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Faculty Working Papers
VALUES OF INFORMATION AND LIQUIDITY PREFERENCE: A COMMENTARY NOTE
Takeshi Murota
#169
College of Commerce and Business Administration
University of Illinois at Urbana-Champaign
FACULTY WORKING PAPERS College of Commerce and Business Administration University of Illinois at Urbana-Champaign March 6, 1974
VALUES OF INFORMATION AND LIQUIDITY PREFERENCE: A COMMENTARY NOTE
Takeshi Murota
#169
VALUES OF INFORMATION AND LIQUIDITY PREFERENCE: A COMMENTARY NOTE
by Takeshi Murota
1 . Introduction
In his recent article [2] Professor Arrow analyzed the Bayesian problems of decision making under uncertainty by casting them into an information theo- retic framework and proposed the concepts of the value of and demand for in- formation. As a possible direction of extending his far-reaching ideas this note is intended to develop the following three aspects of importance in his article.
At first, we show that his definition of the value of information con- tains one logical slip, more precisely, a still remaining confusion of com- paring the utility of income with the cost, the very same point that he keenly criticized in reference to other authors' preceding contributions in economic and statistical studies oi iaformation. In order to improve his re- sult, we redefine the concept of the value of iitformation in such a way that we can revive the essence of J. Marschak's proposal [10] of operationally referring the value of information as a demand price. We also obtain its precise formulation in the Arrow's special context of logarithmic utility function .
Secondly, we present a concept of the value of information in the supply sense to clarify the dual nature of the values of information viewed from its buyer's and seller's standpoints. In this regard the information
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is assigned the attribute of an object for interpersonal which flows in and out among individuals in a given economy under uncertainty.
Thirdly, we analyze the problem of a characterization of liquidity pref erence as behavior towards imperfect information, which seems to be intrinsi< in his models of risk -bearing [1, 2], This problem was once quantified by Marschak [9] as early as 1949 and taken up again by Radner [12] and Hirsh- leifer [6] rather recently, while its thorough investigation is not availabl yet in the current economic literature. In the context of portfolio selec- tion theory of Markowitz [8] and Tobin [14] one is supposed to reveal his preference about a given variety of risky assets in terms of his mean-vari- ance utility, which is a derivative from his utility function of income and probability distributions of stochastic returns of assets. But in order fo our analysis of liquidity to be consistent with the conventional framework of finite-state general equilibrium models under uncertainty, we do not fol- low this mean-variable approach. Starting directly from an individual's utility function of income in Arrow's model, we attempt to illustrate his be havior pattern towards his imperfect knowledge on an uncertain nature from the angle of his optimal liquidity holding.
Though very primitive our results are in this note, they might serve one to initiate a construction of more general models which may capture im- portant problems in the economics of uncertainty that have been outside of the scope of traditional literatures.
2. Basic Model
Let us summarize Arrow's model [2 J in the following manner. A decision maker's uncertain economic environment is assumed to be completely described
Mathematical structure of this transformation from one utility to an- other was elaborately investigated by Richter [13] and Chipman [4].
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by a finite probability space (Q, P) and a random variable X:JJ + R on this space, where
(1) ft ■ {1, ..., S}: an index set of finite S possible states of nature.
(2) P a {(p , ..., p«) ; £p. ■ 1, p. > 0; i e ft}: a decision maker's prior
i (objectively known or subjective) probability dis- tribution on the occurrence of each state in ft, where p. is the probability that state i occurs.
(3) X = (Xj , . . . , Xq) : a piven structure of monetary returns from each one
dollar bet on the occurrence of each state in ft.
This amounts to saying that the decision maker who
bets one dollar on state i receives X. dollars and
nothing otherwise. The decision maker is characterized by his initially held monetary resource which is normarized to the value 1 and his von Neuman-Morgenstern utility function of income:
(4) U: R •*■ R, where U(y) is assumed to be monotone increasing, differ-
entiate and of diminishing marginal utility in
income y. His action in this economy is confined to choosing an (S + 1) dimensional decision vector a which is restricted to the feasible set A of decisions defined as
(5) A - {a = (a,, . ,., ac}j a4 + b ■ 1, a. > 0 for all i £ ft, b > 0},
where a. is the amount bet by him on the occurrence of state i and b is the amount retained uninvested in a liquid form out of his initial response.
1 Each X. can be considered as a reciprocal of unit price of each i-th security, provided that Arrow regards this model as a further development of his now classic paper [1].
