An Alternative to the Feltham-Ohlson Valuation Framework: Using q-Theoretic Income to Predict Firm Value
Miles B. GietzmannCass Business SchoolCity University
Adam Ostaszewski
Dept. of Mathematics
London School of Economics
May 2003, altgo-conf.tex
Abstract
In this model we provide a theoretical justi¯cation for why the functional relationshipbetween earnings and value will be non linear. Moreover in our stylized model we derivea closed form for the relationship 英國留學生dissertation網and show why earnings response coe±cients are lowerfor ¯rms that are contracting or expanding relative to those ¯rms that are maintaininga steady investment strategy. We extend earlier research which posits a simple convexrelationship based upon ¯xed abandonment values and also generalize research which usesreal options valuation models based upon the assumption that ¯rms only ever exercise
one real investment option and then are committed to that strategy ad in¯nitum. Inparticular, since in some empirical settings the special case of `¯xed' abandonment willnot apply, we show how the form of convexity changes. Secondly, in our model ¯rmsare allowed to dynamically change investment strategies, for instance expanding in oneperiod followed by contraction in the subsequent period. Given an objective of derivingcomparative statics results for earnings response coe±cients, our dynamic model is ableto capture more accurately real investment behavior than a model in which ¯rms only
ever decide to expand or contract once. Our model provides both an alternative rationalefor accounting measures having information content and an alternative framework for theempirical speci¯cation of tests of `accounting value relevance' based upon ¯nite mixture(regime-switching) distributions. Our model shows how one can view equity value ascomprising opening cash, q-revalued opening stock, current q-income and future q-income.We are grateful for comments from Nick Bingham, Graham Brightwell, Patricia Dechow,Mike Kirschenheiter, Russ Lundholm, John O'Hanlon, and from seminar participantsat the University of Lancaster, EIASM Madrid Economics and Accounting workshop,and at King's College London.
1
1 INTRODUCTION
In this section we discuss the established Feltham-Ohlson (FO) valuation model and brie°yreview the main ¯ndings and some related research. We argue that, since the FO approach
has no transparent role for management, the approach excludes consideration of important realoptions that typically arise empirically when investment decisions are undertaken. In addition,we present a simple example that shows that the traditional residual income number is not theonly accounting measure which admits valuation equivalence to the discounted dividend stream.Another well-known criticism, following Peasnell (1982), of the clean surplus class of models,such as FO, is that the models do not give rise to any structural implications for the applicationof accounting rules. That is, it may be hard to argue that the models present a justi¯cation foraccrual accounting when there is little evidence of the need for accrual adjustments. Exploitingthis equivalence type result we show that a di®erent form of residual income valuation doesgive rise to a reasonably tractable method for analyzing optimal investment decisions anddevelops an approach to go beyond the general equivalence result and identify a restricted setof accounting measures that meet a certain `axiomatic' property, as follows. When consideringcandidate earnings numbers with the intention of predicting ¯rm value, we require that asthe chosen earnings number increases, this rationally results in higher estimates of future ¯rmvalue. We thus propose that this simple monotonicity property should be satis¯ed by candidate#p#分頁標題#e#
earnings measures, on the grounds that investors will question any measurement methods of anearnings number for which current higher earnings, can mean lower ¯rm value in the future.Initially, one may suspect that satisfaction of this seemingly quite mild axiomatic conditionwill not be particularly discriminating and that many earnings measures will satisfy the axiom.However, interestingly, we ¯nd thatthe established earning measure used in the literature(residual income) fails to satisfy the axiom and show how an alternative income measure basedupon the established q-theory of income does satisfy the axiom. Clearly, with one simple axiomwe cannot provide a way to discriminate between all possible earnings measures. However, wesuggest that unlike the FO approach which provides no discrimination, our analytical approachis amenable to testing the satisfaction of additional well-speci¯ed axiomatic requirements andso o®ers the ability to re¯ne the number of candidate earnings numbers that satisfy a chosen
set of axioms1.
In addition to providing a theoretical means to discriminate between alternative earningsmeasures, our approach also contributes to empirical issues. In particular, since our approach isbased upon multi-period optimization, we are able to derive comparative statics results whichexplain in a constructive way, why for instance, earnings response is non linear. In particularwe show why for ¯rms that are expanding aggressively, the earnings response coe±cient may bequite low. Since our model is based upon optimizing behaviour we believe we may o®er a superiorexplanation for the role of earnings in estimating future value in such settings than thoseresearchers who simply conclude that a low earnings response coe±cient may be interpreted1We believe this to be of some importance at this time, given the active debate concerning the overalldesirability of comprehensive and other earnings measures.as evidence of the lack of usefulness of earnings numbers and rush to explore the explanatorypower of non ¯nancial performance measures.
1.1 Real Options and the Feltham-Ohlson ModelIn our model management need to evaluate real options embedded within typical investmentdecisions. We review an established model in section two which derives the q-theory of investmentin such a setting. In section three we introduce a new investment model in whichreal options naturally arise and can be solved for optimally. This analysis allows us to makeprecise statements about expected ¯rm future value and leads us naturally to think about analternative measure of income based upon q-theory. In section four we then consider how an
investor could utilize alternative income measures to forecast future ¯rm value. We show thatestimates based on residual income are subject to `hysteresis e®ects', and expected future ¯rm
value can take multiple values for a given reported residual income number. We then showthat our proposed income measure, q-theory pro¯t, is not subject to this same problem. Wesubsequently show how residual income can be shown to be equivalent to q-theory pro¯t in arestrictive setting. We present concluding remarks in section ¯ve.#p#分頁標題#e#
We also note that there exists a number of review papers of the FO (Ohlson (1995) andFeltham and Ohlson (1995)) approach, such as Lo and Lys (2000) and Walker (1997), whichthoroughly review the model and provide critiques of the approach. However, having subjectedthe model to a critique, those papers do not provide constructive alternative valuation approaches.
In contrast we try to mount a constructive response to the identi¯ed limitations ofthe FO approach by developing a new model designed to overcome the lack of a well-de¯nedfunction for management with respect to project selection in FO. In the following sections wederive a valuation model in which management has a role to play via real options in project
selection2.
The FO model is normally developed by ¯rst recalling a well-known transformation of thetraditional discounted future dividend valuation model:
1X
¿=1
°¿Et(dt+¿ ): (1)
at date t; where dt = dividends paid at the end of each period t; ° = (1 + r)¡1 the discountrate and Et = the expectations operator. Before considering the transformation, there aretwo natural interpretations of (1). The ¯rst has expectations computed using an equivalentmartingale measure for the equity price (a modelling assumption is that such exists on the2To the best of our knowledge only two other authors consider a similar modelling approach. Yee (2000)also incorporates project selection but in a very di®erent way from our model. In Yee ¯rms facing poor returnscan switch out of existing projects as other exogenous projects are available. By contrast, in our model weare concerned with the expansion and contraction path of an investment in place, that is, the ¯rm does not
completely abandon a project when things are bad, they ¯rst need to manage a contraction or later expansionon an ongoing basis. The other paper, much closer to ours in spirit, is Zhang (2000) which is discussed at theend of the subsection.
3grounds of no arbitrage opportunities), and then the discount rate r is interpreted as theriskless rate. Alternatively, if the returns on equity Wt are modelled as independently and
identically distributed (i.i.d.; assuming such a belief on the part of investors), then the physicalprobability for the distribution of equity price may be used as an equivalent procedure, in whichcase the discount rate becomes the constant expected rate of return, and that of necessity isset equal to the `required rate of return' for the given class of risk. Our model is based on the
latter premise; that is to say, the model assumes that management control economic activitiesso that expected return is set equal to the `required rate of return'. The precise signi¯cance ofthis rule is studied in later sections, and involves recognition of elements of irreversibility. Thestudy of such settings through identi¯cation of embedded investment call and put options isstandard in the real-options approach to investment.Equation (1) requires a technical assumption3. From this equation, and also subject to asimilar kind of technicality4, appealing to the clean surplus identity#p#分頁標題#e#
Bt = Bt¡1 + yt ¡ dt (2)
(where Bt = book value of equity at t; yt = earnings at the end of period t ) leads to the
residual income according to the identity:
St = Equity value at time t = Bt +
1X
¿=1
°¿Et(eyt+¿ ); (3)
where residual income, or `abnormal earnings' as it is alternatively called, is de¯ned by
eyt ´ yt ¡ rBt¡1:
The most attractive feature of this approach is that it links valuation to observable accounting
data. The ability to re-express (1) in a way that gives accounting centre stage via (3) has been
well-known for a considerable time. Ohlson's particular contribution was to set out a speci¯c
proposal for how eyt+¿ evolves. In particular he posited that
eyt+1 = !eyt + xt + "t+1; (4)
where 0 · !; xt = value relevant information not yet captured by accounting and "t+1 is a
zero-mean disturbance term. In turn he assumed
xt+1 = gxt + ´t+1; (5)
where g < 1 and ´t+1 is a zero-mean disturbance term. Together (4) and (5) imply that abnormal
earnings follow an AR(1) process. It is apparent immediately that the Ohlson approach
3The `no bursting bubble' assumption °¿E[W¿ ] ! 0 as ¿ ! 1 is required here.
4Namely: °¿B¿ ! 0 as ¿ ! 1; i.e. book value does not grow faster than the riskless or required rate of
return (whichever is appropriate).
4
presents an opaque model of management, since nowhere does the Ohlson model consider managerial
project selection or opportunities. Similarly the Feltham-Ohlson (FO) extension, which
allows for conservative accruals, is silent with respect to project opportunities and the real options
that these create. Thus, while the FO approach does establish a dependence of abnormal
earnings on book value, it does so via a simple (decision opaque) mechanistic formulation. Lo
and Lys (2000) pick up this point and comment in detail on links with the Gordon dividend
growth model, pointing out that the assumption of an AR(1) process, although perhaps viewed
initially as quite benign, implies very real restrictions on the economic settings in which the
FO model can justi¯ably be applied.
Remark 1: The Feltham - Ohlson model is not well suited to applications where
¯rms adopt °exible investment strategies. One of our principal objectives is to
derive an alternative model framework which puts at center stage a valuation model
based upon ¯rm's period-by-period observed decision on whether or not to expand,
contract or maintain investment.
That is, a signi¯cant limitation of the FO approach is that it is essentially a static strategic
theory of investment in which once management make an investment they implicitly ignore the
type of strategic new investments and divestments opportunities that typically characterize the#p#分頁標題#e#
rich empirical setting in which investment decisions are taken in practice. A central part of our
model will be to identify a ¯rm's optimal dynamic investment strategy. That is, in our model
we will consider how management dynamically adjust their investment strategy in response to
time-varying stochastic conditions. We suggest that our model provides a more natural bridge
upon which to structure empirical observations of ¯rms that routinely switch from contracting,
shutting down, maintaining or expanding investment projects.
Remark 2: We show that an alternative accounting measure also provides an equivalence
to valuation resulting from discounting dividend streams via (1), and furthermore
that this alternative measure has a `desirable' feature.
Furthermore, we shall later argue that because of the decision-opaque nature of the FO approach,
it is under-speci¯ed in terms of what role-informational asymmetries are being assumed,
if any. When the possibility for asymmetries is allowed for, we then suggest one imposes5 a
regularity requirement which provides a simple test for what seems to be a `reasonable property'
for an accruals system, namely, that when using an income measure to predict future ¯rm
value there exist a functional relationship between the two. We show the FO residual income
model may fail this test, and so fails to be a `satisfactory measure' upon which to condition
forecasts of future ¯rm value. Again anticipating an argument that will be made more formally
5In later sections we shall provide a preliminary consideration of the issue of what constitutes a \good"
accounting accruals measurement system. At this early stage we are just highlighting that our methodology
can at least lead to some discrimination between alternative accruals processes unlike FO. We stress that at
this early stage we are not claiming to be in a position to identify optimality of accruals measurement, simply
that we can provide a partial ranking unlike the total inability to provide rankings under the FO approach.
5
in subsequent sections, this arises because we can show how the FO measure is subject to \hysteresis
e®ects". Speci¯cally, we show that given the same level of FO residual income eyt for two
¯rms, the prediction of optimal future ¯rm value must be conditioned upon whether the ¯rm
is expanding or contracting its investment set. That is, if one ¯rm is expanding while the other
is contracting, even though the residual income ¯gures are identical6, our theory predicts that
di®erent valuations be attached to the respective ¯rms. Put di®erently, simple linear extrapolation
of future ¯rm value based upon current residual income omits important features central
to characterizing the empirical nature of ¯rms' investment settings.#p#分頁標題#e#
The approach of Zhang (2000) also considers how to revise the FO approach to include real
option e®ects. In that respect the initial starting point of his approach and ours is identical.
However, the Zhang model is essentially a one shot model in which ¯rms only ever once decide
whether to expand, maintain or contract investment7. That is, after the one time decision
they are locked into that decision ad in¯nitum. In contrast, our model is dynamic in the sense
that for instance in three successive periods a ¯rm may expand, contract and then maintain
investment. On the surface one may at ¯rst believe that the Zhang approach, although o®ering
a simpli¯cation, may be able to capture most of the essential pertinent features of investment
behaviour. However, since the model is essentially one shot, empirical issues of coping with
over- or under-investment in the previous periods are not captured, that is, the Zhang model
is not history dependent. We develop a model that is history dependent in the sense that
we introduce an additional variable, opening capital stock, use of which management need to
optimise given stochastic input prices. In contrast the Zhang approach depends only upon a
stochastic e±ciency factor (which partly mirrors our price variable) while capital stock levels
change according to a simple exogenous assumption. Thus at its simplest our model is a two
variable investment model (a stochastic price or e±ciency parameter, and a history dependent
opening investment stock parameter) whereas the Zhang model considers only the ¯rst variable.
In terms of empirical implications our model potentially provides an explanation for why two
¯rms which, according to the Zhang model, would both expand investment may be seen to
adopt di®ering maintenance and expansion strategies respectively given that one of them had
\over-invested" in the previous period. That is, our approach allows a richer empirical model
to be ¯tted to data8 in which capital stocks, as well as e±ciency (or price variability), have an
important explanatory e®ect.
In order to give an initial °avour9 of our approach, we will introduce a simple two-period
6The informal intuition is as follows. Two ¯rms could have the same residual income, with one ¯rm making
high revenues and expanding and purchasing signi¯cant additional amounts of capital, while the other ¯rm has
only intermediate level revenues but can achieve the same overall pro¯t ¯gure by contracting and running down
capital stocks.
7Zhang (2000) makes this point clearly in the text arguing that the assumptions are made to insure tractability.
Hence one of our contributions is to maintain tractability for a more realistic investment setting in which
¯rms vary their investment startegies through time.#p#分頁標題#e#
8Another important di®erence between our approaches is that rather than our focus upon dynamic optimization,
Zhang's focus is upon the links bewteen valuation and `arbitrarily' biased accounting numbers.
9Although the di®erence presented in the subsection below may be considered by some readers as small,
we actually introduce a far more signi¯cant change in emphasis on income measures away from the traditional
6
model which illustrates how we choose to account for values in our general model setting.
1.2 An Example of Equivalence with an Alternative Measure of
Residual Income
We motivate our discussion by a simple two period model10. The returns technology is assumed
to follow a simple square-root formulation so that period pro¯t from applying x units of capital
into production gives the ¯rm a return of 2px: From this the purchase cost of the capital
px needs to be deducted in order to determine pro¯t. We shall assume that the ¯rm expects
the input price of capital to rise before the next period in which another production decision is
taken and the ¯rm actually chooses11 to commence with x+u units of capital at t = 0 purchased
at p0 a unit12. The ¯rm plans to use x of the units in the ¯rst period and u of the units in the
second period with the square-root returns function operating in both periods. Thus:
opening net assets B0 = p0(u + x):
We compute the two periods' respective earnings and residual incomes under the historic
cost convention as:
B1 = 2px + p0u B2 = (1 + r)2px + 2pu
B1 ¡ B0 = y1 B2 ¡ B1 = y2
y1 = 2px ¡ p0x y2 = 2pu ¡ p0u + 2pxr
ey1 = y1 ¡ rp0(u + x) ey2 = y2 ¡ r(2px + p0u)
ey1 = 2px ¡ (1 + r)p0x ¡ rp0u ey2 = 2pu ¡ (1 + r)p0u:
Note that the revenue 2px included in B1 is assumed to arise at the end of the ¯rst period
(i.e. time t = 1) for discounting purposes. Since we will want to show valuation equivalence
with another method of calculating residual income, we note that under the above historic cost
assumptions the value of the ¯rm at time t = 0 is given by opening book value plus the sum of
discounted (historical) residual incomes:
B0 +
ey1
1 + r
+
ey2
(1 + r)2
residual income focus in sections three and four.
10This initial model is presented for paedagogic purposes. Many of the most interesting dynamic features are
absent so as to ¯rst alert the reader's attention to pure accounting valuation issues before formally considering
the investment optimality dynamics, which complicate the analysis, but adds important empirical richness to
the setting.
11Clearly one of the tasks of subsequent sections will be to show, when this is optimal and when it is not,
to identify the optimal policy. The intuition here is that given the future value of the stock is expected to#p#分頁標題#e#
increase, the fact that the price is stochastic, means there is an economic value associated with not committing
to purchase all resource needs in advance. That is the fact that prices could fall as well as rise leads to some
value of waiting.
12Assume this is ¯nanced by the owners initial equity investment.
7
= p0(u + x) +
2px ¡ (1 + r)p0x ¡ rp0u
1 + r
+
2pu ¡ (1 + r)p0u
(1 + r)2
= p0u +
2px ¡ rp0u
1 + r
+
2pu ¡ (1 + r)p0u
(1 + r)2
=
2px
1 + r
+
2pu
(1 + r)2 :
Finally the key thing to note from this simple example is that during intermediate periods
(e.g. t = 1), calculating residual income requires one to keep track of both investment stock
used up in the period (x) and investment stock carried forward (u) for future use in some other
period, that is:
ey1 = 2px ¡ (1 + r)p0x ¡ rp0u; ey2 = 2pu ¡ (1 + r)p0u: (6)
Now in contrast, rather than track historic-cost accounting income, as in the F-O framework,
we shall instead track current-value accounting income adding an adjustment for per-period
holding gains denoted HG (we thus include both realized and unrealized gains). That is, we
shall assume that any physical stock valued at u which remains unused during a period is valued
at u(1+r) at the end, just as with any (banked) cash receipts generated in the previous period.
