Do interest rates decline when there is ambiguity about growth?1 Johannes Gierlinger2 Universitat Aut`onoma de Barcelona, Barcelona GSE & MOVE Christian Gollier3 Toulouse School of Economics, University of Toulouse-Capitole January 24, 2017

1 This

research benefitted from the financial support of the Chairs ”Risk Markets and Value Creation” (SCOR) and ”Sustainable finance and responsible investment” at TSE. Gierlinger acknowledges support from the Spanish Ministry of Education, through grant ECO2009-09847, the Spanish Ministry of Economy and Competitiveness, through the Severo Ochoa Programme for Centres of Excellence in R&D (SEV-2011-0075), and a FIR Finance and Sustainability research grant. 2 Department of Economics, Campus UAB, Edifici B, 08193 Bellaterra, Spain. Email: [email protected] 3 Toulouse School of Economics, 21 All´ ee de Brienne, 31000 Toulouse, France. Email: [email protected]

Abstract This paper studies whether ambiguous beliefs about consumption growth decrease interest rates. Various ambiguity preferences are shown to potentially increase rates. We distinguish two effects. The first acts like a pessimistic belief distortion that satisfies the monotone likelihood ratio property. It decreases rates for multiplier preference from robust control theory, but not necessarily for smooth or maxmin preferences. Second, we identify an additional “ambiguity-prudence” effect for smooth preferences. It is negative if and only if absolute ambiguity aversion is decreasing. The term structure is shown to be qualitatively different from expected utility in analytical examples.

Keywords: Precautionary saving, ambiguity aversion, robustness concerns, bond pricing, sustainable development.

JEL classifications: D81, E43, D91, H43.

1

Introduction

Interest rates in most industrialised economies have remained below their historical average for an extended period. As of January 29, 2016, an estimated 23% of globally traded treasury bonds paid negative nominal interest.1 The decline has been particularly pronounced for long-term bonds, where nominal yields on secondary markets are close to, or below, zero. That is, households receive little, if any, compensation for delaying their consumption. These phenomena are frequently linked with perceived uncertainty about consumption growth. A recent survey about long-term interest rates by the US president’s Council of Economic Advisers (CEA) proposes that “[t]he presence of large potential consumption declines with hard-to-estimate likelihoods can greatly increase the subjective variability of future consumption growth, inducing substantial precautionary saving that pushes real interest rates down.”2 Indeed, the 2008 financial crisis coincided with a rise in personal saving rates in the US from 3 percent in 2007 to 6.1 percent in 2009. Moreover, in 2010, precautionary purposes became the primary motive to save according to the Survey of Consumer Finances (Bricker, et al., 2012). When testifying before congress in 2009, the Chair of the CEA, Christina Romer, likened the reluctance to consume to a similar phenomenon during the Great Depression: “[T]he main effect of the crash of the stock market in 1929 on spending operated not through the direct loss of wealth, but through the enormous uncertainty it created.[...] This 1

The Financial Times, 29 January 2016. Available: http://on.ft.com/1JLV1x3/ Council of Economic Advisers (2015) “Long-term interest rates: A survey”, July, p.57. 2

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volatility led consumers and firms to be highly uncertain about what lay ahead. I found narrative and statistical evidence that this uncertainty led to large drops in consumption and investment spending.”3 These theories suggest the distinction between situations of risk –where the likelihood of all future events is known with certainty–and situations of ambiguity (Knightian uncertainty) to be relevant for saving decisions. In particular, ambiguity may create an additional precautionary motive which decreases interest rates since households behave as if they adopted a “conservative” view on the likely growth distribution. This paper provides a unified formal analysis of the suggested mechanism for various uncertainty preferences from the Financial Economics literature. We consider a representative household whose beliefs about consumption growth are ambiguous and whose preferences belong to one of three classes: multiple-priors expected utility (MEU) preferences (Gilboa and Schmeidler, 1989) evaluate choices according to a minimzing prior, smooth ambiguity preferences (Klibanoff, Marinacci and Mukerji, 2005) are based on a certainty equivalent over expected utility levels, and multiplier preferences (Hansen and Sargent, 2001; Strzalecki 2011) incorporate robustness concerns about a potentially misspecified prior through a worstcase prior which is “close” in terms of relative entropy. We identify two novel effects beyond the classical determinants of bond prices. The first can be expressed in terms of a monotone-likelihood-ratio (MLR) deterioration of a reference belief. Respectively, ambiguity-averse households behave as if they assumed the worst prior in a convex set 3

Romer, C. D., “From Recession to Recovery: The Economic Crisis, the Policy Response, and the Challenges We Face Going Forward.” Testimony before the Joint Economic Committee, October 22, 2009. See also Romer (1990).

2

(MEU), as if an objective second-order belief were distorted in favor of low-expected-utility priors (smooth preferences), or as if low-consumption states became more likely (multiplier preferences). We find that this pessimism reduces the interest rate for multiplier preferences since it emphasises states where additional consumption is particularly valuable. By contrast, pessimism may increase the interest rate under MEU and smooth preferences since a distribution which is worse in terms of expected utility need not increase expected marginal utility. Given our general finding, we identify situations in which the effect of pessimism on rates is unambiguously negative. To do so, we provide joint restrictions on household beliefs and preferences which accommodate various specifications in existing literature. For instance, with minimal assumptions on preferences, rates always decrease if the set of priors can be ordered by stochastic dominance. Smooth preferences are shown to cause an additional shift in the marginal utility from future consumption which cannot be expressed as a belief distortion. This effect decreases rates if and only if the second-order utility function exhibits decreasing absolute ambiguity aversion (DAAA), defined by the usual Arrow-Pratt index. The intuition can be given by analogy to the effect of decreasing absolute risk aversion (DARA), which implies that a uniform increase in a lottery’s payoffs by one Dollar raises its certainty equivalent by more than one Dollar. Similarly, under DAAA, when future consumption increases, the change in the “certainty-equivalent” expected utility exceeds the mean change in expected utility across priors. In particular, we show that DAAA characterises a decrease in real rates when households are neutral to risk but averse to ambiguity. Similar to prudence, which characterises positive saving in the absence of wealth and

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impatience effects (Leland, 1968; Dr`eze and Modigliani, 1972), we refer to this property as ambiguity prudence. Building on our qualitative results, we determine the term structure of interest rates in analytical examples. We derive ambiguity-corrected versions of the classical Ramsey pricing rule (Ramsey, 1928) for various stochastic processes from previous literature. Absent ambiguity, the standard risk-adjusted Ramsey rule corresponds to the sum of the rate of impatience, a wealth effect, and an additional precautionary term if the agent is prudent (see Hansen and Singleton, 1983; Gollier, 2002; Weitzman, 2007a). Under the assumption of efficient markets, this rate can also be interpreted as the socially efficient discount rate for risk-free projects with corresponding maturity. Our generalisation with power utility functions shows that ambiguity introduces an additional negative precautionary term. Its magnitude is increasing with the degree of relative ambiguity aversion, the level of ambiguity, and with the time horizon. In particular, we find that the term structure of efficient rates may be qualitatively different from the expected utility benchmark. If the stochastic process is reverting to a mean which is not known with certainty, then the term structure under expected utility becomes flat as the time horizon increases. Instead, under ambiguity aversion, the term structure decreases also in the long run. The remainder of the paper is organised as follows. In Section 2, we introduce the two-periods saving problem and the equilibrium rates. Section 3 derives sufficient conditions for uncertainty to reduce interest rate. Section 4 quantifies the effect of the time horizon in analytical examples. Section 5 concludes and discusses the relation to the literature.

