Public Information and E¢ cient Capital
Investments: Implications for the Cost of
Capital and Firm Values
Peter O. Christensen
Department of Finance
Copenhagen Business School, Denmark
Hans Frimor
Department of Accounting and Auditing
Copenhagen Business School, Denmark
Preliminary and incomplete
December 11, 2015
Abstract
In a standard …nancial economics model of asset pricing and value maximizing …rms, we
show better public information about …rm-speci…c and economy-wide events facilitates a
more e¢ cient allocation of capital among …rms as well as more e¢ cient aggregate capital
investments. Thus, investor welfare increases as the informativeness of public information
increases. On the other hand, we also show that improvements in investor welfare due to
better public information are often associated with a higher cost of capital and with lower
…rm values.
Keywords: Public Information; E¢ cient Resource Allocation; Investor Welfare; Cost of
Capital; Firm Value.
1
Introduction
A well-functioning capital market should, through the equilibrium price system, (a) facilitate an e¢ cient sharing of the risks in the economy among individually optimizing investors/consumers and (b) allocate the real resources in the economy to their most e¢ cient
use (over time, among …rms, and within …rms) when …rms chose their actions to maximize
…rm value (cf. Debreu 1959). In single-person decision making settings, better pre-decision
information is valuable if, and only if, this information leads the decision maker to change his
actions (cf. Blackwell’s Theorem). In other words, there is no value of just knowing earlier
(with time-additive preferences); but better information about the future consequences of
current actions may facilitate more e¢ cient action choices, i.e., the decision-facilitating role
of information. Prior literature has established the same result holds in a capital market
setting if beliefs are homogeneous, preferences are time-additive, and the capital market is
(e¤ectively dynamically) complete (see, e.g., Kunkel 1982, Hakansson, Kunkel, and Ohlson
1982, and Feltham and Christensen 1988). In this setting, better public information (such
as accounting or macro economic information) is only valuable to investors if the change in
posterior beliefs and the associated equilibrium price system lead value maximizing …rms to
change their capital investments.
Feltham and Christensen (1988) show …rm-speci…c productivity information is valuable
to investors trading in well-diversi…ed portfolios because it facilitates a more e¢ cient allocation of aggregate capital investments among …rms. Firm-speci…c windfall information
has no value to investors even though the market prices of individual …rms depend on this
information, because it does neither a¤ect the marginal productivity of invested capital nor
the value of well-diversi…ed portfolios. Economy-wide information a¤ecting the beliefs of
future aggregate production/consumption is valuable to investors whether it pertains to the
marginal productivity of invested capital or windfall states a¤ecting only the level of future
aggregate production/consumption. A change in expectations about the level of future production/consumption a¤ects the expected marginal rates of substitution between future and
current consumption and, thus, the state prices. This leads to changes in aggregate capital
investments by value maximizing …rms, i.e., a more e¢ cient resource allocation over time.
We ask the question: Can we empirically evaluate public information system changes
(such as changes in …nancial reporting regulation) in terms of the informativeness and the
e¢ ciency of the resource allocation by investigating the capital market reactions to such
changes, for example, changes to …rms’ cost of capital or changes to …rm values at the
announcement date? The answer is negative: the impact on shareholder wealth and on the
required rate of return are both poor indicators of the impact on investor welfare.
1
The capital market reaction depends on the type of information, the investors’ preferences and on the production function. Even in the most basic consumption-based capital
asset pricing model (CCAPM) with time-additive, constant relative risk aversion (CRRA)
preferences and log-normally distributed aggregate consumption (conditional on the capital
investment), the …rms’cost of capital and their value may increase or decrease following the
announcement of a more informative public information system. For example, we show that
more informative …rm-speci…c productivity information generally increases the …rms’cost of
capital, while the ex ante …rm values decrease (increase) for relative risk aversion parameters
above (below) one. More informative economy-wide productivity information also increases
the …rms’ cost of capital unless the relative risk aversion parameter is slightly above one,
while the ex ante …rm values decrease (increase) for relative risk aversion parameters above
(below) one as for …rm-speci…c productivity information.
The key to understanding these results is that, in a general equilibrium setting, changes in
the capital market information structure a¤ect not only value maximizing capital investments
but also the equilibrium interest rates and the pricing kernel determining the equilibrium
risk-adjustments to expected future payo¤s. We apply the standard consumption-based
capital asset pricing model (see, e.g., Rubinstein 1976, and Breeden 1986) in which …rm
value (and the price of any other asset) is determined by discounting the expected future
payo¤s minus a risk adjustment for the covariance between future payo¤s and the pricing
kernel with the zero-coupon interest rates from the term structure of interest rates. Hence,
the impact of better public information on ex ante …rm values and, eventually, on the implied
cost of capital given …rm value and expected future payo¤s, is determined by the impact
on expected future payo¤s, the risk adjustment to expected future payo¤s, and on the term
structure of interest rates. In the CCAPM with constant relative risk aversion, equilibrium
interest rates are increasing in expected aggregate consumption growth and decreasing in
the riskiness of aggregate consumption growth, and the pricing kernel is a measure of the
scarcity of future aggregate consumption in di¤erent states.
Consider …rst an economy in which there are only diversi…able …rm-speci…c risks such
that the representative investor faces no risks. In this setting, there are no risk-adjustments
to equilibrium interest rates and expected future payo¤s. Hence, the ex ante …rm values
are solely determined by the expected future payo¤s discounted by equilibrium zero-coupon
interest rates.
As noted above, better information about additive …rm-speci…c windfall gains or losses
has no consequence for e¢ cient capital investments and, therefore, no impact on ex ante
…rm values before such signals become publicly available. On the other hand, better information about …rm-speci…c productivity events, which a¤ects both the level of an individual
2
…rm’s future payo¤ and its marginal productivity of invested capital, allows for a more ef…cient allocation of aggregate capital investments among …rms. In other words, increasing
the …rm-speci…c productivity informativeness yields higher future aggregate payo¤s for the
same level of aggregate capital investments, i.e., the productivity of aggregate capital investments increases. For …xed aggregate capital investments, this implies that the marginal
rate of substitution between future and current consumption decreases (which is the price
of the zero-coupon bond). Consequently, the interest rate and, thus, the hurdle rate for an
additional dollar invested to be valuable, increases.
A more e¢ cient allocation of aggregate capital investments among …rms not only increases
future aggregate payo¤s for …xed aggregate investments but also the marginal productivity
of an additional aggregate dollar optimally invested in …rms. Hence, there is a trade-o¤
between smoothing the higher aggregate payo¤s over time by reducing aggregate capital investments (i.e., the income e¤ect) and taking advantage of the higher marginal productivity
of invested capital by increasing aggregate capital investments (i.e., the substitution e¤ect).
The equilibrium outcome of this trade-o¤ depends on the investors’preferences and, in particular, on the investors’elasticity of intertemporal substitution (EIS). With CRRA utility,
the inverse EIS is equal to the constant relative risk aversion parameter ( ). In a somewhat similar setting, Merton (1969) shows consumption smoothing is the dominant concern
when is above one (EIS below one), while the marginal productivity gain is the dominant
concern when is below one (EIS above one). Consistent with this result, we show the
equilibrium aggregate capital investments are decreasing (increasing) in the informativeness
of …rm-speci…c productivity information if is above (below) one. This implies that the
impact on equilibrium aggregate capital investments “dampens” (“ampli…es”) the impact
on the equilibrium interest rate if > 1 ( < 1) compared to a setting in which aggregate
capital investments are kept …xed as the informativeness of the …rm-speci…c productivity
information increases. In the dividing case with log-utility ( ! 1), the concerns for consumption smoothing and the marginal productivity gain are perfectly o¤setting such that
the equilibrium aggregate capital investments are independent of the …rm-speci…c productivity informativeness. In all cases with > 0, the equilibrium interest rate increases as the
informativeness of …rm-speci…c productivity information increases.
Stepping back one period before the …rm-speci…c information becomes publicly available
(i.e., to the ex ante date), aggregate and expected …rm-speci…c capital investments decrease
(increase) for > 1 ( < 1) as the informativeness of the …rm-speci…c productivity information system increases. Hence, the equilibrium spot interest rate for the …rst period increases
for > 1 ( < 1) due to higher (lower) aggregate consumption growth in the …rst period.
The higher equilibrium interest rates in both periods and the less e¢ cient use of the
3
productivity gains (by reducing aggregate capital investments) for > 1 imply that the ex
ante …rm values decrease, i.e., there is a negative stock market reaction to the announcement
of a more informative …rm-speci…c productivity information system. The ex ante expected
utility of the representative investor increases, since less is invested for higher future payo¤s
and since the consumption stream is more “smooth” over time. The …rms’ex ante implied
cost of capital is a (somewhat complicated) weighted average of the equilibrium spot interest
rate and the equilibrium ex post interest rate (from the capital investment date until the
date of the future payo¤s), where the weights are determined by the expected dividends at
each date. Since both interest rates increase with …rm-speci…c productivity informativeness,
the ex ante implied cost of capital also increases unless the weight on the lower of the two
interest rates increases too much. In other words, the impact on the ex ante cost of capital
may, in general, depend on the shape on the equilibrium term structure of interest rates,
which in turn may depend on the pattern of future consumption growth arising from other
sources than capital investments.
The more e¢ cient use of the productivity gains (by increasing aggregate capital investments) with < 1 more than o¤sets the increase in the second-period equilibrium interest
rate such that the ex ante …rm values increase. In this case, there is a positive stock market
reaction to the announcement of a more informative …rm-speci…c productivity information
system and, furthermore, the stock market reaction is positively related to the change in
the representative investor’s ex ante expected utility. Hence, the interpretation of a stock
market reaction to an announcement of a change in information system in terms of welfare gains/losses is highly dependent on the size of the investors’relative risk aversion (or,
equivalently, the inverse EIS). The …rms’ implied cost of capital generally increases with
…rm-speci…c productivity informativeness as when > 1. This makes sense, since an improved allocation of aggregate capital investments among …rms and over time, in general,
lowers the investors’marginal utility of an additional dollar of future consumption and, thus,
the hurdle rate for an additional dollar invested to be valuable must increase.
Secondly, consider an economy in which there are both diversi…able …rm-speci…c risks
and economy-wide productivity risks. The comparative statics with respect to increasing the
informativeness of …rm-speci…c productivity information are basically the same as discussed
above for an economy without economy-wide risks. Contrary to …rm-speci…c information,
increasing the informativeness of economy-wide productivity information has an impact on
both the risk premia in equilibrium interest rates and on the pricing kernel, which are
both stochastic in the second period contingent on the speci…c economy-wide signal. The
economy-wide productivity signal is modeled as the normally distributed posterior mean of
the log of the productivity parameter, which is multiplied on a concave power function of
4
the capital investments, such that aggregate future payo¤s are lognormally distributed given
the productivity signal and the capital investment.
This implies that the impact of a higher economy-wide productivity signal is similar to the
impact of increasing the informativeness of the …rm-speci…c productivity information, i.e., a
higher economy-wide productivity signal is associated with both a higher level of expected
future aggregate payo¤s as well as a higher expected productivity of invested capital. Hence,
the equilibrium aggregate capital investments are decreasing (increasing) in the economywide productivity signal when
> 1 ( < 1) and more so, the more informative these
signals are. In other words, more informative economy-wide productivity information reduces
(increases) the risk in future payo¤s and, thus, the risk in aggregate consumption when
> 1 ( < 1). Since the risk of future aggregate consumption is positively related to the
representative investor’s incentive for precautionary savings in zero-coupon bonds, it follows
that the equilibrium interest rates increase (decrease) when > 1 ( < 1).
The risk-adjustment to expected future payo¤s in the determination of ex ante …rm values is determined by the ex ante covariance between the pricing kernel and the …rms’future
payo¤s. The countercyclicality (procyclicality) of equilibrium aggregate capital investments
with the economy-wide productivity signal reduces (increases) the variation in the pricing
kernel (as well as the variation in future aggregate payo¤s) such that the risk-adjustment is
reduced (increased) when > 1 ( < 1) as the informativeness of the economy-wide productivity information increases. For both > 1 and < 1, the impact on the risk-adjustment
dominates the impact on the expected payo¤s such that the risk-adjusted expected future
payo¤s increase (decrease) when > 1 ( < 1).
The impact of more informative economy-wide productivity information on equilibrium
interest rates and risk-adjusted expected future payo¤s thus go in opposite directions for the
determination of the ex ante …rm values for both > 1 and < 1. We show that the impact
on the equilibrium zero-coupon interest rates are the dominant force such that the equilibrium
ex ante …rm values decrease (increase) when > 1 ( < 1) as the informativeness of the
economy-wide productivity information increases. Hence, as with …rm-speci…c productivity
information, the stock market reaction to the announcement of a more informative economywide productivity information system is negatively (positively) associated with the change
in the investors’equilibrium ex ante expected utility (welfare) when > 1 ( < 1).
We show the …rms’ex ante implied cost of capital increases as the informativeness of the
economy-wide productivity information increases unless the relative risk aversion is slightly
above one. Hence, even if equilibrium interest rates decrease for < 1, the aggressive and
procyclical use of the productivity signals in determining the equilibrium aggregate capital
investments implies that the associated increase in the implied rate of return risk premium
5
more than o¤sets the reduction in interest rates. When is somewhat larger than one, the
impact of the less aggressive and countercyclical use of the productivity signals in determining
the equilibrium aggregate capital investments on the implied rate of return risk premium
is not large enough to o¤set the increase in interest rates. This changes when is slightly
above one such that the ex ante implied cost of capital decreases as the informativeness of
the economy-wide productivity information increases.
Our analysis has important implications for the usefulness of empirical studies of capital
market reactions to changes in public information systems (such as changes in …nancial
reporting regulation, e.g., IFRS adoption) to assess changes in the informativeness and the
associated e¢ ciency of the resource allocation in the economy among …rms and over time.
The …rms’ex ante implied cost of capital generally increases as the e¢ ciency of the resource
allocation increases, but there are exceptions; for example, when the investors’relative risk
aversion is slightly above one and we are considering economy-wide productivity information.
The stock market reaction in terms of changes in ex ante …rm values is negatively (positively)
associated with the e¢ ciency of the resource allocation when the investors’ relative risk
aversion is above (below) one. Of course, the problem is that we do not know the investors’
relative risk aversion (or their elasticity of intertemporal substitution). At the very least,
one should be very careful in how the results from such empirical studies are interpreted;
for example, reductions in the …rms’cost of capital are generally a sign of less informative
public information systems and less e¢ cient resource allocation, while a positive stock market
reaction must be interpreted similarly when the relative risk aversion parameter is above one
(which seems to be the more relevant case empirically).
Related literature
In this section there will be some critical comments on the information and cost of capital literature focussing on the ex post cost of capital in (a) pure exchange economies possibly with
a constant riskfree interest rate, (b) production economies with essentially a single consumption date, (c) production economies with several substituting generations of investors forced
to sell their assets at the end of each period and an exogenous riskfree interest rate. When
investors are risk averse and alive for more than one period, and there are economy-wide
risks, risk premia and equilibrium interest rates are intimately related through the investors’
…rst-order conditions for intertemporal portfolio choice...
6
2
The Economy
We consider a large economy with two periods in which investors consume at three dates
t = 0; 1; 2; and trade long-lived assets at dates t = 0 and t = 1. There are three types of longlived assets in the economy: N J risky assets/…rms in unit supply, and two zero-coupon bonds
in zero net-supply maturing at t = 1 and t = 2, respectively. For simplicity, we assume that
all N …rms of type j = 1; :::; J share the same production technology such that within type
j …rms variations in payo¤s are solely due to variations in …rm-speci…c investments and/or
…rm-speci…c risks. The …rms’dividends are denoted dtjn ; t = 0; 1; 2; j = 1; :::; J; n = 1; :::; N .
The dividends at t = 0 are exogenously given, i.e., d0jn = db0j , while the dividends at t = 1
are determined as the di¤erence between exogenous payo¤s from …rst-period production db1j
(which we, for simplicity, assume are riskless) and capital investments qjn for second-period
production, i.e., d1jn = db1j qjn . Each …rm has an unmodelled manager (an automaton)
who chooses capital investments at t = 1 to maximize the cum-dividend market value of the
…rm at that date. The …rms’dividends at t = 2 depend on the capital investments made
at t = 1, an uncertain economy-wide state of nature se 2 Se as well as on an uncertain
…rm-speci…c state of nature sjn 2 Sjn = Sj , i.e., d2jn = fj (se ; sjn ; qjn ), where the production
function fj ( ) is a twice di¤erentiable, strictly increasing and strictly concave function of
N
J
qjn . All uncertainty is depicted by the set of states S = Se
j=1 n=1 Sjn , over which
investors have homogeneous prior beliefs given by the probability density function (s),
s 2 S. For simplicity, we assume that all elements of the states are independent, i.e., (s) =
J
N
1
e (se )
j=1 n=1 j (sjn ); s 2 S.
At t = 0, investors and managers hold the beliefs (s) over states s 2 S and at t = 2
the state is revealed. At t = 1, the investors’ and managers’ beliefs over states s 2 S
depend on a common public signal y 2 Y and is represented by the conditional probability density function (sjy). The public signal consists of an economy-wide signal ye 2 Ye
pertaining to the economy-wide state se 2 Se and …rm-speci…c signals yjn 2 Yjn = Yj
pertaining to the …rm-speci…c state sjn 2 Sjn = Sj for each …rm such that (sjy) =
J
N
e (se jye )
j=1 n=1 j (sjn jyjn ); s 2 S; y 2 Y . We conjecture (and later verify) that in
a large e¢ cient economy, value-maximizing …rm-speci…c capital investments at t = 1 depend only on the economy-wide signal ye 2 Ye and the …rm-speci…c signal yjn 2 Yjn , i.e.,
qjn (y) = qj (ye ; yjn ); j = 1; :::; J; n = 1; :::; N; y 2 Y .
Since our focus is strictly on the role of public information for the e¢ cient amount of
aggregate capital investments and the allocation of this investment among …rms, and not on
1
Feltham and Christensen (1988) consider the more general setting in which the …rm-speci…c states (and
signals) are only required to be independent conditional on the economy-wide state (and signal).
