Abstract
We find equilibrium stock prices and interest rates in a representative-agent model where dividend growth is uncertain, but gradually revealed by dividends themselves, while asset prices reflect current information and the potential impact of future knowledge. In addition to the usual premium for risk, stock returns include a learning premium, which reflects the expected change in prices from new information. In the long run, the learning premium vanishes, as prices and interest rates converge to their counterparts in the standard setting with known dividend growth. If both relative risk aversion and elasticity of intertemporal substitution are above one, the model reproduces the increase in price-dividend ratios observed in the past century, and implies that—in the long run—price-dividend ratios may increase a further forty percent above current levels.
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Notes
For example, see Campbell and Shiller [1], Breen et al. [2], Fama and French [3], Glosten et al. [4], Lamont [5], Baker and Wurgler [6], Lettau and Ludvigson [7], Campbell and Vuolteenaho [8], Polk et al. [9], Ang et al. [10], Binsbergen et al. [11], Chen et al. [12], Kelly and Pruitt [13], Van Binsbergen et al. [14], Li et al. [15], Da et al. [16] and Martin [17].
In detail, \(d\widehat{W^D}_t = \frac{\mu ^D - \widehat{\mu ^D}_t}{\sigma ^D} dt + dW_t.\)
The trivial exception is \(\gamma = 1\), which leads to \(S_t = D_t /\beta \), whence learning has no effects on prices, both with anticipative utility and with rational expectations.
A similar but more technical calculation with Epstein–Zin isoelastic preferences confirms that the prices still diverge, except in the case of unit EIS (elasticity of intertemporal substitution) that nests logarithmic utility and implies that \(S_t = D_t/\beta \).
Formally, consider a measurable space \((\Omega , \mathcal F, \mathbb {P})\) supporting a uniform random variable \(P\sim U[0,1]\) and an IID sequence \((X_t)_{t\ge 1}\), \(X_t\sim B(P)\), where B(P) denotes the Bernoulli distribution with parameter P. Additionally, define the filtration generated by the observations of \(X_t\). Let \(\mathcal F_t = \sigma (X_1, \ldots , X_t),~t\ge 0\), which is the filtration used for Bayesian updating.
This assumption can be relaxed to \(P \sim Beta({\eta }_0, \beta _0)\), with \({\eta }_0,\beta _0>0\).
Recall that time-additive power utility with risk aversion \(\gamma \) recovers from the Epstein–Zin setting \(\gamma =\rho \) and \(\theta =1\), and using the transformation \(V_t = \frac{U_t^{1-\gamma }}{(1-\gamma )(1-\delta )}, \delta = {\text {e}}^{-\beta }\), whence (5) becomes \(V_t = \frac{C_t^{1-\gamma }}{1-\gamma } + {\text {e}}^{-\beta } \mathbb {E}_t\left[ V_{t+1}\right] \).
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Work is partially supported by NSF (DMS-1736414), and by the Acheson J. Duncan Fund for the Advancement of Research in Statistics. Partially supported by the ERC (279582), NSF (DMS-1412529), and SFI (16/SPP/3347 and 16/IA/4443).
Appendices
Appendix: Transitory versus permanent learning
This subsection demonstrates the difference between transitory and permanent learning by examining in detail the asset pricing implications of a model of transitory learning, and by contrasting them with the ones obtained in the prologue for permanent learning.