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2

Equilibrium interest rates

In this section, we determine the competitive price of a risk-free bond in tree economies `a la Lucas (1978). A representative household is endowed with a random output e ct of a perishable consumption good at dates t = 0, 1, 2, .... At the initial date, t = 0, investing α ∈ R into a bond which matures in t periods increases consumption in t by αRt , which is certain and corresponds to an annualised real interest rate rt = 1t ln(Rt ) in continuous time. Households know Rt with certainty but they entertain ambiguous beliefs about the process {e ct }t=0,1,2,... . More concretely, they behave as if there were an unobserved parameter θ in a set Θ ⊆ R which characterises the process. Conditional on parameter θ, the tree generates a random payoff e ctθ ∼ Ftθ in period t. The limit case without uncertainty can be represented by a time-separable expected utility criterion. Accordingly, after investing α in a bond with maturity t, the discounted expected utility in t becomes Z βt Eu(e ctθ + αRt ) = βt

u(c + αRt )dFtθ (c).

The discount factor βt ∈ (0, 1] corresponds to a constant rate δ = − 1t ln(βt ) of pure preference for the present in continuous time. Throughout, we assume u : R → R to be three times differentiable, increasing, and concave. Whenever the household is neutral to ambiguity, decisions are made on the basis of a best estimate θb ∈ Θ. In this case, the optimal investment maximises u(c0 − α) + βt Eu(e ctθb + αRt ).

(1)

Otherwise, the information in Θ is aggregated in line with the household’s ambiguity preferences. We retain the assumption that the optimal

5

investment in bonds with maturity t can be analysed in terms of a timeseparable objective function

u(c0 − α) + βt Vt (α),

(2)

where βt Vt (α) represents the discounted future utility seen from period 0 after investing α in the bond with maturity t. By the above, whenever Vt is concave, the necessary and sufficient condition for optimality becomes u0 (c0 − α∗ (Rt )) = βt Vt0 (α∗ (Rt )), where α∗ (Rt ) maximises the objective (2) for a given interest Rt . Finally, we obtain the equilibrium price Rt∗ through the market-clearing condition α∗ (Rt∗ ) = 0: Rt∗ =

u0 (c0 ) βt Vt0 (0).

(3)

Absent ambiguity concerns, (3) yields the classical consumption-based bond pricing formula Rt (θ) = [u0 (c0 )/βt Eu0 (e ctθ )].4 Accordingly, any randomness in e ctθ reduces Rt (θ) relative to a deterministic process if and only if Eu0 (e ctθ ) is larger than u0 (Ee ctθ ), which holds if and only if u satisfies prudence (see Leland, 1968; Dr`eze and Modigliani, 1972; or Kimball, 1990).

2.1

Motivating Example: Multiple-priors EU

Consider the “maxmin” criterion (Gilboa and Schmeidler, 1989). We assume that the saving choice can be analysed on the basis of the one-step4

See for example Cochrane (2001).

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ahead criterion, Vt (α) = min Eu(e ctθ + αRt ). θ∈Θ

(4)

That is, the trade-off from consuming in periods 0 or t is analysed by evaluating future payoffs according to the minimiser in a set Θ which represents a convex set of priors {Ftθ }θ∈Θ . Accordingly, expanding set Θ acts like an increase in ambiguity aversion. Conversely, when Θ is a singleton, then Vt represents expected utility preferences. Consider now a situation in which household beliefs are ambiguous while an analyst makes his predictions based on an unambiguous belief Fbt ≡ Ftθb. b systematically bt ≡ Rt (θ) We wish to determine whether the prediction R overestimates the market-clearing interest Rt∗ . Together with the concavity of u, the concavity of the min-function guarantees that Vt is concave. As a result, the interest rate decreases if the minimising beliefs increase marginal expected utility, Eu0 (e ctθ ) ≥ Eu0 (e ctθb) for all θ ∈ arg minθ∈Θ Eu(e ct ). However, the following example shows that taking a worst-case view on the likely distribution of growth may discourage saving. Example 1. Let u(x) = ln x and βt = 1. The initial income is c0 = 2 while e ctθ assigns probability (θ/2) to extreme realisations and the remaining probability to an intermediate realisation: e ctθ ∼ ( 2θ , 1; (1−θ), 2; 2θ , 5), where θ ∈ Θ ≡ [0, 1]. Taking account of the market clearing condition, α = 0, the expected utility Eu(e ctθ ) is minimised at θ = 0. Accordingly, markets are cleared with Rt∗ = 1. However, Eu0 (e ctθ ) = 0.1θ + 0.5 is increasing in θ. Therefore, basing the prediction on any interior θb ∈ (0, 1) b  bt = 5/(5 + θ). strictly underestimates the rate Rt∗ > R 7

The example illustrates that there is no systematic upward bias in the predicted interest rate when ambiguity is ignored.5 This raises the question about the determinants of the effect of ambiguity and whether realistic restrictions of the economic environment give rise to an unambiguous effect.

3

Decomposing the effect on interest rates

3.1

Pessimism and the equilibrium interest rate without ambiguity

Later on in this paper, we show that ambiguity-averse agents behave as if their beliefs correspond to a distorted version of a reference probability estimate. While the exact nature of this distortion depends on the preference at hand, they all conform to a well-defined notion of pessimism, captured by the monotone likelihood ratio (MLR) property. Without loss of generality, we assume that conditional expected utility

Ut (θ) ≡ Eu(e ctθ )

is increasing on Θ ⊆ R. Definition 1. Let q and q ∗ be two probability mass functions (probability density functions) on Θ ⊆ R. We say that q ∗ is more pessimistic than q if they satisfy the monotone likelihood ratio property: for all θ, θ0 ∈ Θ, q ∗ (θ0 ) q ∗ (θ) ≥ . θ > θ =⇒ q(θ) q(θ0 ) 0

5

To see that ambiguity can indeed decrease the interest rate, assume e ctθ ∼ (θ, 2; 1 − θ, 4) in Example 1. Now θ = 1 minimises Eu(e ctθ ) and we obtain Rt∗ = 1. For any θb < 1, bt = 2(1 + θ) > 1 = Rt∗ . the predicted interest rate is biased above R

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Since MLR is a special case of first-order stochastic dominance (FSD), it implies that any decreasing function f : Θ → R generates a larger expected value of f under the pessimistic probability. Suppose now that f (θ) = Eu0 (ctθ ) is decreasing in θ, i.e. that a prior generating a larger expected utility also reduces the expected marginal utility. Because ambiguity aversion deteriorates the distribution of θ in the sense of MLR-pessimism, the assumption that Eu0 (ctθ ) is decreasing in θ implies that ambiguity aversion raises the unconditional expected marginal utility, thereby it tends to raise saving. However, it is not true in general that a change in beliefs that raises expected utility always reduces expected marginal utility.6 In other words, it is not true in general that u and v = −u0 rank lotteries in the same way. In the remainder of this section, we examine some restrictions in which a von Neumann-Morgenstern utility function v agrees with u that distribution F is preferred to G: Z

b

Z u(x)d(F − G)(x) ≥ 0

a

b

v(x)d(F − G)(x) ≥ 0.