7
the role of public information for e¢ cient risk sharing and implementation of the investors’
e¢ cient intertemporal consumption plans, we assume, for simplicity, there are N J investors,
who are identical in all respects such as endowments, beliefs, and preferences.2 Hence, we
may assume there is no net-trading in equilibrium and, thus, each investor receives at each
date the fraction N1J of the dividends on the market market portfolio of …rm shares in nonzero net supply, i.e., the equilibrium consumption of a representative investor at each date
P P
t = 0; 1; 2 is determined as N1J Jj=1 N
n=1 dtjn . Our speci…cation of the economy-wide and
…rm-speci…c states and information, and the strong law of large numbers for independent
random variables, then implies that for a large economy (in which N ! 1), the equilibrium
consumption for a representative investor can be expressed as3
J
1Xb
d0j
c0 =
J j=1
c1 (ye ) =
J
1Xb
d1j
J j=1
(1a)
d0 ;
J
1X
E [qj (ye ; yjn )jye ]
J j=1
J
1X
c2 (se ) =
E [fj (se ; sjn ; qjn ) jse ]
J j=1
d1
J
1X E
q (ye )
J j=1 j
J
1X E
f (se )
J j=1 j
f o (se ):
d1
q o (ye );
(1b)
(1c)
Equations (1b) and (1c) re‡ect the fact that if investors hold well-diversi…ed portfolios in a
large economy, then their consumption plans only depend on the economy-wide information
ye at t = 1, and the economy-wide state se at t = 2, i.e., the impact of …rm-speci…c information and states is diversi…able. In order words, the representative investor’s equilibrium
consumption at t = 1 and at t = 2 depend only on the average of the expected dividends
conditional on the economy-wide signal and state, respectively, for the J …rm types.
We assume the investors’ preferences are represented by identical time-additive utility
functions on the form
2
X
u(c0 ; c1 ; c2 ) =
ut (ct ):
t=0
where ut ( ) is twice di¤erentiable, strictly increasing, strictly concave, and has in…nite marginal utility for consumption at the minimal level of consumption ct , i.e., limct #ct ut (ct ) = 1.
It then follows from the investors’…rst-order conditions for optimal portfolio choice at dates
t = 0 and t = 1 that there exists a pricing kernel fm t g such that the equilibrium ex-dividend
2
Feltham and Christensen (1988) consider the role of dynamic trading in a su¢ ciently varied set of
well-diversi…ed portfolios as a mechanism to achieve e¢ cient risk sharing, which in turn ensures Gorman
aggregation with CRRA preferences.
3
See Malinvaud (1972, 1972), Berninghaus (1977) and Feltham and Christensen (1988) for de…nitions of
equilibria and e¢ ciency in large economies.
8
price of any long-lived asset with a sequence of dividends fd1 ; d2 g is given by
vt =
2
X
2
X
B t Et [m t d ] =
=t+1
=t+1
B t fEt [d ] + Covt [m t ; d ]g ;
t = 0; 1;
(2)
where
mt=
u0 (c )
;
Et [u0 (c )]
B t = exp [
t
Et [m t ] = 1;
(
t)] =
Et [u0 (c )]
;
u0t (ct )
and t is the zero-coupon interest rate at date t for maturity , and Et [ ] and Covt [ ]
denote the conditional expectation and covariance, respectively, given information at date
t. Using that the investors’equilibrium consumption plans are measurable with respect to
the economy-wide signal/state, cf., equations (1b)–(1c), the pricing kernel is also measurable
with respect to the economy-wide signal/state and, thus, the equilibrium prices can also be
expressed as
vt =
2
X
e
B t Et [m t E [d ]] =
2
X
=t+1
=t+1
B t fEt [d ] + Covt [m t ; Ee [d ]]g ;
t = 0; 1;
(3)
where Ee [ ] and Cove [ ] denote the conditional expectation and covariance, respectively,
given the economy-wide signal/state at date > t. In other words, the equilibrium pricing
is such that only variations in future dividends due to economy-wide events command a risk
premium— variations in future dividends due to …rm-speci…c events are fully diversi…able.
The literature uses a varied set of cost of capital measures. One measure is the dollar
risk premia, i.e.,
RP t
Covt [m t ; Ee [d ]] ;
t = 0; 1; = 1; 2:
(4)
This is the generally preferred concept for risk premia in valuation theory, since it is always
well-de…ned (when there are no arbitrage opportunities in the capital market) and is consistent with contemporaneous multi-period asset pricing theory (see, e.g., Rubinstein 1976
and Christensen and Feltham 2009). Another concept, which is more directly aligned with
required hurdle rates of returns for …rms’capital investments, is the expected rates of return
on the …rms’stocks (whenever it is well-de…ned), i.e.,
t+1;t
ln
Et [dt+1 + vt+1 ]
;
vt
9
t = 0; 1:
(5)
Christensen et al. (2010) denote 21 the ex-post cost of capital and show that it depends
on both the informativeness of the public information system at t = 1 and on the realized
public signal itself and, thus, it is generally a stochastic variable as seen from t = 0. The
expected rate of return at t = 0, i.e., 10 , is similarly denoted the pre-posterior cost of
capital, and it generally depends on the informativeness of the public signals at t = 1. The
rate of return risk premia are de…ned as the di¤erence between the expected rates of returns
and the riskless spot interest rate, i.e.,
! t+1;t
t+1;t
t+1;t ;
t = 0; 1:
(6)
Christensen et al. (2010) show that in a perfectly competitive exchange economy with
homogeneous beliefs and time-additive preferences, the impacts of the informativeness of the
public information systems on the ex-post cost of capital and the pre-posterior cost of capital
are perfectly o¤setting in the sense that the ex ante cost of capital at t = 0 is independent of
the informativeness of the public information system at t = 1. The ex ante cost of capital is
de…ned as the discount rate which makes the discounted expected future dividends equal
to the equilibrium price of the risky asset at t = 0, i.e., is the discount rate solving the
equation
2
X
exp [
] E0 [d ] :
(7)
v0 =
=1
The ex ante cost of capital is also known as the implied cost of capital at t = 0. Since the
stochastic dividends are …xed in an exchange economy, no impact of the public information
system on the ex ante cost of capital is equivalent to no impact of the public information
system on the ex ante asset prices.
In this article, we ask the question whether more informative public reporting at t = 1
yields more e¢ cient capital investments and, thus, dividends streams (in terms of the ex
ante expected utility of the representative investor) and if so, what will be the impact on
the cost of capital and …rm values? The measure of the e¢ ciency of the public information
system and the associated equilibrium capital investments is the certain annuity consumption
stream CE over the three consumption dates, which gives the representative investor the
same utility as the equilibrium ex ante expected utility EU , i.e., CE is a generalized ex ante
certainty equivalent for our multi-period setting solving the equation
EU =
2
X
ut (CE):
(8)
t=0
Our primary cost of capital measure will be the ex ante cost of capital. It re‡ects the
10
constant required expected rate of return on capital raised from the capital market at t = 0
and used for capital investments or dividends at t = 1. It follows from Christensen et al.
(2010) that any impact of the public information system on this measure is strictly due to
the impact on the e¢ cient allocation of capital investments among …rms and over time in
our production economy. We do not exclude the possibility that new capital is raised (as
negative dividends) at t = 1, but that is a zero-NPV activity in a competitive capital market.
Even though uncertainty about future consequences of current capital investments may have
been reduced due to public information at t = 1, you get what you pay for. Similarly, the
impact of the public information system on …rm values will be measured by the impact on
the ex ante equilibrium …rm values at t = 0 before any signals are released from the public
information system.
3
E¢ cient Resource Allocation
In this section we study the impact of public information on the e¢ cient allocation of capital
among …rms and on the optimal aggregate capital investments in the economy. We impose
additional structure on the production functions and the set of states to clearly distinguish
between states that a¤ect the productivity of invested capital and states which merely a¤ect
the level of future payo¤s independently of the amount of capital invested at t = 1. That is,
we expand the description of the states to encompass both productivity and windfall states
(see Feltham and Christensen 1988).
De…nition 1 e 2 e and jn 2 j constitute windfall states, and e 2 e and jn 2 j
constitute productivity states if se = ( e ; e ) and sjn = jn ; jn ; j = 1; :::; J; n = 1; :::N;
and
(a)
(se ; sjn ) =
e ( e ) e ( e ) j ( jn ) j ( jn );
(b) fj (se ; sjn ; qjn ) = gj ( e ;
(c)
e
and
jn
jn )
+ hj ( e ;
i.e., independence,
jn ; qjn );
i.e., separable production impact,
in‡uence both the level and slope of hj ( ) as a function of qjn .
Feltham and Christensen (1988) show that public information at t = 1 about …rm-speci…c
productivity states ( jn ), economy-wide productivity states ( e ), and economy-wide windfall
states ( e ) improves the e¢ ciency of capital investments in the economy, while information
about …rm-speci…c windfall states ( jn ) has no impact on e¢ cient capital investments. In
this article, we impose a particular structure on the production functions to illustrate more
11
concretely how public information a¤ects e¢ cient capital investments, the cost of capital,
and …rm values. In particular, we make the assumption that
gj ( e ;
hj ( e ;
jn )
jn ; qjn )
= aj + b j
e
= exp
+
j
+ cj
j e
jn ;
+
bj
j jn
0; cj
0;
k
qjn
;
k 2 (0; 1) ;
0;
j
0;
j
where all states are standard normally distributed, i.e., e ; jn ; e ; jn N (0; 1): Given normally distributed states, we model (without loss of generality) the public signals at t = 1 as
the posterior means of the associated states, i.e.,
e jy e
N (y e ;
2
1 e ); jn jy jn
N (y
jn ;
2
1 j );
e jy e
N (y e ;
2
1 e ); jn jy jn
N (y
jn ;
2
1 j ):
Hence, the public signals have pre-posterior distributions:
y
e
N (0;
2
0 e ); y jn
N (0;
2
0 j );
y
e
2
0 e ); y jn
N (0;
N (0;
2
0 j );
with the sum of the pre-posterior and the posterior variances being equal to the prior variances, i.e.,
2
0 e
+
2
1 e
= 1;
2
0 j
+
2
1 j
2
0 e
= 1;
+
2
1 e
= 1;
2
0 j
+
2
1 j
= 1:
The informativeness of the public signals can thus be measured as the pre-posterior variances
of the signals 20( ) for a given prior variance (equal to one without loss of generality).4
Lemma 2 Consider two public information systems 0 and 00 for which the pre-posterior
variances of the signals 20( ) are higher for 00 than for 0 . In this setting, 00 is more informative than 0 .
3.1
E¢ cient allocation within …rm types
In this section we determine the e¢ cient allocation of a …xed per …rm aggregate capital
investment qj among …rms of the same type for a given public signal y 2 Y . The key
quantity for determining an e¢ cient allocation of capital investments among …rms is the
marginal rate of transformation, i.e.,
MRTj ( e ;
4
jn ; qjn )
@hj ( e ; jn ; qjn )
= k exp
@qjn
All proofs are in Appendix C.
12
j
+
j e
+
j jn
k 1
qjn
:
Note that the ratio of marginal rate of transformations for …rms of the same type do not
depend on the economy-wide state. Hence, we can determine the optimal allocation of qj
independently of the economy-wide information and state. Since …rm-speci…c risks do not
command a risk premium (see equation (3)), the optimal allocation of qj is such that the
conditional expected marginal rate of transformation is the same for all …rms of type j given
the …rms’…rm-speci…c productivity signals y jn . Solving for the optimal capital investment
qjy (qj ; y jn ) of …rm n of type j in terms of the common conditional expected marginal rate of
transformation for type j …rms, and using the fact that for a well-diversi…ed portfolio of type
j …rms, the average optimal capital investment must be equal to qj , we get the following
result.
Lemma 3 Assume the per …rm aggregate capital investment in type j …rms, j = 1; :::; J, is
equal to qj .
(a) The optimal investment in …rm n of type j given the public …rm-speci…c productivity
signal y jn is given by
qjy (qj ; y
jn )
= exp
j
y
1
2
jn
j
1
k
2
0 j
= (1
k) qj ;
(9)
and it is increasing in the …rm-speci…c productivity signal y jn . It does not depend on
economy-wide productivity information or on windfall information given qj .
(b) The payo¤ from capital investments hj ( ) in a well-diversi…ed portfolio of type j …rms
is given by
hE
j ( e ; qj ) = exp
j
+
j e
+
1
2
2
j
1+
2
0 j
k
1
k
qjk ;
and it is increasing in the informativeness of the …rm-speci…c productivity signals,
(10)
2
0 j.
Not surprisingly, …rms receiving higher …rm-speci…c productivity signals invest more than
…rms receiving lower …rm-speci…c productivity signals— this is e¢ cient allocation of capital
investments among …rms at work! The more e¢ cient allocation of capital investments among
…rms of the same type increases the productivity of aggregate capital investments with a
factor exp 12 2j 20 j 1 k k and, thus, more so when the …rm-speci…c productivity signals are
more informative, i.e., when the pre-posterior variance 20 j increases.
13
3.2
E¢ cient allocation among …rm types
In this section we determine the e¢ cient allocation of a …xed aggregate capital investment
P
q 0 per …rm among the J …rm types, i.e., q o = J1 Jj=1 qjy (q o ): Having determined the e¢ cient
allocation of capital investments within …rm types, the marginal rate of transformation for
…rms of type j is
MRTj ( e ; qj )
@hE
j ( e ; qj )
= k exp
@qj
j
+
j e
+
2
j
1
2
1+
2
0 j
k
1
k
qjk 1 :
To simplify the analysis, we consider throughout a setting in which the …rm types have
the same sensitivity with respect to the economy-wide productivity state e , i.e., j =
; j = 1; :::; J. This implies that the ratio of marginal rate of transformations for …rms of
di¤erent types do not depend on the economy-wide state and, thus, the e¢ cient allocation
of investments among …rm types is independent of the economy-wide information and state.
Hence, the marginal rate of transformation is common to all …rm types. Solving for the
optimal capital investment qjy (q o ) for …rms of type j in terms of the common marginal rate
of transformation, we get the following result.
Lemma 4 Suppose the aggregate capital investment per …rm is equal to q o and
1; :::; J:
j
= ;j =
(a) The optimal capital investment in …rms of type j is
qjy (q o ) =
j
2
0 j
qo;
(11)
where the expression for j 20 j is given in the proof. The fraction of aggregate
capital investments invested in type j …rms j 20 j does not depend on the economywide and …rm-speci…c signals, but it is an increasing function of the informativeness
of the …rm-speci…c productivity signals, 20 j .
(b) The payo¤ from capital investments hj ( ) at t = 2 to the representative investor is
h( e ; q o ) = exp [ +
k
e] qo ;
(12)
where the expression for is given in the proof. The payo¤ h( e ; q o ) is increasing in
the economy-wide productivity signal e , and in the informativeness of the …rm-speci…c
productivity signals 20 j ; j = 1; :::; J:
Increasing the informativeness of …rm-speci…c productivity information for a given …rm type
increases the productivity of investments within that …rm type and, thus, this …rm type
14
gets allocated a larger fraction of the aggregate investments in the economy. Increasing the
informativeness of the …rm-speci…c productivity information for all …rm types increases the
overall productivity of aggregate investments in the economy.
3.3
E¢ cient allocation over time
The public productivity information at t = 1 a¤ects the e¢ cient allocation of aggregate
capital investments among …rms. When aggregate capital investments are allocated more
e¢ ciently, the future payo¤s increase for a …xed aggregate capital investment, and the productivity of an additional aggregate dollar invested increase. In this section we determine
how these facts a¤ect the optimal aggregate investments and, thus, the optimal intertemporal allocation of aggregate consumption possibilities. An optimal aggregate investment
maximizes the representative investor’s conditional expected utility given public information
at t = 1, i.e.,
i
h
q o (ye ) = arg max U1 (ye ) = u1 (cy1 ) + E u2 (cy2 )jy e ; y e ; q o ;
(13)
q o 2R+
where cyt denote consumption at date t = 1; 2 given an optimal allocation of the aggregate
capital investment q o among …rms given in Lemma 3 and Lemma 4, i.e.,
cy1 = d1
qo;
cy2 = g( e ) + h( e ; q o );
P
P
where g( e ) = a + b e ; a = J1 Jj=1 aj ; and b = J1 Jj=1 bj . The …rst-order condition for an
optimal aggregate investment q o (ye ) given the economy-wide public signal ye = (y e ; y e ) is
exp [
21 (ye )]
E
@h
@h
jye ; q o (ye ) + Cov m21 ;
jye ; q o (ye )
@q o
@q o
= 1;
(14)
where
m21 (ye ) =
u02 (c2 )
;
E [u02 (c2 )jye ; q o (ye )]
exp [
21 (ye )] =
E [u02 (c2 )jye ; q o (ye )]
:
u01 (c1 (ye ))
Comparing to equation (3), we obtain the following result.
Lemma 5 Suppose j = ; j = 1; :::; J: The optimal aggregate capital investment q o (ye )
maximizes the cum-dividend market value of the market portfolio taking the equilibrium riskless interest rate 21 (ye ) and pricing kernel m21 (ye ) as given. In turn, the equilibrium riskless
interest rate 21 (ye ) and pricing kernel m21 (ye ) both depend on the optimal aggregate capital
investment q o (ye ).
15
(a) The optimal aggregate capital investment q o (ye ) and its allocation among …rms do not
depend on the …rm-speci…c windfall signals and their informativeness.
(b) The optimal aggregate capital investment q o (ye ) does not depend on the …rm-speci…c
productivity signals, but may depend on the informativeness of these signals through
the impact on the e¢ ciency of its allocation among …rms.
(c) The optimal aggregate capital investment q o (ye ) may depend on both the economy-wide
windfall and productivity signals and on their informativeness.
Firm-speci…c windfall information a¤ects neither the allocation of aggregate capital investments among …rms nor the equilibrium consumption of a representative investor holding a
fraction of the market portfolio and, thus, has no impact on the optimal aggregate capital
investments. Firm-speci…c productivity information facilitates a more e¢ cient allocation of
aggregate capital investments among …rms and, thus, increases the productivity of aggregate capital investment as well as the level of aggregate future payo¤s for a given aggregate
capital investment. The former e¤ect calls for higher aggregate capital investments, i.e.,
a substitution e¤ect, while the latter e¤ect calls for lower aggregate capital investments in
order to smooth the productivity gains over the two periods, i.e., an income e¤ect.
Economy-wide windfall information does not a¤ect the allocation of aggregate capital
investments among …rms, but it a¤ects the beliefs about future aggregate consumption possibilities. A favorable economy-wide windfall signal yields higher conditional expected future
aggregate consumption possibilities, and more informative economy-wide windfall signals reduce the posterior variance of future economy-wide windfall gains. Both e¤ects calls for
reduced aggregate capital investments. The former e¤ect is due to smoothing of expected
windfall gains over the two periods, i.e., an income e¤ect, while the latter e¤ect is due to
a reduced demand for precautionary savings. Economy-wide productivity information have
similar e¤ects in addition to its impact on the productivity of aggregate capital investments,
i.e., the substitution e¤ect.