Consider the case of an unobservable dividend drift that follows an Ornstein–Uhlenbeck process with known coefficients. As before, the dividends themselves are still observable and can be used to estimate the current drift. Let the dividends again grow geometrically, i.e.,
However, let now the growth rate \(\mu _t\) follow a hidden Ornstein–Uhlenbeck process
which means that \(\mu _t\) fluctuates around its long-term mean \({\bar{\mu }}\). Denoting by
it follows that
Moreover, \({\mathcal F}^D_t = \sigma \left( (D_u)_{0\le u\le t}\right) = \sigma \left( (R_u)_{0\le u\le t}\right) = {\mathcal F}^R_t.\) The Kalman–Bucy filter \({\hat{\theta }}_t = \mathbb {E}\left[ \theta _t\vert {\mathcal F}^R_t\right] \) and its variance \(\gamma (t) = \mathbb {E}[({\hat{\theta }}_t-\theta _t)^2]\) satisfy
Let \(\gamma _{\pm } = -\kappa \sigma _D^2 \pm \sigma _D\sqrt{\kappa ^2 \sigma _D^2 + \sigma _\mu ^2}\), be the two roots of the quadratic of the right hand side of (13). Assuming again that \(\theta _0 \sim \mathbb {N}(\mu _0, \sigma _0^2)\), and setting \({\hat{\theta }}_0 = \mu _0, \gamma (0) = \sigma _0^2,\) then the solution to the Kalman–Bucy filter is
and the Brownian motion under \({\mathcal F}_t^R\) is \(\widehat{W^D}\), defined as
so that
Thus
Recall the price process in (3), with the state-price density \(M_t\) is proportional to the marginal utility of consumption \({\text {e}}^{-\beta t} D_t^{-\gamma } \). Thus
Note that, for \(s\le t\), it holds that
which, combined with \({\hat{\theta }}_t ={\hat{\theta }}_s {\text {e}}^{-\kappa (t-s)} + \int _s^t\frac{\gamma (u)}{\sigma _D} {\text {e}}^{-\kappa (t-u) } d\widehat{W^D}_u\), yields
Hence,
Therefore,
As in the long run \(\gamma (u)\) converges to \(\gamma _{+}\), it follows that for large s, t
where
Hence,
This equality in turn implies that
As \(\kappa >0\), the expression \(H(\tau )\) grows at most linearly in \(\tau .\) As a result, for \(\beta >0\) large enough, the above expression is finite. (There is no closed form solution, even in the stationary case \(\gamma (u)= \gamma _{+}\) for all \(u>0\).) Thus, in contrast to the setting of permanent learning, described in the main text, this model of transitory learning gives rise to finite prices, at least for sufficiently large discount rates.
The same argument carries over to Epstein–Zin preferences. Denoting the aggregator by
so that the indirect utility \(V_t\) satisfies
In order to find \({\bar{\sigma }}_v(t)\), recall that
where
As the only source of randomness of the utility comes from the consumption, and both the consumption and the utility processes are linear in consumption, \(\sigma _v(t) = \sigma ^D V_t.\) The transformation to an equivalent normalized utility process is \(\bar{U} = U \circ \phi \), where \(\phi (v) =\int {\text {e}}^{\int A(x)dx} dv,\) which in this case, is \(\phi (v) = \frac{v^{\eta }}{{\eta }}.\) From Itô’s formula, it follows that
It then follows that \({\bar{f}}\) indeed equals (16), \(\bar{A} =0,\) and
As dividends coincide with consumption, i.e. \(C_t=D_t\), the state-price deflator \(M_t\) is [45]
Using the fact that
it follows that
Therefore,
and, substituting (16) and (17), yields
where
Therefore
From (14),
Together with (18) it now follows that
Thus, similarly to (15),
where
Thus
We now calculate
Again, for the same reasons as in the additive utility case above, this expression is finite for a discount rate \(\beta \) large enough, which confirms the claim that prices remain finite even for Epstein–Zin preferences.
B Proofs
Proof of Lemma 3.1
For \(n=1\), it follows from the definition of \(X_1\) that its distribution is \(X_1 \sim BetaBin (1,1,1).\) For \(n>1\), we calculate the posterior distribution. Recall that
where \(f_P\) is the pdf of P and L is the log-likelihood. Hence, \(P\vert X_1,\ldots , X_{n-1} \sim Beta(\sum _{i=1}^{n-1} X_i+1, n-1-\sum _{i=1}^{n-1} X_i+1)\), and thus \(X_n\vert X_1, \ldots , X_{n-1} \sim BetaBin(1,\sum _{i=1}^{n-1} X_i+1, n-1-\sum _{i=1}^{n-1} X_i+1).\) Moreover, given the observations \(X_1, \ldots ,X_{n-1}\) and using (19), for \(n\ge 1\) it holds that
\(\square \)
We formulate this section for the general case of Epstein-Zin utility. The case of power utility corresponds to \(\theta =1\). First, we show the Epstein–Zin utility is well defined, i.e., the infinite-horizon limit in (6) exists. See also (Pennesi [46], Theorem 1) for a related result.
