approximations by using a co-state formulation. There are two essentially equivalent interpretations of these co-states. One is they function as a set of Lagrange multipliers on the state evolution equations. The other is that they are partial derivatives of value functions. The co-state equations are forward-looking, linking next period’s co-state vector to this period’s co-state vector. Given the recursive utility structure, we must include the implied value functions in the computations as they enter the relations of interest.
The first-order conditions for
where
In addition, we solve a forward-looking co-state equation given by
The approximation formulas, (8.22), (8.23), (8.28), (8.29), that we deduced previously for
From recursive utility updating equation:
Dividing by both sides of the equation by
From the relation, we see that
8.7.1. An example economy with long-run uncertainty#
Consider an AK model with recursive utility and adjustment costs.
The exogenous state dynamic capture both long-run uncertainty in the mean growth rate and the overall volatility in the economy.
The state variable,
With these exogenous dynamics, we obtain the following zero and first-order approximations:
and
We impose the resource constraint:
The endogenous state dynamics are given by:
where
Express the first-order conditions for the consumption-capital and investment-capital ratios as:
It is convenient to rewrite the first-order conditions for the consumption-capital ratio as:
which in turn implies that
More generally, we will seek to approximate
Notice that these first-order conditions do not depend on the co-state process
When
8.7.2. First-order approximation when #
We obtain the following zero and first-order approximations for the exogenous dynamics:
and
The first-order conditions for
Solving for
which is independent of the state, as should be expected since
The order one approximation is then:
Stochastic volatility, as in the [Bansal and Yaron, 2004] model of consumption dynamics, will be present in the second-order approximation.
8.7.3. Second-order approximation when #
We next consider the second-order approximations. The second-oder approximation for
where we previously noted that
The combined approximation for
The approximate dynamics for the exogenous states remains the same for
8.7.4. Shock elasticities#
We use the shock elasticities to explore pricing implications of this recursive utility specification. In what follows, we use exponential/linear/quadratic implementation by [Borovička and Hansen, 2014] and by [Borovička and Hansen, 2016] with the parameter configuration given in Table 1 of [Hansen et al., 2024]. This latter reference combines inputs from other sources including [Schorfheide et al., 2018] and [Hansen and Sargent, 2021].
Fig. 8.1 gives the shock exposure elasticities for consumption to each of the three shocks. They can be interpreted as nonlinear local impulse responses for consumption (in levels not logarithms). The elasticities for the growth rate shock and the stochastic volatility shock start small and increase over the time horizon as dictated by the persistence of the two exogenous state variable processes. The elasticities for the direct shock to capital are flat over the horizon as to be expected since the shock directly impacts log consumption in a manner that is permanent. Notice that while elasticities for the volatility shock are different from zero, their contribution is much smaller than the other shocks. Stochastic volatility does induce state dependence for the other elasticities as reflected by the quantiles.[5]

Fig. 8.1 Exposure elasticities for three shocks.
Fig. 8.2 gives the shock price elasticities for growth-rate shock when

Fig. 8.2 Price elasticities for three shocks.
Fig. 8.3 and Fig. 8.4 provide the analogous plots for

Fig. 8.3 Price elasticities for three shocks.

Fig. 8.4 Price elasticities for three shocks.

Fig. 8.5 Price elasticities for three shocks.
8.7.5. An alternative model of intertemporal substitution/complementarity.#
We now extend the preference specification of the consumers in the AK model of Section 8.7.1 to explore implications of time nonseparability in preferences. We are motivated to do so by rather substantial previous literetaure. An important earlier contributor is [Ryder and Heal, 1973], who solve a social planner’s problem with stochastic growth. [Sundaresan, 1989], [Constantinides, 1990], [Heaton, 1995], and [Hansen et al., 1999] consider asset pricing implications with internal habit persistence. These papers essentially explore decentralizations of the planner’s problem. The latter paper considers simultaneously a recursive robustness specification as we will do here, although for a different model specification.
To accomplish this, we introduce an additional state variable, which we will call the habit stock,
where
Notice that the sum of the coefficients on the right side of the evolution equation sum to one. This allows us to interpret
We write evolution equation in logarithms as
For numerical purposes, we transform this equation to be:
where we treat
We impose the output constraint as:
where
where
What enters the utility function each date inside the recursive utility preference specification is the CES aggregate:
for
For computational purposes, we divide both sides of (8.38)`by
Consistent with the habit-persistence literature, we allow the parameter
Remark 8.9
The asset pricing literature often features the computationally more challenging case in which
For simplicity, consider the case in which
Consider now the special case in which
More generally,
This depicts the marginal value of
In our calculations, we approximate
Figure Fig. 8.6 explores the median exposure elasticities for the investment-capital ratio as a function of the parameter

Fig. 8.6 Investment/capital exposure elasticities for the growth and capital shocks with different values of

Fig. 8.7 Investment/capital exposure elasticities for the growth shock with different values of
We next consider the shock price elasticities for the growth shock. Fig. 8.8 reports the elasticities for three values of

Fig. 8.8 Price elasticities for the growth shock for

Fig. 8.9 Price elasticities for the growth shock for

Fig. 8.10 Price elasticities for the growth shock for
Overall, we see that investment/capital and consumption/capital impulse responses are noticeably sensitive to the choice of
[Pollak, 1970] introduced a version of external habit persistence in consumer demand function. The stock
8.7.6. External habits model#

Fig. 8.11 Investment/capital exposure elasticities for the growth and capital shocks with different values of
Please note that the axes in the left panel on the graph below are scaled 0.01 times relative to the right panel.