The net-impact on optimal aggregate capital investments of the substitution, the income,
and the precautionary savings e¤ects following from changes in the informativeness of …rmspeci…c productivity information, the speci…c economy-wide signals, and their informativeness depends on the investors’preferences. In Section 4, we consider standard time-additive
CRRA preferences and study these trade-o¤s explicitly depending on the investors’common
relative risk aversion parameter.
16
3.4
Ex ante welfare
In the preceding subsections, we have characterized the e¢ cient allocation of resources among
…rms and over time, and how that allocation may depend on speci…c signals and their
informativeness. More informative …rm-speci…c productivity signals increase payo¤s on welldiversi…ed portfolios for the same aggregate capital investment and are, thus, valuable to
investors from an ex ante perspective if they prefer more to less. At the same time, the
optimal aggregate capital investments may depend on both the informativeness of …rmspeci…c productivity signals and the speci…c economy-wide signals and their informativeness.
Since the optimal aggregate capital investment maximizes the conditional expected utility
of the representative investor at t = 1 (see equation (13)), we can apply Lemma 2 and
the Blackwell Theorem to shown that the investors’ ex ante expected utilities are weakly
increasing in the pre-posterior variances of the public signals, i.e., public information is
valuable to investors if, and only if, this information a¤ects optimal capital investment
choices.
Proposition 6 Consider two public information systems 0 and 00 for which the informativeness of the signals (i.e., pre-posterior variances 20( ) ) are higher for 00 than for 0 . In
this setting, the investors’ ex ante expected utilities with 00 are at least as high as with 0 .
In particular, the investors’ex ante expected utilities
2
0 j; j
(a) do not depend on the informativeness of the …rm-speci…c windfall signals
1; :::; J,
(b) increase in the informativeness of the …rm-speci…c productivity signals
2
0 j; j
(c) increase in the informativeness of the economy-wide windfall signals
2
0 e,
=
= 1; :::; J,
and
(d) increase in the informativeness of the economy-wide productivity signals 20 e , unless the
substitution, income, and precautionary savings e¤ects balance such that the optimal
aggregate capital investment is independent of the economy-wide productivity signals.
4
Implications for the Cost of Capital and Firm Values
In this section, we examine the implications for the cost of capital and ex ante …rm values of
the impact of the informativeness of public information on e¢ cient capital investments. As
shown in the preceding section, the e¢ cient allocation of aggregate investments among …rms
do not depend on the investors’preferences, but the optimal aggregate capital investment
does depend on these preferences. Closed-form expressions for equilibrium interest rates and
17
risk premia can only be calculated for certain classes of utility functions and distributions
of future payo¤s. We use the standard model in consumption-based asset pricing with timeadditive power utility, i.e.,
ut (ct ) = exp [
t] 1 1 ct1 ;
ct > 0;
> 0;
6= 1:
Combined with lognormal distributions for consumption, this utility function yields closedform solutions for asset prices. To preserve a lognormal distribution of t = 2 consumption
for the representative investor given public information at t = 1, we assume throughout the
following analysis that the average …xed components of t = 2 dividends is equal to zero, i.e.,
J
1X
a=
aj = 0;
J j=1
and that there is no economy-wide windfall risk, i.e.,
cj = 0;
j = 1; :::; J:
As noted above, the impact of economy-wide windfall information is to facilitate improved
consumption smoothing through an inverse relationship between the economy-wide windfall
signal and the aggregate capital investments. That role of economy-wide windfall information
is shared with economy-wide productivity information which, in addition, is informative
about the productivity of aggregate investments. Hence, there is not much loss of generality
in assuming that there is only economy-wide productivity information. In addition, the
aggregate t = 2 dividends can only be lognormally distributed if the sensitivity of the
economy-wide productivity state is the same for all …rm types, i.e., we assume throughout
the following analysis that
j = 1; :::; J:
j = ;
Even though we make assumptions to ensure that the representative investor’s t = 2 consumption is lognormally distributed given information at t = 1, the prior distribution at
t = 0 of the representative investor’s t = 2 consumption is not lognormal if the equilibrium
aggregate investments at t = 1 varies with the economy-wide productivity signal at t = 1
(which is generally the case). Since there is no closed-form solution for equilibrium capital
investments even with lognormally distributed aggregate t = 2 dividends given information
at t = 1, these investments as well as the ex ante asset prices and the ex ante expected
utility of the representative investor must be calculated numerically.
We illustrate our results using a setting with three …rm types primarily distinguished
18
by the informativeness of their …rm-speci…c productivity informativeness. The base case, in
which all …rms have identical production functions and information systems, is given by the
parameters in Table 1.
Exogenous payo¤s
db0j
db1j
29
50
Production functions
Informativeness
2
0 j
2
0 e
0.00 0.00 1.10 0.20 0.25 0.75 0.50 0.50
0.50
aj
bj
j
j
k
2
0 j
Table 1: Base case parameters for exogenous payo¤s, production functions,
and informativeness of information systems.
We assume throughout that the rate of time preference is = 2%, but we vary the values
of the relative risk aversion parameter . Note that with time-additive power utility, the
relative risk aversion parameter is also a measure of the inverse elasticity of intertemporal
substitution, i.e., a high relative risk aversion for stochastic variations in consumption at a
given date also re‡ects a strong desire for smoothing consumption across periods.
We start our analysis in Section 4.1 considering the impact of public information on
e¢ cient capital investments and the ex post cost of capital, while Section 4.2 examines how
the informativeness of forthcoming public information a¤ects the ex ante cost of capital and
…rm values.
4.1
E¢ cient capital investments and ex post cost of capital
In this section, we examine the impact of increasing the informativeness of the public signal
y = y e ; fy jn ; y jn gj=1;:::;J;n=1;:::;N at t = 1 on the equilibrium capital investments and the
…rms’ex post cost of capital. Consider …rst a given average aggregate capital investment q o
at t = 1. Given an optimal allocation of the aggregate capital investment q o among …rms,
the representative investor’s consumption at t = 1 and t = 2 is given by (see Lemma 4)
cy1 = d1
qo;
cy2 = exp [ +
k
e] qo :
Conditional on the economy-wide productivity signal y e and the average aggregate capital
investment q o , the representative investor’s t = 2 consumption is lognormally distributed,
i.e.,
ln[cy2 ]jy e ; q o N ( + y e + k ln q o ; 2 21 e ):
(15)
19
The second-period spot interest rate y21 (y e ; q o ) at t = 1 is determined as the marginal rate
of substitution between t = 1 and t = 2 consumption (see equation (2)), i.e.,
y
21 (y e ; q o )
=
ln
"
E[u02 (cy2 )jy e ; q o ]
u01 (cy1 )
#
:
Calculating the conditional expected marginal utility of t = 2 consumption and simplifying
yields the following result.
Lemma 7 Conditional on the economy-wide productivity signal y e and the average aggregate capital investment q o , the equilibrium second-period spot interest rate y21 (y e ; q o ) at t = 1
is determined as
y
21 (y e ; q o )
= +
= +
i
h
E ln[cy2 ]jy e ; q o
+ y e + k ln[q o ]
ln[cy1 ]
ln[d1
1 2
Var
2
qo]
i
h
ln[cy2 ]jy e ; q o
1 2 2 2
1 e:
2
(16)
Hence, the second-period spot interest rate is determined as the rate of time preference
plus the inverse elasticity of intertemporal substitution times the (percentage) expected
consumption growth minus a risk premium. The risk premium is determined as one-half
times the relative risk aversion parameter
squared times the variance of (percentage)
consumption growth. The intuition for the risk premium in the second-period spot interest
rate is that more uncertainty about t = 2 consumption increases the precautionary demand
for sure dollars at t = 2 (i.e., the riskless security) and, thus, in equilibrium, the spot interest
rate must be lower.
Since an increase in the …rm-speci…c productivity informativeness 20 j increases the average productivity parameter of capital investments (see Lemma 4), the conditional expected
consumption growth also increases, while the conditional variance of consumption growth
is una¤ected. Hence, the second-period spot interest rate y21 (y e ; q o ) increases as the informativeness of …rm-speci…c productivity information 20 j increases for any of the …rm
types j = 1; :::; J (conditional on y e and q o ). Similarly, the conditional expected consumption growth increases in both the economy-wide productivity signal y e and in the average
aggregate capital investment q o , while the conditional variance of consumption growth is
una¤ected by these quantities. Hence, the second-period spot interest rate y21 (y e ; q o ) is an
increasing function of both y e and q o . Furthermore, the second-period spot interest rate
y
21 (y e ; q o ) is an increasing function of the informativeness of the economy-wide productivity
2
signal, i.e., 20 e = 1
1 e , due to a reduced uncertainty about t = 2 consumption and, thus,
a lower demand for precautionary savings and, hence, the risk premium in the second-period
20
spot interest rate is reduced.
The second-period pricing kernel my21 ( e ; y e ; q o ) determining the ex post risk premia in
equilibrium security prices at t = 1 is determined as the ratio between the marginal utility
of t = 2 consumption and the expected marginal utility of t = 2 consumption conditional
on the economy-wide productivity signal y e and the average aggregate capital investment
q o (see equation (2)), i.e.,
my21 ( e ; y e ; q o ) =
u02 (cy2 ( e ; q o ))
E[u02 (cy2 )jy e ; q o ]
:
Using the lognormal distribution property of t = 2 consumption and completing the calculations yields the following result.
Lemma 8 Conditional on the economy-wide productivity signal y e and the average aggregate capital investment q o , the equilibrium pricing kernel my21 (y e ; q o ) at t = 1 is determined
as
my21 ( e ; y e ; q o ) = exp
= exp
h
h
i
E ln[cy2 ]jy e ; q o
(y
e)
e
1
2
ln[cy2 ( e ; q o )]
)2
(
2
1 e
1 2
Var
2
:
h
ln[cy2 ]jy e ; q o
ii
Since the investors’t = 2 consumption is increasing in the economy-wide productivity state
y
e , the pricing kernel m21 ( e ; y e ; q o ) is decreasing in e and, thus, the pricing kernel is a
measure of the “scarcity” of consumption at t = 2 with an expected value equal to one
conditional on the the economy-wide productivity signal y e . Note that the pricing kernel is independent of the average aggregate investment q o in our “lognormal setting” and,
thus, the equilibrium pricing kernel with an optimal average aggregate capital investment is
m21 ( e ; y e ) = my21 ( e ; y e ; q o ).
y
The equilibrium ex-dividend price v1jn
of …rm n of type j at t = 1 given the …rm-speci…c
signals (y jn ; y jn ), the economy-wide productivity signal y e , and the average aggregate
capital investment q o is given by equation (3) as
y
y
v1jn
= v1j
(y
= exp[
y
jn ; y jn ; y e ; qj (
y
21 ]
h
))
E[dy2j jy jn ; y jn ; y e ; qjy (
21
)]
y
RP21jn
i
;
(17)
where the conditional expected t = 2 dividend is
E[dy2j jy
y
jn ; y jn ; y e ; qj (
= aj + c j y
)] = aj + cj y
jn + exp
j
+
j
jn
+ E[hyj jy
y
jn +
1
2
y
jn ; y e ; qj (
2
1 j
j
+
)]
ye+
1
2
qjy ( )
2
1 e
k
;
(18)
and where the equilibrium capital investment function qjy ( ) (for …xed q o ) is given by (9) and
(11) in Lemma 3 and Lemma 4 as
qjy (y
jn ; y e ; q o )
= exp
y
j
1
2
jn
j
1
k
2
0 j
= (1
k)
j
2
0 j
qo:
y
The conditional dollar risk premium RP21jn
is given by equation (4) as
y
y
RP21jn
= RP21j
(y jn ; y jn ; y e ; qjy ( ))
h
h
= Cov my21 ; E dy2j jy jn ; y jn ;
i
y
;
q
(
)
jy
e j
y
jn ; y jn ; y e ; qj (
i
) ;
(19)
and the rate of return risk premium ! y21jn is given by equations (5) and (6) as
! y21jn = ! y21j (y
y
jn ; y jn ; e ; qj (
y
y
21j (y jn ; y jn ; e ; qj (
)) =
))
y
21 (y e ; q o )
i3
2 h y
y
E d2j jy jn ; y jn ; e ; qj ( )
5
= ln 4 y
v1j (y jn ; y jn ; y e ; qjy ( ))
y
21 (y e ; q o ):
(20)
Lemma 9 Conditional on public information at date t = 1, it holds that:
y
(a) The dollar risk premium RP21j
for …rm n of type j at t = 1 is given as
y
RP21jn
= 1
2 2
1 e
exp
E[hyj jy
y
jn ; y e ; qj (
(21)
)]:
(b) The one-period ahead expected rate of return y21jn of …rm n of type j at t = 1 is given
as
y
y
y
21jn = 21 (y e ; q o ) + ! 21jn ;
where the rate of return risk premium ! y21jn is given as
! y21jn
= ln
"
aj + c j y
aj + c j y
jn
jn
+ exp
+ E[hyj jy
(c) The one-period ahead expected rate of return
22
y
jn ; y e ; qj ( )]
y
y
2 2
1 e E[hj jy jn ; y e ; qj (
y
21
)]
#
:
of the market portfolio of …rms at
t = 1 is given as
y
21
=
y
21 (y e ; q o )
+ ! y21 ;
where the rate of return risk premium ! y21 is given as
! y21 =
2 2
1 e:
The …rms’ex post dollar risk premia depend on both the average aggregate capital investment, q o , and on the …rm-speci…c and economy-wide productivity signals, y jn and y e , and
is, thus, non-trivial random variables as seen from an ex ante perspective at t = 0. This is
because the level of dividends covarying with the economy-wide productivity state at t = 2
depend on and are increasing in these signals. However, the …xed component of t = 2 dividends aj and the …rm-speci…c windfall signal y jn do not a¤ect the dollar risk premium,
since the latter is independent of the economy-wide productivity state at t = 2 and is, thus,
fully diversi…able for investors.
On the other hand, the …rms’ex post rate of return risk premia depend on both the …xed
component of t = 2 dividends aj and the …rm-speci…c windfall signal y jn . Since
0 and
cj 0, the ex post rate of return risk premium of …rm n of type j decreases as aj increases
and as y jn increases. This re‡ects the fact that as the fraction of expected t = 2 dividends
which does not add to the dollar risk premium increases, the lower is the required expected
rate of return. Hence, this is a similar impact on the required expected rate of return as the
impact on the required rate return on a stock when leverage is reduced. The …rm-speci…c
productivity signal y jn , on the other hand, increases the required expected rate of return,
since it increases the dollar risk premium. Appendix A develops this result further expressing
a …rm’s required expected rate of return as a weighted average of required expected rates of
returns on …rm-speci…c “windfall claims”and “productivity claims.”
Since we assume that the average …xed component of t = 2 dividends is equal to zero, i.e.,
a = 0, and …rm-speci…c windfall risks are fully diversi…able, this “leverage e¤ect”on the rate
of return risk premia for individual …rms has no impact on the t = 2 dividends of the market
portfolio (nor on a well-diversi…ed portfolio of type j …rms) and, therefore, no impact on the
rate of return risk premium of the market portfolio (or a well-diversi…ed portfolio of type j
…rms). In particular, the rate of return risk premium of the market portfolio does neither
depend on …rm-speci…c signals nor on economy-wide signals or on the average aggregate
capital investment. This implies that the equilibrium rate of return risk premium on the
market portfolio for an equilibrium average aggregate capital investment is ! 21 = ! y21 .
23
y
21
Combined with Lemma 7, the one-period ahead expected rate of return
portfolio of …rms at t = 1 is given as
y
21
= +
+ y e + k ln[q o ]
ln[d1
qo] +
2 2
1 e
1
2
1
of the market
(22)
:
Hence, the one-period ahead expected rate of return y21 of the market portfolio of …rms at
t = 1 is increasing in the informativeness of the economy-wide productivity signal 20 e =
2
1
> 2, ceteris paribus.
1 e if
However, the equilibrium average aggregate capital investment q o also depends on the
informativeness of the economy-wide productivity signal 20 e and on the economy-wide productivity signal y e itself. The equilibrium average aggregate capital investment q o is by
Lemma 5 determined as the value maximizing investment for the market portfolio taking
the second-period spot interest rate and the pricing kernel as given, i.e., q o is determined
implicitly by equation (14) as
exp [
21 (y e )] = exp
h
y
21 (y e ; q o
= k (q o (y e ))k
1
i
(y e ))
exp [ ] [E [exp [
e ] jy e ]
+ Cov [m21 ; exp [
e ] jy e ]] :
(23)
Using the expression for the interest rate in Lemma 7, and using the expression for the
pricing kernel in Lemma 8 to compute the last term on the right-hand side of (23) yield the
following result.
Lemma 10 Conditional on public information at date t = 1, the equilibrium average aggregate capital investment q o (y e ) is determined implicitly by the equation
21
(y e ) = ln [k]
(1
k) ln [q o (y e )] +
+
ye+
1
2
(1
2 )
2
1 e
;
(24)
where
21
(y e ) = +
+ y e + k ln[q o (y e )]
ln[d1
q o (y e )]
1 2 2 2
1 e:
2
(25)
Having determined the equilibrium spot interest rate, 21 (y e ), the equilibrium rate of return
risk premia, ! 21jn , the equilibrium average aggregate capital investment, q o (y e ), and its
allocation among …rms, the following two subsections examine the impact on these quantities
of varying the informativeness of …rm-speci…c and economy-wide productivity information,
respectively.
24
4.1.1
Firm-speci…c information
Note from Lemma 10 that the informativeness of …rm-speci…c productivity information
neither a¤ects the risk premium in the equilibrium interest rate ( 12 2 2 21 e ) nor the riskadjustment to the expected marginal productivity of equilibrium average aggregate investments ( 12 2 (1 2 ) 21 e ). Hence, the impact of …rm-speci…c productivity informativeness
on the equilibrium interest rate and the risk-adjusted expected marginal productivity of average aggregate investments is exclusively through the impact on the average productivity
parameter of aggregate capital investments.
As noted above, increasing the informativeness of …rm-speci…c productivity information
2
0 j for any of the …rm types j = 1; :::; J improves the allocation of aggregate capital investments among …rms as re‡ected by an increase in the average productivity parameter of
capital investments (see Lemma 4). For …xed aggregate capital investments, an increase in
the average productivity parameter of capital investments increases the expected consumption growth and, thus, increases the interest rate. Improving the allocation of given aggregate
capital investments among …rms increases the level of future consumption but leaves current
consumption unchanged. This suggests that aggregate investments should be reduced in
order to smooth the productivity gains in consumption over the two periods (as re‡ected by
the inverse elasticity of intertemporal substitution ). However, improving the allocation of
aggregate capital investments also increases the marginal return of an additional aggregate
dollar invested, which suggests that aggregate investment should be increased. The latter
(substitution e¤ect) suggests that the equilibrium interest rate increases further compared to
a setting with …xed aggregate investments, while the former (income e¤ect) suggests that the
impact on the equilibrium interest rate is reduced compared to a setting with …xed aggregate
capital investments.