=⇒

(5)

a

In particular, whenever v = −u0 , then condition (5) says that an increase in expected utility must lower marginal expected utility. Consider first the general case in which F, G may be arbitrary distributions. Clearly, condition (5) holds in any such case if v is an affine transformation of u. A second polar case obtains when u, v are allowed to be arbitrary increasing functions. Then (5) holds whenever F and G can be ranked by FSD. Alternatively, we can relax FSD to second-order 6

If that would be true, a good news would always raise the equilibrium interest rate in the absence of ambiguity. In fact, a good news can depress the interest rate, as already seen in example 1. See also Abel (2002).

9

stochastic dominance (SSD) at the expense of adding more structure on functions u, v. That is, if u, v are increasing and concave, (5) holds if F dominates G according to SSD. Notice that the previous conditions are sufficient but not necessary for property (5) since each implies that u also agrees with v while we only need the converse. Jewitt (1989) characterises such a one-sided agreement when v is more concave than u. Definition 2. F dominates G according to Jewitt’s order if (5) holds for any pair (u, v)of increasing and concave functions, with v more concave than u. Jewitt (1989) shows that, if F does not SSD-dominate G, then the condition holds if and only if there exists a threshold, k ∈ (a, b), such that Z

x

(G(z) − F (z))dz ≥ 0 for all x ∈ [a, k], Z

a k

(G(z) − F (z))dz = 0,

(6)

a

F ≥G

on [k, b].

Contrary to FSD and SSD, Jewitt’s order applies to trade-offs between “risk and return” since F may be dominant despite having a lower mean than G. Condition (6) holds, in particular, when F and G satisfy the single-crossing property (see Jewitt, 1987). To summarise, we obtain the following sufficient conditions for a decrease in marginal expected utility when expected utility increases. Lemma 1. Let u be increasing, concave, and differentiable on X ⊆ R. Functions u and (−u0 ) agree on the ranking of two distributions F, G on X in any of the following cases: 10

1. u satisfies constant absolute risk aversion (CARA); 2. F FSD-dominates G; 3. F SSD-dominates G and u satisfies prudence (u000 ≥ 0); 4. F Jewitt-dominates G and u satisfies non-increasing absolute risk aversion (DARA). Proof. It suffices to show that (5) holds for v = −u0 . Part 1. follows from the fact that CARA implies −u0 (x) = Au(x) for some nonnegative A. Part 2. obtains since u and −u0 are increasing functions by assumption. If u000 ≥ 0 holds in addition, then 3. follows from u, −u0 being increasing and concave. 4. follows since DARA is equivalent to −u0 being a concave transformation of u, together with the definition of Jewitt’s order. The subsequent sections build on Lemma 1 to determine the effect of pessimism for various ambiguity preferences.

3.2

Multiple-priors expected utility

Consider again the MEU criterion from (4). The model does not specify an objective second-order distribution over Θ to discipline the best estimate b However, given that Θ represents a closed and convex set of priors θ. {Ftθ }θ∈Θ , we can obtain any prior θb ∈ Θ which is consistent with the set of beliefs by an implicit probability distribution over a subset of Θ. The smallest subset of this kind is formed by the extreme points of the set of priors.7 Let, K ⊆ Θ describe the set of the extreme points {Ftθ }θ∈K . We refer to the members of K as the extreme priors. In many applications, 7

a ∈ A is an extreme point of a nonempty and convex set A if it cannot be obtained from a convex combination of the remaining elements A \ {a}. Equivalently, a is an extreme point if A \ {a} remains convex.

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the extreme priors are immediate. If Θ represents uncertainty about a single parameter, then K may reduce to the bounds of an interval. For instance, our Example 1 from Section 2.1 is based on two extreme priors {0, 1} = K ⊂ Θ = [0, 1], and θb ∈ Θ corresponds to applying second-order b θ). b weights (q(0), q(1)) = (1 − θ, The following proposition shows that, subject to appropriate assumptions on u, ambiguity aversion increases the willingness to save if one of the extreme priors is stochastically dominated by all other extreme priors. To illustrate, notice first that any Ut (θ) is indeed minimised at an extreme prior θ∗ ∈ K since the conditional expected utility under any θ can be obtained from a convex combination of the levels Ut (k) with k ∈ K. In particular, whenever the minimising θ is unique, then it must belong to K. A minimising extreme prior θ∗ ∈ K can be expressed by a second-order distribution q ∗ : K → [0, 1] which has a single atom at θ∗ . Moreover, if Fbt lies in the interior of the set {Ftθ }θ∈Θ , then it can be obtained by strictly positive second-order probabilities q : K → (0, 1]. As a result, the MLR property holds trivially since

q ∗ (θ) q(θ)

= 0 for all k 6= θ∗ . Hence, whenever

conditional expected marginal utility is decreasing in θ, then ambiguity aversion decreases interest rates. Thanks to Lemma 1, we can ensure that u and −u0 agree on the ranking of extreme priors. Proposition 1. Let preferences belong to the MEU class with strictly increasing and strictly concave u, and let K ⊆ Θ represent the set of exbt ≥ R∗ = treme priors. Ambiguity aversion reduces the interest rate, i.e., R t minθ∈Θ Rt (θ), if any of the following conditions holds: 1. u exhibits constant absolute risk aversion; 2. One of the extreme priors is FSD-dominated by all others: ∃θ∗ ∈ K 12

such that Ftθ F SD Ftθ∗ for all θ ∈ K; 3. u satisfies prudence and one of the extreme priors is SSD-dominated by all others: ∃θ∗ ∈ K such that Ftθ SSD Ftθ∗ for all θ ∈ K. Proof. Due to the concavity of V , we obtain the result if the minimisers of Ut maximise expected marginal utility, where θ∗ ∈ arg minθ∈Θ Ut (θ) implies θ∗ ∈ arg maxθ∈Θ Eu0 (e ctθ ). By part 1. of Lemma 1, CARA guarantees that θ∗ maximises expected marginal utility in Θ. To prove 2.–3., notice that the strict monotonicity and strict concavity of u guarantee that θ∗ is the unique minimiser among the extreme priors, Eu(e ctk ) > Eu(e c∗t ) for all remaining extreme priors k ∈ K with k 6= θ∗ . Moreover, since the expected utility under any alternative prior Ftθ is a convex combination of the priors in K, θ∗ must be the unique minimiser of Ut . Finally, by parts 2.–3. of Lemma 1, θ∗ must be the unique maximiser of conditionally expected marginal utility, which proves the result. If the analyst makes his predictions on the basis of CARA utility, then he overestimates the interest rate. This restriction on preferences can be relaxed if the set K contains a stochastically dominated extreme prior. Frequently, Θ represents uncertainty about the correct specification of a normal distribution. If K consists of normal distributions with a common variance, then the extreme priors can be ranked by FSD. Similarly, if it consists of normal distributions with a common mean, then the extreme priors can be ranked by SSD. In order to make use of Jewitt’s ranking of priors, we need to ensure that the dominated prior θ minimises expected utility. To illustrate that said property may fail, reconsider the log-utility Example 1. The minimising prior θ∗ = 0 dominates all alternative priors according to Jewitt’s order. 13

Indeed, this parameter also minimises conditional marginal utility in θ ∈ Θ and thereby maximises Rt (θ). Proposition 2. Let preferences belong to the MEU class with an increasing and concave function u that satisfies DARA. Ambiguity aversion debt , if the unique minimising prior θ∗ = creases interest rates, Rt∗ ≤ R arg minθ∈K Ut (θ) is dominated by the remaining extreme priors {θ ∈ K : θ 6= θ∗ } according to Jewitt’s order. Note that the DARA property is widely accepted in the economics literature. In particular, it is implied by the empirical regularity that the amount invested in risky assets increases with a household’s wealth.