Lemma B.1
Fix an admissible consumption \(C\in \mathcal {L}_\delta \). Then
It follows that the limit in (6) is well defined [hence so is \(U_t(C)]\). Moreover, such \(U_t(C)\) is the unique solution to the recursive equation
with the asymptotic condition
where for any consumption streams \({\tilde{C}}, {\hat{C}}\), the modified process \({\hat{C}}^{{\tilde{C}},n}\) is defined as
Proof of Lemma B.1
Fix any \(N\ge t.\) Then \(U_N^N(C) =0.\) Similarly, if \(N\ge 1\), then \(U_{N-1}^N(C) = (1-\delta )^{\frac{1}{1-\rho }} \mathbb {E}_{N-1}[C_{N-1}].\)
By (backward) induction, assume that (20) is true for \(t=k+1\), and show it for \(t=k\). The induction assumption and Jensen’s inequality imply that
Then
where the first inequality follows from the induction step, and the second from Jensen’s inequality, proving the induction step. It follows that the limit in (6) is well defined, as \(\{U^N_t\}\) for fixed t is an increasing sequence in \(N\ge t\). Hence, \(U_t^N(C)\) is well defined for every N, and thus so is its limit \(U_t(C)\) in (6).
Additionally, (21) now follows by continuity, after taking the limit \(N\rightarrow \infty \) in \( U^N_t(C)= \left\{ (1-\delta ) C_t^{\frac{1-\gamma }{\theta }} + \delta \left( \mathbb {E}_t[ (U^N_{t+1})^{1-\gamma }]\right) ^{\frac{1}{\theta }} \right\} ^{\frac{\theta }{1-\gamma }}\). Whereas the uniqueness of the solution follows from the uniqueness of \(U_t^N\) and the fact that \(U_t(C^{0,N}) = U_t^N(C).\)\(\square \)
Next, set
Define
In view of (26), optimizing over the consumption at time \(s\ge t\) without any constraint on initial wealth, leads to the problem
For convenience, denote \(P_t\) the cum-dividend price, defined as
To complete the description of the market, define the price at time t of a bond maturing at \(t+1\) as
and the interest rate as
Next, to define an equilibrium in this market it remains to define admissible consumption plans. Let \(X_t\) be total the wealth of the representative agent at time t (before any consumption takes place).
Definition B.2
The wealth process X starting from time \(t_0\) is admissible, if \(X_t \ge 0\) for all times \(t\ge t_0\). For a given consumption stream \(C_t\ge 0,~t=t_0, t_0+1,\ldots \) set value of the consumption stream starting from time \(t\ge t_0\) as
Lemma B.3
Let \(t_0\ge 0\), then for any admissible consumption \(C_s,~s\ge t_0\),
Moreover, if \(X_{t_0} = P_{t_0}\) any admissible consumption C is dominated by D, in that \(W_{t_0}(C) \le W_{t_0}(D)\).
Proof
At any time, the agent can invest in two assets, the bond and the stock. Assume that at time at time t the portfolio is valued at \(X_t\). The dividend is paid out first. Then the portfolio can be rebalanced, to include \({\phi }_t\) shares of stock and \(\psi _t\) cash. Thus \(X_t = {\phi }_t(P_t -D_t) + \psi _t\), since the stock price \(P_t\) is cum-dividend, and whence \(\psi _t = X_t - {\phi }_t(P_t -D_t).\) After which the consumption \(C_t\) happens. Then at the next period \(t+1\), the portfolio is worth \(X_{t+1}\), which is comprised of \({\phi }_t P_{t+1}\) wealth invested in stock and \((\psi _t-C_t)(1+r_t)\) cash, i.e.,
Because \(P_t =D_t + \mathbb {E}_t\left[ m_{t+1,t} P_{t+1}\right] \) for any t, from (29) it follows that
Repeating this argument, (31) follows.
By admissibility of \(X_{T+1}\) and the non-negativity of m it follows that \(X_{t_0} \ge \sum _{t=t_0}^T\mathbb {E}_{t_0} \left[ m_{t,t_0} C_{t} \right] \) for all \(T\ge t_0.\) Thus, (32) follows by letting \(T\rightarrow \infty \):
\(\square \)
1.1 B.1 Additive power utility
Proof of Theorem 4.1
The closed form formula for the stock price \(P_n\) at time n is as follows
where
Now, changing the order of the summation
where the second equality uses the identity \( \sum _{j=k}^\infty q^j \, \left( {\begin{array}{c}j\\ k\end{array}}\right) = \left( 1- q \right) ^{-k-1} q^k \) with \(q= {\text {e}}^{ \left( (1-\gamma ){\eta }- \beta \right) } \), showing (7).