The following result shows that the equilibrium ex post cost of capital on the market
portfolio increases as the …rm-speci…c productivity information becomes more informative
in any of the J …rm types, while the impact on the equilibrium average aggregate capital
investment depends on the investors’inverse elasticity of intertemporal substitution .
Proposition 11 Conditional on public information at date t = 1, it holds that:
(a) The equilibrium interest rate 21 at t = 1 is increasing in the informativeness of …rmspeci…c productivity information 20 j for …rm type j = 1; :::; J.
(b) The equilibrium rate of return risk premium on the market portfolio ! 21 is independent
of the informativeness of …rm-speci…c productivity information 20 j for …rm type j =
1; :::; J.
25
(c) The equilibrium one-period ahead expected rate of return 21 of the market portfolio of
…rms is increasing in the informativeness of …rm-speci…c productivity information 20 j
for …rm type j = 1; :::; J.
(d) The equilibrium average aggregate capital investment q o is strictly decreasing (increasing) in the informativeness of …rm-speci…c productivity information 20 j , j = 1; :::; J,
if, and only if, the inverse elasticity of intertemporal substitution is larger (smaller)
than one.
2 2
The rate of return risk premium on the market portfolio ! 21 =
1 e is independent of
…rm-speci…c productivity informativeness and, thus, the impact of …rm-speci…c productivity
informativeness on the one-period ahead expected rate of return on the market portfolio
21 (y e ) = 21 (y e ) + ! 21 (i.e., the equilibrium ex post cost of capital on the market portfolio)
is exclusively through the equilibrium interest rate 21 .
The optimal trade-o¤ in determining the optimal aggregate capital investments between
the impact of more informative …rm-speci…c productivity information on the level and on
the marginal productivity of capital investments, i.e., between consumption smoothing and
increased returns on an additional aggregate dollar invested, depends on the investors’inverse
elasticity of intertemporal substitution. The optimal trade-o¤ is such that optimal capital
investments increase in the informativeness of …rm-speci…c productivity information if the
incentives for intertemporal consumption smoothing is relatively minor (i.e., < 1), while
the opposite result obtains if the incentive for consumption smoothing is a main concern (i.e.,
> 1). The dividing case is log-utility (i.e., lim !1 1 1 ct1 = ln(ct )) in which case there
is no impact of the informativeness of …rm-speci…c productivity information on equilibrium
aggregate capital investments.5
Even though the equilibrium aggregate investments may decrease as the …rm-speci…c productivity information becomes more informative (due to the investors’consumption smoothing incentives), the impact on the equilibrium expected consumption growth and, thus, on
the equilibrium interest rate is still positive. That is, the productivity gains from more
e¢ cient allocations of aggregate capital investments among …rms more than o¤set the negative impact on future payo¤s of the reduction in equilibrium aggregate capital investments.
This is illustrated in Figure 1. While the equilibrium aggregate capital investments may decrease or increase (depending on whether > 1 or < 1), the equilibrium one-period ahead
expected rate of return on the market portfolio increases as the …rm-speci…c productivity
informativeness increases due to a more e¢ cient allocation of aggregate capital investments
5
These results are closely related to the income and substitution e¤ects of an exogenous increase in the
expected return on a risky asset for the optimal portfolio choice with a riskless and a risky asset …rst analyzed
by Merton (1969) for power utility and lognormally distributed payo¤ on the risky asset.
26
23.50
Equilibrium aggregate investment
Ex post interest rates and ERR MP
21%
19%
17%
15%
13%
11%
9%
7%
5%
0.00
23.00
22.50
22.00
21.50
21.00
20.50
20.00
0.00
0.20
0.40
0.60
0.80
1.00
Firm-specific productivity informativeness
Interest rate: RRA = 2
ERR MP: RRA = 2
Interest rate: RRA = ½
ERR MP: RRA = ½
0.20
0.40
0.60
0.80
Firm-specific productivity informativeness
RRA = 2
RRA = ½
1.00
Figure 1: Second-period interest rate and ex post one-period ahead expected rate of return on
the market portfolio (ERR MP) (left panel) and equilibrium aggregate capital investments
(right panel) as functions of …rm-speci…c productivity informativeness ( 20 j ) for identical
…rm types given by the parameters in Table 1. Both panels show the results for = 0:5
and for = 2. In all cases, the economy-wide productivity signal is equal to its prior mean
(y e = 0).
among …rms and, thus, a higher equilibrium second-period interest rate. Of course, the rate
of return risk premium on the market portfolio ! 21 increases as the relative risk aversion
increases.
4.1.2
Economy-wide information
It follows from Lemma 10 that the equilibrium average aggregate capital investment q o
depends on the informativeness of the economy-wide productivity signal 20 e and on the
economy-wide productivity signal y e itself. The following proposition establishes this dependence and the consequences for the equilibrium interest rate and the ex post cost of
capital on the market portfolio.
Proposition 12 Conditional on public information at date t = 1, it holds that:
(a) The equilibrium interest rate 21 at t = 1 is increasing in the economy-wide productivity
signal y e , and if the inverse elasticity of intertemporal substitution is larger than
one-half also in the informativeness of economy-wide productivity information 20 e .
(b) The equilibrium rate of return risk premium on the market portfolio ! 21 is independent
of the economy-wide productivity signal y e and the aggregate capital investment q o , but
it is decreasing in the informativeness of economy-wide productivity information 20 e .
(c) The equilibrium one-period ahead expected rate of return 21 of the market portfolio of
…rms is increasing in the economy-wide productivity signal y e , but it may be decreasing
or increasing in the informativeness of economy-wide productivity information 20 e .
27
Equilibrium aggregate investment
Ex post interest rates and ERR MP
25%
20%
15%
10%
5%
0%
-0.50
23
22
21
20
19
-0.50
-0.25
0.00
0.25
0.50
Economy-wide productivity signal
Interest rate: RRA = 2
ERR MP: RRA = 2
Interest rate: RRA = ½
ERR MP: RRA = ½
-0.25
0.00
0.25
Economy-wide productivity signal
RRA = 2
0.50
RRA = ½
Figure 2: Second-period interest rate and ex post one-period ahead expected rate of return
on the market portfolio (ERR MP) (left panel) and equilibrium aggregate investments (right
panel) as functions of the economy-wide productivity signal for identical …rm types given by
the parameters in Table 1. Both panels show the results for = 0:5 and for = 2.
(d) The equilibrium average aggregate capital investment q o is decreasing (increasing) in the
economy-wide productivity signal y e if, and only if, the inverse elasticity of intertemporal substitution is larger (smaller) than one. The equilibrium average aggregate
capital investment q o is decreasing in the informativeness of economy-wide productivity
information 20 e .
It follows from Lemma 10 that an increase in the economy-wide productivity signal y e has the
same impact on interest rates and equilibrium aggregate capital investments as an increase in
the average productivity parameter of aggregate capital investments. Figure 2 shows how
the equilibrium second-period interest rate, the one-period ahead expected rate of return on
the market portfolio (left panel), and the equilibrium aggregate investments (right panel)
depend on the economy-wide productivity signal for > 1 and for < 1. The expected
marginal productivity of investments and the expected level of t = 2 payo¤s of all …rm types
increases with the economy-wide productivity signal. Hence, an increase in the economywide productivity signal y e has the same impact on interest rates and equilibrium aggregate
investments as an increase in the informativeness of the …rm-speci…c productivity signals
(compare to Figure 1). While the dollar risk premium on the market portfolio increases with
the economy-wide productivity signal, the rate of return risk premium is independent of this
signal (see Lemma 9(a) and (c)).
Figure 3 shows how the equilibrium second-period interest rate, the one-period ahead
expected rate of return on the market portfolio (left panel) depends on the economy-wide
productivity informativeness for an exogenous aggregate capital investment equal to the
equilibrium aggregate investment for 20 e = 0 (shown in the right panel). The economywide productivity signal is equal to its prior mean (y e = 0) in both panels.
28
Equilibrium aggregate investment
Ex post interest rates and ERR MP
23.00
30%
25%
20%
15%
10%
5%
0%
0.00
0.20
0.40
0.60
0.80
1.00
Economy-wide productivity informativeness
Interest rate: RRA =½
ERR MP: RRA =½
Interest rate: RRA = 2
ERR MP: RRA = 2
Interest rate: RRA = 4
ERR MP: RRA = 4
22.50
22.00
21.50
21.00
20.50
20.00
0.00
0.20
0.40
0.60
0.80
Economy-wide productivity informativeness
RRA = ½
RRA = 2
1.00
RRA = 4
Figure 3: Second-period interest rate and ex post one-period ahead expected rate of return
on the market portfolio (ERR MP) (in the left panel) for an exogenous aggregate investment
(q o = q o ( 20 e = 0)) (shown in the right panel) as functions of the economy-wide productivity
informativeness for identical …rm types given by the parameters in Table 1. Both panels
show the results for = 0:5, = 2, and for = 4. The economy-wide productivity signal is
equal to its prior mean (y e = 0) in both panels.
The equilibrium second-period interest rate is increasing in the economy-wide productivity informativeness for all values of the relative risk aversion due to a lower demand for
precautionary savings as the second-period uncertainty decreases (see Lemma 7). On the
other hand, the rate of return risk premium ! y21 on the market portfolio decreases in the
economy-wide productivity informativeness (independently of the economy-wide productivity signal y e ) such that the one-period ahead expected rate of return on the market portfolio
may decrease (increase) for low (high) values of the relative risk aversion < 2 ( > 2),
while the two e¤ects are exactly o¤setting for = 2 (see equation (22)).
Figure 4 shows the equilibrium second-period interest rate, the one-period ahead expected
rate of return on the market portfolio (left panel), and the equilibrium aggregate capital
investment (right panel) as functions of the economy-wide productivity informativeness (for
an economy-wide productivity signal equal to its prior mean (y e = 0)). The equilibrium
aggregate investment decreases as the economy-wide productivity informativeness increases
for all values of the the relative risk aversion (Proposition 12(d)). This is due to a reduction
in the uncertainty about t = 2 consumption and, thus, a lower expected marginal utility of
t = 2 consumption ceteris paribus (which is re‡ected in a higher second-period interest due
to a lower demand for precautionary savings, see Figure 3, left panel). The lower expected
marginal utility of t = 2 consumption shifts the investment incentives in order to increase
consumption at t = 1 and decrease consumption at t = 2, i.e., decrease aggregate investments
at t = 1.
29
23.00
14%
Equilibrium aggregate investment
Ex post interest rates and ERR MP
16%
12%
10%
8%
6%
4%
2%
0%
0.00
0.20
0.40
0.60
0.80
1.00
Economy-wide productivity informativeness
Interest rate: RRA =½
ERR MP: RRA =½
Interest rate: RRA = 2
ERR MP: RRA = 2
Interest rate: RRA = 4
ERR MP: RRA = 4
22.50
22.00
21.50
21.00
20.50
20.00
0.00
0.20
0.40
0.60
0.80
Economy-wide productivity informativeness
RRA = ½
RRA = 2
1.00
RRA = 4
Figure 4: Second-period interest rate and ex post one-period ahead expected rate of return
on the market portfolio (ERR MP) (in the left panel) for equilibrium aggregate investments
(q o ( 20 e )) (shown in the right panel) as functions of the economy-wide productivity informativeness for identical …rm types given by the parameters in Table 1. Both panels show the
results for = 0:5, = 2, and for = 4. The economy-wide productivity signal is equal to
its prior mean (y e = 0) in both panels.
The reduction in equilibrium aggregate investments “dampens” the impact of an increased economy-wide productivity informativeness on the equilibrium second-period interest
rate compared to a setting with an exogenous aggregate investment. For = 12 , the impact
is completely neutralized, while the equilibrium second-period interest rate still increases in
the economy-wide productivity informativeness for > 12 (Proposition 12(a)). The equilibrium rate of return risk premium ! 21 on the market portfolio is una¤ected by changes in
equilibrium aggregate investments (Proposition 12(b)). The “dampening” e¤ect of reduced
equilibrium investments on the equilibrium ex post interest rate is such that the equilibrium
one-period ahead expected rate of return on the market portfolio, i.e., the equilibrium ex
post cost of capital for aggregate investments 21 , is decreasing in the informativeness of
the economy-wide productivity information 20 e for all values of the relative risk aversion
shown in Figure 4 contrary to the setting with an exogenous aggregate investment shown in
Figure 3.
It may seem puzzling that the equilibrium aggregate investment decreases at the same
time as the ex post cost of capital for aggregate investments 21 decreases when the economywide productivity information becomes more informative. The reason is that increased
economy-wide productivity informativeness decreases the remaining uncertainty about the
economy-wide productivity state revealed at t = 2. Since the payo¤s at t = 2 is a convex
function of the economy-wide productivity state, the expected payo¤s at t = 2 on the market
portfolio decreases as the remaining uncertainty about the economy-wide productivity state
is reduced. Hence, there is both a denominator e¤ect (on the ex post cost of capital) and
30
a numerator e¤ect (on the expected payo¤s) of changes in the economy-wide productivity
informativeness.
For a …xed aggregate investment, the ex ante distribution of t = 1 and t = 2 payo¤s
are una¤ected by the informativeness of the economy-wide productivity signals. From the
above comparative statics, it follows that the equilibrium aggregate investments depend not
only on the informativeness of the economy-wide productivity signals but also on the signal
itself. For a given economy-wide productivity signal, the aggregate investment decreases as
the informativeness of the signal increases, however, the distribution of signals also changes,
which makes the impact on ex ante expected t = 1 and t = 2 payo¤s unclear. That is, the
impact of altering the informativeness of the economy-wide productivity information system
on, for example, equilibrium capital investments is best assessed from an ex ante perspective
at t = 0.
4.2
Ex ante cost of capital and …rm values
Most of the literature on information and the cost of capital has taken a pure exchange economy as a starting point, i.e., an economy in which capital investments (and other production
choices) do not change as the informativeness of the information system changes. In Appendix B, we reconcile our analysis with the key result of this literature: any reduction in the
ex post cost of capital due to more economy-wide information is perfectly o¤set by an equal
increase in the pre-posterior cost of capital leaving the ex ante cost of capital and …rm values
unchanged (see Christensen et al. 2010). Hence, any impact on the ex ante cost of capital
and …rm values must be through changes in the equilibrium capital investment policies as
the informativeness of the information system changes. In this section, we …rst analyze the
e¤ects of increased …rm-speci…c productivity informativeness on the ex ante cost of capital
and …rm values and, subsequently, we analyze the e¤ects of increased informativeness of the
economy-wide productivity information on the ex ante cost of capital and …rm values.
4.2.1
Firm-speci…c information
The equilibrium spot and zero-coupon interest rates at date t is determined by the expected
marginal rate of substitution between future and current equilibrium consumption, i.e.,
t
=
Et [u0 (c )]
=(
ln
u0t (ct )
t) :
Since equilibrium consumption at t = 1 and at t = 2 are not lognormally distributed given
the prior beliefs when capital investments are based on economy-wide information, we cannot
31
explicitly calculate the expected marginal utility of t = 1 and t = 2 consumption. However,
Proposition 12 contains information su¢ cient for determining the impact of altered informativeness of …rm-speci…c productivity information on spot and zero-coupon interest rates
at t = 0:
From Proposition 12(d) we know that equilibrium investments q o (y e ) are decreasing (increasing) in the informativeness of …rm-speci…c productivity information 20 j ; j = 1; :::; J
if, and only if, the inverse elasticity of intertemporal substitution is larger (smaller) than
one. As the distribution over the economy-wide signals, y e ; is una¤ected by the precision
of …rm-speci…c information, the changes to equilibrium investments implies that the equilibrium spot interest rate 10 is increasing (decreasing) in the informativeness of …rm-speci…c
productivity information if > 1 ( < 1).
The fact that more is invested for any economy-wide signal when < 1 implies more is
available for consumption in any state at t = 2: This in turn implies that the zero-coupon rate
< 1:
20 is increasing in the informativeness of …rm-speci…c productivity information when
0
0
From Proposition 12(a) we know that 21 (y e ) = ln(u1 (d1 q o (y e )) ln(E [u2 (c2 )jy e ]) is increasing in the informativeness of …rm-speci…c productivity information regardless of the size
of . When > 1, less is invested at t = 1 as the informativeness of …rm-speci…c productivity
information increases for any economy-wide signal y e . This implies that ln(u01 (d1 q o (y e ))
is decreasing, but as 21 (y e ) is increasing, ln(E [u02 (c2 )jy e ]) must be decreasing even more. In
sum, this implies that E [u02 (c2 )jy e ] is decreasing in the informativeness of …rm-speci…c productivity information for any y e and, thus, so is E [u02 (c2 )] = E [E [u02 (c2 )jy e ]], since the prior
distribution of the economy-wide signals y e is una¤ected by the precision of …rm-speci…c
information. This implies that the zero-coupon rate 20 is increasing in the informativeness
of …rm-speci…c productivity information when > 1:
Proposition 13 At the ex ante date t = 0, the comparative statics for the equilibrium
interest rates with respect to the informativeness of …rm-speci…c information are as follows:
(a) The equilibrium spot interest rate 10 is increasing (decreasing) in the informativeness
of …rm-speci…c productivity information 20 j ; j = 1; :::; J, if > 1 ( < 1).
(b) The equilibrium zero-coupon interest rate 20 is increasing in the informativeness of
…rm-speci…c productivity information 20 j ; j = 1; :::; J.
Given the equilibrium ex-dividend price of a type j …rm at t = 0, the equilibrium implied
cost of capital at t = 0 for type j …rms j is determined by equation (7) as the discount rate
solving the equation
v0j = E d1j jqj ( ) exp
j
32
+ E d2j jqj ( ) exp
2
j
and, in particular, the implied cost of capital on the market portfolio
discount rate solving the equation
v 0 = E[d1
q o (y e )] exp [
is determined as the
(26)
] + E[h( e ; q o )] exp [ 2 ] :
Having determined the equilibrium capital investments at t = 1, we must …rst determine the
equilibrium ex ante …rm values, v0j and v 0 ; in order to determine the implied cost of capital
measures. For brevity, we only consider the implied cost of capital on the market portfolio. The ex ante price of the market portfolio is the risk-adjusted expected net-dividends
discounted with the risk-free interest rates:
v 0 = exp [
10 ]
E d1
+ exp [ 2
20 ]
q o (y e ) + Cov0 m10 ; d1
(27)
q o (y e )
E h( e ; q o ) + Cov0 m20 ; h( e ; q o )
:
Studying equation (27), we …nd the following result.