Thus let us de¯ne current value accounting income that incorporates holding gains as:
yCV
t = (Bt + HGt) ¡ (Bt¡1 + HGt¡1) + dt (7)
= BCV
t ¡ BCV
t¡1 + dt (8)
where
eyCV
t = yCV
t ¡ rBCV
t¡1 and BCV
t = Bt + HGt:
For our setting above, the current-value accounting values are given by:
HG1 = r:p0u HG2 = 2rpx
BCV
1 = 2px + p0u(1 + r) BCV
2 = (1 + r)2px + 2pu
BCV
1 ¡ BCV
0 = yCV
1 BCV
2 ¡ B1 = yCV
2
yCV
1 = 2px ¡ p0x + p0ur yCV
2 = 2pu ¡ p0u(1 + r) + 2pxr
eyCV
1 = yCV
1 ¡ rp0(u + x) eyCV
2 = yCV
2 ¡ r(2px + p0u(1 + r))
= 2px ¡ (1 + r)p0x = 2pu ¡ (1 + r)2p0u:
Next we note that, under the above current value cost assumptions, the value of the ¯rm at
time t = 0 is given by opening book value plus the sum of discounted (current-value) residual
incomes, which is identical to the above valuation with pure historic costs:
B0 +
eyCV
1
1 + r
+
eyCV
2
(1 + r)2
8
= p0(u + x) +
2px ¡ (1 + r)p0x
1 + r
+
2pu ¡ (1 + r)2p0u
(1 + r)2
=
2px
1 + r
+
2pu
(1 + r)2
= B0 +
ey1
1 + r
+#p#分頁標題#e#
ey2
(1 + r)2 ;
and thus from an investor-valuation perspective at t = 0 the two methods are equivalent.
However, look at the two current-value residual incomes:
eyCV
1 = 2px ¡ (1 + r)p0x; eyCV
2 = 2pu ¡ (1 + r)2p0u:
Letting
bt = (1 + r)ptx;
we see immediately that the current value residual incomes can simply be written as
eyCV
1 = 2px ¡ b0x; eyCV
2 = 2pu ¡ b1u; (9)
and hence unused stock in each period does not need to be included in the determination of
current-value residual income as is the case in (6). It is important to recognize these
expressions naturally lead to use of replacement-cost accounting. That is, given that
we wish to consider whether intermediate-period residual income is useful for predicting future
¯rm value, we shall ¯nd it simpler to characterize current value residual incomes as illustrated
in (9).
Remark 3: Like the FO traditional historic cost residual income measure, our
current-value residual income measure is equivalent to the discounted dividend
stream.
Having shown an alternative decomposition of accounting income, we next return to the issue
of the AR(1) process that FO employ. The reason why FO make this assumption in their model
is because they need some method to predict how residual income is generated. In contrast to
their mechanistic formalization, we assume that residual income results explicitly from ¯rmbased
microeconomic optimization. In the dynamic investment setting that we consider here,
this corresponds to a requirement of solving for the optimal value function of the ¯rm,
which when added to book-value at any point in time, following a stochastic realization of
a parameter, provides the appropriate valuation of the ¯rm conditional upon optimal
decision making13. Thus, provided we can solve for the optimal value function, we can critically
appraise the question concerning how well an accounting measure, such as residual income,
performs at predicting ¯rm value. Indeed, one can directly refer to the relationship between the
accounting-based measure and the optimal value function.
13As with the earlier discusion in this section we are trying to maintain an element of intuitive informality
before subsequently introducing formal technical arguments.
9
Given that the identi¯cation of the optimal value function underpins our analysis, the
following two sections are concerned with developing the optimization procedures required to
determine the optimal value function. Section 2 presents a selected overview of a well-known
general model which explains most succinctly why the implicit optimization of traditional static
investment analyses, such as that of FO, is found to be de¯cient. The model shows that since the
call and put options embedded in investment expansion and contraction options are omitted,#p#分頁標題#e#
these traditional approaches do not form the basis for identi¯cation of optimal investment
decision making.
Remark 4: Attempting to show empirically how FO residual income relates to
expected ¯rm value can be misguided because if managers actually used FO residual
income to rank projects, this would imply an element of sub-optimization on the
part of managers.
We now turn to consider how to characterize optimal (dynamic) investment behavior.
2 The Real-Options Approach to Investment Valuation
We commence our discussion of the real options approach by brie°y reviewing the work of Abel,
Dixit, Eberley and Pindyck (1996) -hereinafter referred to as ADEP - which presents an easily
accessible introduction to the literature and clearly demonstrates the above-outlined limitation
with the FO model. After setting out the ADEP model we discuss various extensions which
lead in a natural way to the speci¯cation of our alternative model.
In a simple two-period setting the model considers the problem of whether a ¯rm should
add to or reduce its opening (¯rst-period) stock of capital K0 which is purchased at a unit price
of b0: This is to be determined given the following three complications: the future (period one)
purchase price of capital bH may exceed its current price (costly expandability; bH > b0); the
future resale price of capital bL may be less than its current price (costly reversibility; bL < b0)
and ¯nally second-period revenues from employing capital are stochastic. The stochastic element
is introduced as follows14. In the ¯rst period total revenue from installed capital is r(K0);
in the second period the revenue, denoted R(K; a); has a stochastic component determined by
the realization of a: Subsequently in the second period after a has been revealed the ¯rm adjusts
the capital stock to a new optimal level denoted K1(a): Di®erentiating the revenue function
with respect to K; the following two critical values of a are identi¯ed:
RK(K0; aL) ´ bL and RK(K0; aH) ´ bH:
That is, the optimal (marginal) decision rule is:
- when a < aL it is optimal to sell capital to the point that RK(K1; a) = bL;
- when aL · a · aH it is optimal to neither purchase nor sell capital, that is K1(a) = K0;
14For brevity we are not including details of all the regularity conditions since they can be found in the original
text.
10
- when a > aH it is optimal to purchase capital until RK(K1; a) = bH;
and so the present value of net cash °ows V (K0) accruing to the ¯rm commencing with capital
stock K0 in period zero with inter period discount rate °; is given by
V (K0) = r(K0) + °
Z aL
¡1fR(K1(a); a) + bL[K0 ¡ K1(a)]gdF(a) (10)
+°
Z aH
aL
R(K0; a)dF(a) + °#p#分頁標題#e#
Z 1
aH fR(K1(a); a) ¡ bH[K1(a) ¡ K0]gdF(a):
Thus the period-one decision faced by the ¯rm is
K0 = argmax V (K0) ¡ b0K0;
and the Net Present Value Rule can be interpreted from the ¯rst-order condition as requiring
V 0(K0) ´ r0(K0) + °bLF(aL) + °
Z aH
aL
R0(K0; a)dF(a) + °bH[1 ¡ F(aH)] (11)
= b0:
This equates the period-one and onwards marginal return to capital to the initial marginal cost;
note that the terms after r0(K0) which take into account the optimal change in capital stock
in the following period. An alternative interpretation is also available. ADEP point out that
equation (11) can be interpreted using Tobin's q-theory of the marginal value of capital. In this
instance the marginal value of capital is
q ´ V 0(K0);
and so the optimal investment rule can be identi¯ed by management if they determine q.
With respect to implementing this rule ADEP (p 761) comment that this (theoretically
correct) rule can be di±cult to apply in practice because \for a manager contemplating adding
a unit of capital, it requires rational expectations of the path of the ¯rm's marginal return to
capital through the inde¯nite future" and thus in practice the most commonly used proxy for
the correct NPV \treats the marginal unit of capital installed in period 1 as if the capital stock
is not going to change again". In this case the marginal value of V 0(K0) is approximated by:
e V 0(K0) ´ r0(K0) + °
Z 1
¡1
RK(K0; a)dF(a); (12)
and ADEP describe this replacement for the left-hand side of (11) as yielding the naive NPV
rule. At this point it is very helpful to note that the di®erence between e V 0(K0) and V 0(K0) is
given precisely by the embedded put and call options present in the problem. To see this we
can rewrite (10) as
V (K1) = r(K1) + °
Z 1
¡1
R(K0; a)dF(a) (13)
+°
Z aL
¡1f[R(K1(a); a) ¡ bLK1(a)] ¡ [R(K0; a) ¡ bLK0]gdF(a)
+°
Z 1
aH f[R(K1(a); a) ¡ bHK1(a)] ¡ [R(K0; a) ¡ bHK0]gdF(a);
11
or more succinctly as
V (K0) = e V (K0) + °P(K0) ¡ °C(K0); (14)
where
e V (K0) ´ r(K0) + °
Z 1
¡1
R(K0; a)dF(a);
P(K0) ´
Z aL
¡1f[R(K1(a); a) ¡ bLK1(a)] ¡ [R(K0; a) ¡ bLK0]gdF(a);
C(K0) ´
Z 1
aH f¡[R(K1(a); a) ¡ bHK1(a)] + [R(K0; a) ¡ bHK0]gdF(a);
here e V (K0) is the expected present value over both periods keeping the capital stock ¯xed at
K0, i.e. not allowing expansion or contraction of the capital stock. Now
P0(K0) =
Z aL
¡1fbL ¡ R0(K0; a)gdF(a) = E[maxfbL ¡ R0(K0); 0g]:#p#分頁標題#e#
is the value of a (marginal) put15 on the marginal product of capital with exercise price bL
corresponding to selling back. Similarly C0(K0) is the value of a (marginal) call on the marginal
product of capital with exercise price bH:
C0(K0) =
Z aL
¡1f¡bH + R0(K0; a)gdF(a) = E[maxfR0(K0) ¡ bH; 0g]
Thus, given (11), to capture the incentives to invest and divest we can decompose the marginal
value into three components:
q = V 0(K0) = e V 0(K0) + °P0(K0) ¡ °C0(K0):
Notice that the present value of expansion requires additional outlay (hence the negative term),
whereas contraction generates additional income (hence the positive term).
To summarize, in the ¯rst period optimality requires management to choose K0 so that
e V 0(K0) = b0 ¡ °P0(K0) + °C0(K0): (15)
That is, under the naive rule in which management set e V 0(K0) = b0, management are
ignoring (strategic) option values to contract or expand in the second period and hence typically
would choose K1 suboptimally.
15The put corresponds to the option to reduce the capital stock K1 by selling k of the existing stock at bL
whenever a < aL: Thus the realized value of the ¯rm when the realization a is below aL is to ¯rst order
r(K1) + °(R(K1 ¡ k) ¡ R(K1) + bLk)
= r(K1) + °k(bL ¡ R0(K1)):
12
Moreover it is straightforward to show16 that the FO model is an implementation of the naive
investment rule which ignores the options to expand and contract available in most real-options
settings and hence accounting valuation theory based upon that approach is unlikely to be
able to capture how accounting valuation impinges upon the ¯rm's actual dynamic investment
strategy (including both expansion and contraction possibilities).
The objective of the next section is to develop a simple model which overcomes this de¯ciency
in that management formally need to evaluate options to expand and contract each period and
moreover it extends the two period ADEP model to more realistic investment horizons of N > 2
¯nite periods17. After setting out the revised ¯nite-horizon investment model, we then return
to consider accounting valuation issues in the following section.
3 Optimal Investment by Management: An Endogenous
Regime-Switching Model of Investment
Our model speci¯cation is somewhat di®erent from that of ADEP. Before concentrating on the
di®ering interpretation over speci¯c variables it is important to establish from the outset that
our general methodological goal is also di®erent. Whereas ADEP were able to identify general
statements concerning the conditions that optimal investment strategy should satisfy and how
that leads one naturally to consider embedded put and call options, they did not actually#p#分頁標題#e#
characterize the functional form for the rewards from adopting an optimal investment strategy.
That is, their analysis is not of direct use when trying to assess whether an accounting measure
does, or does not, allow users to predict (optimal) future ¯rm value. We depart from their
approach by introducing speci¯c functional forms to characterize the basic investment setting
with the hope of being able to identify how optimal future ¯rm value depends parametrically
upon decision variables that management face.
The following quite technical section shows that within our model speci¯cation we can in
fact identify future ¯rm value as the optimal value function for the dynamic investment strategy
adopted by management and that this takes a quite intuitive form18.
Remark 5: In our model setting, future ¯rm value V () is given by the sum of
expected future period-by-period (optimized) indirect pro¯ts, plus the valuation
of the existing stock of investment at its expected marginal value, which is the
q-theoretic income.
Recalling the original Ohlson motivation for introducing an AR(1) process as a means for
16See Lo and Lys (2000). The FO approach simply assumes constant expansion (as in the Gordan growth
model) rather than period-by-period expansion or contraction as will be allowed for in the model developed
below.
17This is not the only di®erence between the two models. As we shall see in the following section there are a
number of other di®erences, the most signi¯cant perhaps being that, in our model setting, depreciation occurs
through use rather than at a constant rate, or alternatively not at all as in the ADEP model.
18The precise statement is given towards the end of this section by equation (31).
13
dealing with the need to model how expectations evolve, it may at ¯rst seem that we too are
now in exactly the same situation - needing to impose a model of how expectations, albeit of
future ¯rm pro¯tability rather than residual income, evolve over time. Appreciation of how
we respond to this point provides the critical conceptual distinction between our approach and
that of FO. In particular, working with the indirect pro¯t function19 we are able to show in this
section how the period t (indirect) pro¯t is functionally determined by the most recent observed
investment input price bt. That is, we show that when attempting to form expectations upon
future values of the indirect pro¯t function, this requires expectations to be formed over how
the stochastic input price bt evolves. We state our assumption formally in equation (16) below.
So have we simply replaced the FO, AR(1) assumption just with some other equally restrictive
assumption? We would argue not, for the following reasons. Our distributional assumption is#p#分頁標題#e#
imposed upon an input price process which arises before any managerial action is taken. This
is in contrast to Ohlson, who imposes a distributional assumption directly on the evolutionary
path of residual income, and hence - as we have seen earlier - this imposes very real constraints
upon the implied investment settings where this could logically be assumed to have followed
from rational managerial behavior. Expressed alternatively, we would argue that it is less
restrictive to impose a distributional assumption on an input than it is to impose one upon an
output that results from managerial actions being applied to inputs. To summarize, it is our
contention that the necessary distributional assumption that needs to be applied to compute
expectations in any model of future ¯rm value, is applied at too late a stage in the model
of managerial behavior in the Ohlson approach. Applying the distributional assumption to
expected residual income necessarily restricts attention to only a subset of real-world decision
scenarios that management may face in practice. For instance, as our earlier discussion makes
clear, the FO model simply does not apply in a setting where a ¯rm has good and bad years.
By contrast, in our model the `good' or `bad' realizations of the stochastic input price are at
centre stage and the evaluation of the induced management's performance is e®ectively in terms
of an assessment of their ability to exercise correctly the embedded growth, maintenance and
or contraction options that come `into the money'.
Having outlined methodologically what we want to achieve in general terms, let us now turn
to the detailed speci¯cation. However, just before doing so, we draw the reader's attention to
the fact that there exists a di®erence in our model and that of ADEP in the way in which capital
is utilized. In particular we develop a model of (installed) capital in which capital depreciates
through use (as directed by management), rather than at a constant rate, or not at all, as in
the ADEP model. We make this assumption to allow for the possibility that the net book value
of an investment asset after subtracting accumulated depreciation could in principle be equal
to the economic value of the asset to the organization. In contrast in the ADEP framework, the
asset is assumed never to depreciate. In addition, we extend the investment planning horizon
beyond a simple two-period framework to a general ¯nite-horizon setting. In order to introduce
the di®erence in speci¯cation as transparently as possible, we ¯rst consider a two-period model
variant of the ADEP model.
19See for instance Varian (1992) for a discussion of the use of the indirect pro¯t function.
14
3.1 The two-period model
In reality, ¯rm investment is subject to multiple sources of uncertainty. In the ADEP model,#p#分頁標題#e#
the source of uncertainty is the price of ¯nished output. By contrast, in our model we focus
upon the input price of capital as the principle source of uncertainty20. Our objective here
will be to characterize V (K0); the optimal value function for capital usage. As we shall see,
by making certain functional assumptions for the operating environment, we will be able to
go further than ADEP, since not only can we identify equivalent optimality conditions to (14),
but moreover we can solve for the conditions once we have derived the functional form for the
optimal value function V (K0):
We now develop our model via direct comparison to the ADEP approach. Simplifying the
output-return side21 we take the time t = 0 revenue to be r(K) = 2pK and the time t = 1
revenue to be R(K; a1) = 2a1pK; where a1 > 0 represents the unit sale price of the output at
time t = 1: To further simplify the analysis, since in our model the input price is the prime
source of uncertainty, we shall take a1 = 1. Concentrating upon the source of uncertainty, we
shall allow b1; the input purchase price of capital at time t = 1 (corresponding to the constant
bH considered by ADEP), to be stochastic. In addition, we assume that the resale price of
the input is b1Á1 at time t = 1 (instead of bL in ADEP notation), where the discount factor
Á1 < 1 re°ects the partial irreversibility of earlier investment. For clarity of exposition, Á1 is
deterministic in this model, but the model can be adapted to allow Á1 to be stochastic. The
fractional value of Á1 is assumed to result from the input not being freely tradeable, and this
creates a fundamental incompleteness in the specialist capital-input market. This has important
implications for the valuation of the ¯rm; the assumptions of the standard martingale approach
in real-option theory posit the existence of a `traded twin security' perfectly correlated with the
real asset. In our case the real asset is the additional capital, for which the purchase and sale
prices diverge at time t = 1 by the factor Á1; so that it is no longer possible to hold long and
short positions at one price. Furthermore, the `input asset' most de¯nitely has a `convenience
yield' on account of its productive value - it is not held purely for trade. We therefore abandon
the simple martingale approach22, and instead adopt the standard `private values' dynamic
programming approach23 for valuation using the physical distribution of the input price b1:
20Our focus here is with capital input hedging possibilities that may exist. For instance see Hopp and
Nair (1991). A generalised version of our model in which both the input price and the output price are
stochastic is available from the authors. The two sources of uncertainty complicate the analysis by requiring
consideration be centered around the ratio of output to the input price without changing the general nature of#p#分頁標題#e#
results substantively.
21In general, we need not restrict attention to a square-root formulation: all we need is concavity. The role
of the square root speci¯cation is to maximize the simplicity of the presentation. The reader should be warned,
however, that the Cobb-Douglas revenue function can generate an arbitrarily large return, albeit only for small
enough inputs. In general, a revenue function would exhibit a bounded return as input vanishes.
22A related situation is that of a four-state model in which prices of a traded asset move up or down and
an investor receives a partially correlated preference shock to buy, sell (or even hold). This single-risky-asset
model is evidently incomplete but presents two obvious martingales, one for `expansion' and one for `contraction'
corresponding to an interpretation of the appropriate buy:sell ratio of the four-state model as a resale discount.
23See Dixit and Pindyck (1994) for an extended discusssion of this point.
15
This is the approach also taken by Abel and Eberley (1995) in their continuous-time in¯nite
horizon model.
Commencing at time t0; we assume that a ¯rm has u0 = ut0 (ut0 ¸ 0) units of capital in
stock24. Given that the ¯rm can purchase some more capital in the next period, the decision of
how to allocate capital stock optimally between the current and latter period will ceteris paribus
be driven by the capital input price process. We shall denote the `one-period-appreciated' 25
price of capital by bt:
3.1.1 Price of inputs
Although in general we use a sequence of times and corresponding prices that evolve geometrically,
the price is nevertheless presented as though it evolves continuously as a geometric
Brownian motion. Such an approach is dictated purely by mathematical convenience; the
mathematics of optimization is much streamlined by the assumption that at each time, price
is distributed continuously rather than multinomially; the presence of interperiod prices is not
referred to in any way because we have periodic management decision making. The price bt
has positive drift (anticipated growth) ¹b > 0, and is presented in the traditional stochastic
di®erential form:
dbt = bt(¹bdt + ¾bdWb(t)); (16)
where Wb(t) is a standard Wiener process. It is assumed that °Á1e¹b < 1 and that °e¹b > 1;
i.e.
