Mark J. Machina CARDINAL PROPERTIES OF "LOCAL UTILITY FUNCTIONS"

Transcription

1 Mark J. Machina CARDINAL PROPERTIES OF "LOCAL UTILITY FUNCTIONS" This paper outlines the carinal properties of "local utility functions" of the type use by Allen [1985], Chew [1983], Chew an MacCrimmon [1979], Dekel [1985], Epstein [1985], Fishburn [1984], Karni an Safra [1985], Machina [1982,1983, 1984], Yaari [1985] an others. 1. SMOOTH PREFERENCES AND LOCAL UTILITY FUNCTIONS Consier the set D[a,b] of all cumulative istribution functions F( ) over some interval [a,b], an an iniviual whose preference ranking over this set can be represente by a real-value preference functional V(F), in the sense that the istribution F*( ) is weakly preferre to F( ) if an only if V(F*) V(F). Note that V( ) is orinal in the sense that some other functional V*( ) will represent the same preference ranking if an only if V*( ) = φ(v( )) for some increasing function φ( ). If the iniviual is an expecte utility maximizer, we know that V( ), or some increasing transformation of it, will take the linear form V(F) U(x)F(x), where U( ) is known as the iniviual's von Neumann-Morgenstern utility function. In such a case we know that the iniviual will rank the istributions F*( ) an F o ( ) accoring to the sign of V(F*) - V(F o ) = U(x) [F*(x)-F o (x)] (1) Expecte utility theory has provie us with many results linking the von Neumann-Morgenstern utility function U( ) with properties of risk preferences. For example, the iniviual will prefer all first orer stochastically ominating shifts if an only if U(x) is increasing in x, an will be risk averse, i.e. averse to all mean preserving increases in risk, if an only if U(x) is concave in x. We also know that the von Neumann-Morgenstern utility 339 B. R. Munier (e.), Risk, Decision an Rationality, by D. Reiel Publishing Company.

2 340 M. J. MACHINA function U( ) is carinal in the sense that the linear preference functional V*(F) U*(x)F(x) will represent the same preferences as V( ) if an only if U*( ) = β U( ) + γ for some constants β > 0 an γ. Consier now some V( ) which is not of the linear (i.e. expecte utility) form. In such a case there exists no von Neumann-Morgenstern utility function U( ). However, if V( ) is at least ifferentiable (i.e. "smooth"), we know that it will be locally linear in the sense that we may take a first orer Taylor expansion about any istribution F o ( ): V(F*) - V(F o ) = U(x;F o ) [F*(x)-F o (x)] + o( F*-F o ) (2) where is the L 1 metric F* F o F*(x)-F o (x) x, an the term o( ) enotes a function which is zero at zero an of higher orer than its argument. Comparing the first orer term in this expansion with equation (1), it is clear that the iniviual will evaluate alternative ifferential shifts of probability mass from the istribution F( ) precisely as woul an expecte utility maximizer with von Neumann-Morgenstern utility function U( ;F o ). We refer to the function U( ;F o ) as the iniviual s local utility function at the istribution F o ( ). 2 The technique of "generalize expecte utility analysis" (e.g. Machina [1982,1983]) essentially consists of exploiting the linear approximation (2) to generalize the funamental tools, concepts an results of expecte utility analysis to the case of nonlinear but smooth preferences over probability istributions. Thus, equation (2) implies that V( ) will prefer all ifferential first orer stochastically ominating shifts from F o ( ) if an only if the local utility function U( ;F o ) is increasing, an V( ) will be averse to all ifferential mean preserving increases in risk about F o ( ) if an only if U( ;F o ) is concave. To exten these results to non-ifferential changes, consier any path {F( ;) [0,1]} from the istribution F o ( ) = F( ;0) to the istribution F*( ) = F( ;1) which is "smooth" enough so that the erivative [ F( ;)-F( ; ) ]/ exists at = for all [0,1], as for example with the "straight line" path F( ;) F*( ) + (1 ) F o ( ). By (2) we therefore have V(F( ; )) = [ ] U(x;F( ; ))F(x; ) (3) an the Funamental Theorem of Integral Calculus yiels