Proposition 14 The ex ante value of the market portfolio is decreasing in …rm-speci…c
productivity informativeness when > 1, and increasing in …rm-speci…c productivity informativeness when < 1.
Changes in …rm-speci…c productivity informativeness lead to changes in capital investments,
to changes in interest rates, to changes in risk premia, to changes in …rm values, and eventually to changes in the implied cost of capital. There is no one-to-one relationship between
the ex ante …rm values and the implied cost of capital, but when the inverse elasticity of
intertemporal substitution is larger than one, the implied cost of capital on the market
portfolio is indeed increasing in …rm-speci…c productivity informativeness.
Proposition 15 When
> 1, the implied cost of capital on the market portfolio
increasing in …rm-speci…c productivity informativeness, 20 j .
is
The reason is that a more e¢ cient allocation of capital across …rms allows for less to be
invested for a larger e¤ect. Thus, when > 1, consumption is increasing at both t = 1 and
at t = 2, which at t = 0 leads to a lower willingness-to-pay for future dollars/consumption.
As demonstrated in Proposition 14, the reduced willingness-to-pay for future consumption
dominates the e¤ect of increased future risk-adjusted expected dividends and, as a result,
the ex ante value of the market portfolio is decreasing in …rm-speci…c productivity informativeness. Simply put: as the implied cost of capital is the discount rate which when used
for discounting future expected dividends returns the equilibrium market price, that rate
33
Interest rates and implied cost of capital
Interest rates and implied cost of capital
12%
10%
8%
6%
4%
2%
0%
0.00
0.20
0.40
0.60
0.80
1.00
Firm-specific productivity informativeness
Spot interest rate
Zero-coupon interest rate
Impiled cost of capital
One-period ahead exp return
8%
7%
6%
5%
4%
3%
2%
1%
0%
-1%0.00
0.20
0.40
0.60
0.80
1.00
-2%
Firm-specific productivity informativeness
Spot interest rate
Zero-coupon interest rate
Impiled cost of capital
One-period ahead exp return
Figure 5: Ex ante interest rates, implied cost of capital, and one-period ahead expected
returns as functions of …rm-speci…c productivity informativeness ( 20 j ) for identical …rm
types given by the parameters in Table 1. The left panel shows the results for = 2, while
the right panel shows the results for = 0:5.
must be increasing— price is falling while expected dividends at both t = 1 and t = 2 are
increasing.
When < 1, the ex ante value of the market portfolio, v 0 , is increasing in …rm-speci…c
productivity informativeness as is the expected aggregate t = 2 dividend, E h( e ; q o (y e )) =
E (q o (y e ))k exp (y e + 12 21 e ) : Contrary to this, the expected aggregate t = 1 dividend, E[d1 q o (y e )]; decreases, since q o (y e ) is increasing in 20 j for any y e when < 1.
Thus, we have a situation in which the ex ante price increases, while dividends at t = 1 and
t = 2 move in opposite directions making the impact on the implied cost of capital unclear.
Now, since consumption at t = 1, d1 q o (y e ); is decreasing, while second-period consumption, E h( ) ; is increasing, intuition might indicate that the implied cost of capital could
decrease if the spot interest rate is higher than the zero-coupon rate— more weight is placed
on the low interest rate. However, we have not been able to generate examples in which
the implied cost of capital on the market portfolio is decreasing in …rm-speci…c productivity
informativeness.
Figure 5 illustrates that the implied cost of capital for the market portfolio is increasing
in …rm-speci…c productivity informativeness given the parameters in Table 1, both when
> 1 and when < 1. The same type of comparative statics hold for …rm j (and for …rms
0
j 6= j) when we only vary the …rm-speci…c productivity informativeness of …rm j, 20 j .
In a setting in which there is no economy-wide information revealed at t = 1, the oneperiod ahead expected rate of returns at t = 0 (see equation (5)) on individual …rms and
on the market portfolio are all equal to the spot interest rate and, thus, 10j = 10 = 10
is increasing in …rm-speci…c productivity informativeness 20 j when > 1 and decreasing
in 20 j when < 1 (see Proposition 13(a)). However, when economy-wide information is
34
30.25
Ex ante firm value
54.50
29.75
54.00
53.50
29.50
53.00
52.50
0.00
Ex ante firm value
30.75
58.50
30.50
58.25
30.25
58.00
30.00
57.75
29.75
57.50
29.50
57.25
57.00
0.00
29.25
0.20
0.40
0.60
0.80
Firm-specific productivity informativeness
58.75
1.00
Ex ante certainty equivalent
0.20
0.40
0.60
0.80
Firm-specific productivity informativeness
Ex ante firm value
Ex ante certainty equivalent
30.00
55.00
Ex ante certainty equivalent
55.50
Ex ante firm value
56.00
29.25
1.00
Ex ante certainty equivalent
Figure 6: Ex ante …rm values and certainty equivalents as functions of …rm-speci…c productivity informativeness ( 20 j ) for identical …rm types given by the parameters in Table 1. The
left panel shows the results for = 2, while the right panel shows the results for = 0:5.
released at t = 1 and aggregate capital investments vary with this information, the oneperiod ahead expected rate of return di¤ers from the spot interest rate. This makes it
di¢ cult to determine analytically the impact on the one-period ahead expected rate of return
on an individual …rm or on the market portfolio of an increase in …rm-speci…c productivity
informativeness. The problem is that neither cum-dividend prices nor consumption are
lognormally distributed when there is economy-wide information released at t = 1 and,
thus, calculating the expected cum-dividend price or the dollar risk premium analytically
is impossible. Nevertheless, Figure 5 shows that the comparative statics for the setting
with no economy-wide information carry over to our numerical examples with economywide risks at t = 1: the one-period ahead expected rate of return on the market portfolio
and on individual …rms tend to move in sync with the spot interest rate when …rm-speci…c
productivity informativeness changes. The reason is that the …rst-period dollar risk premium
is small relative to the expected cum-dividend …rm value and, thus, the …rst-period rate of
return risk premium is small as well.
We now return to the t = 0 value of the market portfolio, and compare it to the investors’
ex ante expected utility as measured by their ex ante certainty equivalent de…ned in equation
(8). Figure 6 illustrates the impact of …rm-speci…c productivity informativeness in a setting
with identical …rm types characterized by the parameters in Table 1. The …gure depicts
the equilibrium ex ante value of the market portfolio and the equilibrium ex ante certainty
equivalent as a function of the informativeness of the …rm-speci…c productivity information
2
= 2 and in the right panel for = 0:5). As noted in Proposition
0 j (in the left panel for
14, the ex ante value of the market portfolio is decreasing in the informativeness of the
…rm-speci…c productivity information when > 1, while the ex ante value of the market
portfolio is increasing in the informativeness of the …rm-speci…c productivity information
35
when < 1. The intuition is that even though a more e¢ cient allocation of capital leads
to increasing future payo¤s, it also reduces the willingness-to-pay for an additional future
dollar due to the investors’decreasing marginal utility of consumption. When > 1, this
reduced willingness-to-pay for future consumption dominates the e¤ect of increased future
dividends and, consequently, ex ante prices fall. The opposite obtains when < 1.
As a more e¢ cient allocation of capital among …rms and across time can be achieved
when …rm-speci…c productivity information is improved, it is clear that the ex ante expected
utility is increasing in the informativeness of …rm-speci…c productivity information regardless
of the speci…cs of the investors’preferences (see Proposition 6). Hence, the ex ante change in
market prices is a poor indicator of welfare implications of information system changes, and
especially so when > 1. In other words, “shareholder wealth”and “shareholder welfare”are
two fundamentally di¤erent concepts. In a general equilibrium analysis of information system
choice in a production economy, the distributions of future payo¤s change but the basic state
prices used to value these payo¤s also change. Investors always bene…t from a more e¢ cient
use of scarce resources in the economy, but the resulting drop in the willingness-to-pay for an
additional future dollar imply that current wealth decreases when > 1. The reduced wealth
is inconsequential to the representative investor holding a constant fraction of the market
portfolio (or, more generally, to individual investors holding equilibrium endowments, see,
e.g., Feltham and Christensen 1988).
4.2.2
Economy-wide information
When …rm-speci…c productivity informativeness changes capital investments, expected dividends at t = 1 and t = 2 also change. One thing that does not change though is the
uncertainty pertaining to the economy-wide signals and states. That is, the distribution
over economy-wide posteriors and the conditional distribution of economy-wide states are
una¤ected by …rm-speci…c productivity information, and so is the ex post rate of return risk
premium on the market portfolio. This obviously ceases to hold when we consider changes in
the informativeness of economy-wide productivity information. Nevertheless, the comparative statics for ex ante …rm values with respect to …rm-speci…c productivity informativeness
carry over unchanged to the setting with varying economy-wide productivity informativeness.
Proposition 16 The ex ante value of the market portfolio is decreasing in economy-wide
productivity informativeness when > 1, and increasing in economy-wide productivity informativeness when < 1.
Since analytical results are di¢ cult to obtain when aggregate consumption is not log-normally
distributed with prior beliefs, the following discussion of the economics behind Proposition 16
36
is based on numerical investigations (for a large set of varying parameters). The discussion
is structured around the fundamental asset pricing relation (3): the ex ante value of the
market portfolio is equal to the sum of future risk-adjusted expected aggregate dividends
discounted with zero-coupon interest rates.
As demonstrated in Proposition 12(d), equilibrium capital investments given the economywide signal, q o (y e ), decrease as the economy-wide productivity informativeness increases
(when 6= 1).6 In our numerical examples, this implies that the expected capital investment,
E [q o (y e )] ; is decreasing and, thus, expected …rst-period consumption, E d1 q o (y e ) , is
increasing in economy-wide productivity informativeness— this is not a foregone conclusion
as the prior distribution of the economy-wide signals y e is changing when we change the
economy-wide productivity informativeness. Though the expected capital investment is reduced, this does not imply that the expected second-period aggregate dividend, E[h( e ; q o )],
is decreasing as well. In our numerical examples, it holds that E[h( e ; q o )] is increasing when
< 1 and decreasing when > 1:7
When is large ( > 1), the investors are relatively risk averse and, thus, with timeadditive preferences they also have a strong desire for consumption smoothing. The investors’
desire for consumption smoothing dominates the desire to exploit the production opportunities e¢ ciently such that the aggregate capital investment is decreasing in productivity, i.e.,
dq o (y e )=dy e < 0 (see Proposition 12(d)). In other words, a high (low) productivity signal,
which means that the level of expected future consumption and the productivity of capital
investments are both high (low), leads to lower (higher) capital investments and, thus, a
reduction (an increase) in t = 2 dividends. From an ex ante perspective, this optimal capital investment policy implies that the uncertainty in second-period consumption is reduced
(compared to a setting with …xed capital investment) and more so, the more informative is
the economy-wide productivity signals about the economy-wide productivity states at t = 2.
As a consequence, the dollar risk premium for t = 2 dividends decreases, and it decreases
more than the expected dividend such that the risk-adjusted expected dividend increases as
the economy-wide productivity informativeness increases. The variation in …rst-period dividends also increases, but the impact on the …rst-period dollar risk premium is not su¢ cient
to dominate the increase in expected …rst-period dividends due to lower expected capital
investments. Hence, the risk-adjusted expected dividends increase in both periods as the
economy-wide productivity informativeness increases.
6
As the capital investment contains a (large) precautionary savings motive (when is large), capital
investments are more sensitive to a reduction in ex post uncertainty the larger is :
7
As argued below, the reason for this is that dq o (y e )=dy e < 0 when > 1, and dq o (y e )=dy e > 0
when < 1 (see Proposition 12(d)). That is, lower expected investments are invested “less e¢ ciently”when
> 1— this naturally leads to lower expected t = 2 payo¤s— and “more e¢ ciently” when < 1.
37
Since risk-adjusted expected dividends on the market portfolio increase in both periods,
when the informativeness of the economy-wide productivity signals increases, the ex ante
value of the market portfolio would also be increasing if there were no impact on interest rates.
However, a high risk aversion ( > 1) also implies that the desire to save for precautionary
reasons is strong such that there is a large positive impact on interest rates of reduced
uncertainty in future consumption. The impact of the reduction in the uncertainty of t = 2
consumption, following from the optimal capital investment policy, on the equilibrium t = 2
zero-coupon interest rate, and the impact of increased expected consumption growth in the
…rst period, imply that the increase in both the spot and the t = 2 zero-coupon equilibrium
interest rates are strong enough to swamp the increase in risk-adjusted expected dividends
at t = 1 and at t = 2, i.e., the ex ante value of the market portfolio is decreasing in economywide productivity informativeness when > 1.8 These e¤ects are larger, the larger is the
inverse elasticity of intertemporal substitution, .
When is large, the drop in the ex ante value of the market portfolio v 0 due to rapidly
increasing interest rates, as informativeness increases, is large relative to i) the increase
in …rst-period expected dividends and ii) the drop in second-period expected dividends.9
Studying equation (26), it is easy to see this requires the implied cost of capital to be
increasing. On the other hand, when is larger than one but not too large, interest rates are
increasing more modestly when informativeness is increased and, thus, the drop in v 0 and the
increase in expected …rst-period dividends are modest relative to the drop in expected secondperiod dividends. Our numerical investigations suggest that there is a cuto¤ for the relative
risk aversion z > 1 for which the implied cost of capital is increasing as the informativeness
of the economy-wide productivity signals increase when > z , and decreasing when 2
(1; z ).
We illustrate these e¤ects in Figure 7 for a setting with identical …rm types characterized
by the parameters in Table 1. The …gure depicts the equilibrium spot and zero-coupon
interest rates as well as the implied cost of capital on the market portfolio as functions of
the informativeness of the economy-wide productivity information 20 e — in the left panel
for = 2 and in the right panel for = 4: When = 2, the implied cost of capital on
the market portfolio is decreasing in the informativeness of the economy-wide productivity
information and when = 4, the implied cost of capital on the market portfolio is increasing
8
For some parameters, the risk-adjusted expected …rst-period dividend is decreasing— this does of course
not lead to an increasing t = 0 value.
9
From the proof of Proposition 17 in Appendix B, it can be seen that if q0 is …xed, then d(E[d2 ]
RP )=d 20 e = (E[d2 ] RP )( 12
) 2 while dv 1 =d 20 e = v 1 21 2 (1
)2 < 0: That is, as uncertainty is
reduced, the risk-adjusted expected dividend increases for > 21 . However, price still drops and the drop in
price is larger, the larger is : This carries over to v 0 ; but only when > 1:
38
Interest rates and implied cost of capital
Interest rates and implied cost of capital
9%
8%
7%
6%
5%
4%
3%
2%
1%
0%
12%
10%
8%
6%
4%
2%
0%
0.00
0.20
0.40
0.60
0.80
1.00
Economy-wide productivity informativeness
Spot interest rate
Zero-coupon interest rate
Impiled cost of capital
One-period ahead exp return
0.00
0.20
0.40
0.60
0.80
1.00
Economy-wide productivity informativeness
Spot interest rate
Zero-coupon interest rate
Impiled cost of capital
One-period ahead exp return
Figure 7: Ex ante interest rates, implied cost of capital, and one-period ahead expected
returns as functions of economy-wide productivity informativeness ( 20 e ) for identical …rm
types given by the parameters in Table 1. The left panel shows the results for = 2, while
the right panel shows the results for = 4.
in the informativeness of the economy-wide productivity information. From the two panels,
it is also evident interest rates are increasing much more rapidly in the informativeness of
economy-wide productivity information when = 4 than when = 2. Note also that the
one-period ahead expected rate of return on the market portfolio follows the spot interest
rate closely due to a low …rst-period dollar risk premium as in the setting with varying
…rm-speci…c productivity informativeness.10
When < 1, the investors are not very risk averse and, importantly, they have a weak
demand for consumption smoothing. In this setting, the investors’desire for consumption
smoothing is dominated by the desire to exploit the production opportunities e¢ ciently
such that capital investments are increasing in productivity, i.e., dq o (y e )=dy e < 0 (see
Proposition 12(d)). This has two consequences. First, lower expected capital investments
are invested “more e¢ ciently”leading to higher expected t = 2 dividends. Second, from an
ex ante perspective, the optimal capital investment policy ampli…es the the uncertainty in
second-period consumption (compared to a setting with …xed capital investment) and more
10
If we vary economy-wide productivity informativeness for the parameters in Table 1, then the implied
cost of capital is decreasing when is between one and roughly 2:75 and increasing for higher levels of risk
aversion. Our examples suggest that the distribution of marginal utilities determine whether the implied cost
of capital is increasing or decreasing in economy-wide productivity informativeness. One way of adjusting
marginal utilities is through but another is through the marginal productivity of capital parameter, k.
If k is reduced from 0:75 to 0:40, interest rates are low— partly due to low expected consumption growth
but also due to a “high” probability of low t = 2 dividends. In this setting, if economy-wide productivity
informativeness is increased, v 0 drops rapidly relative to the increase in E[d1 q o ] and the drop in E[h( e ; q o )].
In that case, the implied cost of capital is increasing in economy-wide productivity informativeness also when
= 2. Hence, it would seem that the choice of parameters determine whether the implied cost of capital is
increasing or decreasing in economy-wide productivity informativeness when > 1. This may be so, but it
should be added that generating examples of increasing implied cost of capital becomes increasingly arduous
as falls below two.
39
Interest rates and implied cost of capital
Interest rates and implied cost of capital
6%
5%
4%
3%
2%
1%
0%
0.00
0.20
0.40
0.60
0.80
1.00
-1%
7%
6%
5%
4%
3%
2%
1%
0%
0.00
0.20
0.40
0.60
0.80
1.00
Economy-wide productivity informativeness
Spot interest rate
Zero-coupon interest rate
Impiled cost of capital
One-period ahead exp return
Economy-wide productivity informativeness
Spot interest rate
Zero-coupon interest rate
Impiled cost of capital
One-period ahead exp return
Figure 8: Ex ante interest rates, implied cost of capital, and one-period ahead expected
returns as functions of economy-wide productivity informativeness ( 20 e ) for identical …rm
types given by the parameters in Table 1. The left panel shows the results for = 0:5, while
the right panel shows the results for log utility.
so, the more informative is the economy-wide productivity signals about the economy-wide
productivity states at t = 2. As a result, the dollar risk premium for t = 2 dividends
increases, and it increases more than the expected dividend such that the risk-adjusted
expected dividend decreases as the economy-wide productivity informativeness increases.