¹b + ln ° > 0 > ¹b + ln ° + ln Á1;
so that in particular per-period the expected rise in input price rises above the required return
on capital and the resale price drops below it. For t > s; we let Q(btjbs) denote the (log-normal)
cumulative distribution of bt given bs and we also let Qn(b) = Q(btnjbtn¡1 = 1) denote the (lognormal)
cumulative distribution of bn = btn given that btn¡1 = 1: When the context permits,#p#分頁標題#e#
we drop the subscript n:The development of the model depends on the multiplicative nature
prices - the distribution of the ratio bt+1=bt is independent of bt:
3.1.2 Optimal investment
http://www.ttc-switch.com/dissertation_writing/In the simplest model the manager observes the price at discrete times, in this case at times t0
and t1; and can purchase/resell capital at these discrete moments in amounts which we shall
denote z0 = zt0 and z1 = zt1 : In order to track the stock of capital carried forward between
periods we shall denote the period t0 opening capital stock as vt0, or just v0, and closing stock
24Note u0 in our notation corresponds to K0 in ADEP notation. We do not adopt their notation because of
the di®erent way in which capital is \consumed" in the two models.
25By `one-period-appreciated', we mean that if the asset is purchased for pn = ptn at the commencement of
the time interval [tn; tn+1), then the unit opportunity cost of funds tied up in the asset are pn(1+r) = bn; where
bn stands for btnand r is the one-period interest rate. Alternatively, one can regard the supplier as rationally
recognising that if payment for delivery from stock is to be delayed a period, then the price payable at the end
of the period needs to include the cost of funds tied up in inventory.
16
as ut0 , or just u0. Let us now consider how to determine the optimal amount of capital ut0 to
carry forward to the next period given the amount purchased in the period is unrestricted, so
that in this case zt0 ¸ 0:
The manager now needs to maximize over both z0 (¸ 0) and x0 the pro¯t26
2px0 ¡ b0z0 + °V0(v0 + z0 ¡ x0; b0):
Here V0(u0; b0) denotes the optimal future expected value given the current price b0 and the
capital stock carried forward u0 paid for in a previous period. (Thus V0 is an increasing concave
function). Equivalently, letting u0 = v0 + z0 ¡ x0 we maximize over x0 and u0
2px0 ¡ b0(u0 + x0 ¡ v0) + °V0(u0; b0): (17)
Then when choosing optimally the closing stock of capital u0 the ¯rst-order condition (if u0 > 0)
from (17) gives:
°V 0 0(u0; b0) = b0 (18)
and27
x0 =
1
b20
; (19)
where the prime denotes the derivative @V (u; b0)=@u . Note that (18) implies that for investment
u0 to be chosen optimally, the unit marginal return needs to be equated to the constant return
1 + r; i.e.
V00(u0; b0)
b0
= °¡1 = (1 + r);
and so the return on u0; namely [V0(u0; b0) ¡ b0u]=b0u; is greater28 than r:
3.1.3 Optimal divestment
A ¯rm planning to divest, i.e. taking z0 < 0; faces a similar problem. If the resale discount
is Á0 the ¯rm considers the corresponding problem: maximize over both z0 (< 0) and x0 the#p#分頁標題#e#
pro¯t29
2px0 ¡ Á0b0z0 + °V0(v0 + z0 ¡ x0; b0);
26That is choice of the variables to maximise the sum of current pro¯t plus the optimal value function V (:)
re°ecting future optimal period payo®s.
27This very simple nature of this result is why we utilise the square-root speci¯cation.
28To see this ¯x b > 0 to be any price at time t = 0 and r an interest rate. Let the non-negative, concave,
di®erentiable function g(u) represent a deterministic value receivable at time t = 1 and assume that
limu!1 g0(u) < b(1 + r): Let u¤ maximise the pro¯t g(u) ¡ b(1 + r)u: De¯ne the rate of return on g(u) to be
R(u); where 1 + R(u) = g(u)=(bu). Evidently if ¹u > 0 satis¯es g(¹u) = (1 + r)b¹u then u¤ < ¹u and R(¹u) = r:
Now the rate is decreasing for u < ¹u; indeed bR0(u) = ¡g#(u)=u2 and g#(u) is an increasing function (since
Dug#(u) = ¡g00(u) ¸ 0); but g#(¹u) = g(¹u) ¡ ¹ug0(¹u) = ¹u[b(1 + r) ¡ g0(¹u)] > 0: Hence R(u¤) > r: Notice that
R(0+) is either unbounded (if g(0) > 0) or g0(0+)=p:
29That Á < 1 is standard in the literature, otherwise if Á = 1 we would have the possibility of simple portfolio
rebalancing.
17
or equivalently, letting u0 = v0 + z0 ¡ x0 ,
2px0 ¡ Á0b0(u0 + x0 ¡ v0) + °V0(u0; b0):
Thus the ¯rst order condition for u0 (again assuming u0 > 0) is
°V00(u; b0) = Á0b0; (20)
and for x0 is
x0 =
1
Á20
b20
: (21)
3.1.4 Tobin's q and normalized inputs
We return to the investment version and let u = bu(b0) denote the solution to equation (18).
Remark 6: Formal identi¯cation of the two-period optimal value function shows it
is made up of three components conditioned upon whether the ¯rm is expanding,
maintaining or contracting investment.
We note that in our two-period model we have
V0(ujÁ1; b0) =
Z eb
L
0
(
1
b1
+ b1u)dQ(b1jb0) + 2pu
Z eb
L=Á1
eb
L
dQ(b1jb0) (22)
+
Z 1
eb
L=Á1
(
1
Á1b1
+ Á1b1u)dQ(b1jb0);
where Á1 is the resale rate for the second period and eb
L = 1=pu: The three integrals classify
investment by the corresponding three input price policy ranges30 according to the ranges of
integration, as follows:
(U) The under-invested range (0 · b1 · eb
1), in which additional investment in capital is made.
Here as in (17) one maximizes over x1 ¸ 0 the second-period pro¯t
2px1 ¡ b1x1
with required input x1 = 1=b21
made available through the purchase of x1 ¡ u at a price b1 and#p#分頁標題#e#
net revenue 2=b1 ¡ b1(x1 ¡ u) = 1=b1 + b1u: Clearly the extreme case is zero purchase when
u = 1=b21
; whence the limit of integration b1 =eb
L:
30Equivalent to the three output price ranges in the ADEP model.
18
(IO) The (endogenously) irreversible31 over-invested range (eb
L · b1 ·eb
L=Á1), where all remaining
capital (excess from period 0) is optimally applied between current and future production.
(RO) The reversible over-investment range (eb
1=Á1 · b1 · 1), where some excess capital is
resold. Here one maximizes over x1 · u the second-period pro¯t
2px1 + Á1b1(u ¡ x1)
obtained by reselling an amount u ¡ x1 of the capital stock. The required input is x1 =
1=(Á1b1)2; yielding net revenue 2=(Á1b1) + Á1b1(u ¡ x1) = 1=(Á1b1) + Á1b1u: The extreme case
is u = 1=(Á1b1)2; giving the limit of integration b1 = 1=(puÁ1): Thus
V00(u; b0; Á1) =
Z eb
L
0
b1dQ(b1jb0) +eb
L
Z eb
L=Á1
eb
L
dQ(b1jb0) +
Z 1
eb
L=Á1
Á1b1dQ(b1jb0):
In general, the resale factor Á1 will not be known at time t0 and so one should take expectations
over Á1leading to an average version V¹00(u; b0) of V00(u; b0; Á1): For presentational purposes
we will usually avoid this additional expectation and pretend Á1 is deterministic.
A critical interpretation of the marginal value V00 of capital is now possible with reference to
Tobin's q: Consider a policy of investment triggered by input prices below a threshold level of
B: The average marginal bene¯t of such a strategy corresponds to the value of Tobin's marginal
q. This we may compute from the last formula by writing B in place of eb
L (so by implication
we are setting B = 1=pu); obtaining the function:
q(B; b0) =
Z B
0
b1dQ(b1jb0) + B
Z B=Á1
B
dQ(b1jb0) +
Z 1
B=Á1
Á1b1dQ(b1jb0):
At this point it is important to note that our assumption of a Cobb-Douglas type technology
gives rise to the following homogeneity property:
q(B; b0) = b0q(B=b0; 1);
which we will wish to apply. An inductive argument shows that this homogeneity property
extends to all periods in the context of a Cobb-Douglas production function (see Appendix D).
The function q0(B) =def q(B; 1) is of course Tobin's marginal quotient, q; namely, the
expected future return on an additional unit of capital measured in ratio to the market value
(replacement cost) of the additional capital. This motivates our notation. Indeed we have
lim
h!0
V0(u + h; b0) ¡ V0(u; b0)
hb0
=
1
b0#p#分頁標題#e#
V 0(u; b0) =
q(B; b0)
b0
= q0(1=(b0pu)):
31Endogenous in the sense that though reversal is possible, it is never optimal in this setting to choose it and
hence the ¯rm acts as if the situation was irreversible.
19
We note that the expression 1=(b0pu) is likewise a marginal quotient: f0(u)=b0;is it is the
marginal return of an investment u if it were currently consumed in production (`current' q
rather than future q).
For a further insight into this equation, observe an important second homogeneity property
(true here by inspection, but preserved also in a multi-period setting, as we show in Appendix
E), namely that:
V0(u; b0) =
1
b0
V0(ub20
; 1):
A parallel derivation of the marginal return on investment starts from the remark that
V0(u + h; b0) ¡ V0(u; b0)
b0h
=
V0((u + h)b20
; 1) ¡ V0(ub20
; 1)
b20
h
;
and so, if we put ~u = ub20
; we see that Tobin's marginal quotient is
lim
h!0
V0(u + h; b0) ¡ V0(u; b0)
b0h
= lim
~h
!0
V0(~u +~h; 1) ¡ V0(~u; 1)
~h
=
@V0(~u; 1)
@~u
:
The transformation B = 1=p~u (noted earlier) shows the latter quotient to be q0(B) = q0(1=p~u) =
q0(1=(b0pu):The change of variable used here, ~u = ub20
; is natural, as it arises from solving the
equation f0(~u) = f0(u)=b0 in which the input quantity u is scaled to ~u - and has the equivalent
marginal return corresponding to a unit input price. We will refer to ~u as a normalized
input quantity.32
The behavior of q(B; 1) is indicated in Figure 1: a strictly increasing function (a property
that is characteristic for multiple period models also)33.
Place Figure 1 Here
3.1.5 Censor equation
The importance of the function q0 stems from the induced decomposition of the solution of (18)
into two steps. The ¯rst step is to solve for eb
1
°q(eb1; b0) = b0; (23)
and the second is to solve eb
1 = eb
L = 1=pu for u; to obtain
bu(b0) = 1=(eb
1)2:
32In the general Cobb-Douglas case the transformation of a quantity v to its normalization is given by
~v = vb1=®
0 :
33We note that q00
(B) =
R B=Á1
B dQ(b1) > 0: A similar calculation is shown later for the multiperiod qn:
20
We call (23) the censor equation. The solution exists and is unique if and only if
infb1 q(b1; b0) < (1 + r)b0 < supb1 q(b1; b0); that is34,
E[Á1]E[b1jb0] < (1 + r)b0 < E[b1jb0];
and, since we have assumed for simplicity that Á1 is deterministic, this amounts to
Á1E[b1jb0] < (1 + r)b0 < E[b1jb0]:
This is a proviso that while (discounted) prices are expected to rise the resale price is nevertheless#p#分頁標題#e#
expected to fall35, a condition that is akin to absence of arbitrage opportunities. We
assume this to hold. We call the value of the price b1 given by eb
1, i.e. solving (23) above, the
censor. Clearly eb
1 is a function of ¹b; ¾; °:
It may be shown that
eb
1(b0) = b0^g1(1 + r); (24)
where ^g1 (the dynamic multiplier factor) is a function of ¹ = ¹b+ln ° and of ¾: The intuition for
this result may be traced to the fact that in our model the price b1 is log-normally distributed,
so that ln b1 has mean ln b0 + (¹b ¡ 1
2¾2); it thus makes sense to scale price b1 not only by b0
but also by the compounding factor (1 + r):
To see why the result is valid note that, since q(eb
1; b0) = b0q(eb
1=b0; 1); the censor equation
may be written in equivalent form as
°q(B; 1) = 1;
where B =eb
1=b0: Shifting the drift from ¹b to ¹ = ¹b+ln ° (which is positive, by the assumption
that °e¹b > 1) and letting the function corresponding to q(B; 1) for this drift be denoted by
~q(G; 1); we have36
q(B; 1) = (1 + r)~q(°B; 1):
Now we may simplify the equation to
1 = °q(B; 1) = (1 + r)°~q(°B; 1);
34If we assume the resale rate is independent of the sale price, then infB q(B; 1) = E[Á1]:
35For simplicity we assume that Á1 is independent of b1; as the inter-period is assumed to be unity, we have
E[b1jb0] = e¹ and the condition amounts to °e¹ > 1 > °E[Á1]:
36Proof: Writing b1 = (1 + r)g1; B = (1 + r)G and ~Q(g1) = Q((1 + r)g1j1) we have
q(B; 1) = q((1 + r)G; 1) =
Z G
0
(1 + r)g1d~Q(g1) + (1 + r)G
Z ÃB
B
d~Q(g1) +
Z 1
ÃB
Á1(1 + r)g1d~Q(g1)
= (1 + r)[
Z G
0
g1d~Q(g1) + G
Z ÃB
B
d~Q(g1) +
Z 1
ÃB
Á1g1d~Q(g1)]
= (1 + r)~q(G; 1) = (1 + r)~q(°B; 1):
21
and so the ¯nal form of the equivalent censor equation reads
~q(g; 1) = 1;
where g = °B = °eb
1=b0: If we denote the solution of this last equation by ^g1 then this quantity
is evidently a function of ¹ and ¾ and we have as claimed
eb
1 = b0^g1(1 + r):
Thus, in particular
bu(b0) =
°2
(bg1b0)2 : (25)
It is of interest to point out that there is a critical value of Á1 = Ácrit for which it is the case
that37 bg1(1 + r) = 1; and so bg1(1 + r) > 1 i® Á1 < Ácrit. In the case that bg1(1 + r) > 1 the
advance purchase is lower than the current demand.
The corresponding problem for divestment calls for the solution of
°q(eb
Á0 ; b0) = Á0b0;
°q(eb
Á0=b0; 1) = Á0; (26)#p#分頁標題#e#
and this will have a solution if and only if
Á0 > inf
B
°q(B; 1) = °E[Á1];
or, if the discount factor Á1 is assumed deterministic, exactly when
Á0 > inf
B
°q(B; 1) = °Á1:
The intuition is simple: if there is no solution, then there is no resale possible in that period38.
Here again we note that
Á0 = °q(eb
Á0=b0; 1) = (1 + r)°~q(°eb
Á0=b0; 1);
so that
eb
Á0 = (1 + r)b0^g1(Á0)Á0;
where ~q(Á0^g1(Á0); 1) = Á0:
37Regarding q as a fuction of Á1 we see that for B ¯xed q or ~q is increasing in Á1 as e.g. dq=dÁ1 = E[b1j1]:Note
that now for Á1 = 1 we have q(B; 1) = E[b1j1] and for Á1 = 0 we have the irreversible case for which evidently
it is the case that ^g1(1 + r) > 1:
38If we assume the resale rate is independent of the sale price, then infB q(B; 1) = E[Á1]:
22
3.1.6 The embedded options
Comparing (10) and (22), we can make the same re-arrangement as ADEP, to give
V0(u; Á1; b0) = 2
q
x(b0) + °[2pu
Z 1
0
dQ(b1jb0)
+
Z eb
L
0
(
1
b1
+ b1u ¡ 2pu)dQ(b1jb0) +
Z 1
eb
L=Á1
(
1
Á1b1
+ Á1b1u ¡ 2pu)dQ(b1jb0)]:
Thus re-de¯ning their notation - rather than introducing new notation (since we will not use
their representation again) - we have similarly to (14)
V0(u) = e V0(ujb0) ¡ °P(u;eb
1jb0) + °C(u;eb
1jb0);
where
e V0(ujb0) ´ 2
q
x(b0) + °2pu
Z 1
0
dQ(b1jb0);
P(u;eb
1jb0) ´
Z eb
L
0
2pu ¡ (
1
b1
+ b1u)dQ(b1jb0);
C(u;eb
1jb0) ´
Z 1
eb
L=Á1
(
1
Á1b1
+ Á1b1u ¡ 2pu)dQ(b1jb0);
with eb
L = 1=pu; and where, just as before, e V0(eb
1) is the expected present value over both
periods keeping the capital stock carried forward ¯xed at u: (Note that in view of the reciprocal
relation between the a and b variables, the put and call have switched roles vis μa vis ADEP.)
As before, looking at the ¯rst-order conditions39 we have, now writing eb
1 for eb
L,
V 0 0(ujÁ1; b0) =
Z eb
1
0
b1dQ(b1jb0) +eb
1
Z eb
1=Á1
eb1
dQ(b1jb0) +
+Á1
Z 1
eb
1=Á1
b1dQ(b1jb0)
= eb
1
Z 1
0
dQ(b1jb0) ¡
Z eb
1
0
(eb
1 ¡ b1)dQ(b1jb0)
+Á1
Z 1
eb
1=Á1#p#分頁標題#e#
(b1 ¡eb
1=Á1)dQ(b1jb0)
= eb1 ¡ E[max(eb
1 ¡ b1; 0)] + E[max(Á1b1 ¡eb
1; 0)] (27)
= e V 0 0(eb
1jb0)=° ¡ P0(eb
1jb0) + C0(eb
1jb0)
= q:
39With due consideration for the Leibniz Rule.
23
Comparison of (27) and (15) yields the key insight that the ¯rm should evaluate the embedded
investment call and put options with strike price given by the censor. In this respect the censor
eb
1 determines the e®ective `future' unit price (e®ective expected next-period price) of inputs,
and thus delivery at that price requires the planner to: (i) receive compensation / revenue
against that price for surrender of expansion potential, and (ii) pay additionally to that price
a compensation / cost for the right of contraction potential40.
Remark 7: The optimal investment rule is determined by evaluating the optimal
investment or divestment such that the marginal bene¯t of capital (Tobin's q) is
equal to the naive NPV together with the value of the marginal (short) put and
(long) call options which have a strike price given by the optimally chosen censor.