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The decision maker will then face the problem:
(6) Given (ft, P) , X, U and A, maximize, with respect 'to a decision vector a e A, the expected value of utility;
EU = Zp. U(a.X. + b).
r \ .1 11
iPi > The situation which Arrow is concerned with is as follows. Suppose
tnat the decision maker is presented an opportunity of acquiring a certain
estimate on a true state of nature in the form of a message in a finite set
given as
(7) ft* = {1*, ..., S*}: an index set of S possible messages, where message
j* implies the estimate, "State j will prevail." Since there seems to be no danger of confusion, we shall use the notation j 2 j* as ]ong as j eft*.
Mora analytically speaking, the acquisition of an estimation becomes possible rough a discrete communication channel describable by means of S x S matrix u *»»." ~h is defined as a channel matrix
(83 Q * [qj'll; £q.- « 1 and q.. > 0 for all i e ft and j e ft*, J j J1 31 -
where the i-th row and j-th column entry q.. of this matrix Q signifies the conditional probability that message j e ft* is sent from the channel while
a true state is i.
1 The word, "channel," does not have to be understood literally in the narrow sense of mathematical communication theory. We may regard it as an operational tool of quantitatively describing degrees of accuracy in any kind of predictive activities such as research, sampling, human verbal con- versation and the like which yield certain estimation on a true state in the stochastic nature.
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The unconditional and conditional probability distributions {p.} in (2) and {q_. . } in (8) will then define the unconditional message probability dis- tribution {q.} and the conditional probability distribution {p..} as
(9) q. ■ Z p.q..; Zq_. - 1, q. > 0 for all j £ ft*
1 i *iMji' .j .1 3 =
(10) p.. - p.q, . /q.; Zp. . = I, p.. > 0 for all i £ ft and j £(!*,
where q. is the probability that message j is sent from the channel and p. . is the conditional probability that the true state is i when message j is sent. Given an information service in the form of a channel Q = |q..| thus characterized, the decision maker previously ignorant on a true state up to his prior probability distribution P can now take advantage of the con- ditional probability distribution {p.,} to form (S + 1) dimensional S deci- sion schedule vectors a(j)'s conditioned by each received message j £ ft*,
within the restriction of his feasible sets A.*s of decision schedules de- ll
= {a, - (a.Cj), ..., as(j), b(j))
fined as (ID A,
E ai(j) + b(j) = 1, ai(j) > 0, b(j) > 0 for all i £ ft}, i
for all j £ ft*.
The components a.(j) and b(j) in a vector a. signify the scheduled amount of money for bet on the occurrence of state i and amount of money retained in a liquid form as functions of a transmitted message j. The decision maker's problem (t) then assumes a new form:
(12) Given (ft, P), X, U, Q, and A. 's, maximize, with respect to S decision schedule vectors a £ A.. ; all j £ ft*, the expected value of the condi- tional utility
E (EU) = 2 q, E p., U[a.(j)X< + b{j)J.
t<U> <*«> i 3 i '
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Our interest is then the evaluation of potential advantage of decision scheme
(12) over the scheme (6), relative cf a given additional information throu the channel (8). We are also concerned with the problem of how the level of optimal liquidity changes as the configuration of information changes. In this regard we need to introduce
DEFINITION I: With respect to the solution vectors a and a. 's in the prob- lems (6) and (12 ) , we define a component b in a as the optimal liquidity and E P>(j)] - Eq.b(j) for-b(j)'s in a. 's as the optimal average liquidity.
3. Demand Value of Information Arrow has inclined to define the arithmetic difference between the maxi- mands (5) and (II) --which has the dimension of utility of income — as the value of information, which presumably has the dimension of money units. This ap- proach is hardly justifiable except for the case where a utility function is linear in income. Following the fully correct approach to this problem by La Valie [7], Hirshleifer [6] and Marschak and Radner [12] we introduce
DEFINITION II: Given (E, P), X, U, A, Q, A.. ss and the payment scheme of re- quiring to pay for information service from the final outcome of the decision
maker, a real number V which satisfies the equations
max £q l p U[a.(j)X. + b(j) - V]
(13) a. £ A.; j e fi* j J i 1J 1 1
= max Zp. U(a.X. + b) a £ A i x x 1
is defined as the demand value of information with posterior payment.
The operational significance of the value of information thus posed re sides in that it is the least upper bound of the buying price of information
It is plain from an elementary result of convex analysis that the prob' lems (6) and (12) have solutions because of the assumptions we imposed on the utility function U in (4) .
