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Wealth and the value of generalized lotteries
JOURNAL
OF ECONOMIC
Wealth
THEORY
15,
54-71 (1977)
and the Value of Generalized
Lotteries
JOHN P. DANFORTH* University
of Minnesota,
Minneapolis,
Minnesota
55455
Received January 27, 1975; revised October 26, 1976
In two now famous articles, Arrow [2] and Pratt [7] demonstrate the existence of correspondencesbetween certain simple risk aversion functions and the demand for and valuation of random economic prospects. Each of these authors simplifies his analysis by assuming that an individual’s utility may be written as a function of his money or wealth. While this assumption appears fairly restrictive, Arrow notes that it is consistent with individual preferences which have as their domain a multidimensional commodity space. Specifically he observes that in perfect competition if prices are constant one may interpret money or wealth as a well-defined composite good. Recently Stiglitz and Deschamps have commented explicitly on the implications ot the composite good interpretation suggestedby Arrow. Stiglitz [IO] analyzes the relation between commodity demand functions and the Pratt-Arrow measures of risk aversion for a von Neumann-Morgenstern utility of money function. Deschamps [4] pursues this same general topic with particular emphasis placed on problems arising from the choice of numeraire. Spence and Zeckhauser [9] also discuss interpreting von NeumannMorgenstern utility of money functions as indirect representations of preferences for goods purchasable with money holdings. They argue that the protasis of the composite good theorem is generally contradicted in an uncertain environment. That is, prices are not generally constant over states of the world, and even if they were the pattern of intertemporal evolution of random economic processesmight make free trading of commodities in each state of the world impossible. * I have benefited greatly from Berm Stigum’s criticism during the preparation of this paper. Also, Leonid Hurwicz and the other participants in the Center for Economic Research Bag Lunch Seminar at the University of Minnesota, and an anonymous referee, provided helpful comments on earlier drafts. None of these individuals should be held responsible for any errors which may remain. 54 Copyright All rights
0 1977 by Academic Press, Inc. of reproduction in any form reserved.
ISSN
00224531
WEALTH
AND GENERALIZED
LOTTERIES
33
The aim of the present paper is to elucidate someinterconnections between an individual’s utility function and his valuation of random economic processeswhen the realizations of these processesare not necessarily representable as composite goods. We shall distinguish between two types of risky ventures in the analysis. First, a random process with all of its realizations lying on a single wealth-consumption expansion path1 is termed a simple lottery or S-L. Second, any probabilistic scheme for the distribution of prizes which is not an S-L is termed a generalized lottery or G-L. The bulk of the paper is devoted to the derivation of a result which is a partial extension of an important theorem due to Pratt [7]. His theorem2 establishesan if and only if relationship between the form of an individual’s utility of money function and his valuation of fixed lotteries for varying levels of initial wealth. In Theorem 1 I present a similar “if and only if” relationship for many commodity lotteries when preferences are restricted so as to be of the additive von Neumann-Morgenstern variety.3 The multicommodity lottery is then viewed as a representation of the random consumption sequenceobtained by employing some intertemporal strategy in a stochastic environment. The current money value of such strategies is related to an individual’s utility function and initial wealth endowment in Theorem 2. This theorem again posits an additive utility function; however, an essentially identical result is establishedin Theorem 3 for a classof nonadditive utility functions. All quasi-concave utility functions homogeneous of degree h E (0, 1) form a proper subset of this class. The last few pages contain extensions of two results provided by Stiglitz [lo] on the relation between attitudes toward risk and demand functions. Stiglitz demonstrates that certain restrictions on the form of an individual’s commodity demand functions may be implied by assumptions pertaining to his valuation of simple lotteries. We show in Corollaries 1, I’, and 2 that certain revealed attitudes toward arbitrary simple lotteries tell us a great deal about the individual’s valuation of generalized lotteries. One intriguing implication of these results is that an individual who is risk neutral toward arbitrary simple lotteries will display positive and decreasingabsolute risk aversion toward all generalized lotteries.4 1Let h: R;+l ---fR;” be the ordinary demandfunction associated with maximizinga utility functionu: R,” + R’. A set of points,B CR,“, is a wealth-consumption expansion path if thereexistsa p ER,” suchthat B = (x E R,.n : x = h(p, A) for some A > 0). e See [7,
Theorem 2, p. 1301.
3For a discussion of the implicationsof this assumption see [6]. 4 We say an individual is risk neutral (risk auerse) toward a lottery if he is indifferent between that lottery and a dollar amount equal to (less than) the expected market value of the lottery prizes. An individual is decreasingly absolute risk auerse if the dollar amount he would be just willing to exchange for a lottery increases with his wealth.
56
JOHN ADDITIVE
PREFERENCES
P. DANFORTH AND
THE
VALUE
OF
LOTTERIES
Throughout this paper we shall be dealing with random prospects which are referred to as lotteries. Formally, a lottery is a quadruple, (Sz, 9, Q, L(.)), where (Sz,S, Q) is a probability spaceand L: Sz + R+T is a function which is measurablewith respect to 9. We shall usually omit reference to an underlying probability space and identify particular lotteries by numerical superscript, i.e., L1, L2, etc. Tn addition, we shall limit our attention to lotteries for which E(L)
= s, L dQ
exists and is finite. We shall denote the classof such lotteries as 9. Individual preferences are assumedto be representable by a twice continuously differentiable, increasing, and strictly quasi-concave function V: R,T + R1. In particular, L1 is preferred (indifferent) to L2 by the individual if and only if E(Vo C) > (=) E(Vo L2). We shall be concerned initially with additive V functions. 1.
DEFINITION
V is additive if there exist functions ui , i = 1, 2,..., T
such that V(x)
=
01 +
i
U$(XJ
Vx = (x1 , x2 ,..., xT) E R+T.
i=l
CONDITION 1. V is additive. Note that for T > 1 ui’(xi) > 0 for all i and xi b 0, since V is increasing. Also, at least (T - 1) of the ui are strictly concave throughout because of the strict quasi-concavity of V. Condition 2 extends this property to all T of the ui’s. CONDITION 2. Each ui is a strictly concave function of xi , i = 1, 2,..., T. Given our assumptionson V, the demand functions given in the following definition are continuous. DEFINITION 2. h(A, p) E R+T is the x E {x E R+T: p . x < A} which maximizes V on this set. It is easily established that for any x’ E R+T there exists an (A’, p’) E ~ x R+T such that x’ = h(A’, p’). DEFINITION 3. For any L E 3’ and p E R,T, A(L, minimize A subject to V (/?(A, p)) 3 EC V 0 L).
p) is the solution to:
WEALTH DEFINITION
4.
For any
AND
GENERALIZED
57
LOTTERIES
L E 9 and p E R+T
JW, P) = P . E(L) - 4-b P). A(L, p) is called the dollar value of lottery L evaluated at commodity prices p. fl(L,p) is called the risk premium associated with lottery L and prices p. From Condition 2 it is immediate that 17(.) is nonnegative for any (L, p) and strictly positive for nondegenerate lotteries. The notation provided in the next definitions will be employed in the statement and proof of Theorem 1. DEFINITION
5.
For any p E R+T and a > 0, dh(A,p,
a) will denote the
vector
Conditions
1 and 2 imply that &(A,
DEFINITION
6.
Let
p, a) > 0, i.e., no commodity
is inferior.
L E 9 and X E R+=. The lottery L + X is the function
on Q for which
(L + X)(w) = L(w) + x
VOJEQ.
As one might well expect, the individual’s attitudes toward risk are a prime determinant of the dollar value he places on a given lottery. A measure of “single commodity risk aversion” is useful in describing these attitudes when preferences are additive. DEFINITION
7. ruj(xj)
=
-u:)(xj)/uj’(xj)
is referred to as the measureof single commodity risk aversion of the function uj evaluated at xj . For T = 1 this measure is simply the well known Pratt-Arrow measure of absolute risk aversion. Pratt’s Theorem 2 provides a complete description of the relation between initial wealth and an individual’s dollar evaluation of a given lottery for T = 1 and monotonic r,(.). Specifically, his theorem asserts(in our notation) that
for all nondegenerate one-dimensional lotteries L E 3 and any (p, a) E R+l x R+l if and only if r,(x) is a decreasing (nonincreasing) function of x on R+'. The same implications hold if “increasing” is substituted for decrasing and the inequality is reversed.