3.3

Smooth ambiguity preferences

In contrast to MEU, the smooth model (Klibanoff, Marinacci and Mukerji, 2005) is based on an second-order belief q : Θ → [0, 1] over the set of b For ease of notation, we assume the set parameters which disciplines θ. of parameters to be finite, |Θ| = n. Our results extend directly to the case where θ is a continuous variable. The degree of ambiguity aversion is captured by the concavity of a second-order utility function,

Vt (α) = φ−1

n X

! q(θ)φ (Eu(e ctθ + αRt )) ,

(7)

θ=1

where φ : R → R is strictly increasing. Typically, the weight q(θ) is interpreted as the objective probability that θ is the true parameter. If φ is linear, then Vt (0) collapses to an expected utility criterion based Pn ˆ b on beliefs Fbt = θ=1 q(θ)Ftθ , which defines θ in such a way that Ft = Ftθˆ. If φ is concave, then the agent displays ambiguity aversion: Vt (0) ≤ Pn ctθ ) = Eu(e ctθb). θ=1 q(θ)Eu(e 14

Similar to its counterpart in risk, we refer to A(U ) = −φ00 (U )/φ0 (U ) as the degree of absolute ambiguity aversion. Klibanoff, Marinacci and Mukerji (2005) show that the constant absolute ambiguity aversion specification φ(U ) = −A−1 exp(−AU ) approaches the MEU criterion from (4) as A tends to infinity. However, this limit case also represents an extreme degree of ambiguity aversion with MEU preferences, where Θ need not contain all plausible priors. Whenever Vt is concave, we obtain that the interest rate decreases under ambiguity if Eu0 (e ctθb) ≤ Vt0 (0). However, concavity is not implied by the assumptions, not even when u and φ are taken to be strictly concave. The following result shows that concavity holds when ambiguity tolerance −(φ0 /φ00 ) is a concave function. Lemma 2. Let preferences belong to the smooth class, and consider the conditional expected utility function Ut : Θ → Y , where Y ⊆ R is a convex set. If ambiguity tolerance −(φ0 /φ00 ) is concave on Y , then Vt is concave. Proof. Relegated to the Appendix. The most common specifications of φ, including exponential, logarithmic, and power functions, satisfy concave ambiguity tolerance. For the remainder, we assume this condition to be satisfied. Assumption 1. −φ0 /φ00 is concave on Y . P Compared to the unique prior Fbt = nθ=1 q(θ)Ftθ , the certainty equivalent formulation changes the return on investment through two effects. Differentiating Vt (α) in (7), we obtain Vt0 (0) =

n X θ=1

q(θ)

n X φ0 (Ut (θ)) 0 Eu (e c ) = a q ∗ (θ)Eu0 (e ctθ ), tθ φ0 (Vt (0)) θ=1

15

(8)

where the first effect is a distortion in weights φ0 (Ut (θ)) , 0 τ =1 q(τ )φ (Ut (τ ))

q ∗ (θ) = q(θ) Pn

(9)

and where the second effect is a shift in levels Pn a=

q(θ)φ0 (Ut (θ)) . φ0 (Vt (0))

θ=1

(10)

Consider first the factor a defined in (10). It increases the willingness Pn 0 0 to save, Vt0 (0), if θ=1 q(θ)φ (Ut (θ)) ≥ φ (Vt (0)). The following Lemma shows that this inequality is characterised by decreasing absolute ambiguity aversion (DAAA). Lemma 3. a ≥ 1 holds for all u increasing and concave if and only if φ satisfies decreasing absolute ambiguity aversion (−φ00 /φ0 decreasing). Proof. Let ye represent a lottery whose outcomes yθ = φ(Ut (θ)) obtain with probability q(θ) for θ ∈ Θ. By the definition of Vt in (7), φ(Vt (0)) = Ee y . Denote by ψ the transformation function which satisfies ψ ◦ φ = φ0 . Pn 0 By the definitions of Vt , ψ, and ye, the inequality θ=1 q(θ)φ (Ut (θ)) ≥ φ0 (Vt (0)) can be rewritten as Eψ(e y ) ≥ ψ(Ee y ). Applying Jensen’s inequality, the inequality holds for all random variables ye if and only if ψ is convex. Further, the chain rule implies ψ 0 ◦ φ = φ00 /φ0 . Since φ0 is nonnegative by assumption, we get ψ 0 increasing if and only if φ00 /φ0 is increasing. In the domain of risk preferences, decreasing absolute risk aversion is widely accepted. Consider now a household who enjoys unambiguous consumption e ct that results in expected utility U = Eu(e ct ). DAAA implies that an increase in U makes the agent more likely to accept a random change in conditional expected utility u e ∼ (q(1), u(1); ...; q(n), u(n)), where the latter results in Ut (θ) = U + u(θ). 16

To illustrate the effect of the factor a, consider a risk-neutral household with linear u. In this case, by equation (8), ambiguity increases the willingness to save if and only if a ≥ 1. More generally, DAAA can be characterised by the following property: controlling for any belief distortion in (9), ambiguity about future consumption induces positive precautionary saving. Similar to prudence, which characterises an increase in marginal expected utility in the presence of risk, we refer to the marginal-utility shift (10) as the ambiguity prudence effect. Consider now the distortion of second-order probabilities in (9). The change in second-order probabilities q ∗ (θ)/q(θ) is proportional to φ0 (Ut (θ)). That is, if φ is concave, then q ∗ emphasises parameters θ which induce low expected utility. In particular, note that concave probability corrections induce an MLR deterioration (see e.g., Lehmann, 1955; Quiggin, 1995). We obtain the analogous result for the second-order correction in (9). Lemma 4. Let Ut : Θ → R be increasing and let φ : Y → R be twice differentiable. The following conditions are equivalent: 1. φ is concave on Y . 2. The distorted belief q ∗ from (9) is more pessimistic than q: θ0 ≥ θ ⇒

q ∗ (θ) q(θ)

≥

q ∗ (θ0 ) . q(θ0 )

Proof. To show that φ concave implies 2., note that

q ∗ (θ) q(θ)

= φ0 (Ut (θ)).