The variation in …rst-period dividends also increases, and the impact on the …rst-period
dollar risk premium is larger than the increase in expected …rst-period dividends due to
lower expected capital investments. Hence, risk-adjusted expected dividends decrease in
both periods as the economy-wide productivity informativeness increases.
The increased variation in both t = 1 and t = 2 aggregate dividends induced by the
optimal capital investment policy, also increase the demand for precautionary savings. The
impact of the increased demand for precautionary savings on equilibrium interest rates is
larger than the impact of increased consumption growth for both periods such that the
equilibrium spot and t = 2 zero-coupon interest rates both decrease. The drop in equilibrium
interest rates is again the dominating force (as for > 1) such that the ex ante value of
the market portfolio is increasing in economy-wide productivity informativeness when < 1
even though risk-adjusted expected dividends decrease in both periods. The increase in the
ex ante value of the market portfolio v 0 is modest compared to the increase in expected
t = 1 and t = 2 aggregate dividends, implying the implied cost of capital is increasing as the
informativeness of the economy-wide productivity signals increase. This e¤ect is stronger
the more risk tolerant is the representative investor (or the lower is )
We illustrate these e¤ects in Figure 8 for a setting with identical …rm types characterized
by the parameters in Table 1. The left panel of the …gure depicts the equilibrium spot and
zero-coupon interest rates as well as the implied cost of capital on the market portfolio as
40
functions of the informativeness of the economy-wide productivity information 20 e when
= 0:5: When = 0:5, the implied cost of capital on the market portfolio is increasing in
the informativeness of the economy-wide productivity information, while interest rates are
decreasing. For comparison, the right panel depicts the associated results for the limiting
case for ! 1 (i.e., for log utility) in which interest rates and the implied cost of capital
are all independent of the informativeness of the economy-wide productivity information. Of
course, the key for these results is that the equilibrium capital investments are independent
of the economy-wide productivity signals in this setting (see Proposition 12(d)) and, thus,
also independent of the informativeness of these signals. In other words, we are e¤ectively
back to an exchange economy in which there is no impact of public information on ex ante
…rm values and the cost of capital.
Until now we have only considered the impact on the market portfolio of altering the
informativeness of economy-wide productivity information. If the structure of individual
…rms are similar to the market portfolio, the impacts on individual …rms are similar to
the impact on the market portfolio. However, a closer look at individual …rms reveals
that the value and the implied cost of capital on a …rm is in‡uenced signi…cantly by the
…xed component of dividends associated with that …rm. If the …xed components are large
compared to the stochastic components of the …rm’s payo¤s at t = 1 or t = 2, then the
…rm’s implied cost of capital will tend to move in the same direction as interest rates, i.e.,
decreasing for < 1 and increasing for > 1: This has the surprising consequence that the
implied cost of capital may be increasing (decreasing) for all …rm types (with a = 0) but
decreasing (increasing) for the market portfolio!
Similarly, even though the one-period ahead expected rate of return on the market portfolio at t = 0 follows the equilibrium spot rate closely (for both < 1 and > 1), this does
not necessarily hold for the individual …rm types. Take for example the zero-coupon bond
maturing at t = 2; at t = 1 the value of this bond, v1J+2 (y e ); is decreasing in the economywide signal y e (see Proposition 12(a)). As consumption at t = 1 is increasing (decreasing) in
the economy-wide signal y e when > 1 ( < 1), the covariance between the t = 1 value of
the zero-coupon bond and the pricing kernel, m10 = u01 (c1 )=E0 [u01 (c1 )], is positive (negative)
when > 1 ( < 1). This implies the one-period ahead expected rate of return at t = 0,
10J+2
=
10
+ ln
E v1J+2
E[v1J+2 ]
+ Cov u01 (d1 q 0 )=E u01 (d1
q 0 ) ; v1J+2
!
;
contains a negative (positive) rate of return risk premium when > 1 ( < 1). Furthermore,
the absolute value of the rate of return risk premium is increasing in the informativeness of
the economy-wide productivity information. This implies the …rst-period rate of return risk
41
Ex ante firm value
57.87
29.74
57.86
54.26
29.73
54.24
54.22
29.72
54.20
29.71
54.18
29.70
54.16
54.14
0.00
30.07
Ex ante firm value
57.85
57.84
30.06
57.83
30.05
57.82
30.04
57.81
29.69
0.20
0.40
0.60
0.80
Economy-wide productivity informativeness
30.08
57.80
0.00
1.00
Ex ante certainty equivalent
0.20
0.40
0.60
0.80
Economy-wide productivity informativeness
Ex ante firm value
Ex ante certainty equivalent
29.75
Ex ante certainty equivalent
54.28
Ex ante firm value
54.30
30.03
1.00
Ex ante certainty equivalent
Figure 9: Ex ante …rm values and certainty equivalents as functions of economy-wide productivity informativeness ( 20 e ) for identical …rm types given by the parameters in Table 1.
The left panel shows the results for = 2, while the right panel shows the results for = 0:5.
premium on the zero-coupon bond maturing at t = 2 is moving in the opposite direction of
the spot interest rate. In our numerical examples, the impact of increased informativeness on
the rate of return risk premium dominates the impact on the spot interest rate such that the
one-period ahead expected rate of return on a zero-coupon bond is increasing (decreasing)
in the informativeness of economy-wide productivity information when < 1 ( > 1). This
result carries over to risky assets provided the …xed component of the …rm’s second-period
payo¤s, aj , is su¢ ciently large.
Returning to the ex ante value of the market portfolio, it is impacted by changes in
economy-wide productivity informativeness in a manner similar to how the ex ante value
is impacted by changes in …rm-speci…c productivity informativeness. This is illustrated in
Figure 9 in a setting with identical …rm types characterized by the parameters in Table 1.
The …gure depicts the equilibrium ex ante value of the market portfolio and the equilibrium ex ante certainty equivalent as functions of the informativeness of the economy-wide
productivity signal 20 e ; in the left panel for = 2 and in the right panel for = 0:5:
When > 1, the ex ante value of the market portfolio is decreasing in the informativeness
of the economy-wide productivity signal and when < 1, the ex ante value of the market
portfolio is increasing in the informativeness of the economy-wide productivity signal. Of
course, the equilibrium ex ante certainty equivalent is increasing in the informativeness of the
economy-wide productivity signal for all values of . Again, ex ante market values of …rms
are poor indicators of welfare implications of information system changes, and especially so
when > 1.
42
5
Conclusion
We have shown this and that, but all our results will probably change dramatically if we
allow for...
Appendices
A
Expected Rates of Returns and Firm-speci…c Risks
y
We can express the equilibrium ex-dividend price v1jn
of …rm n of type j at t = 1 in equation
(17) as the sum of the equilibrium values of two claims, a “windfall claim”and a “productivity
claim”with only the latter commanding a dollar risk premium, i.e.,
y
v1jn
= exp[
y
21 ]E[gj jy jn ]
+ exp[
y
21 ]
h
E[hyj jy
y
jn ; y e ; qj ( )]
y
RP21jn
The “windfall claim”has a one-period ahead expected rate of return
interest rate, i.e.,
gy
gy
y
21jn = ln E[gj jy jn ]=v1jn = 21 ;
i
gy
hy
v1jn
+ v1jn
:
gy
21jn
equal to the spot
whereas the “productivity claim”has a one-period ahead expected rate of return hy
21jn equal
y
to the spot interest rate plus the rate of return risk premium ! 21 on the market portfolio,
i.e.,
y
hy
y
y
hy
y
21jn = ln E[hj jy jn ; y e ; qj ( )]=v1jn = 21 + ! 21 :
De…ning one-period ahead gross rates of returns as R21 d2 =v1 , we may therefore write the
one-period ahead expected gross rate of return of …rm n of type j at t = 1 as
y
E1 [R21jn
]
=
gy
v1jn
E[gj jy
y
v1jn
jn ]
gy
v1jn
+
hy
v1jn
E[hyj jy
y
v1jn
y
jn ; y e ; qj (
hy
v1jn
)]
=
gy
v1jn
y
v1jn
f
R21
+
hy
v1jn
y
v1jn
m
E1 [R21
];
f
m
] = exp[ y21 + ! y21 ] is
where R21
= exp[ y21 ] is the gross spot riskless rate of return and E1 [R21
the one-period ahead expected gross rate of return on the market portfolio. De…ning asset
m
m
betas as y21
Cov1 [R21 ; R21
]=Var1 [R21
], we can therefore also write the one-period ahead
expected gross rate of return of …rm n of type j at t = 1 as
y
f
E1 [R21jn
] = R21
+
y
21jn
m
E1 [R21
]
f
R21
;
43
y
21jn
=
gy
v1jn
y
v1jn
0+
hy
v1jn
y
v1jn
1:
20%
19%
19%
Ex post cost of capital
Ex post cost of capital
20%
18%
17%
16%
15%
14%
13%
18%
17%
16%
15%
14%
13%
12%
-2.00
-1.00
0.00
1.00
Firm-specific windfall signal
Product. signal = -1
Product. signal = 0
12%
-2.00
2.00
Product. signal = 1
-1.00
0.00
1.00
Firm-specific productivity signal
Windfall signal = -1
Windfall signal = 0
2.00
Windfall signal = 1
Figure 10: Ex post one-period ahead expected rate of return on an individual …rm as functions
of the …rm-speci…c windfall signal y jn (left panel) and the …rm-speci…c productivity signal
y jn (right panel) for identical …rm types given by the parameters in Table 1. Both panels
show the results for = 2 and an economy-wide productivity signal equal to its prior mean
(y e = 0).
In other words, the beta on …rm n of type j at t = 1 is a weighted average of the beta on the
riskless asset ( f21y = 0), and the beta on the market portfolio ( my
21 = 1). The weights on the
two betas depend on the fraction the “windfall claim”comprises of the total …rm value. That
fraction is increasing in the …rm-speci…c windfall signal y jn (and in the …xed component aj
of t = 2 dividends). Hence, although …rm-speci…c windfall risk does not command a dollar
risk premium, …rm-speci…c windfall information does a¤ect equilibrium one-period ahead
expected gross rates of returns for individual …rms and, thus, these expected rates of returns
are stochastic as seen from a t = 0 perspective (even if the second-period spot interest rate
y
21 were to be deterministic).
We illustrate these results in Figure 10. The left panel illustrates that a …rm’s ex post
cost of capital is decreasing in its …rm-speci…c windfall signal for all values of the …rm-speci…c
productivity signal. This follows from the fact that the fraction of the …rm’s total payo¤s
which does not command a risk premium increases with the …rm-speci…c windfall signal.
The right panel of Figure 10 illustrates that a …rm’s ex post cost of capital is decreasing
(increasing) in its …rm-speci…c productivity signal for negative (positive) …rm-speci…c windfall signals, but it does approach the one-period ahead expected rate of return of the market
portfolio as the …rm-speci…c productivity signal becomes su¢ ciently large. On the other
hand, if the …rm-speci…c windfall signal is equal to its prior mean of zero, the …rm’s ex post
cost of capital is independent of its …rm-speci…c productivity signal and equal to the equilibrium one-period ahead expected rate of return of the market portfolio. Considering the
payo¤s of the …rm as a sum of a “windfall claim”(which does not command a risk premium)
and a “productivity claim” (which has a one-period ahead expected rate of return equal to
44
that of the market portfolio independently of the …rm-speci…c productivity signal), these
comparative statics follows from the sign and the size of the fraction the “windfall claim”
comprises of total …rm value. For example, if the windfall signal is negative, the expected
windfall payo¤s at t = 2 are negative and, thus, the value of the “windfall claim” at t = 1
is negative and, hence, the …rm can be considered as a levered position in the “productivity
claim” implying that the …rm’s ex post cost of capital is larger than the one-period ahead
expected rate of return of the “productivity claim.” If the …rm-speci…c productivity signal
is su¢ ciently low, the value of the …rm at t = 1 becomes negative for su¢ ciently negative
values of the …rm-speci…c windfall signal in which case, the one-period ahead expected rate
of return is not well-de…ned.
B
Exchange Economy
To reconcile our analysis with that of a pure exchange economy, we assume in this appendix
that consumption, trade and investment take place as described but that an additional round
of trading is interjected after t = 1 but before t = 2: Let t = b
1 denote the date at which
this extra round of trading takes place, and assume additional information yb is released after
trading at t = 1 but prior to trading at t = b
1. The additional information feeds the posterior
beliefs such that
ye
e jb
N (b
y e ; b21 e );
y jn
jn jb
N (b
y jn ; b21 j );
y jn
jn jb
with pre-posterior distributions characterized by
yb e jy
e
N (y e ;
2
1 e
b21 e ); yb jn jy
jn
N (y
jn ;
2
1 j
b21 j ); yb jn jy
N (b
y jn ; b21 j )
jn
N (y
jn ;
2
1 j
b21 j ):
This speci…cation allows us to examine the impact of additional public information, 21( )
b21( ) , keeping the equilibrium capital investments qj (y) at t = 1 …xed and, thus, given these
capital investments, our economy from t = 1 to t = 2 is an exchange economy.
Investors trade at t = b
1 in the marketable assets and as no consumption occurs at t = b
1,
only the relative prices of assets are meaningfully de…ned at this date. We choose without
loss of generality the zero-coupon bond maturing at t = 2 as the numeraire such that this
asset has a unit price at t = b
1. Hence, the t = b
1 price of asset n of type j is (see (17) and
45
(21))
vb1jn ( ) = E d2jn jb
y jn ; yb jn ; yb e ; qj (y)
RP2b1jn
2 2
b1 e
= aj + cj yb jn + E[hj yb jn ; yb e ; qj (y) exp
;
the t = b
1 price of a well-diversi…ed portfolio of type j …rms is (by applying the law of large
numbers)
vb1j ( ) =
N
1 P
vb ( ) = Eyb jn ;by jn aj + cj yb jn + E[hj yb jn ; yb e ; qj (y) exp
N n=1 1jn
= aj + exp
(
j
2
0 j
2 2
b1 e
exp
j
+ (b
ye+
q o (y e ))k ;
and the t = b
1 price of the market portfolio is
vb1 ( ) =
1
2
J
1 P
vb ( ) = exp
J j=1 1j
2 2
b1 e
exp
b21 e ) +
1
2
(b
ye+
1
2
2
j
1+
b21 e )
2
0 j
1
k
k
2 2
b1 e
(q o (y e ))k ;
depends only on the …rm-speci…c productivity informativeness at t = 1. That
where
is, …rm-speci…c information arriving after t = 1 is fully diversi…able such that the t = b
1
price of a well-diversi…ed portfolio of type j …rms is una¤ected by the (additional) …rmspeci…c information even though the t = b
1 price on any individual …rm will depend on
this information. Equivalently, the dollar risk premium RP2b1jn and the one-period ahead
expected rate of return 2b1jn on any individual …rm depends on the …rm-speci…c signals and
their precision. But the dollar risk premium RP2b1j and the one-period ahead expected rate
of return 2b1j on a well-diversi…ed portfolio of type j …rms or the market portfolio are left
una¤ected by additional …rm-speci…c information. Contrary to …rm-speci…c information,
economy-wide information does not diversify, and the additional economy-wide information
reduces the posterior uncertainty which serves to reduce the dollar risk premium and, thus,
also the ex post rate of return risk premium ! 2b1 on the market portfolio as illustrated in
Figure 3.
At any point in time, equilibrium prices are determined by the representative investor’s
…rst-order conditions as in (3) and, hence, prices at t = 0 and at t = 1 are una¤ected by
the additional information and trading at t = b
1. In other words, the ex ante equilibrium
prices (and, thus, the ex ante cost of capital) of individual securities are independent of
the informativeness of the information system at t = b
1 (see also Christensen et al. 2010).
Since the capital investments are not a¤ected by the additional information at t = b
1, the
46
equilibrium consumption plan and ex ante expected utility of the representative investor are
una¤ected as well.
In equilibrium, there can be no arbitrage in the periods between any two subsequent
trading dates (or any other two trading dates). At t = 1, the possible events at t = b
1 are the
posteriors yb, and the payo¤s of the securities are the asset prices at t = b
1. It follows there
is no arbitrage in the period from t = 1 to t = b
1 if, and only if, a pricing kernel, mb11 ; exists
such that
v1jn = v1jn (y
h
= exp
jn ; y jn ; y e ; qj (y))
i
1) E vb1jn jy
b
b
11 (y e )(1
jn ; y jn ; y e
+ Cov vb1jn ; mb11 jy
Pricing the t = 2 zero-coupon bond using this relation implies
and pricing asset n of type j implies
v1jn = exp
h
b
21 (y e )=(1
1)
i
aj + c j y
jn
+ E[hj y
b
11 (y e )
jn ; y e ; qj (y)
jn ; y jn ; y e
2 2
b1 e
and, thus, the dollar risk premium between t = 1 and t = b
1 equals
RPb11jn =
Cov vb1jn ; mb11 jy
= E[hj y
2 2
1 e
jn ; y e ; qj (y)
exp
RPb11jn
2 2
1 e
exp
exp
2
(
b21 e )
2
1 e
(y e +
1
2
2
1 e)
(q o (y e ))k exp
2
(
2
1 e
1
b21 e )
for the market portfolio. This implies the one-period ahead expected rate of return
…rm n of type j at t = 1 is
b
11jn
=
b
11 (y e )
+ !b11jn =
b
21 (y e )=(1
where the rate of return risk premium !b11jn is given as
!b11jn
1);
jn ; y jn ; y e
for an individual asset, and
RP b11 = exp
b
21 (y e )=(1
=
exp
:
1) + !b11jn ;
#
)
E vb1jn jy jn ; y jn ; y e ; qj (y)
b 1)
= ln
b
11 (y e ) =(1
v1jn (y jn ; y jn ; y e ; qj (y))
"
#
2 2
aj + cj y jn + E[hj y jn ; y e ; qj (y) exp
b1 e
= ln
=(b
1
2 2
aj + cj y jn + E[hj y jn ; y e ; qj (y) exp
1 e
( "
47
1):
1
b
11jn
of
This yields the following result.