58
JOHN P. DANFORTH
The following theorem provides between lotteries and risk premiums THEOREM 1. If Conditions
some insight for T > 1.
on the correspondence
1 and 2 are satisfied,
for all nondegenerate L E 9, any p E R, T and any a > 0 if and only if rLci(xi) is a strictly decreasing (nonincreasing) ,function of xi on [0, w) for all i = 1, 2 ,..., T. Also,
all nondegenerate L E 9, any p E R,T and any a > 0 only if rvi(xi) a strictly increasing function of xi on [0, CO)for all i z I, 2,..., T.
for
is
Figure 1 illustrates Theorem 1 for T = 2 and pz = 1 when rul and ra2 are strictly decreasing. Note that dh(A(L, p), p, a) has been abbreviated as dh in the diagram.
.?r(L +3h,
p) 1
ix:p.x=A(L,p)+a; ;x:p.x=A(L,p);
-;x:p.x=p.ELI
FIGURE
x,
1
Proof of Theorem 1. First I shall establish the “only theorem. That is, it will be proven that if
if” portion
of the
for all L E 9, anyp E RT and any a > 0, then rthi(x,), i = I,..., T, is decreasing (nonincreasing, increasing) on R+l. As an initial step we construct a lottery Li = (Lli, Lzi,..., LTi) as follows.
WEALTH
Take an arbitrary satisfying
AND
integrable
GENERALIZED
function
LOTTERIES
59
J?~: Q + R+l and solve for the z
E(Ui 0 LJ = 2$(z). Next find an (AZ, fi) pair such that h,(A”,fi) = z + A, where d is an arbitrary positive number. Since A,(.,$) is continuous and h,(O,j) = 0 there exists an A1 (less than A2 since h is increasing in A) with hi(Al, 3) = z. Finally, let L,i = hj(A’, ~5)forj # i and let Lii = Li . By construction, V(h( Al, p)) = E( v 0 Li) and II(Li, j) = j3i[E(Ll)
Now by definition II(L”
-
Z]*
if
+ dh(Al, j, A2 - Al),B)
< (G, >I fl(Li>fi,
then j$ uAMA~, PN < (<, >) C z.&(~",jq) &i
+ E(U~ o (L: + A))
or, equivalently, Ui(hi(A2, j)) = Ui(Z + A) < (<,
>) E(Ui 0 (Li” + 0)).
Since A > 0 and Li were chosen arbitrarily, Pratt’s Theorem 2, cited above, guarantees that rUi(xi) is decreasing (nonincreasing, increasing). We now verify the “if” portion of the theorem. That is, it will be proven that if the r,i(x& i = 1, 2 ,..., T, are decreasing (nonincreasing) on R+l then T;T(L + 44%
P), P, 4 P) < (G) WL, P>
for all L E 9, any p E R,T and any a > 0. The T = 2 case is established first and then the result is extended by induction to T > 2. If T = 2 E( V 0 L) = E(u, 0 L,) + E(u, 0 L,).
60
JOHN
For our arbitrarily
P.
DANFORTH
chosen L E 9, we find Z1 and f, such that i = 1,2.
E(Ui 0 LJ = u,(aJ, By Definition
3 we have
for arbitrarily chosen p E R+=. We now have two possibilities treated as separate cases.
which
will be
Case I. hi(A(L, p), p) = Zi , i = 1,2. For an arbitrarily chosen a > 0 dhi(A(L, p), p, a), henceforth denoted in this proof simply as dhi , is nonnegative for i = 1, 2 and strictly positive for one or both of them due to noninferiority. Pratt’s Theorem 2 ensures that
if rU1 and rUz are decreasing (nonincreasing). A@ + 4
P) > (2,
since V and h( ., p) are increasing.
0
Thus, 4L, P) + a
Therefore,
if rU1 and rU2 are decreasing (nonincreasing,
increasing).
Case II. hi(A(L, p), p) # Z;i , i = 1, 2. The following lemma is instrumental in establishing when these inequalities obtain. LEMMA
u,(f, + 4)
1.
the desired result
If 3i;i# hi(A, p), i = 1, 2, then + u,(f, + 4)
> (2) u,MA,
P) + Ah,) + Q44
P) + &)
if rU1 and rUz are decreasing (nonincreasing) and the inequality is reversed if rU1 and rUz are increasing and hi(A, p) > 0, i = 1, 2. Lemma 1 is graphically illustrated for r,i decreasing in Fig. 2 and is proven in the Appendix to this paper. Notice that decreasing, constant, and increasing absolute risk aversion of the ~0s is associated, respectively, with decreasing, constant, and increasing curvature of the indifference curves as one moves along any wealth-consumption expansion path. Tn a later section this intriguing result is related to some work on the link between attitudes toward
WEALTH
AND
GENERALIZED
LOTTERIES
61
risks and commodity demand functions presented in [4, lo]. In the figure Y = f - W(LP),P). Pratt’s Theorem 2 guarantees that if rul and rut are decreasing (nonincreasing), then
h(NLph p) + Y = R FIGURE
2
Application of Lemma 1 thus yields
whenever rul and ruz are decreasing(nonincreasing). Thus,
for rul and ruz decreasing (nonincreasing) as asserted. Extension of the result to T > 2 is accomplished by induction. We first introduce some new notation which will hopefully simplify the argument:
62
JOHN
is the maximizer
P.
DANFORTH
for
It should readily be apparent that ~+YA,
P) = p;+;
Wb, P) + &z+1(&+1)
subject to b + ~n+lxn+l
d 2,
and that
Next let rpn@, P) = - G(b,
P)/ F,“(b, P).
Neave [5, Theorem 1, p. 461 has proven that if the rui , i = I,..., II, are decreasing (nonincreasing) then r&b, p) is decreasing (nonincreasing) in b. Our induction hypothesis for n > 2, IH . (n), is: “If rui , i = I,..., n, are decreasing (nonincreasing) then F(b, p) = i
E(ui 0 L,)
i=l
implies V”(b + a, p) < (G)
f ECui 0 (L + &“(b, i=l
p, a)))
for a > 0 and at least one nondegenerate Li , i = l,..., n. If no Li is nondegenerate only weak inequality is assured. Now we simply find the xzl such that E&3+1 o L+1)
= %+&n*+1)
and the b* such that i
E(ui o Li) = rn(b*, p).
WEALTH
AND GENERALIZED
LOTTERIES
63
Neave’s result, in combination with Lemma 1, ensures that if rUi , i=l ,..., IZ + 1 are decreasing(nonincreasing) then
implies
This implication together with IH . (n) and Pratt’s Theorem 2 implies IH . (n + I), a restatement or IH . (n) with (n + 1) inserted wherever n appears. IH . (2) has been proven and therefore IH CT) is valid. IH . (T) is simply the “if” assertion of Theorem 1.
ADDITIVE
PREFERENCES AND THE VALUE
OF CHOSEN STRATEGIES
It is quite clear that few if any economic phenomena would conform to the basic specifications of Theorem 1. That is, Theorem 1 provides no options for decision making or choice. Given the opportunity to allocate the increment to wealth, a, it is doubtful that augmenting the initial lottery, L, by dh(A(L, p), p, a) would be optimal. Also note that the value of a specific lottery is compared before and after changes in wealth, thus neglecting the lottery choice problem. In order to discusssomeproblems of individual decision making the concept of a consumption strategy is introduced. A consumption strategy, s, associateswith each state of the world, w E Q, a vector (x1 ,..., xr). Restricting our attention to those strategies which are integrable we have s E 9’. The fact that consumption strategy connotes some notion of maximization does not obviate its status as a lottery. The following definition will be utilized in the remainder of the discussion. DEFINITION 8. 9(A, p) = {s E 8: s is feasible for wealth A and certain prices p}. Reference to “feasible” in Definition 8 is left vague since the restrictions
64
JOHN
P.