3.2 Generalizing to n > 2 Periods and an Alternative to Applying
Equivalence Between Residual Income and Discounted Dividends:
q-theoretic Pro¯t and Discounted Dividends
We now generalize the above simple two-period model for n > 2 and derive an alternative to
the FO residual income valuation equation. The equivalence (3) between discounted dividend
streams and residual income is only one of the possible equivalence relationships that could
be used to demonstrate a role for accounting values in predicting future value. One of our
contributions is to identify another equivalence relationship, namely (34) or (35), where residual
income ceases to be the main focus for valuation. As we shall see when we derive the functional
form for the optimal value function, it becomes natural to consider replacing residual income
by a measure of `indirect pro¯t', which can be interpreted as `optimal operating pro¯t before
40Alternative interpretation: The naive non-linear view is that one unit of capital next period will be wortheb
1
and leads to an inventory of 1=eb
21
but the marginal valuation ignores the present value of the option to expand
when it is cheap to do so (i.e. b1 <eb
1) and this will call for extra outlay (hence the negative sign of this PV)
and also ignores the option to contract when b1 >eb
1=Á so that it is worth selling for Áb1which brings in extra
income. It is possible to use put-call symmetry (parity) to obtain
F01
(u; Á; b0) =
Z eb1
0
b1q(b1jb0)db1 +eb
1
Z eb1=Á
eb1
q(b1jb0)db1 +
+Á
Z 1#p#分頁標題#e#
eb1=Á
b1q(b1jb0)db1
= E[b1] ¡
Z eb1=Á
eb1
(b1 ¡eb
1)q(b1jb0)db1 +
¡(1 ¡ Á)
Z 1
eb1=Á
b1q(b1jb0)db1:
This may be interpreted as comprising ¯rst the naive expected value of holding one unit of stock, secondly
short one limited call (operable in a limited range), and ¯nally (1¡Á) units short of an asset-or-nothing option.
24
extraordinary items', and which we call `q-income' as de¯ned below in this section. (See section
4.2 for its signi¯cance.) The future value is then a discounted sum of the future periods'
`q-income'.
Remark 8: Our model of optimal investment choice by management requires consideration
of the ¯rm's indirect pro¯t which within this setting we describe as
`normal operating pro¯t' regarded as optimal pro¯t before extraordinary items41.
In the next section we will compare the future-value prediction algorithms based upon our
q-theoretic operating pro¯t measure, to those based upon residual income. Let us now turn to
introduce the new equivalence result.
We adopt the following notational assumption in order to minimize the use of subscripting.
If at the end of period t ¡ 1 we have ut¡1 capital stock left over for the commencement of
production in period t; we denote the capital stock at commencement of new production by
ut¡1 = vt:
When the period of analysis is unambiguous we shall drop the time subscript and simply refer
to opening stock v and closing stock u for the period under consideration.
3.2.1 General optimal marginal value formula V0n
Applying this simpli¯ed notation the following general characterization is then possible: for
each n and corresponding time tn there exists a `capital investment / carry-forward function'
u(v; b) = un(v; b); (28)
which solves the equation
(v ¡ u(v; bn+1))¡1=2 = °V0n+1(v; bn; Án+1)
and an input price censor function b(v) = bn(v) and a constant à = Ãn; such that
V0n (v; bn; Án+1) =
Z b(v)
0
bn+1dQ(bn+1jbn) +
Z Ãb(v)
b(v)
(v ¡ u(v; bn+1))¡1=2dQ(bn+1jbn)
+
Z 1
Ãb(v)
Án+1bn+1dQ(bn+1jbn):
Assuming a general concave revenue function f(x) in place of the square-root form, the
presence of an additional period of production, moves the exercise price (trigger) down. Here is
the intuition: the provision for the future is the greater the further the horizon, but the trigger
41In our model setting the only extraordinary item is the opportunity gain or loss from purchasing
the investment stock in advance.
25
varies inversely with quantity so the the trigger is smaller the further the horizon; at the same#p#分頁標題#e#
time the manager is less likely to sell stock back at a discount if he / she has the option to use
that same stock at a later date. The general formula, though daunting, is not much di®erent42.
Assuming un¡1 = vn is carried into the future at time tn¡1 we have:
V0n ¡1(vnjÁn; bn¡1) = eb
n ¡ E[max(eb
n ¡ bn; 0)] + E[max(Ánbn ¡eb
n; 0)]
¡
Z eb
n=Án
eb
n
(eb
n ¡ f0(xn(vn; bn)))dQ(bnjbn¡1)
+
Z hn(eb
n;Án)
eb
n=Án
(f0(xn(vn; bn)) ¡eb
n)dQ(bnjbn¡1);
where the ¯rst line refers to a strategy of not carrying forward capital (with put and call
options referring to expansion and contraction), whereas the lines following refer to option values
resulting from carrying stock forward. Hereeb
n = bn(vn; Án) is the price at which management at
tiem t = tn is indi®erent between carrying-forward stock and selling stock o®, while xn(vn; bn)
is the optimal demand in period n for input, given a stock vn of input and current input price of
bn. Here again for simplicity we have assumed Án is deterministic. The carrying-forward option
in the displayed formula exists in a range from eb
n to hn(eb
n; Án) where the function hn(B; Á) is
the solution to the simultaneous equations
hn(B; Á) = bn(vn; Á); B = bn(vn; 1); (29)
and is further split into two intervals by reference to the point eb
n=Án. In the Cobb-Douglas
case the form of the function hn(eb
n; Án) is determined in Appendix D.
3.2.2 General form of optimal future value Vn
Generalizing the two-period model we can then show43 that the optimal future expected value is
given by a formula incorporating three expected values according to which of its three options -
investment, divestment or mere partitioning of its capital stock between current and future use
- the ¯rm uses. Next we need some notation to denote the choice of the optimal current-period
production plan. Letting G(bn) = (f0)¡1(bn) represent the internal optimal demand for input
that maximizes f(x) ¡ bnx over x; then the exact form of the formula is (see Appendix A for
more detail)
Vn¡1(v; bn¡1) =
Z bn(v;1)
0
[f(G(bn)) ¡ bnG(bn) + bn(v ¡ bun(1; bn)) + Vn(bun(1; bn); bn)]dQn(bn)
42The form of the optimal solution changes as we change the number of periods. As we increase the number
of periods, this increases the range of inactivity since, with more periods to follow (i.e. to act on the volatility),
the chance of eventually experiencing su±ciently good demand conditions to use up existing \excess" stock
increases; correspondingly the bene¯t of selling it at a discount is commensurately reduced.#p#分頁標題#e#
43Technical details are available from the authors upon request.
26
+
Z bn(v;Án)
bn(v;1)
[f(v ¡ un(v; bn)) + Vn(un(v; bn); bn)]dQn(bn)
+
Z 1
bn(v;Án)
[f(G(Ánbn)) ¡ ÁnbnG(Ánbn) + Ánbn(v ¡ bun(Án; bn)) + Vn(bun(Án; bn); bn)]dQn(bn):
Here bn(v; 1) replaces b(v) while bn(v; Á) replaces Ãb(v); whereas v is the opening stock, Án
the resale (discount) factor for the next period, bun(1; bn) is the optimal carry-forward into the
following period when investing, bun(Án; bn) is the optimal carry-forward when divesting, and
un(v; bn) is the optimal carry-forward in the absence of investment or divestment. Under the
integral signs we see period n production income, future costs of additional investment, or
future income from divestments, given that prior period costs incurred purchasing stock are
charged to the period in which the stock was acquired44.
The ¯rst two terms on the right, namely f(G(bn)) ¡ bnG(bn); merit particular attention.
Here G(bn) = (f0)¡1(bn) is an internal optimal demand for input that maximizes f(x) ¡ bnx
over x; let us denote it temporarily by xn. Since bn = f0(G(bn)) = f0(xn) we see that the
indirect pro¯t f(G(bn)) ¡ bnG(bn) can also be written as f#(xn); where45
f#(x) := f(x) ¡ xf0(x): (30)
3.2.3 Future value as q-income stream
Since we will be comparing the ability of di®erent income measures to forecast future ¯rm value,
we shall refer to our new indirect income measure f#(x) with a y-variable notation. This is in
order to follow traditional notation for income. Speci¯cally, we set
Y q(x¤(b)) =def f#(x¤(b)):
An inductive application of the recurrence formula for V (:) (shown earlier), coupled with
some re-arrangements of the other terms, yields the following identity in terms of indirect pro¯ts
for the undiscounted optimal future value of the project given a carried forward capital stock
un. The details are given in Appendix C. That is, instead of working with the equivalence
between (3) and (1) we consider the equivalence between (1) and :
Vn(unjbn) = qnun + E[
XN
m=n+1
°m¡n¡1Y q(x¤m)]; (31)
On the right-hand side we sum the closing capital stock un evaluated at Tobin's q; plus the sum
of all future indirect pro¯ts, where:
44Thus the total value of the ¯rm in time tn money must add to the given formula the cash position which
includes past income and deductions of the historic cost of stock v suitably compounded.See later.
45Thus the function ~ f(bn) = f#(G(bn)) is the Fenchel dual of f: However, we are also concerned with
evaluating f# at other points, eg at G(Ánbn):
27
i)Y (x) = f#(x) = f(x) ¡ xf0(x) denotes the indirect pro¯t function associated with the production#p#分頁標題#e#
function f(x);
ii) um+1 = um+1(umjbn; :::; bm) is the optimal carry-forward from period m to period m+1 given
the price history bn; :::; bm;
iii) x¤m = x¤m(um¡1; bm) is the general optimal demand for input at time m (so that when the
¯rm expands x¤m = G(bm));
iv) qm = qm(um; bm) is the period-m Tobin's marginal q; de¯ned as the average marginal bene¯t
of utilization of a unit of input in period m (given the current value of bm and the closing
stock um of the current period). When um is selected optimally (given opening stock vm) the
discounted value of qm ranges between replacement cost bm and resale cost Ámbm. Indeed when
um takes the value corresponding to optimal expansion, discounted qm is the replacement cost
and similarly when um takes the value corresponding to optimal contraction, discounted qm
takes the value Ámbm.
Rewriting the identity thus
°[Vn(unjbn) ¡ qnun] = E[
NX
m=n+1
°m¡nY (x¤m)] = V #
n (unjbn); (32)
we see that the lefthand-side is the discounted future value less its marginal cost, and we denote
this quantity by °V #
n (unjbn); consistently so, since qn = V0n :
Our analysis of assessing future value shows the importance of Tobin's q; i.e. of marginal
bene¯t, and we stress that this refers to replacement cost, as such, only in the expansion regime.
It is natural therefore to measure current earnings as well by reference to Tobin's q, especially
as both current demand and future demand have equal marginal value at an optimum (after
taking due note of appropriate discounting).
De¯nition: The q-income at time tn is the indirect pro¯t, namely, the revenue less
marginal cost of input, in symbols f(xn) ¡ xnf0(xn); i.e. f#(xn); where xn and un have been
chosen to optimise the expression
f(xn) + cn(xn + un ¡ vn) + °Vn(un; bn);
given vn; and where cn = bn for xn + un ¡ vn > 0 and cn = Ánbn for xn + un ¡ vn < 0:
We note that q as introduced above is characterized along the lines of ADEP as being
composed of:
- a certainty-equivalent price less the put option to expand plus the call option to contract plus
the option to carry forward unused stock, i.e. typically it is of the form
q0 = eb
1 ¡ E[max(eb
1 ¡ b1; 0)] + E[max(Á1b1 ¡eb
1; 0)]
+
Z h(eb
1;Á)
eb
1
(f0(x1(u; b1)) ¡eb
1)dQ(b1jb0) (33)
28
for some function h and so includes the option to expand, to contract and to carry-forward
optimally. Note that the future value of the ¯rm, as measured in time tn+1 values, associated
with the end of the production period [tn; tn+1]; is
Y (x¤n) + °Vn = Y (x¤n) + °qnun + E[#p#分頁標題#e#
NX
m=n+1
°m¡nY (x¤m)]:
So recalling (1) and (3), we now see that we may write down the ¯rm equity value Sn in terms
of its book-value Bn at time tn (i.e. the cash position kn plus historic cost hnvn of opening
stock vn), by means of the following identity:
Sn = Bn + Y (x¤n) + vn ¢ HGn + E[
XN
m=n+1
°m¡nY (x¤m)]; (34)
where HGn denotes the holding gain (per unit) on opening stock, and takes the following
value, given a historic valuation of hn per unit:
(U) HGn = bn ¡ hn; (IO) HGn = °qnbn ¡ hn; (RO) HGn = Ánbn ¡ hn:
Alternatively the equity value may be expressed in terms of the cash position kn at time tn
and the corresponding opening stock position vn as
Sn = kn + °qn ¢ vn + Y (x¤n) + E[
NX
m=n+1
°m¡nY (x¤m)]: (35)
In words: the equity value comprises opening cash, q-revalued opening stock, current q-income
and future q-income V #. We recall that the q-revaluation price of stock is either bn (i.e.
replacement cost) in regime (U), or Ánbn (i.e. resale price) in regime (RO) or an intermediate
value in regime (IO).
To summarize, our approach considers an alternative valuation identity and generalizes the
earlier two-period model to multiple periods. After taking appropriate discounting, the form
of the optimal value function V (:) comprises:
-q adjusted value of the closing capital stock
- plus the expected q-income stream.
Moreover we have established the form of Y (x¤n(bn)) given a Cobb-Douglas technology (see
Appendix B for the general details). It is satisfying that the q-income in this case is proportional
to the revenue. For the square root function speci¯cally:
- the period n indirect pro¯t function Y (x¤n(bn)) takes a notionally simple form; it is 1=bn when
the project is under-invested, 1=(Ánbn) when it is over-invested, and an intermediate value in
the third regime.
29
Thus for our simple square-root returns model we can identify Vn(unjbn) by forming expectations
over the input price process bn: Furthermore forming this expectation simply requires
looking at the appropriately censored integral of the next input price and the censoring value
is the current input price bn times a factor gn; a generalized version46 of (24).
Remark 9: (Existence of an Informational Asymmetry) We have shown how a
manager determines the optimal expected future value of the ¯rm by appropriate
valuation of embedded put and call options. We must now ask why couldn't an
external investor also directly identify the optimal value without the need to refer
to other (subsidiary) information such as accounting income data.
3.2.4 Informational asymmetry
We shall henceforth assume that whereas the internal manager observes bt; the investor does#p#分頁標題#e#
not. Thus at issue is whether some other accounting data may be helpful for the investor trying
to form inferences on the value of Vt(): Before embarking on this route we note that it could be
argued that an investor reading the annual ¯nancial accounts could be able to infer the input
price of capital from the movements in capital items in the accounts. Our response here is as
follows. Recall that in our introduction to the model we simpli¯ed the presentation by assuming
the returns function faced by the ¯rm was a simple square-root function and hence f#(xn) = 1b
,
that is once the q-theoretic operating pro¯t was reported an investor knows exactly the value
of b and then can readily determine the value for V (): Note though that our initial working
assumption of the square root function was simply to ease initial presentation. The important
result we derive above (34) assumes only concavity of the returns function and in that case
observing reported pro¯t (i) does not allow the investor to infer what b was directly, and (ii)
even if the investor had some other means of ¯nding out the true value of b; that would still
not be enough to infer the functional form of V () since in the case of the general Cobb-Douglas
returns function xµ (for which f#(x) = (1 ¡ µ)f(x)), the function V () cannot be recovered
from knowledge of b without knowledge of the technology returns parameter µ:That is, if the
reader feels uneasy about our modelling assumption that b is unobservable to an investor, then
assuming that the general returns / technology factor µ is unobservable to the investor induces
the same desired result that an investor cannot from an observed pro¯t ¯gure disentangle what
the values for b and µ are - hence directly determine the optimal value function.
Fortunately since f#(x) is directly proportional to revenue f(x); the current marginal value
f0(x) is proportional to the ratio of current revenue over current consumption x:So despite the
relevant q-theoretic variable being the current marginal pro¯tability f0(xn)=bn; it is appropriate
for an external investor to take an interest in f#(xn):
This identi¯cation of an asymmetry clearly raises the issue of optimal contract design within
a principal-agent context47. We note that given our dynamic model setting, issues of dynamic
46We will give a speci¯c example in the next section below.
47Our underlying model framework di®ers from that of Dutta and Reichelstein (1999) and Govindarajan
30
commitment and renegotiation will immediately arise and we leave to a following study the
pursuit of these extensions. Our task here is to identify the ¯rst best. In contrast to most
single period agency models where this is a trivial issue, in our setting it is not, evidenced#p#分頁標題#e#
perhaps by the fact that Feltham and Ohlson have been working with a model over the last
decade which clearly has not been ¯rst best (since it ignores option values).
Having identi¯ed how managers can determine the optimal future value of the ¯rm, and on
the assumption that an informational asymmetry exists between the manager and the investor,
we now consider how the investor could use accounting measures to make inferences concerning
the future value of the ¯rm.
and Ramikrishnan (2001) because they assume that the change in cash °ow or earnings is linearly a®ected by
exercise of e®ort on the part of the manager and do not recognise the option component of investment. As we
have seen this options component does not necessarily add to cash °ow or earnings in a linear way.
31
4 Using an Earnings Measure to Infer Firm Value
Given Remark 9 let us now consider how a representative investor could use an earnings measure
to value a ¯rm. Recalling the fundamental FO result (4):
eyt+1 = !eyt + xt + "t+1;
which has been used repeatedly by empiricists to test for the value relevance of accounting
measures, at this stage we summarize our critique of this previous research via two remarks:
Remark 10: (Non-optimality of the FO Model)
Since the FO model does not recognize real options that arise in practice it is hard to know
what (if anything) empirical tests48 using the FO speci¯cation actually mean for the decision
value signi¯cance of accounting measures. For instance, recalling our results on the underlying
naivety of the investment model in the FO approach, we comment in subsection 4.3 that only in
the restrictive case of non stochasticity in the underlying parameter is the FO residual income
model consistent with our optimization model.
Remark 11: (Non-Linearity of the Firm Value)
By explicitly recognizing the real options omnipresent with investment in real assets, we have
shown that the type of linear functional relationship embodied by (4), and used so frequently
by empiricists49, is inappropriate except in special cases. Instead our real-options analysis,
which brings to the fore the three distinct regimes of optimal investment behavior (contraction,
maintenance and expansion), suggests that, rather than testing for accounting value relevance
with a single linear regression model, an approach using ¯nite-mixture (distribution) models
with three regime changes may be appropriate. We stress that the implied linearities are
between our chosen earnings measure and future value50.
At least two questions naturally arise from these remarks. What is the signi¯cance for
earnings based valuation of the non optimality of FO residual income? If a simple linear
regression is not representative of the underlying optimal investment environment what is an#p#分頁標題#e#
appropriate (perhaps approximate) empirical speci¯cation? We will address these questions in
the following two subsections respectively.
48Both Feltham and Ohlson have on occasion raised concerns about the empirical appllications of their model.