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service, or more roughly, the maximum buying price of information in the sens* that a presented information service is worth acquiring if its value V exceeds the cost C of acquiring it. This view revives the essence of Marschak's idea [10] on the reference of the value of information as a demand price, after freeing it from his dimensional problem which Arrow pointed out [2, p. 275]. This Definition II immediately leads us to
PROPOSITION I: If a utility function U in (4) is strictly increasing, con- tinuous and of diminishing marginal utility in income, then the value V of information in accordance with Definition II uniquely exists and it is non- negative. Proof: It is given in Mathematical Appendix at the end of this note.
While the investigation in the general properties of the demand value of information remains as an important problem, our immediate concern in this note is the special case of Arrow, i.e., the case where the sure system of bets exists, i.e., the random variable X satisfies the inequality
(14) E(l/X ) < 1,
i and where
(15) U(y) = log y: the base of logarithm = natural number e. Before proceeding our discussion, we have to note:
REMARK: (Arrow [2, p. 268]). Under the assumptions imposed on U in (4), if U' (0) = +», then the decision maker will invest all his money if and only if there exists a sure system of bets expressed by (14) .
It is obvious that the utility function (14) satisfies the condition in the above Remark. Confining ourselves to this Arrow's special case, we readily obtain
1 These assumptions are slightly different from the ones given by Arrow which are written out in (4) in Section 2.
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PROPOSITION II: Under the assumptions (14) and (15), the value V of infor- mation in accordance with Definition II is an exponential function of the amount of information I conveyed by the channel (8) in the sense of Shannon, more precisely,
(16) V - (1 - e**1)/ I(l/X.),
i where
(17) I = I(Q/P) S - Ep log p ♦ Zq Ep log p *
i1 j 3 i XJ 3
Proof: Using the customary Lagrangean method of maximization, we get the optimal solutions for (6) and (12) under assumptions (14) and (15) as
a. * pi; b = 0
a.(j) - p.,[l - VI (1/X.)] * V/X,; b(j) = 0
1 1J k K 1
fox "1l i £ ft and j e Q*. Evaluating the right and left hand sides of the definitional equation (13) in terms of these solutions, we obtain the
equation
- log[l - mi/\)l ■ - Sp log p ♦ Eq. Zp log p , k i j J i J }
which amounts to the equation (16) in question.
q.e.d. The formula (16) clarifies a rather misleading criticism of Arrow against Marschak. Having observed that the "value of information" in his own arbi- trary definition turns out to be equal to the amount of information itself, Arrow concluded that if the cost of acquiring information is proportionate to the amount of information then there is no way for a decision maker to de- termine how much information or what channel he wants since both the value of
1 Readers who are not familiar with elementary concepts in information
theory can refer to any textbook in this area, such as Ash [3] .
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and cost of information are proportionate to each other [2, p. 275]. But this argument is based on his own dimensional confusion of comparing the utility of income with the cost, i.e., the same kind of comparison which Marschak made [10]. In fact, the value of information properly measured in monetary units in Proposition I is strictly concave in the amount of infor- mation so that his indeterminancy problem of the optimal amount of informa- tion does not occur at least within the conditions which he assumed.
It should be understood, however, that our criticism of Arrow using his own assumptions does not necessarily mean our full acceptance of all of his assumptions either. The proportional cost of information to its amount seems to be a very narrow assumption and there may be many economic situa- tions in which the buyers of information face the price of information that is not quite proportional to its amount . One of the purposes of the next section is to show one such counterexample by investigating a case where the information service is a privately owned, perishable object for interpersonal exchange and which yields a supply price of information strictly convex in its amount rather than proportional to it.
4. Supply Value of Info, mat ion and Other Remarks
Our investigation in the value of information in the demand sense nat- urally leads us to characterize the similar problem from the supply side. Let us consider a decision maker in an environment similar to the one before but who initially owns an information channel with its matrix (7) and who is ready to sell it out to somebody else. Symmetrically as in Definition II we introduce
DEFINITION III j For a similar decision maker as in Definition II who is characterized by (ft, P) , X, U, and A and privately owns a perishable informa-
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tion service Q, a real number W which satisfies the equation
(18) max Eq Zp U[a (j)X + b(j)] a. e A.] j e ft* j 3 i 1J x x
max 2p. U(a.X. + b + W)
— » * ■*- X A
a e A i
is defined as the supply value of information with posterior payment .