Since Ut is increasing by assumption, the result obtains. Conversely, 2. implies concavity of φ since 2. implies φ0 (y) ≥ φ0 (y 0 ) for all y, y 0 ∈ Y with y 0 ≥ y. Thanks to Lemma 4, we are now in a position to determine the effect of the distorted q ∗ on the marginal utility from future consumption. 17

Lemma 5. Let Ut : Θ → R be increasing. The pessimism effect in (9) decreases the interest rate for all concave φ if and only if f (θ) ≡ Eu0 (e ctθ ) is decreasing in θ. P Proof. First, we prove that Eu0 (e ctθ ) decreasing in θ implies θ q ∗ (θ)Eu0 (e ctθ ) ≥ P 0 ctθ ). By Lemma 4, we have that q ∗ is an MLR distortion of θ q(θ)Eu (e q. As a result, any decreasing function f : Θ → R yields lower expectation under q than under q ∗ . Finally, we prove by contradiction that P ∗ P 0 ctθ ) ≥ θ q(θ)Eu0 (e ctθ ) implies Eu0 (e ctθ ) decreasing in θ. Asθ q (θ)Eu (e P P sume instead θ q ∗ (θ)Eu0 (e ctθ ) ≥ θ q(θ)Eu0 (e ctθ ) while Eu0 (e ctθ ) < Eu0 (e ctθ+1 ) for some θ ∈ [1, n − 1]. Among the class of concave functions, take any φ which satisfies φ00 < 0 on the interval (Ut (θ), Ut (θ+1)) and φ00 = 0 otherwise. P −1 In this case, the distortion g(θ) ≡ φ0 (Ut (θ)) ( nτ=1 q(τ )φ0 (Ut (τ ))) and the conditional marginal expected utility function f (θ) are anti-comonotone in θ since (f (θ) − f (θ0 ))(g(θ) − g(θ0 )) ≤ 0 with strict inequality for some θ0 = θ + 1. According to the covariance rule, this implies n X

q(θ)g(θ)h(θ) <

θ=1

n X

! q(θ)g(θ)

θ=1

n X

! q(θ)h(θ) .

θ=1

Pn Pn Pn ∗ 0 ctθ ) while Since θ=1 q(θ)g(θ) = 1 θ=1 q(θ)g(θ)f (θ) = θ=1 q (θ)Eu (e Pn Pn 0 ctθ ), we have a contradiction to the and θ=1 q(θ)f (θ) = θ=1 q(θ)Eu (e P ∗ P initial assumption θ q (θ)Eu0 (e ctθ ) ≥ θ q(θ)Eu0 (e ctθ ). Finally, we are able to provide sufficient conditions for a decrease in the interest rate, characterised by the inequality 0

Eu (e ctθb) ≤ a

n X

q ∗ (θ)Eu0 (e ctθ ).

θ=1

The following result combines Lemmas 1, 3, and 5. 18

(11)

Proposition 3. Let preferences belong to the smooth class with u, φ increasing and concave and let φ satisfy decreasing absolute ambiguity averbt , if any of the following sion. Ambiguity decreases the interest rate, Rt∗ ≤ R conditions hold: 1. u satisfies constant absolute risk aversion; 2. For any θ, θ0 ∈ Θ, Fθ and Fθ0 can be ranked by FSD; 3. For any θ, θ0 ∈ Θ, Fθ and Fθ0 can be ranked by SSD and u satisfies prudence. Proof. By Lemma 3, DAAA implies a positive ambiguity prudence effect, a ≥ 1. By Lemma 5, the pessimism effect increases marginal utility from saving if Eu0 (e ctθ ) is decreasing in θ. Conditions 1.-3. in Lemma 1 prove the result. Thanks to the transitivity of the stochastic dominance relations, conditions 2. and 3. are satisfied whenever the members of Θ can be ordered by FSD and SSD, respectively. Similar to Proposition (2), SSD can be relaxed if those distributions which induce high expected utility are also less risky according to Jewitt (1989). Proposition 4. Let preferences belong to the smooth class with u, φ increasing and concave. In addition, let u satisfy DARA and let φ satisfy bt , if θ0 > θ implies DAAA. Ambiguity decreases the interest rate, Rt∗ ≤ R that Fθ0 Jewitt-dominates Fθ . Proof. The expected-utility ranking of priors coincides with Jewitt’s order by assumption. As a result, by condition 4. in Lemma 1, we obtain that Eu(e ctθ0 ) ≥ Eu(e ctθ ) ⇒ Eu0 (e ctθ ) ≥ Eu0 (e ctθ0 ) if preferences satisfy DARA. The remainder of the proof is analogous to Proposition 3. 19

The cost of relaxing SSD to Jewitt’s order is twofold. First, prudence needs to be strengthened to DARA. Second, in contrast to FSD and SSD, Jewitt-dominant priors need not induce higher expected utility, which requires an additional assumption to this effect.

3.4

Multiplier preferences

Finally, consider the class of multiplier preferences from robust control theory (Hansen and Sargent, 2001; Strzalecki, 2011). For ease of notation, we assume e ct to be a discrete random variable with realisations ct (s), where s ∈ {1, . . . , S}. Households acknowledge that their beliefs about growth P pb ∈ ∆ ≡ {p ∈ RS : p ≥ 0, Ss=1 ps = 1} may be incorrect. To make decisions robust to this uncertainty, they evaluate choices on the basis of a distorted prior p∗ ∈ ∆ which is “close” to pb in terms of a low relative P entropy cost C(p) = Ss=1 ps ln( ppbss ). Given market clearing, α = 0, p∗ is the minimiser in the following expression, " Vt (0) = min

S X

p∈∆

# ps u(ct (s)) + kC(p) ,

(12)

s=1

where a low k ∈ R+ represents a greater concern for model uncertainty. In order to relate the analysis to our previous results, we exploit that multiplier preferences can be represented in terms of a second-order expected utility criterion (see, e.g., Strzalecki, 2011). That is, solving the minimisation, we can express (12) by

Vt (0) = ψ −1

S X

! ps ψ (u(ct (s))) ,

(13)

s=1

where ψ(u) = − exp(− uk ). Similar to condition (11) with smooth prefer-

20

ences, the agent increases her saving if and only if S X

pbs u0 (ct (s)) ≤

s=1

S X

p∗s u0 (ct (s)),

(14)

s=1

where the minimiser of (12) satisfies p∗s ∝ pbs exp(− u(ctk(s)) ). Indeed, p∗ is an MLR deterioration of the reference belief pb. However, p∗ is now a corrected belief about the states s, as opposed to the second-order belief in the smooth model (9). Moreover, since the exponential ψ renders ψ 00 /ψ 0 constant, an ambiguity prudence effect akin to (10) is absent. Hence the following result. Proposition 5. Ambiguity reduces the interest rate with multiplier preferbt . ences, Rt∗ ≤ R Proof. Let cs be increasing in s ∈ {1, 2, ..., S}. Rearranging, we obtain exp(− u(ctk(s)) ) p∗s = PS . u(ct (l)) pbs p exp(− ) l l=1 k Since exp(− u(c) ) is decreasing in c, the distorted beliefs p∗ must be an k MLR-deterioration of pb. Finally, since MLR is a special case of FSD, (14) holds by condition 2. in Lemma 1. In contrast to competing models, multiplier preferences induce pessimism about the growth rates, rather than their likely distribution. As a result, ambiguity always increases the willingness to save since lower consumption growth induces higher marginal utility.