To show (8), recall the definition \(B(t, t+1)\)—the price at time t of a zero coupon bound maturing at time \(t+1\) in (29). As for power utility, (24) becomes \(m_{t+1,t} = {\text {e}}^{-\beta } \left( \frac{D_{t+1}}{D_t}\right) ^{-\gamma }\), (8) readily follows by recalling the definition of the interest rate \(r_{t+1,t}\) in (30).
Recall the with power utility the stochastic discount factor \(\pi \) in (25), (26) is \(\pi _{t,t} = D_t^{-\gamma }.\) Thus, by (24), (26) it follows that \(\mathbb {E}_s\left[ \pi _{t,s}C_t\right] = \mathbb {E}_{t_0}\left[ \frac{1}{1-\gamma }\frac{\partial V_{t_0}(D)}{\partial D_s} C_t\right] ,~s\ge t\ge t_0,\) for any \(C_t\ge 0\) admissible consumption. Recall that from Lemma B.3 for any admissible consumption C with initial portfolio wealth \(P_{t_0}\), \(\sum _{t=t_0}^\infty \mathbb {E}_{t_0}\left[ \pi _{t,t_0} D_{t} \right] \ge \sum _{t=t_0}^\infty \mathbb {E}_{t_0}\left[ \pi _{t,t_0} C_{t} \right] .\) For such consumption it follows that
where the first inequality follows from
which in turn follows from (27).
Assume for convenience that \(t_0=0.\) Note that, if \(X_0 = P_0\), \({\hat{C}}_t = D_t\) and \({\phi }_t = 1\), then (33) implies by induction that \(X_t = P_t\) for all \(t\ge 0\). Now, consider the alternative strategy in which at time t the number of shares changes from 1 to \(1+\varepsilon \) on some \(\mathcal F_t\)-measurable event \(A \subset \{|P_t|<M , D_t > 1/M\}\), with \(M>0\). Note, that after the dividend is paid, the share price is \(P_t-D_t\). Thus consumption correspondingly changes from \(D_t\) to \(D_t-\varepsilon (P_t-D_t)\) and to \(D_s (1+\varepsilon )\) for \(s\ge t+1\). That is, define \({\phi }_s^\varepsilon = {\phi }_s + \varepsilon 1_{\{s\ge t\}\cap A}\) and \(c^\varepsilon _s = D_s -\varepsilon P_t 1_{\{s = t\}\cap A} + \varepsilon D_s 1_{\{s\ge t+1\}\cap A} \), and note that this strategy continues to satisfy (33). (Note that \(\varepsilon \) may be either positive or negative.)
Setting \(u(t,C_t) = {\text {e}}^{-\beta t} \frac{ C_t^{1-\gamma }}{1-\gamma }\), the change in expected utility from (D, 1) to \((c^\varepsilon ,{\phi }^\varepsilon )\) is thus
where the last inequality reflects the assumed optimality of the consumption stream D together with the treading strategy \({\phi }\equiv 1\). By concavity, note that for any \(t, x, y>0\):
Whence, on the event A, for \(s>t\)
Therefore, again on A,
where for the first equality the fact that \(u\) is increasing and concave was used. Likewise,
Hence on A, for \(\epsilon >0\) small enough
In view of (36) and (37), it follows that the respective incremental ratios are dominated by an integrable random variable, uniformly in \(\varepsilon \). Thus, dividing \(\Delta ^\varepsilon \) in (35) by \(\varepsilon \) and passing to the limit as \(\varepsilon \downarrow 0\), Lebesgue’s dominated convergence theorem yields
Analogously, as \(\varepsilon \uparrow 0\) it follows that \(\lim _{\varepsilon \downarrow 0}\frac{\Delta ^\varepsilon }{\varepsilon } \ge 0\), whence the limit must be zero. By the tower property of conditional expectation,
As \(M\uparrow \infty \), the event A spans any element of \(\mathcal F_t\), which implies that
This completes the proof by recalling the definition of the SDF m in (24). \(\square \)
We now adapt this proof to the Epstein–Zin recursive utility case.