DANFORTH
associated with feasibility vary greatly from problem to problem. Such factors as the timing 01 consumption decisions, availability of investment opportunities, and debt constraints would all enter into a definition of feasibility. An additional condition is required for the subsequent discussion. CONDITION 3. If z E R,T and p z < a then, for any A E R,l, p E R,T, and a E s, Z(A, p)’ z C Z(A -t a, p). The condition requires that if an individual’s initial wealth is increased, he may useall or part ot it to finance additional purchases of the T commodities at the prevailing market prices. THEOREM2.
If
Conditions
1, 2, and 3 are satisfied
,Ey(yp, ELW(.Nl = w44
P))
S#hld.P) implies
if ru ,( .) is decreasing
(nonincreasing)
for all i = I,.. ., T.
The interpretation of the theorem is straightforward. An individual is asked to determine the minimum expenditure, at prevailing market prices, required to finance a riskless consumption plan which he would accept in exchange for his current uncertain prospects. The individual is then presented with a dollars. He is now asked to calculate the minimum costs of a riskless consumption plan that he would be willing to trade for the consumption strategy he could pursue with his augmented initial wealth. The theorem assertsthat the cost of procuring an acceptable safe consumption plan will have risen by more than (at least) a dollars if the individual’s single commodity measures of absolute risk aversion are decreasing (nonincreasing). Though each strategy, s, may also be thought of as a lottery, there is a subtle difference between this result and Theorem 1. The individual’s strategy has been chosen by him. It is his plan of action for market activities now and in the future. If his initial endowments were to change he might revise the portfolio of stocks he wants to hold. This may result from a change in opportunities (his original wealth might not have been sufficient to meet the margin requirements of his revised portfolio) or simply becausehis attitudes toward uncertain prospects depend on his initial wealth. Proof of Theorem 2. Since any s E L?(A, p) is also an element of Y, we know from Theorem 1 that if
-W’M~))I = W@,
P))
WEALTH
AND GENERALIZED
65
LOTTERIES
then
WV + WA, P, 4.I > (2) W(A + 0,PN if rui(.) is decreasing (nonincreasing) for all i = I, 2,..., T. Since MAP,
a) > 0,
Condition 3 implies
s + MA,
VALUE
p, a) 6 WA + 4 P).
OF STRATEGIES FOR A CLASS OF NONADDITIVE
PREFERENCES
Up to this point the analysis has rested on the assumption that preferences may be represented by an additive von Neumann-Morgenstern utitility function (Condition 1). Despite the highly restrictive nature of this condition, in many instances it may be a perfectly acceptable simplifying assumption. Nevertheless it would be of value to be able to say something about generalized commodity lotteries when preferences are not additive. Happily there are conditions which, if satisfied by a nonadditive utility function, imply a result analogous in content to Theorem 2 above. These restrictions, unlike Conditions 1 and 2, are price specific. That is, for every price vector, p, there is a set of utility functions VP , which satisfy these new conditions. Specifically, V EVP if and only if, (1) k(A, p) is a linear function of A, and (2) V@(A,p) + J) = V(h(A,p)) y = ( yr ,..., y7) f 0 and a > 0 implies VCV(h(A + a, p) + y) 2 TWA + a, p)). Furthermore, if Condition 3 is replaced by Condition 3’, a result similar to Theorem 2 may be established for a larger set of utility functions. 3’. If q!~E 9 with p . &w) < a a.e. Q on fi then JP(A, p) + 4 c ZLP(A+ a, p). This larger set of functions, ,VP’, is comprised of all V which satisfy property (2) but not necessarily property (1). Notice that the pair of indifference curves pictured in Fig. 2 satisfy property (2). Since V is no longer assumedto be additive the r,<‘s referred to in Theorem 2 are no longer meaningful measures.Thus, in stating our comparable results for 9; and V’,‘, we employ the following measureof risk aversion, CONDITION
5 Recall that in proving Theorem 1, Neave [5] was cited for a proof that if V is additive and Y,; is decreasing, i = I,..., T, then I.“(., p) is decreasing for arbitrary positive p.
66
JOHN P. DANFORTH
THEOREM 3. If Condition satisfied and V E %FD’,
3 is satisjed
and V E 9; or if Condition
3’ is
Max E[ V(s)] = V(h(& p)) ssdLP(A,pj S#h(A,Pl implies
frv(A,
p) is positive and decreasing (nonincreasing).
Proof qf Theorem 3. First let us define the indirect F-R * 1 + xR +T + RI, by the rule R4
PI = V(W,
utility
function
PH.
Since h(A.p) is continuous and increasing in A and V is continuous and increasing in h, Vis continuous and increasing in A. Also from our definition of V
is equal to r,(A, p) defined above. Now let A(s(w)) = min A such that i;‘cA, p) > V(s(w)). Thus if the maximizing strategy in 2’(A, p) is s*, -EL@W*(.)), P)I = V(A, P) implies -E[K@*(~)) + a, PII > (2) UA + a, PI (*) as a direct consequence of Pratt’s Theorem 2 since r,(A, p) is decreasing (nonincreasing). Now, if V E V, property (1) implies M&*(w)),
P, 4 = WA,
P, a),
so by property (2) of ‘9; WCs*(w)) + a, P) < V@*(w) + WA
P, 4).
s = (s” + Llh(A, p, a)) is not necessarily the maximizing s E 2’(A + a, p) but is feasible by Condition 3. Thus the theorem is valid for V E Y> when Condition 3 is satisfied. 0 An anonymous referee suggested the portion of this theorem dealing with Condition 3’ and VP‘.
WEALTH
If V E “ya’ property
AND
GENERALIZED
LOTTERIES
61
(2) implies
and s* + dh(A(s*(w)), p, a) is feasible by Condition 3’. Therefore the theorem is valid for V E Ypl when Condition 3’ is satisfied. As noted above, the restriction that a lottery’s prizes lie along a particular wealth-consumption expansion path rules out many interesting economic phenomena. Thus, measures of attitudes toward wealth or money lotteries like r,(A,p) need not in general convey all the information we would like regarding attitudes toward a larger set of commodity lotteries or strategies. Theorem 3 indicates that this situation is to a large degree ameliorated when V(;(A, p) is an indirect representation of a V E Vi . The relationship between an individual’s attitudes toward risks and his preference ordering of the commodity space has been central to the foregoing analysis. In Theorem 3 restrictions were placed directly on the shape of an individual’s indifference sets in order to investigate the relationship between the dollar value of lotteries and initial wealth holdings. We had earlier found that assumptions on attitudes toward risks imposed severe restrictions on the shape of indifference surfaces when preferences were additively separable. In particular, we obtained the remarkable result (see the statement and proof Lemma 1) that for additive preferences the rate of change of indifference curve slopes at the point where they intersect a common wealth-consumption curve is a positive linear function of the individual’s measures of single commodity risk aversion. This result was a crucial element in the proof of the extension of Pratt’s Theorem 2 to many commodity generalized lotteries in our Theorem 1. The link between the curvature of indifference sets and attitudes toward risk has also been noted by Stiglitz [LO]. He has considered properties of V which are implied by attitudes toward lotteries, s*, of the form s*(o) = h(A(s*(w)), p) for some fixed p and all w E a. He presents the intriguing finding for risk averters’ that in our notation would be stated: If r,(A,p) is a constant function of A E (0,co) for any p > 0 or if A . rv(A, p) is a constant function of A E (0, co) for any p > 0 then V is a homothetic function. One may easily verify that strict quasi-concavity and homotheticity of V imply V E VP for all p.* Thus we obtain for strictly quasi-concave V the two corollaries: 7 This is actually a combination of two results in [IO]. 8 If Vis homothetic, h(A, p) = A/A* . h(A*, p) for A* > 0, p > 0. Also by homotheticity V(x) = V(X + z) =) V(a) = V(ax + CLZ)for OL> 0. By stricty quasi-concavity V(fl@x) -I(1-~)(0l~+az))~~((olu)for0~~<1.Take~>1and(1-~)=1/~andwehave V(CLX+ Z) > V(U). Finally, notice that if V is homogeneous of degree A E (0, l] then it is homothetic and r&4, p) = [(A - 1)/A] *g(p).