Our critique here is with the claimed theoretical validity of empirical research claiming to apply the FO results.
49We shall discuss the extensions proposed by Burgsthaller and Dichev (1997) and others in subsection 4.2.
50Alternatively we could look at the relationship between our measure of current earnings and market value
St by recovering the relationship from (34).
32
4.1 The signi¯cance of the non-optimality of FO residual income for
earnings based valuation
A central feature that di®erentiates our approach from earlier studies of the use of earnings
numbers to predict future ¯rm value is that via (31) we can actually identify exactly the
variable that is being estimated. Hence we can objectively appraise the ability of a chosen
earnings method such as residual income to predict future ¯rm value. Expressed precisely, if we
let ey denote residual income, then we can consider analytically what is the relationship between
the explanatory variable ey and the variable being predicted V (:):
We have derived analytic expressions for the residual income and they are given below.
However, the qualitative features driving the form of the dependence of ey on the unobservable
bi are pictured in the Figure below showing that the residual income ~yi at the end of the period
[ti; ti+1]; as a function of the input price bi; is asymptotically vertical as bi ! 0+ and has a
linear oblique asymptote with positive slope as bi ! +1:Consequently, for each level of residual
income in the range (apart from the minimum) there are at least two corresponding price levels
bi; making the future value of the project ambiguous.
Figure 2 : Graph of ey vs b
To see this we note that the residual income is de¯ned by cases as follows. If we let hi
denote the historic unit cost of the investment asset holding of vi at the beginning of the period
[ti; ti+1] it is straight-forward to show that:
~yi+1 =
8>>>>><
>>>>>:
1
bi
+ vi[bi ¡ (1 + r)hi]; for bi < 1=pvi;
2
bi ¡ hi
1
b2i
¡ rhivi; for 1=pvi · bi · bi(vi; 1);
2
q
xi(vi; bi) ¡ hixi(vi; bi) ¡ rhivi; for bi(vi; 1) < bi · bi(vi; Á);
1
Ábi
(1 ¡ ui(1; Á)) + Ábivi + ui(1;Á)
Á2b2i
hi ¡ (1 + r)vihi; for bi > bi(vi; Á);
where in the ¯rst case the ¯rm is expanding and the last case selling o® some investment assets.
These formulae enable us to produce the required plot of future ¯rm value less historic cost51#p#分頁標題#e#
of investment assets carried forward (Vn(:) ¡ hnun) against residual income ~y(:) as follows:
Figure 3: Hysteresis
This plot shows clearly why it could be misleading to condition expectations of future ¯rm
value solely on accounting residual income. The plot shows that for a given level of residual
income a multiplicity of future ¯rm values may be possible. That is, there does not exist a
functional relationship between residual income and future ¯rm value and hence there is no
51The same qualitative features arise if we simply plot V (:) against ~y(:): We have deducted the historic cost
of the investment assets carried forward so as to capture the accounting convention of matching.
33
theoretical support for the empirical practice of linearly regressing future ¯rm value on residual
income52.
The intuition for this hysteresis e®ect arising is as follows. Compare two ¯rms with identical
residual income, one expanding investment and the other contracting. The reason the two ¯rms
with the same residual income have di®erent future values is that the expanding ¯rm faces a
charge for the additional investment which reduces income whereas the contracting ¯rm is
selling of assets which increases income. That is same the residual income number may result
from two distinctly di®erent investment strategies which in turn imply di®erent future ¯rm
value; ¯rms contracting now are not expected to have the same future value as ¯rms that are
expanding now (holding currently observed residual income constant).
4.2 Considering q - income for earnings based valuation
Our earlier analysis has explained how upon observation of the current input price for the investment
good, management face three investment-strategy regimes; expand investment when
conditions are favorable (over the U under-invested range), neither add to nor sell any investment
good (over the IO endogenously irreversible range) and sell some stock of the investment
good (the RO reversibility range). Thus our ¯rst task is to derive the qualitative features for
our prediction model of V (:) based upon current q-income.
Regime (U): Under-invested in capital stock with v = 0 or v < critical value bv: This
case is where the stock of capital in place is insu±cient and it is optimal for management to
increase the stock.The resulting q-income is Y q = 1=b and we consider the optimal multi-period
behaviour in terms ¯rst of b and then, by substitution, in terms of Y q:
In the multi-period setting, suppose ¯rst, for simplicity, that at the start of business there
is no stock of capital. Via a generalization of (24) and (25) it can be shown that solving
the ¯rst-order condition for the optimal value function to derive the optimal capital purchase
corresponds to requiring that a capital stock be purchased equal to#p#分頁標題#e#
bv0 =
1
b20
+
°2
(eg1;1 ¤ b0)2 + ::: +
°2n
(eg1;N ¤ b0)2 ; (36)
where
egn;m = bgn ¢ bgn+1 ¢ ::: ¢ bgm
and bg1; :::; bgN are the period-by-period price input censor parameters of the model. A special
case53 perhaps illustrates best the e®ect of the input price persistence factors. Assuming in the
52In fact what has been done in the past is even more dubious as Lys and Lo (199*) point out since a truncated
estimate for V (:) is typically used.
53Note in our analysis the values for respective bgt are derived optimally, whereas the assumed values below
are only for illustrative purposes. It need not in general be the case that bgt > 1:See towards the end of section
3.1.
34
special case b0 = 1; ° = 1 that the resale factors are such that bgt > 1 for all t; and taking
bg1 = bg2 = :::bgt = 1:5 for all t; we have
bv0 = 1 +
1
(1:5)2 +
1
(2:25)2 + ::::
that is, enough stock is purchased to meet demand for the current period, 44.4% of current
demand for the following period and 19.75% of current demand for the period following that
etc.
Intuition underlying choice of the optimal q-investment level when following the
(U) expansion strategy
Given a low stock of capital, upon seeing an advantageous purchase price, the optimal strategy
is to purchase additional stock for the current and future periods (expand investment) with the
amount brought forward in this example being less for each period further into the future in
order to guard against over-stocking before waiting to see how the input price evolves in the
future.
To demonstrate how the total investment is planned to be applied across the sum of the
periods we note that formula (36) may be derived as follows:
bv0 =
1
b20
+ bu1(b0)
=
1
b20
+ bv1(b0^g1(1 + r))
=
1
b20
+
°2
b20
^g2
1
+ bv2(b0^g1^g2(1 + r)2):::
=
1
b20
+
°2
b20
^g2
1
+
°4
b20
^g2
1^g2
2
+ :::
Thus the stock is built up \as if" the undiscounted prices in the future were known to be
bm = bg1 ¢ bg2 ¢ ::: ¢ bgmb0:A full derivation is given in Appendix B.
In general at time ti the optimal opening investment stock to purchase is given by
bvi(bi) =
1
b2i
+ bui(bi) =
1
b2i
+ °2
Ã
1
(egi+1;i+1 ¤ bi)2 + ::::: +
°2(N¡i)
(egi+1;N ¤ bi)2
!
: (37)
Given the current optimal pro¯t is given by 1=bi; the optimal expected future ¯rm value is
obtained by following a strategy of increasing the stock to bvi = bv(bi); which results in54#p#分頁標題#e#
V i(^u(bi); bi) =
1
bi
V i(^u(1); 1);
54The following formula is derived in Appendix C
35
where the bar denotes expectation over the future resale factor Ái+1: This homogeneity is derived
in Appendix E. Notationally we may write this optimal expected value in the form
b Vi =
Ci;N
bi
; (38)
for some constant Ci;N = V i(bu(1); 1): That is, qualitatively - the optimal future value of the
¯rm is given by the q-pro¯t Y q
i = 1=bi multiplied by a certain constant55 (which is dependent
on volatility):
b Vi = Y q
i ¢ Ci;N: (39)
Regime (RO): Over-stocked in capital stock with some excess sold o® The case of
a costly divestment is quite similar. For a given price bn there are now two benchmark stock
levels. The ¯rst and lower value is the optimum level bvn (computed as above) below which the
stock level should not fall but there is now a second, larger, upper optimum level bvn(Án; bn);
dependent also on the current resale rate, above which the stock should not rise. The ¯rst
order condition (20) implies that current demand is as though the resale price was the purchase
price so that the q-income is now Y q
n = 1=(Ánbn): Again one considers optimal multi-period
behaviour ¯rst in terms of bn and then, by substitution, in terms of Yq;n:
At time tn the optimal highest stock level worth keeping exists and is given by
bvn(Án; bn) =
1
(bnÁn)2 +
°2
(bnÁn^gn(Án))2
"
1 +
°2
eg2 n+2;n+2
+ ::: +
°2(N¡n)
eg2 n+2;N
#
;
i.e. as though the current price were Ánbn and discounted future prices were to be °bn+1 = Ánbn ¢ ^gn(Án); °2bn+2 = Ánbn^gn(Án)bgn+2; :::; °mbn+m = Ánbn ¢ ^gn(Án)egn+2;n+m; ::: . Corresponding to
bvn(Án; bn) there is an optimal current revenue from production, namely 1=(Ánbn); and an optimal
carry-forward bun(Án; bn) = bvn(Án; bn) ¡ 1=(bnÁn)2; i.e. of the form bun(Án; bn) = bun(Án; 1)=b2
n:
Note that in our notation bun(Án; 1) = bun(Án)=Á2
n to ensure that bvn(Án; bn) = (1+bun(Án))=(Á2
nb2
n):
Intuition underlying choice of the optimal investment level when following the (RO)
expansion strategy
Given a large stock of capital, upon seeing an advantageous resale rate, the optimal strategy
is to sell some of the additional (investment) stock that was planned for use in this and future
periods (contract investment), but the amount sold forward is less for each period further into
the future that we consider; this is because the longer we wait the more chance there is that
the ¯rm could move into an under-stocked position. Hence, analogously to the (U) regime, the#p#分頁標題#e#
future value from carrying-forward is:
Vn(bun(Án; bn); bn) =
1
bn
Vn(bun(Án; 1); 1);
55We note that this last equation also holds in the general Cobb-Douglas case (for an appropriate rede¯ned
constant).
36
(where the bar recalls the expectation over the future resale rate). We can again rewrite the
displayed formula as: b V Á
n = CÁ
n;N
Ánbn ;where CÁ
n;N = ÁnVn(bun(Án; 1); 1): This, as before, follows from
the general formula of Appendix E. Again we have a linear relationship:
b V Á
n = Y q
n ¢ CÁ
n;N; (40)
between future value56 and accounting pro¯t Y q
n = 1=(Ánbn):
Regime (IO): Given Án < 1 and the ¯rm is neither over-invested nor under-invested
Here we are concerned with the intermediate input price range:
bn = bn(v; 1) < bn < bn(v; Án) = bn: (41)
The revenue is, as always,
2
q
xn(vn; bn);
so we set the q-pro¯t to be
Y q
n = Y q(x¤n) =
q
xn(vn; bn);
as required by our formula since f#(x) = px: Thus we have, since xn(vn; bn) = 1=bn
2 and
xn(vn; bn) = 1=bn
2; that
Y q
n (bn) =
1
bn
; Y q
n (bn) =
1
Ánbn
;
so the intermediate input price range corresponds to
1
bn
< Y q
n <
1
Ánbn
;
as 1=bn > 1=bn and 1=bn < 1=bn: In this range the ¯rm neither invests nor divests. It partitions
its stock vn into current optimal consumption xn(vn; bn) and investment carried forward
un(vn; bn); and the cash income in this range is thus
2
q
xn(vn; bn):
The relation between the expected future ¯rm value and q-income is then given57 by:
Vn = Vn(vn ¡ (Y q
n )2; bn(Y q
n ));
where bn = bn(Y q
n ) solves
Y q
n =
q
xn(vn; bn):
Remark 12: (Monotonicity of V #
n in Y q
n )
56Again this last equation holds good in the general Cobb-Douglas case (for an appropriate constant).
57Observe that dVn
dYn
= @Vn
@u (¡2Yn) + @Vn
@b ( dYn
dbn
)¡1 < 0 if @Vn
@b > 0:
37
Provided either the volatility is large enough, i.e. ¾ ¸ ¾¤(Án+1); or, equivalently, provided the
forthcoming discount factor Án+1 is close enough to unity, i.e. Á¤n+1(¾) · Án+1 < 1;the function
V # regarded as a function of Y q
n is monotonic increasing. For instance, in a two-period model,
a su±cient bound is provided by the inequality
Á1 > exp(¡1:65¾):
See Appendix H.
4.2.1 Convexity of future ¯rm value in q-income#p#分頁標題#e#
To compare our model predictions with thosed derived in the literature, we need to consider
留學生dissertation網the equity value of the ¯rm given its current q-income Y q:A plot follows.
Figure 4: Graph of S vs Y q
The graph has three sections correponding to the three regimes considered earlier. We
comment on the qualitative features. For small enough value of Y q (i.e. less than Y q) the
equity value of the ¯rm takes a convex (in fact hyperbolic) form58
vn
Y q ¡ vnh + o(Yq)
in the square-root case (the ¯rst term generalizing to vnY ¡®=(1¡®); in the case of a Cobb-Douglas
index ®). For large values of Y q (i.e. greater than Y q) the equity value is asymptotically linear
and takes the form
(2 + V (^u(1); 1))Y q ¡ vnh + vn(Y q)¡®=(1¡®):
One may take the view that on our de¯nition of income Y q this quantity is unlikely to be very
small and so the vertical asymptote is in itself is irrelevant, but the \convexity" it exhibits is
not out of line with the cluster-plot given in Burgstahler and Dichev (B&D):
To see how this relates to the Burgstahler and Dichev (1997) ¯ndings, recall that in essence
the B&D paper empirically tests the future value of a ¯rm by a two-period model. In the later
of the two periods the earnings E1 predict a possible future earnings stream valued at W1
1 = cE1
(where c is the earnings capitalization factor). Management have the option to switch from
this earnings stream to an alternative activity. That activity also generates a future earnings
stream W2
1 ; known as adaptation value, which is a constant independent of E1 and assumed
¯xed a priori at a value A. The model's empirical proxy for A is the book value B0: The ¯rm
currently has earnings E0; so the current market value of the ¯rm S0 comprises book value B0;
the current earnings E0; and the expected value V0 of the claim: maxfW2
1 ;W2
1 g = maxfcE1;Ag:
In a log-normal setting for the distribution of E1 given E0; the value V0 has the well-known
convex shape of a call-value struck at K = A=c: Our model agrees with the linear valuation
W1
1 = cE1 but only provided E1 is large enough. On the other hand, our model does not give
58In fact, the exact form is Ánbn(vn ¡ ^u(Á; 1)=b2) + 1
bn
V (^u(Á; 1); 1) + 1
Ábn ¡ hnvn
38
management the option to receive a ¯xed income stream A for values of E1 below some strike
value K: Instead the adaptation value depends on the value of E1 and is at best viewed as
piecewise linear in E1 with ranges of linearity endogenously de¯ned by Y q and Y q as given by
(??). Note that B&D also subdivide the earnings range into three intervals in order to verify#p#分頁標題#e#
convexity (by testing whether the slope of the respective best linear ¯t to the data is increasing).
Another interestin point is that presumably in order to ensure large enough subsample sizes,
B&D chose to have equal numbers of observations in each interval. Given the option-valuation
basis for their model, it would have been economically more intuitive had they selected their
middle interval centered on the implied option strike K.
To summarize our ¯ndings are in broad agreement with the stylized facts proposed by B&D
in their option style valuation model. That is our model predicts asymptotic linearity for large
values of current earnings and a convex valuation for low current earnings. However, in our
model the market value can be negatively relatated to (low) earnings as was observed by B&D
in their empirical study. Whereas they could provide no formal explanation our model shows
it may be consistent ¯rm optimization with non constant abandonment value.
Before moving to the concluding section we shall now present a subsection which shows that
residual income is in fact a special case (under restrictive conditions) of q-income and hence
in these special circumstances applying FO residual income is consistent with optimality of
investment behaviour.
4.3 The Equivalence Between Residual Income and q-Theoretic Operating
Pro¯t: Intuition Underlying the Special Case
We start by recalling the example of subsection 2.1. Working with current value residual income
we have from (9) that
eyCV
1 = 2px ¡ b0x; eyCV
2 = 2pu ¡ b1u:
Noting that by suitable rede¯nition of notation and introducing a time script, if
b0(1 + r) = b1
then
eyCV
1 = 2px0 ¡ bx0 = Y (x¤0 ); eyCV
2 = 2px1 ¡ bx1 = Y (x¤1):
That is, provided x0 = x¤0 = ( 1
b0 )2 and x1 = x¤1 = ( 1
b1 )2; the current value residual income is
identical to q-theoretic operating pro¯t. This simple example shows how management focusing
upon residual income is a special case of adopting a focus upon q-theoretic operating pro¯t.
Speci¯cally the equivalence holds in the restricted case that the discounted input prices are
constant through time. This result is just another recurrence of what we have established
earlier: focusing upon residual income does not take into account the put and call, expansion
and contraction options that arise with investment decisions taken in a stochastic environment.
Only in the special case where those options have no value, because the input price is constant
39
(non stochastic), will prediction relative to the two respective measures be equivalent. To see
this recall (35) -
Sn = kn + °qn ¢ vn + Y (x¤n) + E[
NX
m=n+1
°m¡nY (x¤m)];#p#分頁標題#e#
- and that optimization is with respect to current production xn and future stock carried forward
un. What is appealing about (35) is that, since qn includes the value59 of embedded put and call
options and of carrying forward stock, the optimization problem is essentially separable - that
is, after identi¯cation of Tobin's qn; management can think about current period optimization
over x independently of expansion or contraction decisions for the overall level of investment
stock to carry forward un. With reference to equation (34)
Sn = Bn + Y (x¤n) + vn ¢ HGn + E[
XN
m=n+1
°m¡nY (x¤m)];
note that in the case when the input price b is ¯xed, the °qnun term cancels against the market
price paid for the stock (zero holding gains), and
Yn(x¤n(bn)) = eyCV
n (x¤n(bn)):
However, note the converse, when °qnun 6= bn; then Yn(x¤n(bn)) 6= eyCV
n (exn(bn)):
5 Conclusion
For the FO model recall that FO superimpose (4) and (5) on (3). However, as has been argued
extensively above, superimposing this simple AR(1) process on the way residual income grows,
considerably restricts the type of underlying investment behavior that could be consistent with
the model. The objective of the paper has been to establish a more °exible model which
facilitates an alternative representation of the expected income stream of terms Et(eyt+¿ ) based
upon optimal managerial real-options evaluation. These ¯ndings are signi¯cant because the
Feltham-Ohlson valuation framework has been used by empiricists to test the value relevance
of accounting data. Some researchers have criticized how empiricists have used the model to
try to specify appropriate empirical testing procedures for the value relevance of accounting
information. We address both the underlying validity of the FO model and the implications for
speci¯cation of empirical testing routines. With regard to validity, we show how, independently
of speci¯cations issues, the underlying constant growth assumption which is central to the
Feltham-Ohlson framework removes the possibility for management to have a role in deciding
whether or not to exercise expansion and contraction possibilities which do occur with most
investment projects. Given this limitation we develop an alternative valuation framework which
does not su®er from these limitations because the option to expand or contract optimally is
given centre-stage in our model of managerial decision-making. This °exible model which puts
59Recall (33).