2
The \ f W of information thus defined** is the greatest lower bound
of the selling price of information service, or roughly , a minimum selling price information in the sense that an information service is worse selling out if the value K falls short of the revenue R. With respect to Arrow's «oecial circumstance, we obtain
PROPOSITION III: Under the assumptions (14) and (15), the supply value W of information in accordance with Definition III is given by
(19) W - (e1 - 1)/ E(l/X.),
i
where I is the amount of information given in (17) .
The proof of this simple result may be omitted. The formula (19) il- lustrates that there is no universal ground to support the cost of informa- tion proportional to its amount.
So far we have been assuming the posterior payment f°r an information ■-vice a r added on to the final outcomes of the decision.
.: case /here the monetary payment for information is taker, out of ded on to the initial resource, then the sets of decision schedules given by (5) and (11) must be redefined accordingly. With respect to any real numbers V and W, let us define sets Arr(j)'s and Art as
1 An information service may be said to be perishable if it does not
maintain its service for owner's benefits once he sells it away.
* The proof of its unique existence and nonnegativity can be easily done similarly to the proof of Proposition I.
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(20) A^(j) = {I - Caj(j), ..., as(j), b(j));
Ea. (j) + b(j) = 1 - V, a. (j) > 0 for all i e ft, b(j) > 0} i i i
(21) Ag « {a ■ (ftj, ..., ag, b) ;
la. + b + 1 + W, a. > 0 for all i e ft, b > 0}. .i ' i = ' -
i
NOTE: In Definition II, if we restrict decision schedule vectors a.'s to the sets A^(j)ls instead of A. 's, a real number V satisfying the equation
(13) can be defined as the demand value of information with prior payment . Similarly, in Definition III, with the restriction of a decision vector a to Arx instead of A, a real number W satisfying the equation (21) is defined as the supply value of information with prior payment . Under the assump- tions (14) and (15), the thus defined values V and W of information are formulated as
(22) V = 1 - e"1
(23) W = e1 - 1.
Summarizing the special results (16), (19), (22), and (23), we con- clude this section with the follow! ig proposition whose proof may be un- necessary:
PROPOSITION IV: Given (ft, P) , X, U, Q, A, A.'s, A^(j)fs, A^ and assumptions
(14) and (15), the demand and supply values V, V, W, and W of information with posterior and prior payments in accordance with their associated defi- nitions are strictly increasing in the amount of information conveyed by a given channel Q relative to a given prior probability distribution P. They are nonnegative and become equal to zero if and only if the amount of infor- mation is zero, i.e., a given channel is useless. Moreover, the demand
1 A channel is said to be useless if Pa = pi for all i e ft and j e fty For details of classification of channels, see, for example, Ash [3],
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values of information are strictly concave and supply values strictly con- vex in the amount of information. The demand values of information with pos- terior and prior payments become identical functions of the amount of infor- mation if £(1/X.) = 1, and the similar fact also holds for the supply values
i of information.
REMARK: The condition £(1/X.) - 1 has a significant implication in the con-
i text of Arrow [1] if we regard 1/X. as a unit price of i-th security in an
uncertain pure exchange economy of C commodities with S possible states. Arrow demonstrated that the optimal allocation of (S x C) contingent commo- dities, which appears to require to operate (S x C) markets can be achieved by operating only (S + C) markets, i.e., S for securities and C for commo- dities. The above condition excludes the possibility of arbitrage between securities' and commodities' markets so that this economization of markets becomes meaningful enough,
5. Liquidity Preference as Behavior Towards Imperfect Information
Our discussion in the previous two sections was so dependent on Arrow's special case conditioned by the assumptions (14) and (15), especially by (14), that the problem of liquidity preference, which his article [2] rather im- plicitly points out, did not actually arise in our analysis. But it should be understood that Definition I and the optimization problems (6) and (13) in Section 2 of this note have already given us the necessary framework for the analysis of optimal liquidity. In contrast to the traditional character- ization of liquidity preference in terms of the mean-variance of probability distribution of risky assets, we are interested in the analysis which is directly based on a utility function of income from which the portfolio selec- tion theorists supposedly deduce the associated mean-variance utility function
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Admittedly, the discrete description of uncertain returns from an in- vestment may not be so practical and is quite foreign among their familiar continuous descriptions in the portfolio selection theory, except for a very few cases such as Chipman's analysis of the situation of two-point proba- bility distribution [4, p. 181]. However, discrete models may be still in- teresting from a purely theoretical point of view because of their akinness to the general equilibrium models under uncertainty of Arrow-Debrue-Radner type as we mentioned in Section 1.