68
JOHN P. DANFORTH COROLLARY
1. If r,(A, p) is a constant function of A E (0, 03) for any
p > 0 then
implies
COROLLARY
2. r’A
. r,(A, p) is a constantfunction of A E (0, 03)for any
p > 0 then
imp&es
.EZ::.,, E[V(s(.))l >
Wz(A + a, P)).
Finally if one restricts one’sattention to more complex lotteries, Corollary 1 may be somewhat strengthened to: COROLLARY 1’. If rv(A, p) is a constant function of A E (0, co) for any particular p > 0, and if there exists an SE S?(A, p) which maximizes E[V(*)] and Q{u ESz: S(w) # h(A(S(w)), p)for given p} >0 then
and a > 0 implies
The additional restriction in this corollary ensuresthat one of the expected utility maximizing strategies has positive probability of outcomes lying off the wealth-consumption expansion path for p. This restriction yields the desired result, since in Condition 2 for V E VP > replaces > for strictly quasi-concave homothetic functions. Thus, < may be replaced by < in the proof of Theorem 3 establishing the result (refer to the proof of Theorem 3, Eq. (*)I.
SUMMARY
OF RESULTS
We have investigated the relationship between an individual’s wealth and his dollar valuation of multicommodity lotteries in this paper. We began by proving that for additively separable utility functions monotonicity of certain single commodity measures of risk aversion was a necessary and
WEALTH
AND GENERALIZED
69
LOTTERIES
sufficient condition for wealth holdings to be monotonically related to the dollar value of every multicommodity lottery (Theorem I). We then argued that intertemporal strategies in a stochastic environment could be interpreted as multicommodity lotteries. Theorem 1 was then invoked in proving that for additive utility functions, decreasing single commodity measuresof risk aversion imply a positive relation between initial wealth and the difference between the certain dollar value of expected utility maximizing strategies and initial wealth (Theorem 2). These two results depended on an intriguing correspondence which was shown to exist between the shape of a consumer’s indifference surfaces and his single commodity measuresof risk aversion. For instance, we found that if every single commodity measure of risk aversion was decreasing then indifference surfaces “flatten out” as one moves in a direction of increasing satisfaction (Lemma 1). We found that by restricting our attention to a class of utility functions giving rise to indifference surfaces with this “flattening out” property we could eliminate the assumption of additive preferences and prove a result analogous to Theorem 2 (Theorem 3). Finally, we used the fact that all strictly quasi-concave homothetic utility functions are contained in this classto extend two theorems due to Stiglitz. Specifically we demonstrated that if an individual’s indirect utility function displays constant absolute risk aversion or constant relative risk aversion regardlessof commodity prices then this individual’s dollar value equivalent of arbitrary multi-commodity lotteries is nonnegatively related to his initial wealth (Corollaries 1, I’, and 2).
APPENDIX:
PROOF OF LEMMA
1
The proof exploits the relationship existing between the rate of change of the marginal rate of substitution of x1 for x2 and the single commodity measures of risk aversion, rU1 and rUZ. Specifically, along an indifference curve, (~2X2/(w2),1+,24
= -wh’)h’lu,‘) = c&l)
- (~;/~2’~(~,‘/~,32
. (u,‘lu,‘> + r&2)
. b4,‘/~2’)z.
(1)
Equation (1) implies that indifference curves “flatten out” (“curl up,” “are parallel”) in the neighborhood of any isocline as one moves along it away from the origin if rul and ru2 are decreasing (increasing, constant). This local property will now be shown to imply the global curvature result stated in the lemma. Without loss of generality we let 32’,> x1a2.* = CI,(A(f., p), p) and
70
JOHN
P. DANFORTH
2, < xpM = &(A(& p), p). Also, we define the real-valued functions f and g on the interval [x,~, Z,] by the implicit equations
and
respectively. As in the body of the paper, &.
= 4Gw,
P), P, 4,
i = 1,2.
Of course the sets {(x, ,f(x,)): x1 E [xlM, SJ} and {(x1 + dh, , g(xJ): a,]} are just segments of the indifference curves associated with XlE w, (xr”, x2”) and (xlM + dh, , xZM + Ah,), respectively. Our assumptions on the ui’s guarantee, by the Implicit Function Theorem, that both f and g are twice continuously differentiable. We also know that g(x,) 3 f(xJ on
blM, $1 sincegh”) 2 fh”)
andg’b4 > f ‘(4 if gCd = f (x1)(g’h) =
-ul’(xl + dh,)/u,‘( g(xr)), f’(x,) = -u,‘(x,)/u,‘( f(x,)) and u1 is strictly concave). These characteristics off and g in conjunction with (1) are used below in proving that
ml)
< (G >)f’(Xl>
a.e. Lebesgue measure on [x,~, a,] if rul and r,,
are decreasing (nonincreasing,
increasing and xlM > 0).
Gv
Proof of (2). First, recall that (~,~‘,f(x,~)) and (xrM + 4, , g(xlM)) maximize utility subject to parallel budget constraints, hence JC’(xlM) > g’(xlM) with the inequality holding only if xrM = 0. This can be shown to imply, using the mean value theorem, that if f’(xJ < (<) g’(xl) for some Xl E (XI”, ~$1 then there exists an X1 E [xr”, a,] with f’(Z,) < (<) g’&) and f”(ZJ < (--c) g”&). Also, if x1M > 0 and f’(xJ > g/(x1) for some a,], then there exists an X1 E [x,~, ZJ with f’(Q > g’(X,) and Xl E (XP, f”&) > g”(T,). Observe, however, that (1) implies f”(x~> = -r&A
*f’h)
+ r,,(fW
- (fY-G2
and
g”(xJ = -rul(xl + 4) It is thus impossible for f’(Q
* g’(xl> + r&(x1)> - (g’(xlN2.
S (<, 2) g’(j;‘l) and f”&)
< (<, 2) g”(Q
WEALTH
AND
GENERALIZED
71
LOTTERIES
to hold simultaneously for an F1 E [xIM, aI] if rYI and rU2 are decreasing (nonincreasing, increasing) since f(Q < g(X,). This establishes (2). By (2) it is clear that
%I g'(x) d-x <(<,>1s'; dx, sqM Xlf'(x)
or equivalently,
or equivalently, id%) - f(%) -=E(G, >) d--G”) - f(Xl”) if ru, and ruz are decreasing (nonincreasing, Therefore, recalling that f(gI) = 55,we have
= &
increasing,
and xIM > 0).
2, + Ah, > (2, <) g(%), and consequently
if rul and rul are decreasing (nonincreasing, This completes the proof of the lemma.
increasing,
and xIM > 0).
REFERENCES
1. A. A. ALCHIAN, The meaning of utility measurement, Amer. Econ. Rev. (1953), 2650. 2. K. J. ARROW, “Lecture II, Aspects of a Theory of Risk Bearing,” Yrjo Jahnsson Lectures, Helsinki, 1965. 3. J. P. DANFORTH, “Expected Utility, Mandatory Retirement and Job Search,” unpublished paper, June 1973. 4. R. DESCHAMPS, Risk aversion and demand functions, Econometricu 41(1973), 455-466. 5. E. H. NEAVE, Multiperiod consumption-investment decisions and risk preference, J. Econ. Theory 3 (1971), 40-53. 6. R. A. POLLAK, Additive von Neumann-Morgenstern utility functions, Econometrica 35 (1967), 485-494. 7. J. W. PRATT, Risk aversion in the small and in the large, Econometrica 32 (1964), 122-136. 8. A. SANDMO, Capital, risk, consumption and portfolio choice, Econometrica 37 (1969), 586-599. 9. SPENCE AND ZECKHAUSER, The effect of the timing of consumption decisions and the resolution of lotteries on the choice of lotteries, Econometrica 40 (1972), 401-405. 10. J. E. STIGLITZ, Behavior towards risk with many commodities, Economefrica 37 (1969), 660-667.