3 CARDINAL PROPERTIES OF "LOCAL UTILITY FUNCTIONS" 341 V(F*) - V(F o) = U(x;F( ; ))F(x; ) 1 [ ] 0 which shows how the iniviual's ranking of these two istributions epens upon the local utility functions along the path between them. In Machina [1982] such path integrals were use to emonstrate that V( ) will exhibit global first orer stochastic ominance preference an global risk aversion if an only if its local utility functions U(x;F) are respectively nonecreasing an concave at each istribution F( ) D[a,b]. For a more formal an complete evelopment of these concepts as well as applications of this approach, the reaer is referre to Machina [1982,1983] as well as the references cite below. We turn now to the carinal properties of local utility functions. (4) 2. ADDITIVE INVARIANCE Equations (2) an (4) show how the set of local utility functions {U( ;F) F( ) D[a,b]} erive from a preference functional can be use to etermine its ranking of both ifferential an global shifts from any istribution F o ( ). Our first result is that any other set of functions {U*( ;F) F( ) D[a,b]} satisfying U*(x;F) U(x;F) + γ(f) will also generate the same ifferential an global rankings via these equations. In other wors, we can a a ifferent constant γ(f) to each local utility function U( ;F) without changing the original ranking. To see that they generate the same rankings over ifferential shifts, recall that the iniviual will prefer (not prefer) such a shift from F o ( ) if an only if the first orer term in (2) is positive (negative). However, since the term [F*(x) F o (x)] integrates to zero, we have that U*(x;F o) [ F*(x)- F o(x) ] = [ U(x;F o)+ γ(f o) ] [ F*(x)- F o(x) ] = U(x;F o) [ F*(x)- F o(x) ], so that the first orer terms generate by the local utility functions U*( ;F o ) an U( ;F o ) will be ientical for all (5)

4 342 M. J. MACHINA ifferential shifts about all F o ( ) D[a,b]. To exten this equivalence to global rankings, note that [ ] U*(x;F( ; ))F(x; ) = [ [ γ ] ] U(x;F( ; ))+ (F( ; )) F(x; ) [ U(x;F( ; ))F(x; ) ] = so that the right sie of equation (4) will be ientical for any path {F( ;) [0,1]}. (6) 3. MULTIPLICATIVE INVARIANCE Given the set {U( ;F) F( ) D[a,b]} of local utility funcions generate from a preference functional V( ), our secon result is that any other set of functions {U*( ;F) F( ) D[a,b]} obtaine by means of the transformation U*(x;F) β(v(f)) U(x;F) will also generate the same ranking, for any positive continuous function β( ). Note that this implies we can multiply ifferent local utility functions by ifferent positive constants, provie that the local utility functions corresponing to inifferent istributions are multiplie by the same constant. To see this, efine the function V( ) an functional V*( ) by υ φυ ( ) β(s)s an V*(F) φ(v(f)) (7) By the chain rule, we have that the local utility function of V*( ) is given by φ (V(F)) U( ;F) = β(v(f)) U( ;F) = U*(,F), which establishes that the family of local utility functions {U*( ;F) F( ) D[a,b]} from the previous paragraph comes from an increasing transformation φ(v(f)) of the original preference function V( ), an hence represents the same global preference ranking. Finally, we note that while the local utility functions use in some of the following references are erive from

5 CARDINAL PROPERTIES OF "LOCAL UTILITY FUNCTIONS" 343 notions of ifferentiability which make weaker assumptions on the higher orer term (1), these local utility functions can also be shown to exhibit the carinal properties iscusse above. Department of Economics University of California, San Diego NOTES 1. I am grateful to Chew Soo Hong an Joel Sobel for helpful comments. 2. Applying the integration by parts formula from Machina [1982,Lemma 2] (see also Feller [1971,p.150]) yiels that the first orer term in the Taylor expansion is equal to - [F*(x) F(x)] U (x;f) x, which is seen to be precisely the classical variational erivative of the functional V( ) with respect to the function F( ). REFERENCES Allen, B., [1985] Smooth Preferences an the Approximate Expecte Utility Hypothesis, forthcoming in Journal of Economic Theory. Chew, S., [1983] A Generalization of the Quasilinear Mean with Applications to the Measurement of Income Inequality an Decision Theory Resolving the Allais Paraox, Econometrica 52, Chew, S. an K. MacCrimmon, [1979] Alpha-Nu Choice Theory: A Generalization of Expecte Utility Theory, manuscript, Faculty of Commerce, University of British Columbia. Dekel, E., [1985] An Axiomatic Characterization of Preferences Uner Uncertainty: Weakening the Inepenence Axiom, forthcoming in Journal of Economic Theory. Epstein, L., [1985] Decreasing Risk Aversion an Mean-Variance Analysis, Econometrica 53, Feller, W., [1971] An Introuction to Probability Theory an its Applications, Vol. II. (New York: John Wiley an Sons.)

6 344 M. J. MACHINA Fishburn, P., [1984] SSB Utility Theory: An Economic Perspective, Mathematical Social Sciences 8, Karni, E. an Z. Safra, [1985] "Preference Reversal" an the Observability of Preferences by Experimental Methos, forthcoming in Econometrica. Machina, M., [1982] "Expecte Utility" Analysis Without the Inepenence Axiom, Econometrica 50, Machina, M., [1983] Generalize Expecte Utility Analysis an the Nature of Observe Violations of the Inepenence Axiom, in B. Stigum an F. Wenstøp (Es.) Founations of Utility an Risk Theory with Applications (Dorrecht, Hollan: D. Reiel). Machina, M., [1984] Temporal Risk an the Nature of Inuce Preferences, Journal of Economic Theory 33, Yaari, M., [1985] The Dual Theory of Choice Uner Risk, manuscript, Department of Economics, Hebrew University of Jerusalem.