Generally speaking, we would like to know how a decision maker's optimal amount of liquidity holding changes as his knowledge on the uncertain nature changes due to additional information supplies to him under the condition
(24) Z(l/Xi) > 1 i
and without imposing too many assumptions on the properties of his utility function. But this general approach seems to be analytically very difficult. Therefore } we confine the scope of analysis to the case of logarithmic util- ity function (15) as before.
As an illustration of the nature of the problem which we are concerned with, let us consider the following simple numerical example:
fi - Q* = {1,2,3}
(25) (V}il p2, p3.) * (.05, .10, .85) (Xj, X2, Xj) = (5, 2, 2.S)1.
Case 1 : Given these datum and U(*) = log (•), the maximization problem (6) in section 2 yields the following corner optimum solution:
3
1 Note that £ (1/X. ) = 11/10 > 1 so that the condition (24) is met. i=l
|
.75 |
.125 |
.12S1 |
|
,125 |
.75 |
.125 |
|
.125 |
.125 |
.75 / |
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(aa, a2> a3> b) = (0, 0, .75, ,25) where we obtain b = .25 as the optimal liquidity according to Definition I
Case 2 : Let us next consider the case where the costless information ser- vice is acquired in the following form of tertiary symmetric channel with error probability e - .25:
Q.2S "
accompanied by the message probability distribution:
foi» Q-2> °^ s (S/32> 3'16> 21/32). The problem (12) yields the set of optimal solutions:
/a (1) a (1) a (1) b(l)-\ f .2 0 .6 .2
( a, (2) a2(2) aa(2) b(2) J = | 0 7/30 13/30 1/3 ^(3) a2(3) a3(3) b(3)/ V0 0 20/21 1/21,
The average optimal liquidity b in accordance with Definition I is then
calculated as
3 b - £ q.b(j) = 31/320 s .097 < .25 * b.
We notice here that the liquidity holding conditioned by the transmitted message 2 is 1/3 and is larger than the liquidity under no information, i.e., b ■ .25 but the liquidity averaged over message probability distribution is smaller than that value .25.
Case 3: Let us observe what the average liquidity is under the more accurate tertiary symmetric channel with error probability e = .04:
1 ii it
A channel characterized by a S xS matrix Q - llq^H is called a S-ary
symmetric channel with error probability £ if q-n * 1 - e for j = i and
q-ji ■ e/(S-l) for j i i; for all i, j ■ 1, . . . % S.
■15-
|
.96 |
.02 |
.02 |
|
.02 |
.96 |
.02 |
|
.02 |
.02 |
.96 |
Q.04
accompanied with the message probability distribution:
Under this channel the set of optimal solutions can be calculated as
fajCl) a20j a3(l) b(l)\ /l 73/268 0 0 95/268
a1C2) a?(2) a3(2) b(2) = 0 39/57 0 18/57
ajC3D a2(3) a3(3] b(3) / ^ 0 0 314/819 5/819
The average optimal liquidity b is then
b * 259/4000 ■ .065 < .097 * b.
Observation of the above results in Cases 1, 2, and 3 given the initial datum (25) tells us the following facts and conjectures:
Note 1: As' was clearly stated and proved in Arrow [2] , the optimal liquidity holding turns out to be positive when the problem (6) or more generally the problem (12) yields corner optimum, which needs a full application of Kuhn- Tucker Theorem in a differential form for it to be solved. From a technical point of view this difficulty may be one of the reasons why a finite state approach to the liquidity preference theory based on a utility function of income of von Neuman-Morgenstern type has not developed until today. ' Note 2: Even though a decision maker is assured that he will absolutely gain from investing all his money (in the above numerical example, X. > 1 for i = 1, 2, 3), he may still prefer to hold some positive liquidity unless perfect information is given to him.
To qualify this second Note, let us first establish
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LEMMA: Under the assumptions (15) U(ffl) a logCO
(24) Ul/\) > I
i
(26) X. > 1 for all i e ft,
the general solution to the problem (12) in Section 2 is written out as:
For all j e Q* and with respect to index sets H. and K. which are
3 3
subsets of ft,
bU) = (1 - S PM)/U - 2 U/Xu))
h e H. J h e H, 3 3
*hj * Phj " CbCj)/)^); for h e H.