OF ECONOMIC
Wealth
THEORY
15,
54-71 (1977)
and the Value of Generalized
Lotteries
JOHN P. DANFORTH* University
of Minnesota,
Minneapolis,
Minnesota
55455
Received January 27, 1975; revised October 26, 1976
In two now famous articles, Arrow [2] and Pratt [7] demonstrate the existence of correspondencesbetween certain simple risk aversion functions and the demand for and valuation of random economic prospects. Each of these authors simplifies his analysis by assuming that an individual’s utility may be written as a function of his money or wealth. While this assumption appears fairly restrictive, Arrow notes that it is consistent with individual preferences which have as their domain a multidimensional commodity space. Specifically he observes that in perfect competition if prices are constant one may interpret money or wealth as a well-defined composite good. Recently Stiglitz and Deschamps have commented explicitly on the implications ot the composite good interpretation suggestedby Arrow. Stiglitz [IO] analyzes the relation between commodity demand functions and the Pratt-Arrow measures of risk aversion for a von Neumann-Morgenstern utility of money function. Deschamps [4] pursues this same general topic with particular emphasis placed on problems arising from the choice of numeraire. Spence and Zeckhauser [9] also discuss interpreting von NeumannMorgenstern utility of money functions as indirect representations of preferences for goods purchasable with money holdings. They argue that the protasis of the composite good theorem is generally contradicted in an uncertain environment. That is, prices are not generally constant over states of the world, and even if they were the pattern of intertemporal evolution of random economic processesmight make free trading of commodities in each state of the world impossible. * I have benefited greatly from Berm Stigum’s criticism during the preparation of this paper. Also, Leonid Hurwicz and the other participants in the Center for Economic Research Bag Lunch Seminar at the University of Minnesota, and an anonymous referee, provided helpful comments on earlier drafts. None of these individuals should be held responsible for any errors which may remain. 54 Copyright All rights
0 1977 by Academic Press, Inc. of reproduction in any form reserved.
ISSN
00224531
WEALTH
AND GENERALIZED
LOTTERIES
33
The aim of the present paper is to elucidate someinterconnections between an individual’s utility function and his valuation of random economic processeswhen the realizations of these processesare not necessarily representable as composite goods. We shall distinguish between two types of risky ventures in the analysis. First, a random process with all of its realizations lying on a single wealth-consumption expansion path1 is termed a simple lottery or S-L. Second, any probabilistic scheme for the distribution of prizes which is not an S-L is termed a generalized lottery or G-L. The bulk of the paper is devoted to the derivation of a result which is a partial extension of an important theorem due to Pratt [7]. His theorem2 establishesan if and only if relationship between the form of an individual’s utility of money function and his valuation of fixed lotteries for varying levels of initial wealth. In Theorem 1 I present a similar “if and only if” relationship for many commodity lotteries when preferences are restricted so as to be of the additive von Neumann-Morgenstern variety.3 The multicommodity lottery is then viewed as a representation of the random consumption sequenceobtained by employing some intertemporal strategy in a stochastic environment. The current money value of such strategies is related to an individual’s utility function and initial wealth endowment in Theorem 2. This theorem again posits an additive utility function; however, an essentially identical result is establishedin Theorem 3 for a classof nonadditive utility functions. All quasi-concave utility functions homogeneous of degree h E (0, 1) form a proper subset of this class. The last few pages contain extensions of two results provided by Stiglitz [lo] on the relation between attitudes toward risk and demand functions. Stiglitz demonstrates that certain restrictions on the form of an individual’s commodity demand functions may be implied by assumptions pertaining to his valuation of simple lotteries. We show in Corollaries 1, I’, and 2 that certain revealed attitudes toward arbitrary simple lotteries tell us a great deal about the individual’s valuation of generalized lotteries. One intriguing implication of these results is that an individual who is risk neutral toward arbitrary simple lotteries will display positive and decreasingabsolute risk aversion toward all generalized lotteries.4 1Let h: R;+l ---fR;” be the ordinary demandfunction associated with maximizinga utility functionu: R,” + R’. A set of points,B CR,“, is a wealth-consumption expansion path if thereexistsa p ER,” suchthat B = (x E R,.n : x = h(p, A) for some A > 0). e See [7,
Theorem 2, p. 1301.
3For a discussion of the implicationsof this assumption see [6]. 4 We say an individual is risk neutral (risk auerse) toward a lottery if he is indifferent between that lottery and a dollar amount equal to (less than) the expected market value of the lottery prizes. An individual is decreasingly absolute risk auerse if the dollar amount he would be just willing to exchange for a lottery increases with his wealth.
56
JOHN ADDITIVE
PREFERENCES
P. DANFORTH AND
THE
VALUE
OF
LOTTERIES
Throughout this paper we shall be dealing with random prospects which are referred to as lotteries. Formally, a lottery is a quadruple, (Sz, 9, Q, L(.)), where (Sz,S, Q) is a probability spaceand L: Sz + R+T is a function which is measurablewith respect to 9. We shall usually omit reference to an underlying probability space and identify particular lotteries by numerical superscript, i.e., L1, L2, etc. Tn addition, we shall limit our attention to lotteries for which E(L)
= s, L dQ
exists and is finite. We shall denote the classof such lotteries as 9. Individual preferences are assumedto be representable by a twice continuously differentiable, increasing, and strictly quasi-concave function V: R,T + R1. In particular, L1 is preferred (indifferent) to L2 by the individual if and only if E(Vo C) > (=) E(Vo L2). We shall be concerned initially with additive V functions. 1.
DEFINITION
V is additive if there exist functions ui , i = 1, 2,..., T
such that V(x)
=
01 +
i
U$(XJ
Vx = (x1 , x2 ,..., xT) E R+T.
i=l
CONDITION 1. V is additive. Note that for T > 1 ui’(xi) > 0 for all i and xi b 0, since V is increasing. Also, at least (T - 1) of the ui are strictly concave throughout because of the strict quasi-concavity of V. Condition 2 extends this property to all T of the ui’s. CONDITION 2. Each ui is a strictly concave function of xi , i = 1, 2,..., T. Given our assumptionson V, the demand functions given in the following definition are continuous. DEFINITION 2. h(A, p) E R+T is the x E {x E R+T: p . x < A} which maximizes V on this set. It is easily established that for any x’ E R+T there exists an (A’, p’) E ~ x R+T such that x’ = h(A’, p’). DEFINITION 3. For any L E 3’ and p E R,T, A(L, minimize A subject to V (/?(A, p)) 3 EC V 0 L).
p) is the solution to:
WEALTH DEFINITION
4.
For any
AND
GENERALIZED
57
LOTTERIES
L E 9 and p E R+T
JW, P) = P . E(L) - 4-b P). A(L, p) is called the dollar value of lottery L evaluated at commodity prices p. fl(L,p) is called the risk premium associated with lottery L and prices p. From Condition 2 it is immediate that 17(.) is nonnegative for any (L, p) and strictly positive for nondegenerate lotteries. The notation provided in the next definitions will be employed in the statement and proof of Theorem 1. DEFINITION
5.
For any p E R+T and a > 0, dh(A,p,
a) will denote the
vector
Conditions
1 and 2 imply that &(A,
DEFINITION
6.
Let
p, a) > 0, i.e., no commodity
is inferior.
L E 9 and X E R+=. The lottery L + X is the function
on Q for which
(L + X)(w) = L(w) + x
VOJEQ.
As one might well expect, the individual’s attitudes toward risk are a prime determinant of the dollar value he places on a given lottery. A measure of “single commodity risk aversion” is useful in describing these attitudes when preferences are additive. DEFINITION
7. ruj(xj)
=
-u:)(xj)/uj’(xj)
is referred to as the measureof single commodity risk aversion of the function uj evaluated at xj . For T = 1 this measure is simply the well known Pratt-Arrow measure of absolute risk aversion. Pratt’s Theorem 2 provides a complete description of the relation between initial wealth and an individual’s dollar evaluation of a given lottery for T = 1 and monotonic r,(.). Specifically, his theorem asserts(in our notation) that
for all nondegenerate one-dimensional lotteries L E 3 and any (p, a) E R+l x R+l if and only if r,(x) is a decreasing (nonincreasing) function of x on R+'. The same implications hold if “increasing” is substituted for decrasing and the inequality is reversed.