21

4

Efficient interest rates and social discounting with smooth preferences

In the following, we study the term structure of annualised real interest rates rt =

1 t

ln(Rt ). We retain the assumption that the decision problem

can be approximated by the one-step-ahead formulation introduced in (7). In doing so, we abstract from the effects that interim realisations may have on the trade-off between consumption in period 0 and period t. A more general dynamic model on the basis of the recursive version of the smooth model due to Klibanoff, Marinacci and Mukerji (2009) is left for future work.8 Similarly, we consider the rate of pure preference δ = − 1t ln(βt ). When expressed in terms of rt and δ, the pricing rule from above corresponds to the usual consumption-based bond pricing formula when there is no ambiguity9 1 rbt = δ − ln t



Eu0 (e ctθb) u0 (c0 )

 .

(15)

We wish to assess in a similar fashion what determines the behavior of the market-clearing interest under ambiguity aversion, rt∗ , where rt∗

1 = δ − ln t

 P ∗  a θ q (θ)Eu0 (e ctθ ) . u0 (c0 )

(16)

Analogous to Section 3.3, q ∗ and a represent the effects of pessimism (10) and ambiguity prudence (9), respectively. Similarly, given concave ambiguity tolerance, we obtain that ambiguity reduces rates, rbt ≥ rt∗ , if and only 8

Recursive preferences account for the ambiguity over continuation values at each date. Our one-step-ahead formulation can be seen as a limit case in which parameters which lead to low instantaneous utility at date t also induce low continuation values at date 0. 9 See for example Cochrane (2001).

22

if Eu0 (e ctθb) =

P

θ

q(θ)Eu0 (e ctθ ) ≤ a

P

θ

q ∗ (θ)Eu0 (e ctθ ).

Notice that the pricing condition (15) also represents the social discount rate, defined by the critical annualised return on risk-free projects to make them socially efficient. Our results can therefore be reinterpreted to help assess whether delayed benefits should be discounted at a lower rate.

4.1

Analytical results

The power-power normal-normal case Consider economies with power utility where the consumption process satisfies the following properties. 1. The logarithm of consumption is known to be normally distributed ln e ctθ ∼ N(ln c0 + θt, σ 2 t);10 2. However, the trend parameter θ is not known with certainty. Agents believe that it is drawn from a normal distribution θ ∼ N(µ, σ02 );11 3. Preferences satisfy constant relative risk aversion u(c) = constant relative ambiguity aversion φ(U ) =

c1−γ 1−γ

k (kU )1−ηk 1−kη

and

with

00

(U ) η = − |U | φφ0 (U , where k = sign(1 − γ) ensures concavity when U is )

negative. In order to express rt∗ in terms of the parameters, we first use the fact that the Arrow-Pratt approximation is exact with power utility and normally distributed log-consumption. Moreover, we show in the Appendix that the same techniques yield an analytical expression of the certainty 10

This is a discrete version of the geometric Brownian motion d ln ct = θdt + σdW in continuous time. 11 e We consider the natural continuous extension of the discrete distribution for θ.

23

equivalent over conditional expected utility levels. Accordingly, the efficient rate from equation (16) is given by: 1 1 rt∗ = δ + γµ − γ 2 (σ 2 + σ02 t) − η 1 − γ 2 σ02 t. 2 2

(17)

Alternatively, (17) can be expressed in terms of the growth rate of expected consumption g = µ + 0.5(σ 2 + σ02 t): 1 1 rt∗ = δ + γg − γ(γ + 1)(σ 2 + σ02 t) − η 1 − γ 2 σ02 t. 2 2

(18)

The first two terms in formula (18) correspond to the classical Ramsey rule, by which rt∗ increases with impatience and with the rate of growth through a wealth effect. If there is risk, then the third term in (18) captures a precautionary effect since CRRA implies prudence.12 Notice that the two sources of variance accumulate to σ 2 t+σ02 t2 . That is, even absent ambiguity aversion, the precautionary effect increases over time as the parameter uncertainty accumulates. This property has been identified in Weitzman (2007a) and Gollier (2008). The last term in (18) captures the novel effect due to non-expected utility, which is increasing in the degree of relative ambiguity aversion η, the degree of ambiguity σ0 , and the time horizon t. The effect of ambiguity aversion on prices is therefore rbt − rt∗ = 21 η |1 − γ 2 | σ02 t, where rbt is determined by a linear φ with η = 0. To analyse the relative importance of this effect, we consider an economy parametrised by a ”quartet of twos”, as proposed by Weitzman (2007b): The rate of pure preference for the present is δ = 2%, the degree of relative 12

This effect is equivalent to correcting the growth rate of consumption g by the precautionary premium (Kimball, 1990) of 0.5(γ + 1)(σ 2 + σ02 t). Indeed, γ + 1 = −cu000 (c)/u00 (c) is the index of relative prudence of the representative agent.

24

risk aversion γ = 2, the mean growth rate g = 2%, its standard deviation σ = 2%. Applied to our base, we consider the trend parameter θ to be distributed normally with mean µ = 2 and standard deviation σ0 = 1%.13 Our choice of normal distributions and a trend close to 2% is also consistent with the findings in Christensen, Gillingham and Nordhaus (2016), and Gillingham, et al. (2015), who analyse a survey on experts’ beliefs about long-term growth until the year 2100. Feeding the above parameters into the pricing formula (17), we obtain rt∗ = 5.92% − 0.01t(2 + 1.5η)%.

(19)

We discipline our choice of the relative ambiguity aversion η to be consistent with empirically observed ambiguity premia. To do so, we adopt the method proposed in Collard, Mukerji, Sheppard and Tallon (2011). For this purpose, consider a situation in which consumption grows by either 20% (with probability π) or 0% over the next 10 years. The agent holds a diffused prior over the true value of π, i.e., she assumes it is uniformly distributed on [0, 1]. Let CE(η) be the corresponding certainty-equivalent growth rate, implicitly defined by 1−kη Z 1   1−kη  1.21−γ 11−γ (1 + CE)1−γ = k π + (1 − π) dπ. k 1−γ 1−γ 1−γ 0 In the absence of ambiguity aversion (or if π is known to be equal to 50%), the certainty-equivalent growth rate equals CE(0) = 9.1%. Surveying experimental studies, Camerer (1999) reports ambiguity premia CE(0)−CE(η) close to 10% of the expected value for such an Ellsbergstyle uncertainty. This corresponds to a reduction by one percentage point 13

According to this distribution, θ lies between 0% and 4% with probability .95.

25

CE(η)

9.0

8.5

8.0

7.5

7.0

6.5

6.0

5.5 5

10

15

20

25

30

η

Figure 1: The certainty-equivalent growth rate CE (in %) as a function of relative ambiguity aversion η. The true rate is 20% with π or 0% with 1 − π, where π ∼ U(0, 1). ∗ r10 ∗ r30

η=0 5.72% 5.32%

η = 5 η = 10 4.97% 4.22% 3.07% 0.82%

Table 1: The efficient rate at the benchmark “quartet of twos”, with σ0 = 1%. in the certainty-equivalent growth rate. Figure 1 plots CE(θ) for relative risk aversion γ = 2. It suggests a choice of the coefficient of relative ambiguity aversion between η = 5 and η = 10. Table 1 reports efficient rates for bonds maturing in 10 or 30 years. Notice that the column for η = 0 represents rbt . Over the first decade, ambiguity aversion decreases annualised rates by an amount between 0.75 and 1.5 percentage points. The relative importance of this effect increases with ∗ ∗ the time horizon. Comparing r30 with r10 , even a low degree of ambiguity

aversion, η = 5, decreases the annualised rate by almost two percentage points. For η = 10, the rate falls by 4.5 percentage points. With η = 10 the rate would turn negative for bonds with maturity of 35 years and beyond.