2L . = 0; for k e K.
fcj 3
»
where
H. * {i e ft; a. > 0} 3 13
K, * Ci e 0: a. ■ 0} ■ 3 ij j
Proof; It is given its Mathematical Appendix.
From this result we immediately obtain THEOREM: If in the above Lemma the given channel is S-ary symmetric with error probability e, and if e is sufficiently small, then the optimal aver- age liquidity b in accordance with Definition I becomes proportional to the error probability independently of the variation of prior probability dis- tribution (p.) and of system of bets {X.}, more precisely, it is given by
As for the definition of symmetric channel, see the footnote of page 14 of this note.
|
S 1 |
" ?4 |
|
|
£ |
v |
1 |
|
S - 1 |
L. |
1 |
|
i = l 1 |
- X" |
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S b • £ q b(j) -
3=1
l
Proof : It is also given in Mathematical Appendix.
This result captures an interesting behavior pattern of a risk averse decision maker who is characteristic in Arrow's model. Although he is ab- solutely sure to gain (X. > 1 fox all i s ft) by investing ail his .money (= 1) on bets, he keeps a certain amount of liquidity and his liquidity preference ceases only when perfect information (e ■ 0) is given to him under the system of bets (24). In contrast to this, his liquidity holding is always equal to zero regardless of his state of knowledge and it is so even under no information if he is presented a sure system of bets
(£C1/X.) < *)• This rather drastic contrast of his behavior in two differ- i ent situations may be rephrased in such a way that in the former a decision
maker's knowledge on his uncertain environment does not matter at all for
him to choose no liquidity as optimal while in the latter it significantly
matters, and in fact, zero liquidity is chosen only accompanied by perfect
knowledge en the environment.
i.
6 . Summary For the purpose of enriching the hypothetical themes proposed in Arrow's article on the value of and demand for information and of making them opera- tionally workable in economic models of uncertainty, we established the con- cept of the demand values of information as its maximum buying prices. To clarify his seeming attempt to regard information as an object of interper- sonal exchange, we also defined the supply values of information as its minimum selling prices so that one can analyze the roles of information flow among individuals in an uncertain economy both from its buyer's and seller's viewpoints.
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To make sure that these new value concepts are not arbitrary trivia, we proved their existence and nonnegativity under loose assumptions. With respect to Arrow's special example based on a Bemoullian logarithmic utility function of income we obtained functional forms of those values of informa- tion which exponentially increase in the amount of information in the sense of Shannon.
We also noted that his original model intrinsically contains an analy- tical characterization of a rational individual's liquidity preference as be- havior towards imperfect information with somewhat different implications from the one in the traditional portfolio selection theory. By means of simple numerical illustrations and a limit theorem based on Kuhn-Tucker Theorem, we analyzed an interesting behavior pattern of a risk averter in his optimal liquidity holding in a sensitive or nonsensitive response to his state of knowledge on the uncertain environment..
Admittedly, most of the propositions obtained in this note have meanings only for illustrative purposes because of the assumption of logarithmic utility function, and not for a general theory, hy confining our analysis within Arrow's special case, we attempted to capture a few essential prob- lems arising in an uncertain economy, which distinguish themselves from the economic problems in a certain world and which we may easily fail to notice if we enlarge the scope of analysis too broadly.
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References
[1] Arrow, K.J,, "Le role des valeurs boursieres pour la repartition la
meilleure des risques," Econometrie, Colloques International^ du Centre National de Recherche Scientifique. Paris; Imprimerie National, Vol. 11, pp. 41-47. English translation: "The Role of Securities in the Optimal Allocation of Risk-Bearing," appeared as Essay 4 in his Essays cited in 2 below.
[2] , "The Value of Demand for Information," in his Essays in the
Theory of Risk -Searing (Chicago; Markham, 1971), also in Decision and Organization, C.E. McGuire and R. Radner, eds. (Amsterdam; North-Holland, 1972).
[3
[4
[S
[6 [7
18
[9
[10
[11
[12 [13 [14
Ash, R. , Information Theory (New York; Interscience Publishers, 1965).
Chipman, J.S., "The Ordering of Portfolios in Terms of Mean and Variance," Review of Economic Studies , Vol. XL (2), No. 122 (April, 1973), pp. 167-90.
Hirshleifer, J., "Liquidity, Uncertainty, and the Accumulation of Infor- mation," Working Paper No. 168, Western Management Science Institute, University of California, Los Angeles (January, 1971).
, "The Private and Social Value of Information and the
Reward to Inventive Activity," American Economic Review, 61 (September, 1971), pp. 561-74.