40
three investment regimes at center stage also shows that a single linear regression model of the
link between ¯rm value and accounting measures is inappropriate. Instead our model shows
how a regime-shifting speci¯cation (giving rise to a tri-mixture of distributions) would more#p#分頁標題#e#
e®ectively capture the underlying statistical relationships that apply. In the previous section we
have derived the basic regime functional forms needed to implement such testing procedures.
41
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42
6 Appendix A: NPV Rule
In this section we derive the NPV Rule. The notation (see section 3.2) is as follows: Fn(v; b0; Á);
or simply F(v; b0; Á) with time tn suppressed, denotes the discounted future maximum expected
pro¯t ignoring the historic cost of the carry{forward input v: The price at the time tn is here
denoted b0 (sic!) and so that the price at time tn+1 is then b1;the resale rate revealed at time
tn+1 is Á: When the time tn is suppressed ¹ F+(v; b1) denotes expectation over Á0 of Fn+1(v; b1; Á0):
Thus, for example F0(v; b0; Á) = °V0(v; b0; Á) and the corresponding value of the ¯rm ignoring
past costs and revenues is V0(vjb0) = f(G(b0)) ¡ b0G(b0) + F1(v; b0; Á). Now we have
°¡1F(v; b0; Á) =
Z b(v;1)
0
h
f#(G(b1)) + b1(v ¡ bu(1; b1)) + F+(bu(1; b1); b1)
i
dQ1
+
Z b(v;Á)
b(v;1)
h
f(v ¡ u(v; b1)) + F+(u(v; b1); b1)
i
dQ1
+
Z 1
b(v;Á)
h
Áb1(v ¡ bu(Á; b1)) + f#(G(Áb1)) + F+(bu(Á; b1); b1)
i
dQ1:
Hence proceeding formally and applying the Liebniz Rule60
°¡1F0(v; b0; Á) =
Z b(v;1)
0
b1dQ1
+
Z b(v;Á)
b(v;1)
h
f0(v ¡ u(v; b1))(1 ¡ u0) + F+0(u(v; b1); b1)u0
i
dQ1
+
Z 1
b(v;Á)
Áb1dQ1:
But f0(v ¡ u(v; b1)) = F+0(u(v; b1); b1) by de¯nition of u(v; b1): So
F0(v; b0; Á) =
Z b(v;1)
0
b1dQ1 +
Z b(v;Á)
b(v;1)
f0(v ¡ u(v; b1))dQ1 +
Z 1
b(v;Á)
Áb1dQ1:
Hence
°¡1F0(u; b0; Á) =
Z b(u;1)
0
b1dQ1 +
Z b(u;Á)
b(u;1)
f0(x(u; b1))dQ1 +
Z 1
b(u;Á)
Áb1dQ1;
=
Z eb
1
0
b1dQ1 +
Z h1(eb
1;Á)
eb
1
f0(x(u; b1))dQ1 +
Z 1
h1(eb
1;Á)
Áb1dQ1;
= eb
1 +
Z eb1
0
(b1 ¡eb
1)dQ1 +
Z h1(eb
1;Á)
eb
1
(f0(x(u; b1)) ¡eb
1)dQ1
+
Z 1
h1(eb
1;Á)
(Áb1 ¡eb
1)dQ1:
60We do not show cancelling terms.
43
7 Appendix B: The optimal replenishment policy.
We prove the following Recurrence Lemma
^un(bn; Án+1) = G(^bn+1(bn)) + ^un+1(^bn+1(bn); Án+2):
Proof. Recall that the function u = ^un(b; Á) is de¯ned by the equation61
F0n
(u; b) = Áb:
We work inductively. To obtain the solution v = ^un(bn; Án+1) of the ¯rst-order condition#p#分頁標題#e#
F0n
(v; bn) = Án+1bn
we begin by ¯rst solving the censor equation
qn(B; Án+1; ; bn) = Ábn:
We denote the solution62 by^bn+1(bn; Á). Recall that
qm(B; Ám+1; bm) =
Z B
0
bmdQ(bmjbm¡1)
+
Z hm(B;Ám+1)
B
f0(xm(^vm(B); bm))dQ(bmjbm¡1) + Ám+1
Z 1
hm(B;Ám+1)
bmdQ(bmjbm¡1):
Here ^vm(b) = G(b) + ^um(b; Ám+1); and G(b) = ff0g¡1(b): Now for an appropriate function
Bn+1(v) we have
F0n
(v; bn) = qn(Bn+1(v); Án+1; bn);
so we now need to solve
Bn+1(bv) =^bn+1(bn; Á):
But recalling that in general F(v; b) = f(v ¡ u(v; b) + F+(u; b); we have
F0n
(^vn(b); b) = f0(^vn(b) ¡ u(^vn(b); b))(1 ¡ u0) + F0n+1(u; b)u0
= f0(^vn(b) ¡ u(^vn(b); b)) = b:
Thus we have the identity
Bn+1(^vn(B)) = F0n
(^vn(B);Bn+1(^vn(B))) = B:
61Recall the convention that F = °V:
62In the Cobb-Douglas case
^b
n+1(bn) = Ábnbgn(Á)
for some constant bgn(Á):
44
Hence for B =^bn+1(bn; Á) we have identi¯ed that bv = vnB
: In conclusion we have
^un(bn; Á) = G(^bn+1(bn; Á)) + ^un+1(^bn+1(bn; Á); 1)
= G(^bn+1(bn; Á)) + G(^bn+2(^bn+1(bn; Á); 1)) + :::
Corollary. The analysis prescribes an aggregate demand of
DÁ
n(bn) = G(^bn+1) + ::: + G(^bn+i) + :::;
where G(b) = ff0g¡1(b) and the sequence^bn+i is given by the iteration
^b
n+1 = ^bn+1(bn; Á);
^b
n+2 = ^bn+1(^bn+1; 1);
^b
n+3 = ^bn+3(^bn+2; 1);
:::
It is now easy to describe the replenishment programme. Suppose we have n periods remaining
and we have a stock v: The acquisition programme calls for an optimal aggregate demand
to be purchased of
D1n
(bn) = G(^bn+1) + ::: + G(^bn+i) + :::;
(i.e. with Á = 1) and either we have v below this amount in which case we need to top up
to this amount or else we are moderately over-stocked and must carry-forward u¤n(v; Án+1; bn)
without selling, or else we must sell to the point where the stock is DÁn+1
n (bn): Thus
u¤n(v; bn) =
8><
>:
bun(1; bn) bn < bn(v; 1);
u(v; bn) bn(v; 1) < bn < bn(v; Án+1);
bun(Á; bn) bn(v; Án+1) < bn:
8 Appendix C: Derivation of Valuation formula
We study ¯rst the general two-stage situation. The current price is b0 the next period price is
b1 and the resale rate is Á. Our notation in this section for the maximum expected value given
a stock v of inputs and given the knowledge of Á = Á1 is F(v; b0; Á); once b1 is revealed and u
is carried forward into the future, the maximum expected revenue from the period beyond is#p#分頁標題#e#
F+(u; b1); where the bar signi¯es expectation over Á2:
We note that F = °V:
8.1 Step 1. We prove a recurrence
°¡1F(v; b0; Á) = Eb1 [f#(x¤(v; b1)) + F+
#(u¤(v; b1); b1)] + °vq;
where the notation is as in section 3.2 above and is recalled below.
45
Proof. We have as in Appendix A that
°¡1F(v; b0; Á) =
Z b(v;1)
0
h
f(G(b1)) ¡ b1G(b1) + b1(v ¡ bu(1; b1)) + F+(bu(1; b1); b1)
i
dQ1
+
Z b(v;Á)
b(v;1)
h
f(v ¡ u(v; b1)) + F+(u(v; b1); b1)
i
dQ1
+
Z 1
b(v;Á)
h
Áb1(v ¡ bu(Á; b1)) + f#(G(Áb1)) + F+(bu(Á; b1); b1)
i
dQ1:
To understand the ¯rst integral (corresponding to the understocked situation), note that the
additional purchase z is speci¯ed by v + z = G(b1) + bu(1; b1) and so the revenue is f(G(b1)) ¡ b1(G(b1) + bu(1; b1) ¡ v):
Now we reorganize the expression on the right. First note that f#(x) = f(x)¡xf0(x); and
since G is the inverse of f0 we have
f#(G(b1)) = f(G(b1)) ¡ b1G(b1):
Similarly F+¤(x) = F+(x) ¡ xF+0(x): But since u = bu(1; b1) solves
b1 = °V0+
(u; b1) = F+0(u; b1);
we have
F+
#(bu(1; b1); b1) = F+(bu(1; b1); b1) ¡ b1bu(1; b1):
Likewise
F+
#(bu(Á; b1); b1) = F+(bu(Á; b1); b1) ¡ Áb1bu(Á; b1):
Lastly u = u(v; b1) solves
f0(v ¡ u) = °V+0(u; b1) = F+0(u; b1);
hence
F+
#(u(v; b1); b1) = F+(u(v; b1); b1) ¡ u(v; b1)f0(v ¡ u(v; b1))
= F+(u(v; b1); b1) ¡ u(v; b1)f0(x(v; b1));
where x(v; b1) = v ¡ u(v; b1): Of course
f#(x(v; b1)) = f(x(v; b1)) ¡ x(v; b1)f0(x(v; b1))
We thus have, writing x for x(v; b1);
°¡1F(v; b0; Á) =
Z b(v;1)
0
h
f#(G(b1)) + F+
#(bu(1; b1); b1)
i
dQ1 + v
"Z b(v;1)
0
b1dQ1
#
+
Z b(v;Á)
b(v;1)
h
f#(x) + F+
#(u(v; b1); b1)
i
dQ1 +
Z b(v;Á)
b(v;1)
[xf0(x) + u(v; b1)f0(x)] dQ1
+
Z 1
b(v;Á)
h
f#(G(Áb1)) + F+
#(bu(Á; b1); b1)
i
dQ1 + v
"Z 1
b(v;Á)
Áb1dQ1
#
;
46
or just
°¡1F(v; b0; Á) =
Z b(v;1)
0
h
f#(G(b1)) + F+
#(bu(1; b1); b1)
i
dQ1
+
Z b(v;Á)
b(v;1)
h
f#(x) + F+
#(u(v; b1); b1)
i
dQ1
+
Z 1
b(v;Á)
h
f#(G(Áb1)) + F+
#(bu(Á; b1); b1)
i
dQ1
+v#p#分頁標題#e#
"Z b(v;1)
0
b1dQ1 +
Z b(v;Á)
b(v;1)
f0(x)dQ1 +
Z 1
b(v;Á)
Áb1dQ1
#
:
This may be rendered in a more compact way as asserted above, namely
°¡1F(v; b0; Á) = Eb1 [f#(x¤(v; b1)) + F+
#(u¤(v; b1); b1)] + vq;
provided we introduce the notation
x¤(v; b1) =
8><
>:
G(b1) b1 < b(v; 1);
x(v; b1) b(v; 1) < b1 < b(v; Á);
G(Áb1) b(v; Á) < b1;
and
u¤(v; b1) =
8><
>:
bu(1; b1) b1 < b1(v; 1);
u(v; b1) b1(v; 1) < b1 < b1(v; Á);
bu(Á; b1) b1(v; Á) < b1;
where
x(v; b1) = v ¡ u(v; b1)
and
q =
Z b1(v;1)
0
b1dQ1 +
Z b1(v;Á)
b1(v;1)
f0(x(v; b1))dQ1 +
Z 1
b1(v;Á)
Áb1dQ1:
It is convenient to de¯ne a function h1 by the simultaneous equations
h1(B; Á) = b1(v; Á);
B = b1(v; 1);
)
i.e. h1(B; Á) = b1(v1B
; Á); where v = v1B
solves B = b1(v; 1): We identify these functions in the
Cobb-Douglas case in a later section. In conclusion we may de¯ne an important function q0(B)
as follows:
q0(B) =
Z B
0
b1dQ1 +
Z h1(B;Á)
B
f0(x(vB; b1))dQ1 +
Z 1
h1(B;Á)
Áb1dQ1:
The solution for B of q0(B) = b0 is the censor eb
1 =eb
1(b0):
47
8.2 Step 2. Deduction from the Recurrence
We prove
V0(v; b0; Á) = E[
NX
n=1
°n¡1f#(x¤n(bn))] + vq0(b1(v; 1));
or
F0(v; b0; Á) = E[
XN
n=1
°nf#(x¤n(bn))] + °vq0(b1(v; 1)):
Proof. We already know that
°¡1F0(v; b0) = V 0 = q(b(v; 1)) =
Z b(v;1)
0
bdQ(b) +
Z b(v;Á)
b(v;1)
f0(x(v; b))dQ(b) +
Z 1
b(v;Á)
ÁbdQ(b)
(using generic notation), hence we may also write
V ¡ vq = °¡1[F ¡ vq°]
= °¡1[F(v; b0; Á) ¡ vF0(v; b0)]
= °¡1F#(v; b0; Á)
= Eb1 [f#(x¤(v; b1)) + F+¤(u¤(v; b1); b1)]:
Taking expectations over Á; we have
°¡1F¤(v; b0) = EÁ;b1 [f#(x¤(v; b1)) + F+¤(u¤(v; b1); b1)]:
We may now apply this result inductively, the ¯rst steps being
°¡1F#
0 (v; b0; Á) = Eb1 [f#(x¤1(v; b1)) + F1
#(u¤1(v; b1); b1)]
= Eb1 [f#(x¤1(v; b1)) + °Eb2 [f#(x¤2(u¤1; b2)) + F2
#(u¤2(u¤1; b2); b2)]]
= Eb1 [f#(x¤1(v; b1)) + °Eb2 [f#(x¤2(u¤1; b2)) + °Eb3 [f#(x¤3 (u¤2 ; b3)) + F3#p#分頁標題#e#
#(u¤3(u¤2; b3); b2)
= Eb1 [f#(x¤1(v; b1)) + Eb2 [°f#(x¤2(u¤1; b2)) + Eb3 [°2f#(x¤3 (u¤2 ; b3)) + °2F3
#(u¤3(u¤2; b3);
Assuming N steps, so that FN+1 = 0; we obtain on suppressing some notation that
°¡1F#
0 (v; b0; Á1) = E[
XN
n=1
°n¡1f#(x¤n(bn))]:
So
V0(v; b0; Á1) ¡ vq1 = E[
NX
n=1
°n¡1f#(x¤n(bn))]:
48
Rewriting, we obtain the valuation
f#(x¤0(v; b0)) + °V0(v; b0; Á1)
= f#(x¤0(v; b0)) + E[
NX
n=1
°nf#(x¤n(bn))] + v°q0(b1(v; 1));
where
q0(B) =
Z B
0
b1dQ1 +
Z h1(B;Á)
B
f0(x(v1B
; b1))dQ1 +
Z 1
h1(B;Á)
Áb1dQ1:
In the next section we identify the functional form for the Cobb-Douglas case.
9 Appendix D: The Cobb-Douglas case
In this section we give the explicit form of all relevant auxiliary functions needed to compute the
terms in the accounting identity. In particular we recall relevant formulas and results derived
in our CDAM paper apropriate to the case f(x) = x1¡®=(1 ¡ ®) when 0 < ® < 1: Note that
f0 = x¡® has inverse G(b) = b¡1=®; and so
f#(x) =
®
1 ¡ ®
x1¡®;
f#(G(b)) =
®
1 ¡ ®
b¡(1¡®)=®:
The homogeneity property is given by
V 0n(u; bn) = bnV 0n(ub1=®
n ; 1);
and more interestingly by
V 0n(u; bn) = BV 0n(uB1=®; bn=B):
Thus for Á = Án or Á = 1 the solution u = ^u(bn; Á) to V 0n(u) = V 0n(u; bn) = bnÁ is of the
form ub1=®
n = u (i.e. u = ub¡1=®
n ), where u = un(Á) solves
V 0n(un; 1) = Á;
assuming
Á > inf u V 0n(u; 1);
otherwise there is no solution (and therefore no need to sell-back stock).
It may be shown quite generally (see Appendix B) that
^un+1(bn; Á) = G(^bn+1(bn; Á)) + ^un+2(^bn+1(bn; Á); 1)
= (^bn+1(bn))¡1=® + ^un+2(^bn+1(bn); 1);
just as in (36).
49
Now the equation
V 0n(un; 1) = Á;
is equivalent (see Appendix B) under a transformation of variables to
qn(B; bn) = Ábn;
and this in turn to
qn(B=bn; 1) = Á:
So we let g = gn(Á) solve
qn(gÁ; 1) = Á;
with the convention63 that gn+1(Á) = 0 when
Á < inf g qn(g; 1):
Thus the original equation for B is solved by setting
B=bn = gn(Á)Á;
i.e. B = gn+1(Á)(Ábn): We thus have
^un(bn; Á) = (gn+1(Á)Ábn)¡1=® + (gn+2(1)gn+1(Á)Ábn)¡1=®#p#分頁標題#e#
+(gn+3(1)gn+2(1)gn+1(Á)Ábn)¡1=®
and
bvn(bn; Á) = (Ábn)¡1=® + (gn(Á)Ábn)¡1=® + (gn+2(1)gn+1(Á)Ábn)¡1=® + :::
= (Ábn)¡1=®
h
1 + (gn+1(Á))¡1=® + (gn+2(1)gn+1(Á))¡1=® + :::
i
(42)
By (42) we have
bvn(bn; Á) = (·n(Áb))¡1=®;
where
·n(Á)¡1=® = 1 + (gn+1(Á))¡1=® + (gn+2(1)gn+1(Á))¡1=® + ::: : (43)
Note that ·N ´ 1:Thus the solution b = bn(v; Á) to v = vn
Áb is
bn(v; Á) = Á¡1·n(Á)¡1v¡®:
Note the identity
bn+1(^un(bn; Á); 1) = Ágn+1(Á)bn: (44)
63This ensures that the bench-mark stock, above which all is to be sold is in¯nity, in keeping with the idea
that there should be no resale.
50
This is evident if we notice that we have to solve
^un(bn; Á) = bvn+1(bn+1; Á)
= (bn+1)¡1=®
h
1 + (gn+2)¡1=® + (gn+3gn+2)¡1=® + :::
i
= (gn+1(Á)Ábn)¡1=® + (gn+2(1)gn+1(Á)Ábn)¡1=® + ::: :
If the project is overstocked, the carry-forward equation
f0(v ¡ u) = V 0n(u; bn) (45)
may be re-written using homogeneity as
f0(ev ¡ eu(ev)) = V 0n(ub1=®
n ; 1);
where eu = ub1=®
n ; ev = vb1=®
n ; or in standardised form as
f0(ev ¡ eu(ev)) = V 0n(eu; 1);
with solution eun(ev). The solution of (45) is then un(v; bn) = eu(vb1=®
n )b¡1=®
n : Note also
f0([v=u] ¡ 1) = V 0n(1; bnu®)
so the utilization ratio v
u
= 1 + G(V 0n(1; ¸nbn=bn+1(u)))
is a function of the ratio of the current price and the top-up limit. Here ¸n is a constant.