26

An AR(1) process for log consumption with ambiguity about its long-term trend Our base-line assumptions on the stochastic process give rise to a linearly decreasing term structure. In order to accommodate the increasing term structure for bonds maturing within a decade, we allow for serial correlation in the growth process, similar to Collard, Mukerji, Sheppard and Tallon (2011) and Gollier (2008). Consider first an auto-regressive process of order 1 a` la Vasicek (1977). The agent understands that the growth rate of log consumption reverts to its mean µ. However, she is uncertain about the value of µ:

ln ct+1 = ln ct + xt , xt = ξxt−1 + (1 − ξ)µ + εt , εt ∼ N(0, σ 2 ), εt ⊥ εt0 ,

(20)

µ ∼ N(µ0 , σ02 ), where 0 ≤ ξ ≤ 1. The base-line model from above corresponds to ξ = 0. The other limit case arises when shocks are permanent (ξ = 1) and ambiguity about the trend has no effect. Applying the same analytical techniques for power utility functions and normal distributions which gave rise to equation (17), we obtain the following generalisation:

rt∗ = δ+γ

Var [E[Xt | µ]] EXt 1 2 Var [Xt | µ] + Var [E[Xt | µ]] 1 − γ − η 1 − γ 2 , t 2 t 2 t (21)

27

where Xt is defined as t

ξ(1 − ξ t ) X 1 − ξ τ Xt = ln ct − ln c0 = µt + (x−1 − µ) + εt−τ . 1−ξ 1−ξ τ =1 Given our specifications, we can express the expectation and variance of Xt in terms of the model parameters, EXt ξ(1 − ξ t ) = µ0 + (x−1 − µ0 ) , t t(1 − ξ)   t t Var [Xt | µ] σ2 2 ξ(1 − ξ ) ξ(1 + ξ ) = −2 , +σ t (1 − ξ)2 t(1 − ξ)3 1+ξ and Var [E[Xt | µ]] σ02 = t t

 2 ξ(1 − ξ t ) . t− 1−ξ

If the agent is ambiguity averse, then the final term in (21) is nonzero. Moreover, we obtain a qualitative distinction between the limiting behavior of ambiguity-neutral economies and ambiguity-averse economies whenever 0 < ξ < 1. In the former, the yield curve becomes flat due to the transitory nature of shocks. Under ambiguity aversion, however, the rate depends on the term Var [E[Xt | µ]]/t, which does not tend to a constant. Accordingly, in the long run, the slope of the yield curve is nonzero and it tends to ∂rt∗ 1 = − η 1 − γ 2 σ02 . t→∞ ∂t 2 lim

To illustrate, consider parameters δ = 2%, γ = 2, µ0 = 2%, σ = 2%, σ0 = 1%, and x−1 = 1%. Further, following Backus, Foresi and Telmer (1998), we let the persistence parameter be ξ = 0.7, such that the half-life of shocks is 3.2 years. In Figure 2, we plot the term structure of discount rates for different degrees of ambiguity aversion. Only the curve for η = 0 28

rt

η=0

4.5

4.0

η=10 η=5 3.5

t 5

10

15

20

25

30

Figure 2: The term structure of discount rates in the case of an AR(1) with an ambiguous long term trend and δ = 2%, γ = 2, µ0 = 2%, σ = 2%, σ0 = 1%, x−1 = 1%, and ξ = 0.7 . has a horizontal asymptote. An AR(1) process for log consumption with ambiguous persistence Consider the same class of processes, except that the long-term trend is known while the persistence is uncertain:

ln ct+1 = ln ct + xt xt = ξxt−1 + (1 − ξ)µ + εt εt ∼ N(0, σ 2 ), εt ⊥ εt0 ξ ∼ U(ξ, ξ). Since an analytical solution is unavailable, we solve for rt∗ numerically with an initial consumption c0 = 1. Figure 3 plots the term structure using the parameter values from the previous section, except that µ = 2% and ξ ∼ U(0.5, 0.9).14 Again, ambiguity has a negative effect on rates which 14

In order to compute rt as in (16) we need to quantify G = −(γ + kη(1 − γ))E [Xt | ξ] +

Eφ0 (Eu)Eu0 u0 (c0 )

= b(E exp(G)) with


1 2 γ − kη(1 − γ)2 V ar [Xt | ξ] 2

29

rt 4.8

4.6

η=0

4.4

4.2

η=10

η=5

t 5

10

15

20

25

30

Figure 3: The term structure of discount rates in the case of an AR(1) with an ambiguous mean reversion coefficient and δ = 2%, γ = 2, µ = 2%, σ = 2%, x−1 = 1%, and ξ ∼ U (0.5, 0.9). tends to increase over time and which becomes dominant for distant time horizons. In contrast to the previous specification, both ambiguity-neutral and ambiguity-averse economies give rise to a decreasing yield curve in the long run.

5

Discussion and relation to the literature

This paper showed that ambiguity about consumption growth need not reduce interest rates when ambiguity preferences belong to the multiple priors or to the smooth class. Ambiguity aversion acts like a strong form of pessimism. However, pessimism about the likely distribution of consumption growth need not increase precautionary saving. Further, we showed that the certainty-equivalent formulation of smooth preferences induces an additional effect which increases the relative importance of the ambiguous −kη

and φ0 (Vt (0)) = b (E exp (H)) 1−kη with 1 H = (1 − kη)(1 − γ)E[Xt | ξ] + (1 − kη)(1 − γ)2 V ar[Xt | ξ]. 2

30

future if and only if absolute ambiguity aversion is decreasing. Multiplier preferences, on the other hand, always lead to lower interest rates since the induced pessimism acts as if households beliefs were deteriorated according to first-order stochastic dominance. When applied to a representative agent economy, our analytical results suggest a decrease in long-term efficient interest rates under various specifications of parameter uncertainty about the growth process. In particular, we find that the magnitude of the additional precautionary effect due to ambiguity aversion tends to dominate in the long run. Moreover, in contrast to expected utility, we find that ambiguity aversion leads to a decreasing yield curve in the long-run under mean-reverting shocks with an uncertain mean. Thanks to the formal equivalence between the market-clearing interest rate under efficient markets and the social discount rate, our analytical results also show that ambiguity aversion increases the present value of projects with delayed benefits if they arise in an uncertain future. A previous version of this paper presented our comparative statics results in terms of the social discount rate under smooth preferences (Gierlinger and Gollier, 2008). Therein we introduce the ambiguity prudence effect, which, to our knowledge, has not been shown previously. In a recent paper, Osaki and Schlesinger (2013) study precautionary saving in response to zero-mean ambiguity shocks with smooth preferences. They also identify the effect of decreasing absolute ambiguity aversion on saving. However, in contrast to the present paper, their approach is based on a precautionary premium, similar to the methods proposed by Kimball (1990). They provide sufficient conditions which are consistent with our results in Section 3.3. In addition, they show that second-order stochastic dominance can be