La Valle, I.H., "On Cash Equivalents and Information Evaluation in Deci- sions under Uncertainty," Part I, II, and III, Journal of the American Statistical Association, Vol. 63 (March, 1968).
Markowitz, H.M. , Portfolio Selection, Cowles Foundation Monograph 16
(New York; John Wiley $ Sons, 1959).
Marschak, J, "Role of Liquidity under Complete and Incomplete Information," American Economic Review, Papers and Proceedings, Vol. XXXIX, No. 3 (May, I949j"s pp. 182-95.
_ , "Remarks on the Economics of Information," in Contributions
to Scientific Research in Management (Los Angeles; Western Date Processing Center, University of California, 1959), pp. 79-98.
Marschak., J. and Radnor, R., Economic Theory of Teams (New Haven; Yale
University Press, 1972).
Radner, R., "Competitive Equilibrium under Uncertainty," Econometrica, Vol. 36, No. 1 [January, 1968), pp. 31-58.
Richter, M.K., "Cardinal Utility, Portfolio Selection and Taxation," Review of Economic Studies, Vol.. XXVII (3), No. 74 (June, 1960), pp. 152-66.
Tobin, J., "Liquidity Preference as Behaviour Towards Risk," Review of Economic Studies, Vol. XXV (2), No. 67 (February, 1958), pp. 65-86.
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Mathematical Appendix
A. Proof of Proposition I in Section 3.
Let a.* ~ (a .(j)*, . ..j a (j)*, b(j)*); i e ft* be optimal solution vec- 3 *
tors of the problem (12) so that
(A-l) Eq Ep U[a (j)*X ♦ b(j)*] > Eq. Ep U[a (j)X ♦ b(j)] . 3 i i, i i - j 3 ± 13 i i
for all a = (a^j), *,., a^j), b(j)); j e ft*.
And let a* = (a*, . . . , a* b*) be an optimal solution vector of the problem
(6). At first we shall prove the quantity (the expected marginal utility due
to additional information)
(A-2) AEU = Eq Ep U[a (j)*X ♦ b(j)*] - Ep. U(a*X. + b*)
j i • i
is nonnegative. Noting the fact that Eq. , ■ 1 and q.p.. = p.q.. for all i
.13 3 13 1 31
and j, we can rewrite (A-2) into
AEU * . Eq. Ep. . U[a. (j)*X. + b(i)*j - Ep. Eq.. U(a*X. + b*) .3 .13 1 ^ 1 w' J , 1 . 31 11
« Eq Ep U[a.(j)*X ♦ b(j)*] - E q Ep.. U[a (j)#X * b(j)#] j J i 13 j J i -
-# tfhere we artificially introduced vectors a. 's defined as
3
-# # # #
ai = (ax(j) , • .., as(j) , b(j) )
- (a^ , . . . , a„ ,03.
Hence from the inequality (A-l), the right hand side of (A-2) must be non- negative, i.e., AEU > 0,
Let us define the real -valued functions f and g in the following manner:
f(Z; a. e A ; j e ft*) * Eq. Ep, ,.U[a. (j )X. + b(j) - Z] 3 J j 3 ^ iJ 1 1
g(Z) - max f(Z; a. e A.; j efi*), a . s A . ; j e ft* 3 3 3 3
Since U is strictly increasing by assumption, f(Z* a. £ A. ; j £ ft*) is strictly decreasing in Z for each (a. e A.; j e ft*) . Hence g(Z) is strictly decreasing
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in Z, On the other hand, from the already proved, fact EU > 0 we obtain
(A-3) g(0) = max Ep. !J(a.X ♦ b) .
a £ A i x x x
If we let Z be sufficiently large, g(2) can be made not to exceed the right
hand side of the equation (A-3) . If we note that the continuity of U implies
the continuity of g, then from the well-established property of continuous
functions there exists V which satisfies
(A-4) g(V) * max 2p. U(a.X. + b)
a £ A i
and from the strict inonotonicity of g we can conclude that this V is unique. If it is negative, then from the strict decreasingness of g and from (A-3) *e get
g(V) > g(0) - _max Ip. Ufa X. + b) .
a e A i '
Since this contradicts with (A-4) itself, V must be nonnegative.
q.e.d.
B, Proof of Lemma in Section 5.