1.2 B.2 Recursive Epstein–Zin utility
The proof for the general recursive Epstein–Zin utility is more complicated, but the proof that the market is in equilibrium uses the same ideas as in the equivalent part of Theorem 4.1. The major difference is that there is no closed form solution to the price process, as opposed to the one found in Theorem 4.1. Hence, we proceed by finding a power expansion. First, it is more convenient to work with the following equilibrium price candidate P.
To establish the connection between utility U and price P, substitute (24) into (38) to get
Comparing this with (21) it follows that
The proof that condition (22) holds is deferred to Lemma B.12. Next, let \(c_t\) defined by
and attempt to find \(c_t\). In other words \(c_t^{\frac{1-\gamma }{\theta }}\) is the price dividend ratio. Then (39) becomes
Substituting (41) into (21), it follows that
and, using (4),
Note that this is a backward recursion. If \(c_{t+1}\) is known and assuming \({\hat{p}}_t\) is also known, then \(c_t\) can be computed. Additionally, note that (42) can be solved if it is assumed that no more learning takes place, that is if \(c_t = c_{t+1} = c.\) In this case,
It follows that
Thus, define
Next, postulate that
and seek the coefficients \(\alpha _i\) by subsisting into (42). Henceforth, the argument \({\hat{p}}_n\) of \(c_n,c_\infty ^{(n)}, \alpha _n\) is dropped for convenience. The coefficients in this expansion are solved explicitly by inserting (43) into (42). For example, the first one equals
and explicit formulas for higher-order coefficients follow similarly. The next auxiliary lemmas helps to verify the expansion (43).
Lemma B.4
There exists \(\nu _0>0\), such that
Moreover, fix the starting point \(n_0\ge 0\), and assume that
Then for \(n\ge n_0\),
So that
Proof
Set \(D_{n+n_0}^{*}\triangleq D_{n_0} \max \{{\text {e}}^{(n+n_0) {\eta }}, {\text {e}}^{(n+n_0)({\eta }+s)} \} = D_{n_0} {\text {e}}^{(n+n_0)({\eta }+ (s)^{+})} .\) Then \(0< D_{n+n_0} \le D_{n+n_0}^{*}\). Similarly, \(D_{n_0} {\text {e}}^{(n+n_0)({\eta }- (s)^{-})} \le D_{n+n_0}.\) It then follows that (44) is satisfied with \(\nu _0 = {\text {e}}^{ \left| {\eta } \right| + \left| s \right| } \). To show (45), using the fact that U is increasing in consumption, it immediately follows from the definition of \(c_{n_0}\) in (41) that \(c_{n_0}\ge 1\), whence \(U_{n_0}^{*} = U_{n_0}(D^{*})\ge U_{n_0}(D).\) Thus, (21) becomes
where we used the identity \(\mathbb {E}_t[ (U_{n_0+1}^{*}) ^{1-\gamma }] =(U_{n_0+1}^{*}) ^{1-\gamma } \) because the consumption \(D_t^{*} \) is deterministic for \(t\ge n_0.\) Recalling that \(\theta = \frac{1-\gamma }{1-\rho }\) , it follows that for \(V_{n_0}^{*} = \frac{(U_{n_0}^{*})^{1-\rho }}{(1-\rho )(1-\delta )}\)
which is the power utility case, with risk aversion \(\rho \). Hence,
which implies (45) by recalling that \(c_{n_0} = \frac{U_{n_0}}{ (1-\delta )^\frac{\theta }{1-\gamma } D_{n_0}} \le c_{n_0}^{*} = \frac{U_{n_0}^{*}}{ (1-\delta )^\frac{\theta }{1-\gamma } D_{n_0} } = \frac{( (1-\delta )V_{n_0}^{*})^{\frac{1}{1-\rho }}}{ (1-\delta )^\frac{1}{1-\rho } D_{n_0}} =\left( \frac{1}{1-\delta {\text {e}}^{(1-\rho )({\eta }+ (s)^{+}) } } \right) ^{\frac{1}{1-\rho }}=c_{\max }.\) This also shows (46), as \((1-\delta )^{\frac{1}{1-\rho }}c_{\min }D_{n_0}\le U_{n_0}(D)\).