58
JOHN P. DANFORTH
The following theorem provides between lotteries and risk premiums THEOREM 1. If Conditions
some insight for T > 1.
on the correspondence
1 and 2 are satisfied,
for all nondegenerate L E 9, any p E R, T and any a > 0 if and only if rLci(xi) is a strictly decreasing (nonincreasing) ,function of xi on [0, w) for all i = 1, 2 ,..., T. Also,
all nondegenerate L E 9, any p E R,T and any a > 0 only if rvi(xi) a strictly increasing function of xi on [0, CO)for all i z I, 2,..., T.
for
is
Figure 1 illustrates Theorem 1 for T = 2 and pz = 1 when rul and ra2 are strictly decreasing. Note that dh(A(L, p), p, a) has been abbreviated as dh in the diagram.
.?r(L +3h,
p) 1
ix:p.x=A(L,p)+a; ;x:p.x=A(L,p);
-;x:p.x=p.ELI
FIGURE
x,
1
Proof of Theorem 1. First I shall establish the “only theorem. That is, it will be proven that if
if” portion
of the
for all L E 9, anyp E RT and any a > 0, then rthi(x,), i = I,..., T, is decreasing (nonincreasing, increasing) on R+l. As an initial step we construct a lottery Li = (Lli, Lzi,..., LTi) as follows.
WEALTH
Take an arbitrary satisfying
AND
integrable
GENERALIZED
function
LOTTERIES
59
J?~: Q + R+l and solve for the z
E(Ui 0 LJ = 2$(z). Next find an (AZ, fi) pair such that h,(A”,fi) = z + A, where d is an arbitrary positive number. Since A,(.,$) is continuous and h,(O,j) = 0 there exists an A1 (less than A2 since h is increasing in A) with hi(Al, 3) = z. Finally, let L,i = hj(A’, ~5)forj # i and let Lii = Li . By construction, V(h( Al, p)) = E( v 0 Li) and II(Li, j) = j3i[E(Ll)
Now by definition II(L”
-
Z]*
if
+ dh(Al, j, A2 - Al),B)
< (G, >I fl(Li>fi,
then j$ uAMA~, PN < (<, >) C z.&(~",jq) &i
+ E(U~ o (L: + A))
or, equivalently, Ui(hi(A2, j)) = Ui(Z + A) < (<,
>) E(Ui 0 (Li” + 0)).
Since A > 0 and Li were chosen arbitrarily, Pratt’s Theorem 2, cited above, guarantees that rUi(xi) is decreasing (nonincreasing, increasing). We now verify the “if” portion of the theorem. That is, it will be proven that if the r,i(x& i = 1, 2 ,..., T, are decreasing (nonincreasing) on R+l then T;T(L + 44%
P), P, 4 P) < (G) WL, P>
for all L E 9, any p E R,T and any a > 0. The T = 2 case is established first and then the result is extended by induction to T > 2. If T = 2 E( V 0 L) = E(u, 0 L,) + E(u, 0 L,).
60
JOHN
For our arbitrarily
P.
DANFORTH
chosen L E 9, we find Z1 and f, such that i = 1,2.
E(Ui 0 LJ = u,(aJ, By Definition
3 we have
for arbitrarily chosen p E R+=. We now have two possibilities treated as separate cases.
which
will be
Case I. hi(A(L, p), p) = Zi , i = 1,2. For an arbitrarily chosen a > 0 dhi(A(L, p), p, a), henceforth denoted in this proof simply as dhi , is nonnegative for i = 1, 2 and strictly positive for one or both of them due to noninferiority. Pratt’s Theorem 2 ensures that
if rU1 and rUz are decreasing (nonincreasing). A@ + 4
P) > (2,
since V and h( ., p) are increasing.
0
Thus, 4L, P) + a
Therefore,
if rU1 and rU2 are decreasing (nonincreasing,
increasing).
Case II. hi(A(L, p), p) # Z;i , i = 1, 2. The following lemma is instrumental in establishing when these inequalities obtain. LEMMA
u,(f, + 4)
1.
the desired result
If 3i;i# hi(A, p), i = 1, 2, then + u,(f, + 4)
> (2) u,MA,
P) + Ah,) + Q44
P) + &)
if rU1 and rUz are decreasing (nonincreasing) and the inequality is reversed if rU1 and rUz are increasing and hi(A, p) > 0, i = 1, 2. Lemma 1 is graphically illustrated for r,i decreasing in Fig. 2 and is proven in the Appendix to this paper. Notice that decreasing, constant, and increasing absolute risk aversion of the ~0s is associated, respectively, with decreasing, constant, and increasing curvature of the indifference curves as one moves along any wealth-consumption expansion path. Tn a later section this intriguing result is related to some work on the link between attitudes toward
WEALTH
AND
GENERALIZED
LOTTERIES
61
risks and commodity demand functions presented in [4, lo]. In the figure Y = f - W(LP),P). Pratt’s Theorem 2 guarantees that if rul and rut are decreasing (nonincreasing), then
h(NLph p) + Y = R FIGURE
2
Application of Lemma 1 thus yields
whenever rul and ruz are decreasing(nonincreasing). Thus,
for rul and ruz decreasing (nonincreasing) as asserted. Extension of the result to T > 2 is accomplished by induction. We first introduce some new notation which will hopefully simplify the argument:
62
JOHN
is the maximizer
P.
DANFORTH
for
It should readily be apparent that ~+YA,
P) = p;+;
Wb, P) + &z+1(&+1)
subject to b + ~n+lxn+l
d 2,
and that
Next let rpn@, P) = - G(b,
P)/ F,“(b, P).
Neave [5, Theorem 1, p. 461 has proven that if the rui , i = I,..., II, are decreasing (nonincreasing) then r&b, p) is decreasing (nonincreasing) in b. Our induction hypothesis for n > 2, IH . (n), is: “If rui , i = I,..., n, are decreasing (nonincreasing) then F(b, p) = i
E(ui 0 L,)
i=l
implies V”(b + a, p) < (G)
f ECui 0 (L + &“(b, i=l
p, a)))
for a > 0 and at least one nondegenerate Li , i = l,..., n. If no Li is nondegenerate only weak inequality is assured. Now we simply find the xzl such that E&3+1 o L+1)
= %+&n*+1)
and the b* such that i
E(ui o Li) = rn(b*, p).
WEALTH
AND GENERALIZED
LOTTERIES
63
Neave’s result, in combination with Lemma 1, ensures that if rUi , i=l ,..., IZ + 1 are decreasing(nonincreasing) then
implies
This implication together with IH . (n) and Pratt’s Theorem 2 implies IH . (n + I), a restatement or IH . (n) with (n + 1) inserted wherever n appears. IH . (2) has been proven and therefore IH CT) is valid. IH . (T) is simply the “if” assertion of Theorem 1.
ADDITIVE
PREFERENCES AND THE VALUE
OF CHOSEN STRATEGIES
It is quite clear that few if any economic phenomena would conform to the basic specifications of Theorem 1. That is, Theorem 1 provides no options for decision making or choice. Given the opportunity to allocate the increment to wealth, a, it is doubtful that augmenting the initial lottery, L, by dh(A(L, p), p, a) would be optimal. Also note that the value of a specific lottery is compared before and after changes in wealth, thus neglecting the lottery choice problem. In order to discusssomeproblems of individual decision making the concept of a consumption strategy is introduced. A consumption strategy, s, associateswith each state of the world, w E Q, a vector (x1 ,..., xr). Restricting our attention to those strategies which are integrable we have s E 9’. The fact that consumption strategy connotes some notion of maximization does not obviate its status as a lottery. The following definition will be utilized in the remainder of the discussion. DEFINITION 8. 9(A, p) = {s E 8: s is feasible for wealth A and certain prices p}. Reference to “feasible” in Definition 8 is left vague since the restrictions
64
JOHN
P.