We are going to consider the problem: (B-lj Maximize, with respect to a e A , . . . , ae e Ac,
Zq Ip log[a. (j)X. ♦ b(j)].
j j i J
By introducing S Lagrangean multipliers Aj , . .., X _, we rewrite the problem
(B-l) into the problem of maximizing
(B-2) L(a1, ..., a«; \1 , . .», Xc)
* Zq £*>.. log[aiCj)X. + b(j)] ♦ IX [1 - Cla.(j) + b(j))]. j J i "J 2 i
Conditions for maximization are:
?qt each j e ft*
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(B-3) -S-^TTT ■ a eA^J'urA\ " X. < °; Equality holds if a. (3) > 0
(b-4) —j- - e iTfjTr^tircjT - Aj i 0; Eciuaiity hoids if b(j) > o
i- X X
ST
(8-5) jy- - 1 - (Ea^j) + b(j)) < 0; Equality holds if A. > 0.
If we assume that X. 's are non-positive, then this assumption violates the con-
.)
ditions (B-3) and (B-4) under supposedly positive values of p. 's and X. 's and
nonnegative values of a. (j)'s. Hence X. > 0 for all j e U* and then the con-
ditions (B-5) must be read as
(B-5f) 1 - (Ea.(j) + b(j)j * 0 for all j e fi*. i
On the other hand, from one of the results of Arrow (see Remark in Section 3, page 7 of this note) we know that b(j)'s are all positive. Hence the condi- tions (B-4) must be read as (B-4*) E pij
i a^X."^ b(jl = Aj>
where we started to use the notation A, = X./q. .
For convenience we define the index sets H. and K. as
3 j
H. ■ {i E Q; a. (j) > 0}
Ki»{ie ft; a. (j) - 0}.
Then the conditions (B-3) yield
?h ■ x, (B-6) — *l \ ■ f,s ■ A.; for all h e H.
(B-7) |ll a -MJL < A.; for all k e K.
The above equation (B-6) is rewritten as
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cB-g5 ircirrVruT " W for a11 h £ Kj •
Summing up both sides of (B-8) over the set H.j we obtain
h ■ H ah(3jXh * D(J) -1 h z H V
Dividing the first two terms of (B-7) by X, (^ 0) and summing up the results
over the set K, we obtain
(B-10) E _ h.j 1 :: p. .
k e K I (JT^rbTpj ' b(j) k e K KJ '
Since HU K - ft, the equations (B-41), (B-9) and (B-10) amount to:
CB-11) A S Cl/X.) ♦ ^ 2 p = A.. 3 h e H n DUJ k e K K 3
On the other band, from the equations (B-6) we have
(B-12) a^j) - J& - ^iii for all h e H.
j Hi
Hence, by noting (B-S'}
(B-13) Z a, (j) + b(j) a £ a, (j) + £ a (j) + b(j)
ix heHn k £ K K
- (1/A ) E p. . - b(j) Z (1/3L ) * b(j) J h e H n3 h £ H
From the equation (B-1L) we knc
(3-14) b(j) Cl/X.) * b(j) - (1/A.) Z p...
h e H n J k e K KJ
From (B-13) and (B-14) we obtain
(1/A ) Ep = 1, i ^
which amount, to A. - 1 (i.e.. a. •- q.) for all j £ ft* since Zp. . = 1
3 J V - i 13
for all j £ ft*. Therefore, (B-14} yields
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k c K •' h f H ^ (B-1S) bCj) - ~~- — - ~~-" ; j e Q*.
1 - £ (1/X.) 1 - I (1/X.)
h. e H h e H
In terms of 'these solutions for b(j)'s we obtain
p. - b(j)/X. if i e H. 0 if i e K.
I
q.e.d:
C. Proof of Theorem in Section 5.
If a given channel Q « ||q, .|| is S-ary symmetric with error probability
£, i.e., q.. = I - £ for j ■ 1 and q.. ■ e/(S-l) for j f i, then p. . + 1 if ji ji ij
i « j and p.4 •*■ 0 if i j* j as e *► 0 and the sets H. 's in the above proof of 13 3
Lemma shrink to singleton sets {j}'s. Hence from the result (3-15) we get (C-l) b(j) - CI - P^/U - Cl/X.)); for all j £ 0*.
We further obtain
q . (1 - p..) - q, - q . p . , = £ o . q . . - p . q . .
i
■ 2 p.q. * u - E p.
Combining these results with (C-l), we obtain the optimal average liquidity
b as
1 - P- b = I q.b(j) - ~-T 2 r1 .
: Ji* 1 - ~
q.e.d.
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