Fig. 8.12 Investment/capital exposure elasticities for the growth shock with different values of
8.8. Solving models#
In this section, we briefly describe one way to extend the approach that builds directly on previous second-order approaches of [Kim et al., 2008], [Schmitt-Grohé and Uribe, 2004], and [Lombardo and Uhlig, 2018]. While such methods should not be viewed as being generically applicable to nonlinear stochastic equilibrium models, we find them useful pedagogically and often as at least initial steps to understanding models that are arguably “smooth.” See [Pohl et al., 2018] for a careful study of nonlinearity in asset pricing models with recursive utility.[7]
We implement these methods for second-order approximation using the following steps.
Solve for
deterministic model.Take as given first and second-order approximate solutions for
and . Solve for the approximate solutions for andCompute the first-order expansion and solve the resulting equations following the previous literature for
and When constructing these equations, use expectations computed using the probabilities induced by . Substitute the first-order approximation forCompute the second-order expansion and solve the resulting equations following the previous literature. Again use the expectations induced by
. In addition, make another recursive utility adjustment expressed in terms the approximations ofReturn to step 2, and repeat until convergence.
Initialize this algorithm by solving the
See the Appendices that follow for more details and formulas to use in the solution method.
As a second approach we iterate over an
While we discussed the approximation for resource allocation problems with recursive utility, there is a direct extension of this approach to solve a general class stochastic equilibrium models by stacking a system of expectational-type equations expressed in part using the recursive utility stochastic discount factor that we derived. For resource allocation problems, we expressed the first-order conditions for the planner in utility units, which simplified some formulas. Equilibrium models not derived from a planner’s problems typically use stochastic discount factors expressed in consumption units when representing investment choices. The approximation methods described in this chapter have a direct extension to such models.
8.9. Appendix A: Solving the planner’s problem#
Write the system of interest, including the state equations (8.31), the consumption equation and static constraint (8.32), the first-order conditions (8.33), and the co-state evolution (8.34) as:
where
Here, for computational purposes, we use that
Our solution will entail an iteration. We will impose a specification for
8.9.1. Some steady state calculations#
Observe from the recursive utility updating that:
In the steady state we view this as two equations in three variables:
We construct
and
in the steady state, and use them to construct the remaining steady-state equations:
In this equation,
8.9.2. and derivatives#
For the order one, write
To compute this contribution, we equation (8.19) to write
We then construct
To compute
and solve this equation forward by first computing the
For the order two approximation,
Express:
It follows from (8.29) that
which we solve this equation forward under the
The
8.9.3. derivatives#
Form:
It may be directly verified that
In producing these representations, we use that have conditional
8.10. Appendix B: Approximation formulas (approach one)#
Consider the equation:
not including the state evolution equations.
8.10.1. Order zero#
The order zero approximation of the product:
Thus the order zero approximate equation is:
since
8.10.2. Order one#
The order one approximation of the product:
Thus the order one approximate equation is:
where we used the implication that
8.10.3. Order two#
The order two approximation of the product:
The terms
To elaborate on the contributions in the second line,
express
Then
The formula for the first of these terms follows from (8.39) and (8.40), along with fact the third central moments of normals are zero.
We add to this second-order subsystem, the second-order approximation of the state dynamics inclusive of the jump variables. We substitute in the solution for the first-order approximation for the jump variables into both the first and second-order approximate state dynamics. In solving the second-order jump variable adjustment we use expectations induced by
8.11. Appendix C: Approximation formulas (approach two)#
In this approach we use the same order zero approximation. For the order one approximation, we use the formula (8.30) for
From formula (8.39), it follows that under the
and conditional precision:
For the order two approximation, we use:
8.12. Appendix D: Parameter values#
To facilitate a comparison to a global solution method, we write down a discrete-time approximation to a continuous time version of such an economy. (See Section 4.4 of [Hansen et al., 2024] for a continuous-time benchmark model that our discrete-time system approximates. Note that we convert the annual parameters in that paper to quarterly time.) The parameter settings are:
0.025 |
0.01 |
32 |
0.01 |
0.014 |
0.0485 |
The numbers for
we use an observationally equivalent upper triangular representation for most of the results. Finally, the numbers for
For the extension to the habit persistence model in Section 8.7.5, we use a habit persistence of