31

further relaxed to third and higher orders under appropriate assumptions on risk preferences. In going beyond the smooth model, our paper shows that the effect of ambiguity on saving depends more generally on whether a belief distortion towards low expected utility increases marginal expected utility. Smooth and MEU preferences need not induce a distortion which induces a stochastic dominance relation. Hence the indeterminacy in saving predictions. On the other hand, the distorted prior under multiplier preferences is always stochastically dominated, thereby leading to an increase in the return on saving. Baillon (2016) introduces an alternative definition of ambiguity prudence in line with the definitions of higher-order prudence under expected utility. Accordingly, such an agent prefers that a noise in probabilities and a noise in payoffs affect disjoint events, rather than having both affect the same event. However, unlike risk prudence, such a property does not characterise an increase in saving in response to an ambiguity shock. Indeed, he shows that the definition does not coincide with the DAAA property for smooth preferences, which characterises positive precautionary saving under risk neutrality, as shown in our Section 3.3. Gollier (2011) studies the effect of ambiguity aversion on static portfolio choice with smooth preferences. Using similar comparative statics techniques to the ones used in this paper, he provides conditions on risk preferences and on the distribution of excess returns which guarantee that ambiguity-averse agents invest less in a risky asset. In his case, the intertemporal distortion through ambiguity prudence plays no role due to the static nature of the portfolio problem. Accordingly, the effect of ambiguity aversion depends on pessimism alone. Our analytical Section 4 is related to Weitzman (2007a) and Gollier

32

(2008), who study the effect of parameter uncertainty on social discount rates in expected utility economies. Weitzman (2007a) finds that uncertainty about the volatility of the growth process may yield a decreasing term structure in social discount rates which approaches negative infinity. Gollier (2008) shows that, in general, the decreasing nature of the term structure is due to the persistence of shocks on the growth rate of consumption. Our analytical example shows that, even with mean reversion, the effect of parameter uncertainty need not vanish in the long run if there is ambiguity aversion. The ambiguity-averse aggregation of priors for a single agent is also related to the aggregation of heterogeneous beliefs in the literature of multiple agents economies. Jouini and Napp (2010) and Gollier (2007) show that investor disagreement affects the term structure. Accordingly, the discount rate can be increasing first and then decreasing, approaching the lowest among all plausible discount rates in the limit. Our analytical results are also related to the asset pricing literature with smooth ambiguity preferences. Ju and Miao (2012) and Collard, Mukerji, Sheppard and Tallon (2011), for instance, show that moderate degrees of ambiguity aversion may generate a high equity premium alongside low riskfree rates. While this is consistent with our analytical results for normal distributions and power utility functions, our general theory shows that risk-free rates may also increase in response to ambiguity. Traeger (2012) also studies the effect of ambiguity aversion on socially efficient discount rates with power utility functions. The benchmark model in his paper satisfies one of the sufficient conditions for an increase in the willingness to save provided in this paper.

33

A

Appendix

A.1

Proof of Proposition 1.

We need the following Lemma, which is Theorem 106 in Hardy, Littlewood and Polya (1934), Proposition 1 in Polak (1996), and Lemma 8 in Gollier (2001). Lemma 6. Consider a function φ : R → R, twice differentiable, increasing P and concave. Consider a vector (q(1), ..., q(n)) ∈ Rn+ with nθ=1 q(θ) = 1, and a function f : Rn → R, defined as n X q(θ)φ(Uθ )). f (U1 , ..., Un ) = φ−1 ( θ=1 0

) . Function f is concave on Rn Define function T such that T (U ) = − φφ00(U (U )

if and only if T is weakly concave in R. Having established the above, consider two scalars α1 and α2 and denote Uiθ = Eu(e ctθ + αi Rt ). Using the notation introduced in the Lemma, it implies that Vt (αi ) = f (Ui1 , ..., Uin ). By the concavity of u, for all θ and any vector (λ1 , λ2 ) with λi ≥ 0 and λ1 + λ2 = 1,

λ1 U1θ + λ2 U2θ = E [λ1 u(e ctθ + α1 Rt ) + λ2 u(e ctθ + α2 Rt )] ≤ Eu(e ctθ + αλ Rt ) =def Uλθ , where αλ = λ1 α1 +λ2 α2 . Since f is increasing in Rn , the previous inequality implies

Vt (αλ ) = f (Uλ1 , ..., Uλn ) ≥ f (λ1 U11 + λ2 U21 , ..., λ1 U1n + λ2 U2n ). 34

(22)

Assume that −φ0 /φ00 is concave. By the Lemma, this implies that

f (λ1 U11 + λ2 U21 , ..., λ1 U1n + λ2 U2n ) ≥ λ1 f (U11 , ..., U1n ) + λ2 f (U21 , ..., U2n ) = λ1 Vt (α1 ) + λ2 Vt (α2 ).

(23)

Combining equations (22) and (23), we obtain Vt (λ1 α1 +λ2 α2 ) ≥ λ1 Vt (α1 )+ λ2 Vt (α2 ) for any (λ1 , λ2 ), therefore V must be concave in α. 

A.2

The adapted Ramsey rule (17)

Recall the specifications: 1. The logarithm of consumption is known to be normally distributed ln e ctθ ∼ N(ln c0 + θt, σ 2 t); 2. However, the trend parameter θ is not known with certainty. Agents believe that it is drawn from a normal distribution θ ∼ N(µ, σ02 ); 3. Preferences satisfy constant relative risk aversion u(c) = constant relative ambiguity aversion φ(U ) =

k (kU )1−ηk 1−kη

c1−γ 1−γ

and

with η =

00

(U ) − |U | φφ0 (U , and where k = sgn(1 − γ) ensures concavity when U is )

negative. As is well known, the Arrow-Pratt approximation is exact under CRRA and normal risk on the logarithm of consumption. Therefore, conditional on each θ, we have that −1

Eu(e ctθ ) = (1 − γ)



 exp (1 − γ)(ln c0 + θt + 0.5(1 − γ)σ t) . 2

We use the same method to compute the φ-certainty equivalent Vt since

35

φ(Eu(e ctθ )) is an exponential function and the random variable θe is normal    Vt (0) = (1−γ)−1 exp (1−γ) ln c0 +µt+0.5(1−γ)σ 2 t+0.5(1−γ)(1−kη)σ02 t2 . However, in order to solve for the pricing rule (16) we are really interested in V 0 (0). A convenient way to structure the algebra is to decompose V 0 (0), exploiting the Arrow-Pratt approximation once more. We have, on the one hand, 1  Eφ0 (Eu(e ctθ )) 2 2 2 = exp (1 − γ) kησ0 t , φ0 (Vt (0)) 2

(24)

and, on the other hand,  E[φ0 (Eu(e ctθ )) Eu0 (e ctθ )] = exp − γ(ln c0 + µt) − 12 γ 2 (σ 2 t + σ02 t2 ) − 0 Eφ (Eu(e ctθ ))  −(γ(1 − γ)kη)σ02 t2 .(25) Finally, multiplying (24) by (25) and plugging the result into (16), we obtain the desired analytical expression (17): 1 1 rt∗ = δ + γµ − γ 2 (σ 2 + σ02 t) − η 1 − γ 2 σ02 t. 2 2

36

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