\(\square \)
Similarly, any admissible consumption stream admits the following bounds.
Lemma B.5
Let \(n_0\ge 0\) be the initial time. Then there exists a constant \(K_0>0\), independent of \(n_0\), such that \( U_{n}(C) \le K_0 X_{n},\) for any \(n\ge n_0\) and for any admissible consumption process C.
Proof
Using \(\nu _0\) from Lemma B.4 and recalling (40), it follows that \(K_1^{-1} \nu _0^{-(n-n_0)} D_{n_0}\le c_{max}^{1-\rho }\wedge c_{min}^{1-\rho } D_n \le P_n \le c_{max}^{1-\rho }\vee c_{min}^{1-\rho } D_n \le K_1 \nu _0^{n-n_0} D_{n_0}, \) for some constant \(K_1>0\) and \(n\ge n_0\). Hence, it also follows that \(\frac{P_n}{P_{n-1}}\le K_1^2 \nu _0.\) Using the bounds on U from Lemma B.4, for another constant \(K_2>0\) it follows that \(m_{n+1,n}\ge \frac{1}{K_2 \nu _0^{-\rho }},\) which implies that the same bound holds for \(1+r_{n+1,n} \le K_2 \nu _0^{-\rho }.\) Thus for \(\nu _1 = K_2 \nu _0^{-\rho } \vee K_1^2 \nu _0\), it follows that \(C_n \le \nu _1^{n-n_0} X_{n_0},~n\ge n_0.\) A similar calculation as in Lemma B.4 yields the upper bound \(U_{n_0}(C) \le K_0 X_{n_0}\), for some constant \(K>0.\)
\(\square \)
Lemma B.6
Set
and assume that \(0<{\bar{\delta }}<1\), where
Moreover, let the assumptions of Lemma B.4 hold. Then \( \left| \left( c_\infty ^{(n)}(p_n)\right) ^{\frac{1-\gamma }{\theta }} - \left( c_n(p_n)\right) ^{\frac{1-\gamma }{\theta }} \right| = O\left( \frac{1}{n}\right) .\)
Proof
First, note that \(c_{\infty }^{(n)}\) almost satisfy (42), more specifically, for \(n>0\) big enough
Fix n and \(N>n\). The idea is to express the difference between \((c_n({\hat{p}}_n))^{1-\gamma }\) and \((c_{\infty }^{(n)}({\hat{p}}_n))^{1-\gamma }\) using the difference at time \(n+1\), and then recursively repeat the process until time N. Observe that
here \(\zeta _n\), and \({\hat{\zeta }}_n\) are unknown points in the Taylor remainder. Note that both \(\zeta _n\) and \({\hat{\zeta }}_n\) are uniformly bounded, independently of n. Indeed, the point \(\zeta _n\) is located somewhere between \( 1+ \delta {\text {e}}^{{\eta }\frac{1-\gamma }{\theta } } \left( \mathbb {E}_n\left[ (c_{n+1}({\hat{p}}_{n+1}))^{1-\gamma } {\text {e}}^{ (1-\gamma ) s X_{n+1} } \right] \right) ^\frac{1}{\theta }\) and \( \delta {\text {e}}^{{\eta }\frac{1-\gamma }{\theta } } \left( \mathbb {E}_n\left[ (c_{\infty }^{(n+1)}({\hat{p}}_{n+1}))^{1-\gamma }{\text {e}}^{ (1-\gamma ) s X_{n+1} } \right] \right) ^\frac{1}{\theta }+1+ \frac{{\text{ Err }}}{n}\). Both of these quantities are bounded between 1 and \(B_1\) from (47). Similarly, the point \({\hat{\zeta }}_n\) is located between \( \mathbb {E}_n\left[ (c_{n+1}({\hat{p}}_{n+1}))^{1-\gamma } {\text {e}}^{ (1-\gamma ) s X_{n+1} } \right] \) and \( \mathbb {E}_n\left[ (c_{\infty }^{(n+1)}({\hat{p}}_{n+1}))^{1-\gamma } {\text {e}}^{ (1-\gamma ) s X_{n+1} } \right] \), which are bounded by \({\text {e}}^{(1-\gamma )s^{-}}\) and \(B_2\) from (48). Recalling the definition of \({\bar{\delta }}\) in (49), the previous chain of inequalities continues as
Which implies that
Letting \(N\rightarrow \infty \) the claim of the lemma now follows as both \(c_N\) and \(c_{\infty }^{(N)}\) are bounded. \(\square \)
This lemma can be generalized to higher orders. (The corresponding proof is omitted.)