DANFORTH
associated with feasibility vary greatly from problem to problem. Such factors as the timing 01 consumption decisions, availability of investment opportunities, and debt constraints would all enter into a definition of feasibility. An additional condition is required for the subsequent discussion. CONDITION 3. If z E R,T and p z < a then, for any A E R,l, p E R,T, and a E s, Z(A, p)’ z C Z(A -t a, p). The condition requires that if an individual’s initial wealth is increased, he may useall or part ot it to finance additional purchases of the T commodities at the prevailing market prices. THEOREM2.
If
Conditions
1, 2, and 3 are satisfied
,Ey(yp, ELW(.Nl = w44
P))
S#hld.P) implies
if ru ,( .) is decreasing
(nonincreasing)
for all i = I,.. ., T.
The interpretation of the theorem is straightforward. An individual is asked to determine the minimum expenditure, at prevailing market prices, required to finance a riskless consumption plan which he would accept in exchange for his current uncertain prospects. The individual is then presented with a dollars. He is now asked to calculate the minimum costs of a riskless consumption plan that he would be willing to trade for the consumption strategy he could pursue with his augmented initial wealth. The theorem assertsthat the cost of procuring an acceptable safe consumption plan will have risen by more than (at least) a dollars if the individual’s single commodity measures of absolute risk aversion are decreasing (nonincreasing). Though each strategy, s, may also be thought of as a lottery, there is a subtle difference between this result and Theorem 1. The individual’s strategy has been chosen by him. It is his plan of action for market activities now and in the future. If his initial endowments were to change he might revise the portfolio of stocks he wants to hold. This may result from a change in opportunities (his original wealth might not have been sufficient to meet the margin requirements of his revised portfolio) or simply becausehis attitudes toward uncertain prospects depend on his initial wealth. Proof of Theorem 2. Since any s E L?(A, p) is also an element of Y, we know from Theorem 1 that if
-W’M~))I = W@,
P))
WEALTH
AND GENERALIZED
65
LOTTERIES
then
WV + WA, P, 4.I > (2) W(A + 0,PN if rui(.) is decreasing (nonincreasing) for all i = I, 2,..., T. Since MAP,
a) > 0,
Condition 3 implies
s + MA,
VALUE
p, a) 6 WA + 4 P).
OF STRATEGIES FOR A CLASS OF NONADDITIVE
PREFERENCES
Up to this point the analysis has rested on the assumption that preferences may be represented by an additive von Neumann-Morgenstern utitility function (Condition 1). Despite the highly restrictive nature of this condition, in many instances it may be a perfectly acceptable simplifying assumption. Nevertheless it would be of value to be able to say something about generalized commodity lotteries when preferences are not additive. Happily there are conditions which, if satisfied by a nonadditive utility function, imply a result analogous in content to Theorem 2 above. These restrictions, unlike Conditions 1 and 2, are price specific. That is, for every price vector, p, there is a set of utility functions VP , which satisfy these new conditions. Specifically, V EVP if and only if, (1) k(A, p) is a linear function of A, and (2) V@(A,p) + J) = V(h(A,p)) y = ( yr ,..., y7) f 0 and a > 0 implies VCV(h(A + a, p) + y) 2 TWA + a, p)). Furthermore, if Condition 3 is replaced by Condition 3’, a result similar to Theorem 2 may be established for a larger set of utility functions. 3’. If q!~E 9 with p . &w) < a a.e. Q on fi then JP(A, p) + 4 c ZLP(A+ a, p). This larger set of functions, ,VP’, is comprised of all V which satisfy property (2) but not necessarily property (1). Notice that the pair of indifference curves pictured in Fig. 2 satisfy property (2). Since V is no longer assumedto be additive the r,<‘s referred to in Theorem 2 are no longer meaningful measures.Thus, in stating our comparable results for 9; and V’,‘, we employ the following measureof risk aversion, CONDITION
5 Recall that in proving Theorem 1, Neave [5] was cited for a proof that if V is additive and Y,; is decreasing, i = I,..., T, then I.“(., p) is decreasing for arbitrary positive p.
66
JOHN P. DANFORTH
THEOREM 3. If Condition satisfied and V E %FD’,
3 is satisjed
and V E 9; or if Condition
3’ is
Max E[ V(s)] = V(h(& p)) ssdLP(A,pj S#h(A,Pl implies
frv(A,
p) is positive and decreasing (nonincreasing).
Proof qf Theorem 3. First let us define the indirect F-R * 1 + xR +T + RI, by the rule R4
PI = V(W,
utility
function
PH.
Since h(A.p) is continuous and increasing in A and V is continuous and increasing in h, Vis continuous and increasing in A. Also from our definition of V
is equal to r,(A, p) defined above. Now let A(s(w)) = min A such that i;‘cA, p) > V(s(w)). Thus if the maximizing strategy in 2’(A, p) is s*, -EL@W*(.)), P)I = V(A, P) implies -E[K@*(~)) + a, PII > (2) UA + a, PI (*) as a direct consequence of Pratt’s Theorem 2 since r,(A, p) is decreasing (nonincreasing). Now, if V E V, property (1) implies M&*(w)),
P, 4 = WA,
P, a),
so by property (2) of ‘9; WCs*(w)) + a, P) < V@*(w) + WA
P, 4).
s = (s” + Llh(A, p, a)) is not necessarily the maximizing s E 2’(A + a, p) but is feasible by Condition 3. Thus the theorem is valid for V E Y> when Condition 3 is satisfied. 0 An anonymous referee suggested the portion of this theorem dealing with Condition 3’ and VP‘.
WEALTH
If V E “ya’ property
AND
GENERALIZED
LOTTERIES
61
(2) implies
and s* + dh(A(s*(w)), p, a) is feasible by Condition 3’. Therefore the theorem is valid for V E Ypl when Condition 3’ is satisfied. As noted above, the restriction that a lottery’s prizes lie along a particular wealth-consumption expansion path rules out many interesting economic phenomena. Thus, measures of attitudes toward wealth or money lotteries like r,(A,p) need not in general convey all the information we would like regarding attitudes toward a larger set of commodity lotteries or strategies. Theorem 3 indicates that this situation is to a large degree ameliorated when V(;(A, p) is an indirect representation of a V E Vi . The relationship between an individual’s attitudes toward risks and his preference ordering of the commodity space has been central to the foregoing analysis. In Theorem 3 restrictions were placed directly on the shape of an individual’s indifference sets in order to investigate the relationship between the dollar value of lotteries and initial wealth holdings. We had earlier found that assumptions on attitudes toward risks imposed severe restrictions on the shape of indifference surfaces when preferences were additively separable. In particular, we obtained the remarkable result (see the statement and proof Lemma 1) that for additive preferences the rate of change of indifference curve slopes at the point where they intersect a common wealth-consumption curve is a positive linear function of the individual’s measures of single commodity risk aversion. This result was a crucial element in the proof of the extension of Pratt’s Theorem 2 to many commodity generalized lotteries in our Theorem 1. The link between the curvature of indifference sets and attitudes toward risk has also been noted by Stiglitz [LO]. He has considered properties of V which are implied by attitudes toward lotteries, s*, of the form s*(o) = h(A(s*(w)), p) for some fixed p and all w E a. He presents the intriguing finding for risk averters’ that in our notation would be stated: If r,(A,p) is a constant function of A E (0,co) for any p > 0 or if A . rv(A, p) is a constant function of A E (0, co) for any p > 0 then V is a homothetic function. One may easily verify that strict quasi-concavity and homotheticity of V imply V E VP for all p.* Thus we obtain for strictly quasi-concave V the two corollaries: 7 This is actually a combination of two results in [IO]. 8 If Vis homothetic, h(A, p) = A/A* . h(A*, p) for A* > 0, p > 0. Also by homotheticity V(x) = V(X + z) =) V(a) = V(ax + CLZ)for OL> 0. By stricty quasi-concavity V(fl@x) -I(1-~)(0l~+az))~~((olu)for0~~<1.Take~>1and(1-~)=1/~andwehave V(CLX+ Z) > V(U). Finally, notice that if V is homogeneous of degree A E (0, l] then it is homothetic and r&4, p) = [(A - 1)/A] *g(p).