Evidently the special functions eun(ev) need numeric evaluation. They are de¯ned inductively
as follows. The base of the induction is
xN(v; bN) = v;
uN+1(v) = 0;
qN¡1(B) =
Z B
0
bNdQ(bN) +
Z BÃN
B
f0(xN(vB; bN))dQ(bN) + ÁN
Z 1
BÃN
bNdQ(bN);
ÃN = 1=ÁN;
vB = 1=B2;
bN(u; 1) = 1=pu;
WN¡1(u) = u + [qN¡1(bN(u))]¡2;
euN(v) = W¡1
N¡1(v)
uN(v; bN) = euN(vb1=®
N )b¡1=®
N :
The inductive step is very similar:
51
xn(v; bn) = v ¡ eun(v);
qn¡1(B) =
Z B
0
bndQ(bn) +
Z BÃn
B
f0(xn(vnB
; bn))dQ(bn) + Án
Z 1
BÃn
bndQ(bn);
Ãn = Á¡1
n ·n(Án)¡1·n(1);#p#分頁標題#e#
vnB
=
1
·n(1)2B2 ;
bn(v; 1) = ·n(1)¡1v¡1=2;
Wn¡1(u) = u + [qn¡1(bn(u; 1))]¡2;
eun(v) = W¡1
n¡1(v);
un(v; bn) = eun(vb1=®
n )b¡1=®
n :
It is important to notice that the de¯nition of ·n calls for values known from earlier in the
induction namely the numbers gm(Ám) for m > n: (See (43) above.)
However, before one can use these special functions, we need to know just when to apply
them, i.e when and how much stock to resell. With this in mind, recall the de¯nition of the
functions hm given by by the simultaneous equations
hm(B; Á) = bm(v; Á) = Á¡1·m(Á)¡1v¡®;
B = bm(v; 1) = ·m(1)¡1v¡®:
)
Solving, we obtain
hm(B; Á) = Á¡1·m(Á)¡1·m(1)B = ÃmB
= hm(1; Á)B;
so that, as asserted earlier in the ¯nite-horizon section, the dependence on B is linear. Note
that ÃN = Á¡1
N :
As for the carry-forward, we have the explicit forms
u¤n(v; Án+1; bn) =
8><
>:
(·n(1)¡1=® ¡ 1)b¡1=®
n bnv® < ·n(1)¡1;
un+1(v; bn) ·n(1)¡1 < bnv® < Án+1¡1·n(Án+1)¡1;
(·n(Án+1)¡1=® ¡ 1)b¡1=®
n Á¡1
n+1·n(Án+1)¡1 < bnv®;
and
x¤n(v; Án+1; bn) =
8><
>:
b¡1=®
n bnv® < ·n(1)¡1;
xn(v; b1) ·n(1)¡1 < bnv® < Án+1¡1·n(Án+1)¡1;
(Án+1b1)¡1=® Án+1¡1·m(Án+1)¡1 < bnv®:
From here it is a small step to compute the indirect pro¯t Yn(bn) by applying f#(x) =
®
1¡®x1¡® to the formulas above. Thus we have
Yn(bn) =
8><
>:
®
1¡®b(®¡1)=®
n bnv® < ·n(1)¡1;
®
1¡®xn(v; b1)1¡® ·n(1)¡1 < bnv® < Án+1¡1·n(Án+1)¡1;
®
1¡®(Án+1b1)(®¡1)=® Án+1¡1·m(Án+1)¡1 < bnv®:
52
10 Appendix E: Linear dependence of pro¯ts on output
In this appendix we prove in the Cobb-Douglas case that
F(v ¢ G(b0); b0) = F(v; 1)f#(G(b0));
so that in the square-root case we have64
F(vb¡2
0 ; b0) =
1
b0
F(v; 1):
Our main conclusion is the result that
F(bv(b0); b0) =
1
b0#p#分頁標題#e#
F(bv(1); 1);
which asserts that for an optimally carried forward stock, the future expected indirect pro¯ts
are linearly dependent on current indirect pro¯t b¡1
0 .
As for the general Cobb-Douglas situation, if f(x) = x1¡®=(1 ¡ ®), so that f#(G(b)) =
®
1¡®b¡(1¡®)=®, the formula at the head of this section in explicit terms is as follows:
F(vb¡1=®
0 ; b0) =
®
1 ¡ ®
b¡(1¡®)=®
0 F(v; 1):
For notation see section ?? above. Observe that in the under-invested regime, when Yn(bn) =
®
1¡®b(®¡1)=®
n , we have
F(bu(bn); bn) = F(bu(1)b¡1=®
n ; bn) = Yn(bn)F(bu(1); 1);
so the linear dependence on Yn continues to hold.
Proof. For transparency we write the proof in the square-root case.
We again refer to the formula (compare Appendix A):
°¡1F(v; b0; Á) =
Z b(v;1)
0
h
f#(G(b1)) + b1(v ¡ bu(1; b1)) + F+(bu(1; b1); b1)
i
dQ1
+
Z b(v;Á)
b(v;1)
h
f(v ¡ u(v; b1)) + F+(u(v; b1); b1)
i
dQ1
+
Z 1
b(v;Á)
h
Áb1(v ¡ bu(v; b1)) + f#(G(Áb1)) + F+(bu(Á; b1); b1)
i
dQ1:
We begin by assuming inductively the property that for all v > 0
F+(vg¡2b¡2
1 ; gb1) =
1
b1
F+(vg¡2; g);
64Thus H(w; b) = F(1=w2; b) is homogeneous of degree ¡1:
53
and show that for all v we have
F(vg¡2b¡2
0 ; gb0) =
1
b0
F(vg¡2; g):
In the formula above replace b0 by b0g and v by vg¡2b¡2
0 : We also make the substitution h =
b1=(gb0). We now factorize out b¡1
0 using inductive assumptions and some simple manipulations.
To see this done note the following calculations. First note that since b(v; 1) = K=pv (for some
constant K) we have b(v(gb0)¡2; 1) = Kb0=pvg¡2 = b(vg¡2; 1)b0: Next we have
F+(u(vg¡2b¡2
0 ; b1); b1)
= F+(u(vg¡2b¡2
0 ; hgb0); hgb0)
= F+(eu(vg¡2b¡2
0 (hgb0)2)(hgb0)¡2; hgb0)
= F+(eu(vg¡2(hg)2)(hgb0)¡2; hgb0)
= F+(eu(vg¡2(hg)2)(hg)¡2; hg)=b0
= F+(u(vg¡2; hg); hg)=b0:
Similarly,
f(vg¡2b¡2
0 ¡ u(vg¡2b¡2
0 ; hgb0))
= f(vg¡2b¡2
0 ¡ eu(vg¡2b¡2
0 h2g2b20
)h¡2g¡2b¡2
0 )
= b¡1
0 f(vg¡2 ¡ eu(vg¡2h2g2)h¡2g¡2)
= b¡1
0 f(vg¡2 ¡ u(vg¡2; hg))
(since un(v; bn) = eu(vb1=®
n )b¡1=®#p#分頁標題#e#
n ). Finally,
F+(bu(1; gb0h); gb0h)
= F+(bu(1; 1)(gb0h)¡2; gb0h)
= F+(bu(1; 1)(gh)¡2; gh)b¡1
0
= F+(bu(1; gh); gh)b¡1
0 :
We thus obtain (dropping the display of the third term in view of its similarity to the ¯rst)
that
°¡1F(vg¡2b¡2
0 ; gb0; Á)
=
Z b(vg¡2;1)
0
"
1
gb0h
+ gb0h(
1
(gb0h)2 ¡
1
g2b20
bu(1; gh)) + F+(bu(1; gb0h); gb0h)
#
dQ1(h)
+
Z b(vg¡2;Á)
b(vg¡2;1)
h
f(vg¡2b¡2
0 ¡ u(vg¡2b¡2
0 ; b1)) + F+(u(vg¡2b¡2
0 ; gb0h); gb0h)
i
dQ1(h) + :::
=
Z b(vg¡2;1)
0
Ã
1
b0
"
1
gh
+ gh(
1
gh2 ¡ bu(1; gh))
#
+ F+(bu(1; gh); gh)b¡1
0
!
dQ1(h)
54
+
Z b(vg¡2;Á)
b(vg¡2;1)
h
b¡1
0 f(vg¡2 ¡ u(vg¡2; hg)) + F+(u(vg¡2; hg); hg)=b0
i
dQ1(h)
+:::
= b¡1
0
Z b(vg¡2;1)
0
Ã"
1
gh
+ gh(
1
gh2 ¡ bu(1; gh))
#
+ F+(bu(1; gh); gh)
!
dQ1(h)
+b¡1
0
Z b(vg¡2;Á)
b(vg¡2;1)
h
(f(vg¡2 ¡ u(vg¡2; gh)) + F+(u(vg¡2; gh); gh)
i
dQ1(h) + :::
=
1
b0
°¡1F(vg¡2; g; Á):
Taking averages, we obtain the required result.
11 Appendix F: computing residual income with unrealised
holding gains added to book-value
In this section we con¯rm the formula for the residual income ~yCV
i+1, i.e. including realised and
adding unrealised holding gains, at the end of the period [ti; ti+1] as a function of the input price
bi. The residual income is given by cases as follows:
~yCV
i+1 =
8>>>>><
>>>>>:
1
bi
+ bivi ¡ vi(1 + r)hi; for bi < 1=pvi;
2
bi ¡ Rhi
1
b2i
; for 1=pvi · bi · bi(vi; 1);
2
q
x(vi; bi) ¡ Rhix(vi; bi); for bi(vi; 1) < bi · bi(vi; Á);
1
Ábi
(1 ¡ ui(1; Á)) + Ábivi + ui(1;Á)
Á2b2i
(1 + r)hi ¡ (1 + r)vihi; for bi > bi(vi; Á);
where hi is the unit cost of the asset holding vi at the beginning of the period [ti; ti+1] inclusive
of all past holding gains and book-value includes all holding gains.
We study the residual income in the three investment regimes discussed in the last section.
We assume that the time t = ti opening cash and asset position is respectively ci and vi and#p#分頁標題#e#
that the historic unit cost of the asset is hi: Thus
Bi = ci + vihi
is the project's book-value for the last period. We use the notation R = 1+r (so that ° = R¡1):
11.0.1 Under-invested
In this regime we assume the price bi is such the optimal asset holding bvi = xi + ui > vi: Thus
zi = bvi ¡ vi > 0: Here xi = 1=b2i
:
55
First sub-range (a) We assume ¯rst that xi > vi i.e. bi < 1=pvi . Then
Bi+1 = (1 + r)ci +
2
bi ¡ bi(xi + ui ¡ vi) + biui
= (1 + r)ci +
1
bi
+ bivi:
Hence
yi+1 =
1
bi
+ bivi ¡ viRhi:
Note that at the endpoint we have
yi+1 =
2
bi ¡ viRhi:
Also observe that
y0i
+1 = ¡
1
b2i
+ vi < 0
in this range with zero slope at bi = 1=pvi:
In this cases the future pro¯t measured in currency of time ti+1 is
°V (ui; Á; bi) ¡ biui = [°V (ui(1); Á; 1) ¡ ui(1)]b¡1
i :
Second sub-range (b) We assume next that xi = 1=b2i
< vi i.e. 1=pvi < bi:
We assume zi is delivered at time ti at price pi The new cash asset position at time ti+1 is
ci+1 = (1 + r)ci +
2
bi ¡ bi(xi + ui ¡ vi); vi+1 = ui;
and so
Bi+1 = (1 + r)ci +
2
bi ¡ bi(xi + ui ¡ vi) + [(vi ¡ xi)Rhi + bi(xi + ui ¡ vi)]
= (1 + r)ci +
2
bi
+ (vi ¡ xi)Rhi:
Note that (vi ¡ xi) + [xi + ui ¡ vi] = ui: The quantity Bi+1 is the adjusted book-value because
the term (vi ¡ xi)Rhi contains the unrealised holding gain (vi ¡ xi)rhi:
Hence
yi+1 =
2
bi
+ (vi ¡ xi)Rhi ¡ Rhivi
=
2
bi ¡ Rhi
1
b2i
:
This has a bi plot peaking at bi = Rhi:
56
Note that at the left endpoint we have agreement with the formula of the earlier subrange:
yi+1 =
1
bi
+ bivi ¡ viRhi =
2
bi ¡ Rhi
1
b2i
;
ensuring continuity across the two subcases.
Again notice that
y0i
+1 = ¡
2
b2i
+ 2hi
1
b3i
=
2
b3i
(hi ¡ bi) < 0;
provided bi > hi: This will be the case in this subrange provided hi · 1=pvi:
In this case the future pro¯t measured in currency of time ti+1 is
°V (ui; Á; bi) ¡ bi(xi + ui ¡ vi) ¡ (vi ¡ xi)Rhi
= [°V (ui; Á; bi) ¡ biui] ¡ (vi ¡ xi)(Rhi ¡ bi)
= [°V (ui(1); Á; 1) ¡ ui(1)]b¡1
i ¡ (vi ¡ b¡2
i )(Rhi ¡ bi):
11.0.2 Overinvested
In this regime we assume over-stocking so zi = bvi ¡vi < 0: Here xi = 1=(Ábi)2: Now ¡zi is sold#p#分頁標題#e#
at time ti at the current price Ápi: Computing at time ti+1; we have
ci+1 = (1 + r)ci +
2
Ábi
+ Ábi(vi ¡ xi ¡ ui); vi+1 = ui;
Bi+1 = (1 + r)ci +
2
Ábi
+ Ábi(vi ¡ xi ¡ ui) + uiRhi:
Note that Bi+1 is the adjusted book-value at time ti+1 and contains as an addition to historic
value the unrealized holding gain of the assets namely rui giving us the term uiRhi:Here
yi+1 =
2
Ábi
+ Ábi(vi ¡ xi ¡ ui) + uiRhi ¡ Rvihi
=
1
Ábi
+ Ábi(vi ¡ ui) + uiRhi ¡ Rvihi
=
1
Ábi
(1 ¡ ui(1; Á)) + Ábivi +
ui(1; Á)
Á2b2i
Rhi ¡ Rvihi:
Notice that
y0i
+1 = Ávi +
ui(1; Á))
Á2b3i
(Ábi ¡ 2Rhi) ¡
1
Áb2i
;
which is positive for large enough bi so long as vi > 0:
Here the future pro¯t measured in currency of time ti+1 is
°V (ui(bi; Á); Á; bi) ¡ uiRhi = V (ui(1; Á); Á; 1)b¡1
i ¡ Rhiui(1; Á)b¡2
i :
57
11.0.3 Midrange
In this regime opening stock is partitioned between current and future needs. Here the cash/asset
position at time ti+1 is
ci+1 = Rci + 2
q
x(vi; bi); vi+1 = u(vi; bi)
and
Bi+1 = Rci + 2
q
x(vi; bi) + Rhiu(vi; bi):
Again the term Bi+1 is the adjusted book-value which contains the unrealised holding gain
rhiu(vi; bi):So
yi+1 = 2
q
x(vi; bi) + hiRu(vi; bi) ¡ Rhivi
= 2
q
x(vi; bi) + Rhi[vi ¡ x(vi; bi)] ¡ Rhivi
= 2
q
x(vi; bi) ¡ Rhix(vi; bi):
Here the future pro¯t is
°V (ui(vi; bi); Á; bi) ¡ u(vi; bi)Rhi:
58
12 Appendix G: Book-value in the Feltham Ohlson model
It bears remarking here that the framework of the Feltham-Ohlson model takes as its primitive
a notion of accounting valuation, namely the historic book-value (from which `earnings' are
de¯ned once dividends are known). Formerly, implicit in their model is a valuation function
$(:) de¯ning the book value from the portfolio Ht = (c; v0; v1; :::; vt) of ex-dividend cash, c; and
unused investment assets v0; :::; vt where vi was bought at times ti and price pi resulting in the
historic cost book valuation of assets on hand being
Bt = c + v0p0 + ::: + vtpt:
That is, suppressing the information concerning the realized prices known at time t;the valuation
takes the general form:
Bt = $(t;Ht):
However, the realisation of the abnormal earnings stream fNt = feytg; as de¯ned from $(:; :),
is then predicted by a model M of its dynamics, which typically depends upon the current#p#分頁標題#e#
book value as initial condition65, and so implies ¯rst of all a stochastic process fMt = feyM
t g; i.e.
stochastically generated prediction of the realised stream fNt; and then the price of equity via
the identity (3). Thus predicted price of equity is a®ected by the accounting convention (which
is the historic cost convention in the Feltham-Ohlson model). To see this more clearly, suppose
$0 is an alternative accounting convention, yielding the alternative valuations
B0t
= $0(t;Ht);
ey0t
´ yt ¡ rB0t
¡1;
then, provided B0t
also satis¯es the standard technical assumption (concerning the rate of convergence),
we have, by the usual argument
Bt +
1X
¿=1
°¿Et(eyt+¿ ) = Wt = B0t
+
1X
¿=1
°¿Et(ey0t+¿ ):
So, abbreviating the summation of discounted expected values temporarily to Vt; we have
Bt + Vt[fNt] = B0t
+ Vt[fN0t]: (46)
It could therefore be shown that the same model of the earnings stream dynamics M gives a
value
B0t
+ Vt[gM0t];
65By formulae such as
Wt = Bt +
!
R ¡ !
eyt +
R
(R ¡ !)(R ¡ °)
xt:
59
which is perhaps a better predictor of Wt than Bt +Vt[fMt]: Now observed actual discrepancies
from the realization could either deny validity of the AR(1) assumption, or require that any
explanation absorb the discrepancy in a dividend policy consistent with the AR(1) assumption
via (2), i.e.
dMt
= eyM
t ¡ Bt + (1 + r)Bt¡1:
However, an alternative accounting convention could perhaps generate di®erent model predictions
closer to reality despite using the same underlying stochastic dynamics.
Evidently, the technical assumption proving (3), namely that °¿B¿ ! 0 as ¿ ! 1 (i.e.
that book value does not grow faster than the bank yield 1+r), implicitly favours the historic
cost convention (as perpetually unused stock is in the limit discounted to zero). However, the
technical assumption may be satis¯ed by any other convention governing unused production
input assets provided, for instance, that these assets are utilized almost surely within a uniformly
bounded horizon. In reality there is an expiry date for most inputs and this guarantees
that it is optimal to utilize them ahead of the best-before date.
Fortunately no such technicalities arise in a ¯nite horizon; moreover, in that setting there is
a identity corresponding to (3) that includes ¯nal book-value BT (possibly as ¯nal dividend).