Lemma B.7
For any \(k\ge 1\), there exists \(\delta >0\) small enough, such that
Lemma B.8
The interest rate \(r_{t,t+1}\) with Epstein–Zin recursive utility is as in (10).
Proof
Using (41), and (4) the SDF from (24) becomes
Recall the definition of bond price \(B(t, t+1)\) in (29). It follows from (50) that
The desired result (10) follows readily now from the definition of the interest rate \(r_{t+1,t}\) in (30). \(\square \)
Corollary B.9
For \(\delta >0\) small enough, (50) implies that
And an error of \(O\left( \frac{1}{n^{k+1}}\right) ,k\ge 1\) can be achieved if higher order approximation of \(\left( c_\infty ^{(n)}({\hat{p}}_n)\right) ^{\frac{1-\gamma }{\theta }}+\sum _{i=1}^k\frac{\alpha _i({\hat{p}}_n)}{n}\) is used to approximate \(\left( c_n({\hat{p}}_n)\right) ^{\frac{1-\gamma }{\theta }}\).
Proof
The proof follows from the combination of Lemmas B.4, B.6, B.7, B.8. \(\square \)
Corollary B.10
For \(\delta >0\) small enough, note that \(\lim \nolimits _{t\rightarrow \infty } \mathbb {E}_{t_0}\left[ m_{t,t_0} P_{t}\right] =0.\)
Proof
Recall that \(\theta = \frac{1-\gamma }{1-\rho }\). Then, under the assumption that \(\delta >0\) small enough,
for some \(\delta _2<1\). This can be seen by considering different cases. For example, when \(\gamma >1\) and \(\rho <\gamma \), so that \(1-\gamma ,\frac{1-\theta }{\theta } = \frac{\gamma -\rho }{1-\gamma }<0\), using the definition of m in (24) and Lemma B.4 it follows that
Thus (51) holds for \(\delta >0\) small enough. Thus, for any \(t_0\ge 0\),
whence \(\lim \nolimits _{t\rightarrow \infty } \mathbb {E}_{t_0}\left[ m_{t,t_0} P_{t}\right] =0.\)\(\square \)
The next corollary is presented for completeness only. It shows that the two price candidates (38) and (28) in the power utility and Epstein–Zin utility coincide.
Corollary B.11
The price P in (38) equals (28).
Proof
Recall that \(m_{t_0,t_0}=1\). The equality between (38) and (28) follows from Corollary B.10. \(\square \)
So far we have been using the recursion (21). We are now ready to show that the asymptotic condition (22) holds.
Lemma B.12
Let \(U_t(D)\) be as in (41). Then for \(\delta >0\) small enough,
Proof
Observe, that the equivalent of (46) also holds for \(U_t\left( D^{0,N}\right) \), for \(N\ge t+1\). Namely,
Thus, similarly to (51) and using the same \(\delta _2\) we can bound \(0<m_{t+1,t}^N\frac{D_{t+1}}{D_t} \le \delta _2\), where
Then from Lemma B.4 it follows that
which in turn converges to zero as \(N\rightarrow \infty \). \(\square \)
We are now ready for the equilibrium proof for Epstein–Zin utility.
Proof of Theorem 4.2
First, note that we have already proved parts of Theorem 4.2. Specifically, Lemmas B.6 and B.7 show the validity of (9) and (11), and (10) and of (12) follow from Lemma B.8 and Corollary B.9 respectively. The next two steps similar to the ones in the proof of Theorem 4.1 is to show that the consumption D maximizes the utility U subject to the budget constraint and then use this result to show the market is in equilibrium.