68
JOHN P. DANFORTH COROLLARY
1. If r,(A, p) is a constant function of A E (0, 03) for any
p > 0 then
implies
COROLLARY
2. r’A
. r,(A, p) is a constantfunction of A E (0, 03)for any
p > 0 then
imp&es
.EZ::.,, E[V(s(.))l >
Wz(A + a, P)).
Finally if one restricts one’sattention to more complex lotteries, Corollary 1 may be somewhat strengthened to: COROLLARY 1’. If rv(A, p) is a constant function of A E (0, co) for any particular p > 0, and if there exists an SE S?(A, p) which maximizes E[V(*)] and Q{u ESz: S(w) # h(A(S(w)), p)for given p} >0 then
and a > 0 implies
The additional restriction in this corollary ensuresthat one of the expected utility maximizing strategies has positive probability of outcomes lying off the wealth-consumption expansion path for p. This restriction yields the desired result, since in Condition 2 for V E VP > replaces > for strictly quasi-concave homothetic functions. Thus, < may be replaced by < in the proof of Theorem 3 establishing the result (refer to the proof of Theorem 3, Eq. (*)I.
SUMMARY
OF RESULTS
We have investigated the relationship between an individual’s wealth and his dollar valuation of multicommodity lotteries in this paper. We began by proving that for additively separable utility functions monotonicity of certain single commodity measures of risk aversion was a necessary and
WEALTH
AND GENERALIZED
69
LOTTERIES
sufficient condition for wealth holdings to be monotonically related to the dollar value of every multicommodity lottery (Theorem I). We then argued that intertemporal strategies in a stochastic environment could be interpreted as multicommodity lotteries. Theorem 1 was then invoked in proving that for additive utility functions, decreasing single commodity measuresof risk aversion imply a positive relation between initial wealth and the difference between the certain dollar value of expected utility maximizing strategies and initial wealth (Theorem 2). These two results depended on an intriguing correspondence which was shown to exist between the shape of a consumer’s indifference surfaces and his single commodity measuresof risk aversion. For instance, we found that if every single commodity measure of risk aversion was decreasing then indifference surfaces “flatten out” as one moves in a direction of increasing satisfaction (Lemma 1). We found that by restricting our attention to a class of utility functions giving rise to indifference surfaces with this “flattening out” property we could eliminate the assumption of additive preferences and prove a result analogous to Theorem 2 (Theorem 3). Finally, we used the fact that all strictly quasi-concave homothetic utility functions are contained in this classto extend two theorems due to Stiglitz. Specifically we demonstrated that if an individual’s indirect utility function displays constant absolute risk aversion or constant relative risk aversion regardlessof commodity prices then this individual’s dollar value equivalent of arbitrary multi-commodity lotteries is nonnegatively related to his initial wealth (Corollaries 1, I’, and 2).
APPENDIX:
PROOF OF LEMMA
1
The proof exploits the relationship existing between the rate of change of the marginal rate of substitution of x1 for x2 and the single commodity measures of risk aversion, rU1 and rUZ. Specifically, along an indifference curve, (~2X2/(w2),1+,24
= -wh’)h’lu,‘) = c&l)
- (~;/~2’~(~,‘/~,32
. (u,‘lu,‘> + r&2)
. b4,‘/~2’)z.
(1)
Equation (1) implies that indifference curves “flatten out” (“curl up,” “are parallel”) in the neighborhood of any isocline as one moves along it away from the origin if rul and ru2 are decreasing (increasing, constant). This local property will now be shown to imply the global curvature result stated in the lemma. Without loss of generality we let 32’,> x1a2.* = CI,(A(f., p), p) and
70
JOHN
P. DANFORTH
2, < xpM = &(A(& p), p). Also, we define the real-valued functions f and g on the interval [x,~, Z,] by the implicit equations
and
respectively. As in the body of the paper, &.
= 4Gw,
P), P, 4,
i = 1,2.
Of course the sets {(x, ,f(x,)): x1 E [xlM, SJ} and {(x1 + dh, , g(xJ): a,]} are just segments of the indifference curves associated with XlE w, (xr”, x2”) and (xlM + dh, , xZM + Ah,), respectively. Our assumptions on the ui’s guarantee, by the Implicit Function Theorem, that both f and g are twice continuously differentiable. We also know that g(x,) 3 f(xJ on
blM, $1 sincegh”) 2 fh”)
andg’b4 > f ‘(4 if gCd = f (x1)(g’h) =
-ul’(xl + dh,)/u,‘( g(xr)), f’(x,) = -u,‘(x,)/u,‘( f(x,)) and u1 is strictly concave). These characteristics off and g in conjunction with (1) are used below in proving that
ml)
< (G >)f’(Xl>
a.e. Lebesgue measure on [x,~, a,] if rul and r,,
are decreasing (nonincreasing,
increasing and xlM > 0).
Gv
Proof of (2). First, recall that (~,~‘,f(x,~)) and (xrM + 4, , g(xlM)) maximize utility subject to parallel budget constraints, hence JC’(xlM) > g’(xlM) with the inequality holding only if xrM = 0. This can be shown to imply, using the mean value theorem, that if f’(xJ < (<) g’(xl) for some Xl E (XI”, ~$1 then there exists an X1 E [xr”, a,] with f’(Z,) < (<) g’&) and f”(ZJ < (--c) g”&). Also, if x1M > 0 and f’(xJ > g/(x1) for some a,], then there exists an X1 E [x,~, ZJ with f’(Q > g’(X,) and Xl E (XP, f”&) > g”(T,). Observe, however, that (1) implies f”(x~> = -r&A
*f’h)
+ r,,(fW
- (fY-G2
and
g”(xJ = -rul(xl + 4) It is thus impossible for f’(Q
* g’(xl> + r&(x1)> - (g’(xlN2.
S (<, 2) g’(j;‘l) and f”&)
< (<, 2) g”(Q
WEALTH
AND
GENERALIZED
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LOTTERIES
to hold simultaneously for an F1 E [xIM, aI] if rYI and rU2 are decreasing (nonincreasing, increasing) since f(Q < g(X,). This establishes (2). By (2) it is clear that
%I g'(x) d-x <(<,>1s'; dx, sqM Xlf'(x)
or equivalently,
or equivalently, id%) - f(%) -=E(G, >) d--G”) - f(Xl”) if ru, and ruz are decreasing (nonincreasing, Therefore, recalling that f(gI) = 55,we have
= &
increasing,
and xIM > 0).
2, + Ah, > (2, <) g(%), and consequently
if rul and rul are decreasing (nonincreasing, This completes the proof of the lemma.
increasing,
and xIM > 0).
REFERENCES
1. A. A. ALCHIAN, The meaning of utility measurement, Amer. Econ. Rev. (1953), 2650. 2. K. J. ARROW, “Lecture II, Aspects of a Theory of Risk Bearing,” Yrjo Jahnsson Lectures, Helsinki, 1965. 3. J. P. DANFORTH, “Expected Utility, Mandatory Retirement and Job Search,” unpublished paper, June 1973. 4. R. DESCHAMPS, Risk aversion and demand functions, Econometricu 41(1973), 455-466. 5. E. H. NEAVE, Multiperiod consumption-investment decisions and risk preference, J. Econ. Theory 3 (1971), 40-53. 6. R. A. POLLAK, Additive von Neumann-Morgenstern utility functions, Econometrica 35 (1967), 485-494. 7. J. W. PRATT, Risk aversion in the small and in the large, Econometrica 32 (1964), 122-136. 8. A. SANDMO, Capital, risk, consumption and portfolio choice, Econometrica 37 (1969), 586-599. 9. SPENCE AND ZECKHAUSER, The effect of the timing of consumption decisions and the resolution of lotteries on the choice of lotteries, Econometrica 40 (1972), 401-405. 10. J. E. STIGLITZ, Behavior towards risk with many commodities, Economefrica 37 (1969), 660-667.