There is thus an alternative convention directly justi¯able by the de¯nition of residual income
itself. Inspection of an equivalent to the de¯ning equation, namely
eyt = Bt ¡ (1 + r)Bt¡1 + dt; (47)#p#分頁標題#e#
in which old book-value is interest-adjusted before being deducted from current book-value,
suggests a common value rendering of the two book-values. We may therefore justi¯ably use
as alternative accounting valuation the following function $0(:) in accord with the common
value accounting convention, namely
B0t
= $0(t;H) = c + v0(1 + r)tp0 + v1(1 + r)t¡1p1 + ::: + vtpt:
The valuation B0t
thus includes in c the interest on cash in the bank from recorded earlier
revenues and also attracts cost-of-capital charges on top of historic costs.
Thus cost of unused stock recorded in both B0t
and B0t
¡1 on this convention cancel each
other out in the (47) calculation of residual income, allowing treatment of unused `investment
stock in place' just like interest on any earlier cash deposits sitting in the bank. This has two
important consequences:
(i) current value residual income attributable to immediate utilization of investment stock is
increased by comparison to the historic cost convention, which would not include any holding
gains on the investment stock;
(ii) residual income attributable to investment stock that has been in place for multiple periods
is decreased relative to the historic cost convention.
Both these factors properly re°ect return from investment in rewarding the record of profitable
activity from investment and down-playing unpro¯table activity. Note that any unused
stock sold back will also increase the value of residual income as a cash addition.
60
We stress that both conventions must of necessity give rise to the same value of the ¯rm by
(46), and either earnings stream may be interpreted from the other, for instance
Vt[fNt]hist = (Vt[fN0 t ] + B0t
)current ¡ (Bt)hist;
or as
(evt)hist = (ev0t
)current + [Bt ¡ (1 + r)Bt¡1]hist ¡ [B0t
¡ (1 + r)B0t
¡1]current:
However, as each gives a di®erent interpretation to the term `residual income', each o®ers a
di®erent route to predicting managerial activity and predicted residual earnings stream. In
each dividends are left outside the scope of equity-value computation.
We should point out an additional advantage of the modi¯ed convention that well serves
our purposes. If we employ a model of economic activity with constant expected return then
the common value convention automatically gives constant returns to unused stock.
12.0.4 Example: A stylised two period residual income model
Suppose we start with x + u units of capital at t = 0 purchased for p0 a unit66 and we plan to
use x of the units in the ¯rst period and u of the units in the second period67 with a square
root returns function operating in both periods, that is:
opening net assets B0 = p0(u + x):
We assume a square root returns function.#p#分頁標題#e#
Version 1: stylised model under historic cost convention We compute the two periods'
respective earnings and residual incomes under the historic cost convention
B1 = 2px + p0u; B2 = (1 + r)2px + 2pu;
B1 ¡ B0 = v1; B2 ¡ B1 = v2;
v1 = 2px ¡ p0x; v2 = 2pu ¡ p0u + 2pxr;
ev1 = v1 ¡ rp0(u + x); ev2 = v2 ¡ r(2px + p0u);
= 2px ¡ (1 + r)p0x ¡ rp0u; = 2pu ¡ (1 + r)p0u:
Note that the revenue 2px included in B1 arises at the end of the ¯rst period (i.e. time
t = 1). As a check, note the value of the ¯rm at time t = 0 is
B0 +
ev1
1 + r
+
ev2
(1 + r)2
66Assume this is ¯nanced by the owners initial equity investment.
67In order to make the simplest representation we shall assume that u is the dynamically optimal secondperiod
usage; that is, even though the ¯rm could buy or sell more units after observing the second period input
price of capital it is not optimal to buy or sell capital. Our immediate object here is to map the the two models
into a common notation rather than to concentrate on optimization. Once the mapping is established we will
return to optimization issues.
61
= p0(u + x) +
2px ¡ (1 + r)p0x ¡ rp0u
1 + r
+
2pu ¡ (1 + r)p0u
(1 + r)2
= p0u +
2px ¡ rp0u
1 + r
+
2pu ¡ (1 + r)p0u
(1 + r)2
=
2px
1 + r
+
2pu
(1 + r)2 :
Version 2: stylised model under `common values' convention And now we compute
using the common value accounting convention, as given below equation (47):
B01
= 2px + p0u(1 + r); B02
= (1 + r)2px + 2pu;
B01
¡ B00
= v01
; B02
¡ B01
= v02
;
v01
= 2px ¡ p0x + p0ur; v02
= 2pu ¡ p0u(1 + r) + 2pxr;
ev01
= v01
¡ rp0(u + x); ev02
= v02
¡ r(2px + p0u(1 + r));
= 2px ¡ (1 + r)p0x; = 2pu ¡ (1 + r)2p0u:
Here
p0(u + x) +
2px ¡ (1 + r)p0x
1 + r
+
2pu ¡ (1 + r)2p0u
(1 + r)2
=
2px
1 + r
+
2pu
(1 + r)2 :
Observe that B01
includes the current income and the interest-adjusted historic valuation
of unused stock left languishing; hence the residual income ev01
comprises the pro¯t on current
production using stock valued at the interest-adjusted historic valuation (as it was bought one
period ago). Similarly, B02
includes the current cash revenue and deposited cash revenues from
the previous period (compounded up); consequent on the treatement in B01
of unused stock, the
residual income ev02
here equals the pro¯t from ¯nal production using long unused stock valued
at the interest-adjusted historic valuation (bought two periods ago). Recalling#p#分頁標題#e#
(evt)hist = (evt)current + [Bt ¡ (1 + r)Bt¡1]hist ¡ [B0t
¡ (1 + r)B0t
¡1]current;
we have
(B1)hist ¡ (B01
)current = (2px + p0u) ¡ (2px + p0u(1 + r)) = ¡p0ur
(B2)hist ¡ (B02
)current = 0:
62
13 Appendix H: Monotonicity68 of V #
Recall that u(v; b) is the optimal carry-forward when the current resource price is b and the
stock v held is such that no units of resource are acquired nor resold. We need to consider the
marginal valuation
P(b) = F(u(v; b); b) ¡ u(v; b)F0(u(v; b); b)
=
1
b
[F(~u; 1) ¡ ~uF0(~u; 1)]
(note that ~u(vb2) = u(v; b)=b2);or dropping the second variable
P(b) =
1
b
[F(U(b2v)) ¡ U(b2v)F0(U(b2v))]:
which is of central importance to us. It represents the bene¯t of the future value of w = b2v
relative to the current price level b: Here U(w) denotes the solution to the equation
f0(w ¡ U(w)) = F0(U(w)):
We need to know that P(b) is decreasing with b:
13.1 An equivalent formulation
Put b =
q
w=v (i.e. w = b2v), and since w increases with b;write
P(
q
w=v) =
pv
pw
[F(U(w)) ¡ U(w)F0(U(w))]
or since v is constant we ask to show that the following is decreasing with w :
1
pw
[F(U(w)) ¡ U(w)F0(U(w))]:
This is the ratio of future pro¯t to current `pro¯t' 2pw¡w 1 pw = pw in which the current cost
is measured at the marginal value of 1=pw:
This leads to a further simpli¯cation. Put u = U(w); so that w = V (u); where w = V (u) is
the inverse function to u = U(w): Thus V (u) solves the equation
f0(V (u) ¡ u) = F0(u):
(Compare Appendix D.) We therefore consider the ratio
F(u) ¡ uF0(u)
q
V (u)
:
68We gratefully acknowledge the contribution of Graham Brightwell to this appendix.
63
13.2 An integral inequality
Let
¦(u) = F(u) ¡ uF0(u)
=
Z 1=pu
0
(
1
b1
dQ1 +
Z 1=(Á1pu)
1=pu
pudQ1 +
Z 1
1=(Á1pu)
1
Á1b1
dQ1
we consider
¦(u)
q
V (u)
=
¦(u)
pu ¢
s
u
V (u)
:
The left-hand side is an intertemporal comparison of the future use of the apportioned resource
u , against the immediate use of the entire resource V (u). On the right-hand side ¦(u)=pu
compares the future pro¯t from use of u to the immediate pro¯t from the use of u on its own
(which would have led to a bene¯t f#(u) = pu).
We now let
K(u) = def¦(u)=pu
=
1
pu
Z 1=pu
0
1
b1
dQ1 +
Z 1=(Á1pu)
1=pu#p#分頁標題#e#
dQ1 +
1
pu
Z 1
1=(Á1pu)
1
Á1b1
dQ1
=
1
pu
X + Y
where X = X1 + X2 and
X1 =
Z 1=pu
0
1
b1
dQ1; X2 =
Z 1
1=(Á1pu)
1
Á1b1
dQ1; Y =
Z 1=(Á1pu)
1=pu
dQ1:
Similarly, we may consider the ratio of future marginal bene¯t F0(u) to the immediate marginal
bene¯t of using u namely 1=pu:We thus put
L(u) = defpuF0(u)
= pu
Z 1=pu
0
b1dQ1 +
Z 1=(Ápu)
1=pu
dQ1 + Ápu
Z 1
1=(Ápu)
b1dQ1
= puZ + Y;
where Z = (Z1 + Z2) and
Z1 =
Z 1=pu
0
b1dQ1; Z2 = Á
Z 1
1=(Ápu)
b1dQ1
Note that the `apportionment ratio' is
V (u)
u
=
1
u
³
u + F0(u)¡2
´
= 1 + L(u)¡2:
64
The ratio of interest is thus
A(u) =def
K(u)
q
1 + L(u)¡2
=
¦(u)
pu ¢
s
u
V (u)
;
and we wish to show A0(u) < 0:
As a preliminary we compute that
K0(u) = ¡
1
2
u¡3=2
Z 1=pu
0
1
b1
dQ1 ¡
1
2
u¡3=2
Z 1
1=(Á1pu)
1
Á1b1
dQ1 < 0;
so that we have
K(u) =
X
pu
+ Y; ¡K0(u) =
X
2u3=2 :
Likewise
L0(u) =
1
2
u¡1=2
Z 1=pu
0
b1dQ1 +
1
2
u¡1=2Á
Z 1
1=(Ápu)
b1pudQ1
and again
L(u) = puZ + Y; L0(u) =
Z
2pu
:
Now
A0(u) =
K0(u)
q
1 + L(u)¡2
+
K(u)
(1 + L(u)¡2)3=2
L0(u)
L(u)3
=
1
L(u)3 (1 + L(u)¡2)3=2
h
K0(u)(L(u)3 + L(u)) + K(u)L0(u)
i
We thus need to show that69
¡K0(u)(L(u)3 + L(u)) > K(u)L0(u)
or
1
2
u¡3=2X
hpuZ + Y + L3
i
>
Z
2pu
"
X
pu
+ Y
#
;
i.e.
X
u
hpuZ + Y + L3
i
> Z
"
X
pu
+ Y
#
:
69Alternatively, we require that
¡K0(u)L(u)3 > K(u)L0(u) +K0(u)L(u)
and the left-hand side is positive. Thus the condition is satis¯ed in any u interval where KL is decreasing in u:
65
or, on subtracting XZ=pu from each side
X
u
h
Y + L3
i
> Y Z
or with X¤ = X=pu and Z¤ = puZ we require for monotonicity that
X¤
h
Y + L3
i
> Y Z¤: (48)#p#分頁標題#e#
13.3 Veri¯cation
In this section we show that
X¤
h
Y + L3
i
> Y Z¤;
provided Á1 is not too small, namely provided
Á1 > exp(¡1:65396¾); (49)
(so that for a typical annual standard deviation ¾ of 30% we require Á1 > 60%). Alternatively
for a given Á1 this requires that
¾ >
ln 1=Á1
1:65396
:
Recall that
Y =
Z 1=(Á1pu)
1=pu
dQ1 = Q1(1=(Á1pu)) ¡ Q1(1=(pu))
and
X¤ =
1
pu
ÃZ 1=pu
0
1
b1
dQ1 +
Z 1
1=(Á1pu)
1
Á1b1
dQ1
!
Z¤ = pu
ÃZ 1=pu
0
b1dQ1 +
Z 1
1=(Á1pu)
Á1b1dQ1
!
and L = puF0(u) so that
L = puF0(u) = puZ + Y = Z¤ + Y:
The argument divides as L · 1or L > 1 i.e. Z¤ · 1 ¡ Y or Z¤ > 1 ¡ Y:
Remark. The optimal hedge u = ^u is such that
Z 1=pu
0
b1dQ1 +
Z 1
1=(Á1pu)
Á1b1dQ1 +
1
pu
Z 1=(Á1pu)
1=pu
dQ1 = 1;
i.e.
L = p^u;
so the two cases we consider when u = ^u are accordingly p^u < 1 or p^u ¸ 1 respectively.
66
13.4 Case Z¤ · 1 ¡ Y:i.e. L · 1
Now we claim that when L · 1 we have the stronger strict inequality, which evidently implies
(48), that
X¤ > Z¤:
Indeed by Jensen's Inequality KL > 1 so if L · 1 then K > 1 ¸ L so in particular
X¤ + Y > Z¤ + Y:
Remark. We have just veri¯ed that in this case
0 > K(u)L0(u) + K0(u)L(u)
that is KL is decreasing in u:
13.5 Case Z¤ ¸ 1 ¡ Y:i.e. L ¸ 1, or ^u > 1
Here we aim to show that provided
Á1 > exp(¡1:65396¾):
the condition (48) holds.
13.5.1 We prove a tighter condition ...
By Jensen
XZ
=
ÃZ 1=pu
0
1
b1
dQ1 +
Z 1
1=(Á1pu)
1
Á1b1
dQ1
!ÃZ 1=pu
0
b1dQ1 +
Z 1
1=(Ápu)
Áb1dQ1
!
> (1 ¡ Y )(1 ¡ Y )
i.e.
X¤Z¤ = XZ > (1 ¡ Y )2
To satisfy (48), i.e.
X¤(Y + L3) ¸ Y Z¤;
it is equivalent to have
X¤Z¤(Y + L3) ¸ Y (Z¤)2
and thus enough to have
(1 ¡ Y )2(Y + L3) > Y (Z¤)2:
67
13.5.2 A monotonicity argument
We consider
¢ = (1 ¡ Y )2(Y + [Y + Z¤]3) ¡ Y (Z¤)2
Treating Z¤ as a free variable, so that ¢ = ¢(Z¤); and Y ¯xed at it true value (when u is#p#分頁標題#e#
given) observe that this di®erence ¢ is strictly positive (and the strict inequality is true) when
Z¤ = 1 ¡ Y: Indeed
¢(1 ¡ Y ) = (1 ¡ Y )2(Y + 1) ¡ Y (1 ¡ Y )2 = (1 ¡ Y )2:
Recall that L = Z¤ + Y > 1 i.e. Z¤ > 1 ¡ Y: So we check that ¢(Z¤) is increasing in Z¤.
We di®erentiate with respect to Z¤ to obtain
¢0 = (1 ¡ Y )2(3 [Y + Z¤]2) ¡ 2Y Z¤:
If
(1 ¡ Y )2 >
1
6
(i.e. Y < 1 ¡ 1=p6;which we term `the p6 condition') we obtain
¢0 >
1
2
[Y + Z¤]2 ¡ 2Y Z¤ ¸ 0;
since
[Y + Z¤]2 ¸ 4Y Z¤:
We are now done, i.e. the inequality holds in the case that Z¤ > 1 ¡ Y subject to the p6
condition on Y holding and in fact
¢ > (1 ¡ Y )2 >
1
6
: (50)
We now study the p6-condition and show that this is implied by (49).
13.6 The p6condition
By this we mean
Y = P[Á1t · b1 · t] < 1 ¡
1 p6
;
with Á1 < 1 where t = 1=Á1pu. Now let P1(t) = P[b1 · t] then
P[Á1t · b1 · t] = P1(t) ¡ P1(Á1t)
is easy to study. We have
P1(t) = ©((log t ¡m)=¾):
68
For t > 0 the function P1(t) ¡ P1(Á1t) has a maximum when
t = em=
q
Á1
equal to
1 ¡ 2©(
1
2¾
log Á1);
so that for t > 0 the p6-condition holds uniformly in t provided Á1 is large enough, namely
1 ¡ 2©(
1
2¾
log Á1) < 1 ¡
1 p6
or
log Á1 > 2©¡1(
1
2p6
)¾ = ¡1:65396¾
i.e.
Á1 > exp(¡1:65396¾):
13.7 N-fold version
http://www.ttc-switch.com/dissertation_writing/In this section of the appendix we indicate why in the mutiple-period setting it is still true that
provided Á1 is large enough ¦ is decreasing in price b1 and so V # is increasing in q-income. One
identi¯es a condition for monotonicity analogous to that of the two period model and veri¯es
that it holds provided that the forthcoming discount factor Á1 approaches unity. This kind of
argument does not give an explicit bound for Á1 although the condition (49) still needs to hold.
The idea of the proof is to demonstrate that the new condition analogous to (48) reduces back
to the old condition (48) when we ignore certain terms. As the old condition (48) is a strict
inequality and in fact (50) holds, we deduce our result by showing that the additional terms#p#分頁標題#e#
tend to zero as Á1 tends to unity. The additional terms contain as factors the two integrals
Z b(v;Á)
b(v)
x(v; b1)=v ¡ x0(v; b1)
q
x(v; b1)
dQ1;
Z b(v;Á)
b(v)
[x(v; b1)=v ¡ x0(v; b1)]
x(v; b1)3=2 dQ1:
13.8 Postscript: Monotonicity of y(b)
For the record we prove thaty that y(b) is decreasing
We have since y =
q
x(v; b) that
1
y
= Fu(u(v; b); b) = bFu(~u; 1) = bFu(b2(v ¡ y2); 1)
so
¡
1
y2
dy
db
= Fu(~u; 1) + bFuu(~u; 1)(2bu ¡ 2yb2 dy
db
)
69
so Ã
2yb3Fuu(~u; 1) ¡
1
y2
!
dy
db
= Fu(~u; 1) + Fuu(~u; 1)2~u:
The lhs bracket is negative (since F is concave). So we must now prove the rhs is positive.
Here x(u; b1) = u; (in this period no resources can be carried forward), so
Fu(~u; 1) =
Z b1(~u)
0
b1dQ(b1) + f0(~u)
Z b1(~u;Á)
b1(~u)
dQ(b1) + Á
Z 1
b1(~u;Á)
b1dQ(b1)
and
Fuu(~u; 1) = f00(~u)
Z b1(~u;Á)
b1(~u)
dQ(b1):
But f0(~u) = ~u1=2; and 2~uf00(~u) = ¡~u1¡3=2 = ¡~u1=2 thus
Fu(~u; 1) + Fuu(~u; 1)2~u
=
Z b1(~u)
0
b1dQ(b1) + Á
Z 1
b1(~u;Á)
b1dQ(b1)
+[f0(~u) + 2~uf00(~u)]
Z b1(~u;Á)
b1(~u)
dQ(b1)
=
Z b1(~u)
0
b1dQ(b1) + Á
Z 1
b1(~u;Á)
b1dQ(b1) > 0:
70