Let \(\epsilon >0\) and let \(t\ge 0\) be the initial time. Assume the initial wealth is \(X_t = P_t\), so that the consumption stream D is admissible (otherwise, it suffices to scale it). Fix a consumption process C, also admissible for this initial wealth. The first goal is to show that \(U_t(C) \ge U_t(D).\) Without loss of generality assume that \(\sum _{s=t}^{\infty } \mathbb {E}_{t}\left[ m_{s,t} C_s\right] = X_t\). Indeed, \(\sum _{s=t}^{\infty } \mathbb {E}_{t}\left[ m_{s,t} C_s\right] \le X_t\) by Lemma B.3. Thus if the inequality is strict we may increase the consumption, and thereby increase the utility.
The goal now is to show that \(U_t(C) \le U_t(D).\) From (34), there exists \(n\ge t\) such that \( \sum _{s=n+1}^\infty \mathbb {E}_t\left[ m_{s,t} C_{s} \right] \le \epsilon ,\)\(\sum _{s=n+1}^\infty \mathbb {E}_t\left[ m_{s,t} D_{s} \right] \le \epsilon ,\) and hence
Recall the definition (23), which defines the modified consumption process \(D^{C,n}=\left\{ \begin{array} {ll} D_s &{}:s\le n,\\ C_s &{} :s> n. \end{array}\right. \) It then follows from Lemma B.3 that
We next show that \(U_t(D^{C,n}) \le U_t(D) +K_0 \epsilon , \) where \(K_0>0\) is the constant from Lemma B.5. Clearly, we only need to consider the case, when \( U_t(D^{C,n}) \ge U_t(D)\). Then from the concavity of U and
where the third inequality is from Lemma B.5. Then
where the first inequality holds by (34), and the third from (27). Letting \(\epsilon \rightarrow 0\) it follows that \(U_t(C) \le U_t(D)\) and thus D maximizes the utility of consumption from a given initial wealth.
We now proceed in a similar fashion to the proof of Theorem 4.1. Consider the alternative strategy in which at time t the number of shares changes from 1 to \(1+\varepsilon \) on some \(\mathcal F_t\)-measurable event \(A \subset \{ \left| P_t \right| , \langle M , D_t\rangle 1/M\}\), with \(M>0\), while at the next time step \(t+1\) the extra shares now worth \(\varepsilon P_{t+1}\) are consumed in addition to \(D_{t+1}\), and for times \(s\ge t+2\) the consumption remains the same as before \(D_s\). That is, define \({\phi }_s^\varepsilon = {\phi }_s + \varepsilon 1_{\{s= t\}}\cap A\) and \(C^\varepsilon _s = D_s -\varepsilon P_s 1_{\{s = t\}\cap A} +\varepsilon P_s 1_{\{s = t+1\}\cap A} \), and note that this strategy continues to satisfy (33). (Note that \(\varepsilon \) may be either positive or negative.) The change in expected utility from (D, 1) to \((c^\varepsilon ,{\phi }^\varepsilon )\) is thus
where the last inequality reflects the assumed optimality of (D, 1). For any increasing, concave function u(x, y), it holds that
whence, on the event A,
Note that from Lemma B.5 on A, we also have that \(P_{t+1} \le M_1\triangleq \frac{M}{\nu _0}\), and \(D_{t+1} \ge \frac{1}{M_1}.\) Set
Assuming \(0<\varepsilon <1/(2M^2)\), we have that \(D_s,C_s^\varepsilon \ge C_s^M\) for all \(s\ge t\). Thus, from (53)
In view of (54), it follows that the respective incremental ratios are dominated by an integrable random variable, uniformly in \(\varepsilon \). Thus, dividing \(\Delta ^\varepsilon \) in (52) by \(\varepsilon \) and passing to the limit as \(\varepsilon \downarrow 0\), Lebesgue’s dominated convergence theorem yields
Analogously, as \(\varepsilon \uparrow 0\) it follows that \(\lim _{\varepsilon \downarrow 0}\frac{\Delta ^\varepsilon }{\varepsilon } \ge 0\), whence the limit must be zero. By the tower property of conditional expectation,
As \(M\uparrow \infty \), the event A spans any element of \(\mathcal F_t\), and recalling the definition \(m_{t+1,t}\) in (24), we get that that
\(\square \)
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Bichuch, M., Guasoni, P. The learning premium. Math Finan Econ 14, 175–205 (2020). https://doi.org/10.1007/s11579-019-00251-z
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DOI: https://doi.org/10.1007/s11579-019-00251-z