Proposition 17 Conditional on public information at date t = 1, y = y e ; fy
it holds for the market portfolio that11
jn ; y jn gj=1;:::;J;n=1;:::;N
(a) The one-period ahead expected rate of return risk premium between t = b
1 and t = 2 is
2 2
b1 e =(2
! 2b1 =
b
1):
(b) The one-period ahead expected rate of return risk premium between t = 1 and t = b
1 is
!b11 =
2
(
2
1 e
1
b21 e )=(b
1):
(c) The sum of the rate of return risk premia between t = 1 and t = b
1 and between t = b
1
and t = 2 equals the rate of return risk premium between t = 1 and t = 2 :
1
! 21 = !b11 (b
1) + ! 2b1 (2
b
1) =
2 2
1 e:
(d) Conditional on y e , the sum of the dollar risk-premium between t = 1 and t = b
1 and the
expected dollar risk premium between t = b
1 and t = 2 equals the dollar risk-premium
between t = 1 and t = 2 :
RP b11 + E RP 2b1 jy e ; q o (y e ) = RP 21 :
Thus, while increasing the informativeness of the additional economy-wide productivity information at t = b
1, i.e., decreasing the posterior variance b21 e , reduces both the rate of
return risk premium and the expected dollar risk premium between t = b
1 and t = 2, it also
increases the rate of return risk premium and the expected dollar risk premium between
t = 1 and t = b
1 such that the informativeness of the additional information neither impacts
the rate of return risk premium nor the dollar risk premium between t = 1 and t = 2: That
is, the informativeness of the public information at t = b
1 only serves to allocate the total
risk in dividends between the returns in the period preceding the signal (pre-posterior risk)
and the period succeeding the signal (posterior risk).
11
While the additivity of the dollar risk premia in (d) also holds for individual …rms, the additivity of
rate of return risk premia in (c) only holds for individual …rms if the …xed component of t = 2 dividends aj
is equal to zero and there is no …rm-speci…c windfall risk, i.e., cj = 0, in which case (a) and (b) also give the
rate of return risk premia for individual …rms. Nevertheless, the ex ante cost of capital (however measured)
at t = 1 or t = 0 is always una¤ected by the additional information at t = b
1, since the equilibrium prices at
these dates and the dividends are una¤ected by this information.
48
;
C
Proofs
Proof of Lemma 2: Given the independence of the di¤erent types of states and signals,
consider without loss of generality a setting with only economy-wide productivity states and
2
information and two information systems 0 with signals y 0 e
N (0; 00 e ) and 00 with
2
2
2
signals y 00e
N (0; 000 e ) such that 000 e > 00 e . Since e
N (0; 1), we can make
an orthogonal decomposition of e into three zero-mean independent normally distributed
variables such that
e
where "2
N 0;
y 0 e = "1 ;
= "1 + "2 + ";
2
00
0 e
0
00
e jy e ; y e
2
0
0 e
. Since
( e j"1 ; "1 + "2 ) =
=
y 00e = "1 + "2 ;
00
e jy e
( e j"1 + "2 ) =
;
2
2
it follows that 0 is a garbling of 00 , while the converse does not hold when 000 e > 00 e .
It then follows from Marschak and Miyasawa (1968, Theorem 6.2) that 00 is more informative
than 0 in the sense of Blackwell (1953).
Proof of Lemma 3: (a): Given normally distributed states and signals, the conditional
expected marginal rate of transformation for …rm n of type j given the …rm-speci…c productivity signal y jn and the economy-wide productivity state is
E [MRTj jy
jn ; e ; qjn ]
= k exp
j
+
j e
+
j
y
jn
As discussed above the lemma, an optimal allocation of qj ,
+
1
2
j
2
1 j
n
qjy (qj ; y
jn )
k 1
qjn
:
o
, among
n=1;:::;N
…rms of type j is such that there exist a common conditional expected marginal rate of
transformation for …rms of type j, Qj ( e ; qj ), such that
h
E MRTj jy
jn ;
y
e ; qj (qj ; y
i
)
= Qj ( e ; qj ) ;
jn
n = 1; :::; N;
implying that
qjy (qj ; y jn )
=
Qj ( e ; qj )
k
1=(k 1)
exp
j
+
j e
+
j
y
jn
+
1
2
j
2
1 j
= (1
k) :
Since the per …rm aggregate capital investment qj is …xed, the law of large numbers implies
that
h
i
1 X y
y
qjE (qj ) = lim
qj (qj ; y jn ) = E qjn
jqj = qj :
N !1 N
49
Inserting the expression for qjy (qj ; y
…rm-speci…c signal yield
E
h
y
qjn
jqj
i
1=(k 1)
Qj ( e ; qj )
k
=
and calculating the expectation with respect to the
jn )
exp
+
j
+
j e
1
2
j
j
1
2
0 j
k
+
1
2
j
2
1 j
= (1
k)
= qj ;
and, thus,
Qj ( e ; qj )
k
1=(k 1)
exp
j
+
= (1
j e
= exp
j
1
2
k)
j
1
k
2
0 j
+
1
2
Inserting this expression back into the expression for qjy (qj ; y
qjy (qj ; y
jn )
= exp
= exp
1
2
j
j
y
j
1
k
1
2
jn
2
0 j
+
j
1
k
1
2
j
2
0 j
2
1 j
= (1
= (1
k) +
2
1 j
j
jn )
k) qj :
yields
y
j
= (1
jn
+
1
2
j
2
1 j
= (1
k) qj
k) qj ;
which is equation (9). The comparative statics follow directly from this expression.
(b): The payo¤ from capital investments hj ( ) at t = 2 on a well-diversi…ed portfolio
of type j …rms, in which the per …rm aggregate capital investment qj at t = 1 is e¢ ciently
allocated, is given by
hE
j ( e ; qj )
1 X
hj ( e ;
= lim
N !1 N
y
jn ; qj (qj ; y jn ))
= E exp
j
+
j e
+
= E exp
j jn
+k
j
j jn
y
jn
exp k
1
2
h
i
y
= E hjn j e ; qj
j
j
1
50
k
y
2
0 j
jn
1
2
= (1
j
1
k
k)
2
0 j
exp
= (1
j
+
k) qjk
j e
qjk :
2
1 j
The law of iterated expectations and
E exp
j jn
+k
= E E exp
= E exp
"
"
= E exp
= exp
= exp
"
y
j
j jn
y
j
j
1
k
1
2
(1
2
j
1
2
+k
+
y
jn +
2
2
0 j
1+
2
0 j
k)
1
y
+
2
j
1
2
+k
k)
j
1
k
y
j
1
2
k
(1
2
0 j
2
j
1
2
k) jy
= (1
1
2
jn
k)
2
0 j
1
k
1
1
2
2 2
j 1 j
1
2
yield
= (1
jn
2
1 j
j
2
0 j
k
j
1
2
jn
2
j
j
1
2
jn
2
0 j
=1
j
2
0 j
1 k
##
jn
= (1
k)
2
0 j
2
2
j
k
(1
k)
2
2
0 j
#
:
k
Collecting terms yields (10), and the comparative static follows directly from this expression.
Proof
of Lemma 4: (a): As discussed above the lemma, an optimal allocation of q o ,
o
n
y
, among …rm types is such that there exist a common marginal rate of transqj (q o )
j=1;:::;J
formation, Q ( e ; q o ), such that
MRTj ( e ; qjy (q o )) = Q ( e ; q o ) ;
j = 1; :::; J;
implying that
qjy (q o )
Since q o =
=
1
J
Q ( e; qo)
k
PJ
y
j=1 qj (q o ),
1=(k 1)
exp
j
+
e
+
1
2
2
j
k
1
k
= (1
k) :
= (1
k)
we get
1=(k 1)
Q ( e; qo)
k
2
0 j
1+
exp [
e]
=
1
J
qo
PJ
j=1
exp
j
+
1
2
2
j
1+
k
2
0 j1 k
Substituting this expression back into the expression for qjy (q o ), we get (11), where
j
2
0 j
1
J
exp
PJ
j=1 exp
j
+
1
2
j
2
j
+
1+
1
2
2
j
k
2
0 j1 k
1+
k
2
0 j1 k
The comparative statics follow directly from this expression.
51
= (1
k)
= (1
k)
:
:
(b): The payo¤ from capital investments hj ( ) at t = 2 to the representative investor is
given by
J
1X E
h( e ; q o ) =
hj ( e ; qjy (q o ))
J j=1
=
J
1X
exp
J j=1
J
1X
=
J j=1
1
J
which is (12), with
exp [ ]
j
+
e
+
2
j
1
2
1+
2
0 j
k
1
2
0 j
j
k
qo
k
1 2
2
= (1 k)
exp
j + 2 j 1+ 0 j1 k
PJ
k
1 2
2
= (1 k)
j + 2 j 1+ 0 j1 k
j=1 exp
J
1X
J j=1
1
J
k
k
exp [
1 2
k
2
= (1 k)
exp
j + 2 j 1+ 0 j1 k
PJ
1 2
k
2
= (1 k)
j + 2 j 1+ 0 j1 k
j=1 exp
k
k
e] qo
:
Clearly, h( e ; q o ) is increasing in the economy-wide productivity signal e . Di¤erentiating
k
1 2
2
exp [ ] with respect to zj exp
= (1 k) yields
j + 2 j 1+ 0 j1 k
d exp [ ]
d 1
=
dzj
dzj J
1
J
zj
PJ
j=1 zj
k
1
=
J
1
J
z
j
PJ
k
1 1
j=1 zj
=
> 0;
k
J 1 PJ
j=1 zj
J
PJ
k
j=1 zj
zj k
1
J
PJ
1
J
j=1 zj
PJ
j=1 zj
k 1
1
J
2k
P
where the last inequality follows from zj > 0 for all j and, thus, zj = Jj=1 zj 1, and k < 1.
Since zj is increasing in 20 j , h( e ; q o ) is increasing in 20 j for all j = 1; :::; J.
Proof of Lemma 5: It follows from equation (3) that the cum-dividend market value of
the market portfolio (per investor) at t = 1 given the public economy-wide signal ye and
aggregate capital investment q o is
Vm
1 (q o )
= d1
q o + exp [
21 (ye )]
Et g( e ) + h( e ; q o )jye + Cov m21 (ye ); g( e ) + h( e ; q o )jye
;
where the equilibrium interest rate 21 (ye ) and pricing kernel m21 (ye ) are taken as given.
From this expression, it follows directly that the …rst order-condition for a value maximizing
52
aggregate capital investment is identical to the …rst-order condition for an optimal aggregate
capital investment for the representative investor, i.e., (14). On the other hand, both the
equilibrium interest rate and the equilibrium pricing kernel depend on the optimal aggregate capital investment, since the representative investor’s marginal utilities depend on this
investment.
(a): Neither the …rm-speci…c windfall signals nor their informativeness a¤ect the conditional expected marginal rate of transformation of …rms and, therefore, this type of information does not a¤ect the e¢ cient allocation of aggregate capital investments among …rms (see
Lemmas 3 and 4). Furthermore, this type of information does not a¤ect beliefs about equilibrium consumption per capita. Hence, it has no impact on the equilibrium interest rates
and the pricing kernel and, therefore, no impact on optimal aggregate capital investments.
(b): Equilibrium consumption per capital does not depend on the …rm-speci…c productivity signals per se and, therefore, these signals have no impact on optimal aggregate
investments. On the other hand, the informativeness of the …rm-speci…c productivity signals
a¤ects the productivity parameter of aggregate capital investments (see Lemma 4) and,
thus, may a¤ect optimal aggregate capital investments.
(c): Clearly, economy-wide productive information a¤ects the expected marginal rate of
transformation of aggregate capital investments and, thus, it enters directly in the …rst-order
condition (14) for optimal aggregate capital investments. Economy-wide windfall information
does not a¤ect the marginal rate of transformation of aggregate capital investments, but it
a¤ects beliefs about future equilibrium consumption per capital and, thus, it may a¤ect
both the equilibrium interest rate and the pricing kernel and thereby the optimal aggregate
capital investment.
Proof of Proposition ??: (a): The optimal aggregate capital investment is given implicitly by (??). Di¤erentiating both sides of this equation with respect to the informativeness
of the …rm-speci…c productivity information for type j …rms 20 j yields that
@
@
Assume
@
@
21
2
0 j
21
2
0 j
=
(
=
1 @
@ 20
1
@
(q o )k
@
(1
j
k
2
0 j
k)
k 1
(q o ) + k (q o )
@q o
@ 20 j
!
+
1
d1
1 @q o
:
q o @ 20 j
0. Then the right-hand side is also negative and, thus,
@q o
@ 20 j
qo
@
:
(1 k) @ 20 j
53
@q o
q o @ 20 j
)
(28)
Substituting this into the left-hand side yields
@
@
1
21
2
0 j
@
k
@
(q o )
=
+
d1
(
1
1
(q o )
2
0 j
(q o )k + k (q o )k
qo
@
q o (1 k) @ 20
k
qo
@
(1 k) @ 20
1
j
!
j
(q o )k + k (q o )k
1
qo
(1 k)
+
1
d1
qo
q o (1 k)
)
@
@
2
0 j
> 0;
where the last inequality follows from the fact that all terms in the brackets are strictly
positive and the fact that @ =@ 20 j > 0. Hence, a contradiction is obtained to the assumption
that @ 21 =@ 20 j 0 and, thus, @ 21 =@ 20 j > 0, and this proves (a).
(b): Collecting terms including @q o =@ 20 j in (28) yields
(
k (q o )k
=
(q o )
(
1
1
+
k
1
d1
(q o )k
k
(q o )
qo
)
!
+ (1
@
@
2
0 j
= (1
1
k)
qo
)
)
@q o
@ 20 j
1 @
:
@ 20 j
Since all terms in the brackets on the left-hand side are strictly positive and the fact that
1 @
> 0, it follows that @@q2o > (<)0 if, and only if < (>)1, and this proves (b).
@ 20 j
0 j
Proof of Proposition ??: (a): Since there are no risk premia in the equilibrium prices at
t = 1 (see (??)) and at t = 0 (see (??)), it follows directly from (5) and (6) that 10jn = 10 .
It follows from Proposition ??(b) that equilibrium consumption at t = 1, i.e., c1 = d1 q o ,
is increasing (decreasing) in 20 j if > 1 ( < 1) and, thus, it follows from (??) that 10 is
increasing (decreasing) in 20 j if > 1 ( < 1).
(b): Since 20 = ( 10 + 21 )=2, if follows from (a) and Proposition ??(a) that 20 is increasing
in 20 j if > 1. If < 1, q o and the average t = 2 dividend cy2 (q o ) for any given q o are both
increasing in 20 j and, thus, equilibrium consumption c2 at t = 2 is increasing in 20 j . It
then follows from (??) that 20 = + (ln [c2 ] ln [c0 ]) is increasing in 20 j .
(c): Since the dividend ratio is assumed to be positive, the implied cost of capital j is by
Descartes’rule of signs given by (??). Let the quantity Qj be de…ned as
Qj
drj +
q
drj
2
+ 4 exp [
10 ] drj
+ exp [ 2
20 ]
> 0:
If follows from (??) that j is increasing in the …rm-speci…c productivity informativeness
2
2
> 1, it follows from (a) and (b) that @ 10 =@ 20 j > 0 and
0 j if @Qj =@ 0 j < 0. Given
54
@
2
20 =@ 0 j
> 0 and, thus,
(
)
@drj
@drj
1
2drj + 4 exp [ 10 ]
+
@ 20 j 2 Qj + drj
@ 20 j
(
1
@
@
4 exp [ 10 ] 210 drj + 8 exp [ 2 20 ]
@ 0j
@
2 Qj + drj
@Qj
=
@ 20 j
@drj
1
+
2
@ 0j
Qj + drj
<
Since @drj =@
2
0 j
drj + 2 exp [
2
0 j
< 0, it then follows that @Qj =@
drj + 2 exp [
10 ]
> Qj + drj =
q
drj
2
10 ]
20
2
0 j
)
@drj
:
@ 20 j
< 0 if
+ 4 exp [
10 ] drj
+ exp [ 2
20 ]
;
or equivalently (since all terms are positive),
drj
2
+ 4 exp [ 2
10 ]
+ 4 exp [
10 ] drj
> drj
2
+ 4 exp [
10 ] drj
+ exp [ 2
20 ]
:
It then follows from 10 < 20 that @Qj =@ 20 j < 0.
Proof of Lemma 7: It follows from (15) that
y
21 (q o )
=
i3
2 h
y
0
E u2 (c2 )jy e ; q o
5=
ln 4
u01 (cy1 )
2
2
6E
ln 6
4
cy2
cy1
3
3
jy e ; q o 7
7
5
i3
h
i
2 h
y
E
exp
ln
jy
e; qo
E exp
ln[c2 ] jy e ; q o
7
6
7=
4
5
h
i
=
ln 6
ln
4
5
y
y
exp
ln[c1 ]
exp ln c1
h
h
h
i
ii 3
2
exp
E ln[cy2 ]jy e ; q o + 12 2 Var ln[cy2 ]jy e ; q o
5
h
i
=
ln 4
y
exp
ln[c1 ]
h
i
h
i
y
1 2
= +
E ln[cy2 ]jy e ; q o
ln[cy1 ]
Var
ln[c
]jy
;
q
e o
2
2
= +
cy2
+ y e + k ln q o
ln[d1
qo]
55
1 2 2 2
1 e:
2
Proof of Lemma 8: It follows from (15) that
my21 ( e ; y e ; q o )
=
u02 (cy2 ( e ; q o ))
E[u02 (cy2 )jy e ; q o ]
exp[ln[ cy2 ( e ; q o )
=
E[exp[ln[ cy2
]]
=
]]jy e ; q o ]
ln[cy2 ( e ; q o )]]
exp[
ln[cy2 ]]jy e ; q o ]
E[exp[
ln[cy2 ( e ; q o )]]
i
h
i
y
y
1 2
exp[ E ln[c2 ]jy e ; q o + 2 Var ln[c2 ]jy e ; q o ]
h
h
i
i
y
1 2
]jy
;
Var
ln[c
= exp[ E ln[cy2 ]jy e ; q o
ln[cy2 ( e ; q o )]
q
e
o ]
2
2
i
h
1 2
= exp[ ( (y e
Var ln[cy2 ]jy e ; q o ]
e ))
2
h
=
= exp[
(y
exp[
1
2
e)
e
)2
(
2
1 e ]:
Proof of Lemma 9: (a): Since
E
h
dy2j jy jn ; y jn ; e ; qjy (
i
) = aj + c j y
jn
+ exp
+
j
j
y
jn
+
1
2
j
2
1 j
+
e
qjy ( )
k
;
and since my21 only depends on the economy-wide productivity signal and state, it follows
that
y
=
RP21jn
=
=
Since E
h
h
h
Cov my21 ; E dy2j jy
Cov my21 ; exp
j
+
j
i
y
;
q
(
)
jy
e j
jn ;
y
jn
+
1
2
j
i
y
(
)
exp
]
jy
;
q
e j
e
h
Cov my21 ; exp [
my21 jy e ; qjy (
jn ; y
y
RP21jn
=
n h
E my21 exp [
exp
n
= exp
exp
j
+
ye+
j
+
y
e ] jy e ; qj ( )
j
1
2
j
y
jn
+
1
2
2
1 e
y
jn
+
56
1
2
2
1 j
j
i
) = 1, it follows that
i
jn ; y
+
+
j
2
1 j
j
2
1 j
j
i
y
;
y
;
q
(
)
jn
e j
qjy ( )
e
y
jn
h
E exp [
qjy ( )
h
E my21 exp [
qjy ( )
+
1
2
j
k
jy e ; qjy ( )
2
1 j
qjy ( )
io
y
]
jy
;
q
(
)
e
e
j
k
io
y
]
jy
;
q
(
)
e j
e
k
k
:
Inserting the expression in Lemma 8 in the second term in the brackets and simplifying yield
E
h
my21
y
e ; qj (
e ] jy
exp [
i
h
) = E exp
(y
= E [exp [
e)
e
= exp
(1
)y e +
1
2
2
= exp
(1
)y e +
1
2
2
= exp
ye+
= exp
2
1
2
ye+
2
(
1+
y
e ; qj (
e ] jy
exp [
2
1 e
e
1
2
)2
2
1 e
exp
2
2
y
(1
2
1 e
y
+
2
1 e
1
2
e
y
i
)
1
2
e
2
1 e
2
1 e
(1
2 )
(1
2 )
1
2
2
1 e
)
)] jy e ] exp
(1
e
1
2
2
1 e
:
Hence,
y
= exp
RP21jn
ye+
exp
= 1
j
2
1 e
1
2
+
2
exp
exp
j
exp
2
= 1
exp
2
jn
2
1 e
+
= 1
y
j
y
j
exp
+
1
2
j
exp
jn
+
1
2
j
2
1 e
exp
2
1 e
E[hyj jy
j
ye+
1
2
2
1 j
qjy ( )
ye+
1
2
2
1 j
qjy ( )
+
y
j
(1
2 )
2
1 e
k
2
1 e
k
jn
y
jn ; y e ; qj (
1
2
+
j
2
1 j
+
ye+
qjy ( )
2
1 e
1
2
k
)];
which demonstrates (a).
(b): It follows from (18) and (a) that
y
v1jn
= exp[
y
21 ]
h
aj + c j y
jn
2
+ exp
2
1 e
E[hyj jy jn ; y e ; qjy (
i
)] :
The one-period ahead expected rate of return of …rm n of type j conditional on information
at t = 1 is thus given as
y
21jn
2
= ln 4
=
y
21
E[hyj jy jn ; y e ; qjy (
exp[
"
+ ln
aj + cj y jn +
h
y
21 ] aj + cj y jn + exp
aj + c j y
aj + c j y
jn
2
y
jn + E[hj jy
2
+ exp
2
1 e
2
1 e
)]
E[hyj jy jn ; y e ; qjy (
y
jn ; y e ; qj ( )]
E[hyj jy
y
jn ; y e ; qj (
)]
#
3
i5
)]
:
The conditional rate of return risk premium conditional on information at t = 1 is therefore
57
given as
"
! y21jn = ln
aj + c j y
aj + c j y
jn
y
jn + E[hj jy
2
+ exp
y
jn ; y e ; qj ( )]
2
1 e
E[hyj jy
y
jn ; y e ; qj (
)]
#
;
which demonstrates (b).
h
i
(c): Repeating the steps in (a) and (b) with h( e ; q o ) instead of E dy2j jy jn ; y jn ; e ; qjy ( )
demonstrates (c). Alternatively recognize that if for an asset it holds that aj + cj y jn = 0,
then ! y21jn = 2 21 e : The market portfolio is such an asset which establishes (c).
Proof of Proposition 12: (a): It follows from (23) that
h
i
k 1
(y
)
=
ln
k
(q
(y
))
+ ln exp
e
e
21
o
ye+
2
1 e
1
2
e ] jy e ]
+ Cov [m21 ; exp [
:
It follows from the proof of Lemma 9 that
Cov [m21 ; exp [
e ] jy e ]
= exp
ye+
1
2
2
1 e
2
exp
2
1 e
1
and, hence,
21
h
k 1
i
+ ln exp
(y e ) = ln k (q o (y e ))
i
h
= ln k (q o (y e ))k 1 + y e +
i
h
= ln k (q o (y e ))k 1 + y e +
ye+
1
2
1
2
1
2
2
2
1 e
(1
2 )
It now follows from the proof of Proposition ??(a) that @ 21 =@
term does not depend on 20 j .
Furthermore, it follows from (16) and the above that
@
@y e
21
=
= (k
Suppose @
21 =@y e
1
@q o
+
q o (y e ) @y e d1
1
@q o
1)
+ :
q o (y e ) @y e
+k
2
1 e
2
exp
2
1 e
2
1 e
2
0 j
:
> 0, since the last additive
1
@q o
q o (y e ) @y e
< 0: Then
(k
1)
1
@q o
+
q o (y e ) @y e
<0,
58
2
1 e
@q o
> q o (y e ) = (1
@y e
k) :
Substituting this into the above, we get
@q o
1
@q o
1
+
q o (y e ) @y e d1 q o (y e ) @y e
q o (y e )
= (1 k) :
+ k= (1 k) +
d1 q o (y e )
@ 21
=
@y e
+k
>
Since all the terms on the right-hand side are positive, we obtain a contradiction to the
assumption that @ 21 =@y e < 0 and, thus, @ 21 =@y e > 0. To show the last claim in (a) take
2
the derivative of 21 (y e ) wrt. 20 e recognizing that 20 e = 1
1 e
@
@
21
2
0 e
=
(1
k)
1
@q o
q o (y e ) @ 20 e
2 1
(2
):
In (c) we establish that @q o =@ 20 e < 0. Hence, the …rst term is positive. If > 1=2, the
second term is positive as well. This establishes (a).
(b): It follows from Lemma 9 that the rate of return risk premium of the market portfolio
2
! 21 = 2 21 e is independent of 20 j , y e , and q o . Since 21 e = 1
0 e the last claim follows
easily. This establishes (b).
(c): As 21 = 21 + ! 21 = 21 + 2 21 e it follows from (a) and (b) that the one-period
ahead expected rate of return of the market portfolio is increasing in 20 j and y e :
From above it follows that
@
@
21
2
0 e
=
@
@
21
2
0 e
+
@! 21
=
@ 20 e
(1
k)
1
@q o
q o (y e ) @ 20 e
2 1
(2
2
)
:
Now, from (16) and the above it follows that in equilibrium
+ (ln( ) + k ln(q o ) + y
e
ln(d1
1
2
qo )
= ln( k(q o )k 1 ) + (y e +
1
2
= ln( k(q o )k 1 ) + (y e +
1
2
2 2
1 e)
2
2
1 e)
2
1 e
(1
2 )
2
1 e ):
ln(d1
qo )
1
2
De…ne the function g( ) from the equation
g(
2
0 e; qo )
= +
ln( ) + k ln(q o ) + y
e
ln( k(q o )k 1 ) + (y e +
59
1
(1
2
2 )
2
1 e)
2 2
1 e
= 0:
The implicit function theorem then says
2
0
@g=@
dq o
=
d 20 e
e
@g=@q o
:
Now,
@g=@
2
0 e
=
1 2 2
2
2
1
2
+
2
and
@g=@q o = [
=
2
1
2
2
(
2 + 1) =
1
1
]
k+
qo
d1 q o
(k
1)
1
2
2
(1
)2 ;
1
:
qo
Thus,
[ q1 k +
o
Inserting this into the expression for @
@
@
21
2
0 e
=
=
@
@
21
2
0 e
1
2
+
@! 21
= (1
@ 20 e
k)
d1
This implies
sign
qo
(
@
@
21
2
0 e
1
]
d1 q o
2
21 =@ 0 e
qo
)
d1 q o
1
)(d1
+ (1
< 0:
o
yields
)2
(k
o
qo )
k)
sign d1 + (1
=
1) q1
1 2
(1
1
2
1
1
qo [ q k + d q ]
k)(1
(k +
)
)2
(k
(1
o
d1 + (1
2
2
1
2
dq o
=
d 20 e
1) q1
2 1
(2
2
)
o
:
k)(1
)(d1
qo ) :
Now,
d1 + (1
k)(1
)(d1
qo )
(
> 0 for
1<
< 0 for
1<
1
1 k
() (1
)(1
> 0 for < 1;
k) >
1;
This implies
@
@
21
2
0 e
1
1
k
:
For a …xed k one may suspect that @ 21 =@ 20 e > 0 for su¢ ciently large, and in our
numerical examples this turns out to be the case. This establishes (c).
60
(d): Inserting the expression for the interest rate (16) into (23) and simplifying yield
+
+
+
ln [ ] + y e + k ln[q o (y e )] ln[d1 q o (y e )]
h
i
= ln k (q o (y e ))k 1 + y e + 21 (1 2 ) 21
ln [ ] + k ln[q o (y e )] ln[d1
i
h
= ln k (q o (y e ))k 1 + (1
1 2 2 2
1 e
2
,
e
q o (y e )]
)y e +
ln [ ] + k ln[q o (y e )] ln[d1
h
i
= ln k (q o (y e ))k 1 + (1
1
2
2
1
2
2
1+
2
2
2
1 e
)2
2
1 e:
,
q o (y e )]
)y e +
(1
Since the two last additive terms on the right-hand side does not depend on 20 j , and the
other terms are the same as in the setting with no economy-wide productivity risks, it follows
from the proof of Proposition ??(b) that @q o (y e ) =@ 20 j > (<)0 if, and only if, < (>)1.
It follows from the above that
@
@y e
+
ln [ ] + k ln[q o (y e )]
@ h h
ln k
@y e
1
+
k
q o (y e ) d1
1
k
+
q o (y e ) d1
=
(q o (y e ))k
1
i
ln[d1
q o (y e )]
+ (1
)y e +
1
2
2
)2
(1
1
1
@q o
@q o
= (k 1)
+ (1
q o (y e ) @y e
q o (y e ) @y e
1
1
@q o
+ (1 k)
= (1
):
q o (y e ) @y e
q o (y e )
2
1 e
i
,
),
Since all terms in the brackets are positive, we get that @q o =@y e > (<)0 if, and only if
< (>)1. That q o (y e ) is decreasing in 20 e for each y e was demonstrated in the proof of
(c). This establishes (d).
Proof of Proposition 14: The cum-dividend value of the market portfolio at t = 1 is
(see equation (3)):
V 1 (y e ) = d1
q o (y e ) + v 1 (y e ) = d1
+ exp [
= d1
21 (y e )]
q o (y e ) + exp [
q o (y e )
E h( e ; q o (y e )) y
21 (y e )]
e
+ Cov(m21 ; h( e ; q o (y e )) y e )
E m21 h( e ; q o (y e )) y
61
e
:
Using Lemma 8 and simplifying, we get
E m21 h( e ; q o (y e )) y e ; q o ( )
h
= E exp
(y e
e)
= exp
y
e
= exp
y
= exp
ye+
e
1 2 2 2
1 e
2
1 2 2 2
1 e
2
2
2
1 2
1 e
2
1 2
(1 2 )
2
k
e ] (q o (y e )) y
exp [
(q o (y e ))k E exp [ (1
k
(q o (y e )) exp
)
(1
e]
)y e +
e
y
1
2
i
e
2
)2
(1
2
1 e
(q o (y e ))k
2
1 e
and, thus,
V 1 (y e ) = d1
21 (y e )] exp
q o (y e ) + exp [
Taking the total derivative with respect to
dV 1 (y e ) h
=
1 + exp [
d 20 j
21 (y e )] exp
2
0 j
ye+
ye+
1
2
2
(1
2 )
2
1 e
(q o (y e ))k :
yields
1
2
2
(1
i dq (y )
e
o
2
d 0j
d (y e )
(q o (y e ))k 21 2
d 0j
d
(q o (y e ))k 2 :
d 0j
2
1 e
2 )
exp [
21 (y e )] exp
ye+
1
2
2
(1
2 )
2
1 e
+ exp [
21 (y e )] exp
ye+
1
2
2
(1
2 )
2
1 e
k (q o (y e ))k
1
The …rst term on the right-hand side is equal to zero, since q o (y e ) by Lemma 5 maximizes
the cum-dividend value of the market portfolio at t = 1. The last two terms can be rewritten
in terms of the ex-dividend value of the market portfolio v 1 (y e ) such that
dV 1 (y e )
= v 1 (y e )
d 20 j
(
d
d
1
2
0 j
d
21 (y e )
d 20 j
)
= v 1 (y e )
d
d
2
0 j
(ln( )
21 (y e )) :
(1
2
1 e
It follows from the proof of Proposition 12(a) that
ln( )
21 (y e ) =
h
i
ln k (q o (y e ))k 1
ye+
1
2
2 )
and, thus,
dV 1 (y e )
= v 1 (y e )(1
d 20 j
k)
1 dq o (y e )
:
q o (y e ) d 20 j
Since the …rst three terms on the right-hand side are all positive, dV 1 (y e )=d
same sign as dq o (y e )=d 20 j .
62
2
0 j
has the
The ex ante value of the market portfolio is determined by the truncated version of
equation (3) as
u0 (d1 q o (y e ))
v 0 = Ey e 1
V 1 (y e ) :
u00 (c0 )
Since dV 1 (y e )=d 20 j has the same sign as dq o (y e )=d
utility function that
sign
(
d
)
u01 (d1 q o (y e ))
V 1 (y e )
u00 (c0 )
d
2
0 j
2
0 j,
we get by the concavity of the
= sign
(
dq o (y e )
d 20 j
)
for all economy-wide productivity signals y e . Since the informativeness of the …rm-speci…c
productivity information for …rm type j does not a¤ect the distribution of the economy-wide
productivity signals, it then follows from Proposition 12(d) that dv 0 =d 20 j < (>)0 if, and
only if, > (<)1.
Proof of Proposition 15: From Proposition 14, v 0 is decreasing in 20 j when > 1:
Further, Proposition 12(a) yields that 21 (y e ) is increasing in 20 j : As
21 (y e )
= + ln
= + ln
"
"
(q o (y e ))k exp
d1
y
q o (y e )
(q o (y e ))k exp
d1
e
#
(y e +
1 2 2 2
1 e
2
1
2
2
1 e)
q o (y e )
#
2 2
1 e(
1
2
+
2
)
is increasing in 20 j , it follows that expected consumption growth, (q o (y e ))k exp[ (y e +
1
2
2
2
> 1;
1 e )]=(d1 q o (y e )), is increasing in 0 j . Now, as q o (y e ) is decreasing in 0 j when
2
1
2
k
it follows that (q o (y e )) exp[ (y e + 2 1 e )] and d1 q o (y e ) are both increasing and thus
are Ey e [ (q o (y e ))k expf (y e + 12 21 e )g] and Ey e [d1 q o (y e )]: The implied cost of capital
on the market portfolio, , is determined as the discount rate solving the equation
v 0 = Ey e d1
q o (y e ) exp [
] + Ey e
(q o (y e ))k exp
(y e +
1
2
2
1 e)
exp [ 2 ] :
Thus, as v 0 is decreasing while the RHS is increasing in 20 j for …xed , it follows that
is increasing in 20 j :
Proof of Proposition 16: Using (2), the equilibrium ex-dividend value of the market
portfolio at t = 0 can be written as
v0 =
2
X
=1
h
i
B 0 E0 m 0 d ;
63
where
B
0
=
E0 [u0 (c )]
;
u00 (c0 )
m
0
=
u0 (c )
:
E0 [u0 (c )]
In equilibrium, consumption for the representative investor equals average dividends at each
consumption date, i.e., ct = dt ; t = 0; 1; 2. Hence,
v0 =
2
X
=1
i
h
0
E0 u (c )d
u00 (c0 )
2
X
E0 [u0 (c )c ]
=
:
u00 (c0 )
=1
t] 1 1 c1t , yields
Inserting the power utility function ut (ct ) = exp[
v 0 = (c0 )
2
X
E0 exp[
] (c )
c
=1
= (c0 ) (1
)
2
X
E0 [ut (ct )]
=1
= (c0 ) (1
) fEU
u0 (c0 )g :
Since t = 0 consumption c0 > 0 does not depend on the information system at t = 1,
the derivative of the ex ante value of the market portfolio with respect to economy-wide
productivity informativeness 20 e is
d
d
v0
2
0 e
= (c0 ) (1
)
d
d
2
0 e
EU :
Since the investors’equilibrium expected utility is strictly increasing in economy-wide productivity informativeness when optimal aggregate capital investments vary with the economywide productivity signals and their informativeness (see Propositions 6(d) and 12(d)), the
equilibrium ex ante value of the market portfolio decreases (increases) in 20 e if
> 1
( < 1).
Proof of Proposition 17: The rate of return risk premium on the market portfolio
between t = b
1 and t = 2 is
"
#
exp (b
y e + 12 b21 e ) (q o (y e ))k
! 2b1 = ln
2 2
exp
b1 e exp (b
y e + 21 b21 e ) (q o (y e ))k
=
2 2
b1 e :
Similarly, the rate of return risk premium on the market portfolio between t = 1 and t = b
1
64
is
!b11 = ln
= ln
"
"
Eyb e exp
exp
2 2
b1 e
2 2
1 e
exp
exp
2 2
b1 e exp (y e +
2 2
(y e
1 e exp
exp
exp
(y e +
(y e +
1
2
1
2
y ae +
+
2
1 e)
2
1 e)
1
2
1
b21 e ) (q o (y e ))k jy e
2
2
(q o (y e ))k
1 e)
(q o (y e ))k
(q o (y
e
))k
#
2
=
b21 e ):
2
1 e
(
#
(c) follows trivially from (a) and (b). To prove (d) notice the dollar risk premium between
t = 1 and t = b
1 is
RPb11 = exp
= exp
2 2
1 e
(y e +
1
2
exp
(y e +
1
2
2
1 e)
(q o (y e ))k exp
2
1 e)
2 2
b1 e
and the dollar risk premium between t = b
1 and t = 2 is
RP2b1 = exp
(b
ye+
1 2
b )
2 1e
Taking expectations over RP2b1 yields
Eyb e [RP2b1 jy e ] = exp
(y e +
1
2
(q o (y e ))k 1
2
1 e)
2
(q o (y e ))k exp
exp
Summing RPb11 and Eyb e [RP2b1 jy e ] yields the desired result.
2
1 e
exp
b21 e )
2 2
1 e
2 2
b1 e
exp
(q o (y e ))k 1
(
1
;
:
2 2
b1 e
:
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