IS IT THE TALK? OR IS IT THE CAKE?

Transcription

1 IS IT THE TALK? OR IS IT THE CAKE? TASTING THE HYPOTHESES PAUL C. KETTLER PRESENTED TO THE CENTRE OF MATHEMATICS FOR APPLICATIONS DEPARTMENT OF MATHEMATICS UNIVERSITY OF OSLO 10 NOVEMBER 2008

2 CONTENTS 1 PROLOGUE 2 UTILITY THEORY 3 UTILITY COPULA 4 EXAMPLES 5 REFERENCES 6 EPILOGUE

3 PROLOGUE La pâtisserie Prologue S ils n ont plus de pain, qu ils mangent de la brioche. Let them eat cake. Marie Antoinette (probably apocryphal, perhaps Marie Thérèse)

4 PROLOGUE The question On everyone s mind as we arrive is, Are we here for the talk? or are we here for the cake? We could just as easily speak of Wheat and Oats Guns and Butter Mean and Variance Do you see differences among these pairs? I shall explain.

5 PROLOGUE The differences Taking each pair in order we see that Wheat and Oats are substitutes. Both are nutritious grains. Guns and Butter are complements. One cannot eat guns or shoot butter. One prefers higher values of the Mean, but lower values of the Variance. The talk cake example has some characteristics of complements and some of substitutes; therefore it becomes more challenging to analyze.

6 UTILITY THEORY Is utility theory useful? Utility theory is at the foundation of all economic thought. One only need look at the works of Adam Smith The Wealth of Nations and John Stuart Mill Principles of Political Economy. In recent times, however, utility theory has fallen on hard times. Researchers refer to the ordinal nature of utility theory as a drawback, and journals, responding to such pressures, are reluctant to accept papers focusing on utility concepts. The time may be right, given the historical dominance of utility theory, to breathe fresh life into the subject. The fusion with copula theory could provide that inspiration.

7 UTILITY THEORY Utility functions Utility functions in two variables can describe the benefits of choices. Economists for many years have spoken of tradeoffs. In the Mean Variance example we have seen indifference curves and the efficient frontier as concepts describing such functions.

8 UTILITY THEORY Utility function characteristics, one variable The space of utility functions of one variable usually is described with these characteristics. Consider T(x). T(x) is defined on an interval [a, ) t(x) = T (x) > 0 t (x) = T (x) < 0

9 UTILITY THEORY Utility function characteristics, two variables The space of utility functions of two variables usually is described with these characteristics. Consider T(x, y). T(x, y) is defined on a quadrant [a, ) [b, ) T x 2 T x 2 > 0 and T y > 0 < 0, 2 T x 2 < 0, 2 T x y = 2 T y x < 0, and 2 T 2 T 2 T x 2 y 2 x y 2 T y x > 0

10 UTILITY THEORY DMU Diminishing Marginal Utility Money doesn t make you happy. I now have $50 million, but I was just as happy when I had $48 million. Arnold Schwarzenegger, Governor of California, Brewer s Cinema, 1995

11 UTILITY COPULA Utility function, utility measure and the Fundamental Theorem of the Calculus Let T : [a, ) [b, ) R (1) I(x, y) := A x B y Consider A x = [a, x] [b, ), x a and B y = [b, y] [a, ), y b 2 T dx dy = T(x, y) + T(x, b) + T(a, y) T(a, b) > 0 x y Let the restrictions of T(x, y) to its lower bounds be these. T 1 (x) := T(x, b) T 2 (y) := T(a, y) Thus T(a, b) = T 1 (a) = T 2 (b) =: k

12 UTILITY COPULA (2) Domain of the utility copula (1) Let the point (u, v) R 2 + be as follows. by Equation (1) if lim y u = lim y I(x, y) = T 1 (x) T 1 (a) = v = lim x I(x, y) = T 2 (y) T 2 (b) = x a y b dt 1 ds ds > 0 dt 2 dt dt > 0 ( T(x, y) T(a, y) ) = lim x ( T(x, y) T(x, b) ) = 0 This result is ensured by the assumptions, now made, (3) T lim y x = lim T x y = 0

13 UTILITY COPULA Domain of the utility copula (2) Remark This assumption on the limits of the first partial derivatives justifies calling the restriction of T(x, y) to the boundaries y = b and x = a the marginal utility functions T 1 (x) and T 2 (y). That the sufficiency of Equations (3) for u and v to be the restrictions T 1 (x) T 1 (a) and T 2 (y) T 2 (b) is obvious.

14 UTILITY COPULA Domain of the utility copula (3) Further remark Their necessity is also transparent, and follows readily from the assumption that either first derivative bounded above zero by γ for some point x or y leads to a contradiction. In such event, T(x, y) T(a, y) > (x a)γ > 0 or T(x, y) T(x, b) > (y b)γ > 0, independent, respectively, of y or x, insofar as the first derivatives are monotone decreasing. As the same bounds apply in the limits, u and v cannot be the restrictions of T(x, y) to y = b and x = a, as assumed in Equations (2).

15 UTILITY COPULA Construction of the utility copula Referring again to Equation (1), making substitutions for the marginal functions and the constant k, one has (4) C(u, v) := C ( T 1 (x) k, T 2 (y) k ) = I(x, y) = T(x, y) + T 1 (x) + T 2 (y) k = T ( T1 1 (u + k), T 1 2 (v + k)) + (u + k) + (v + k) k C(u, v) = T ( T1 1 (u + k), T 1 2 (v + k)) + (u + v) + k > 0

16 UTILITY COPULA Boundary conditions The utility copula as constructed adheres both to the ground condition and the uniform margin condition. The ground condition... C(u, 0) = C(0, v) = 0 The uniform margin condition... C 1 (u) = C(u, ) = u, and C 2 (v) = C(, v) = v

17 UTILITY COPULA Utility copula measure The utility copula has a measure computed from the utility function measure by a straightforward application of the chain rule. c(u, v) := 2 C u v = 2 T x y dt 1 dx dt 2 dy

18 EXAMPLES I. A logarithmic substitution utility function (1) T(x, y) = log(x + y 1), on [1, ) 2 The margins are T 1 (x) = log x T 2 (y) = log y In this example k = 0, and the utility copula (5) C(u, v) = u + v log(e u + e v 1), on [0, ) 2 by Equation (4).

19 EXAMPLES Substitution Utility Copula Perspective and Level Curves Substitution Utility Copula Substitution Utility Copula Level Curves C(u,v) v u v C(u,v) u

20 EXAMPLES I. A logarithmic substitution utility function (2) One may extend this copula to a one-parameter family. [ ( C θ (u, v) = u θ + v θ log exp ( u θ) + exp ( v θ) 1 θ 1)], on [0, ) 2, where θ (0, ).

21 EXAMPLES II. A Gaussian utility function (1) T(x, y) = (2π) 1 g(x, y), on [1, ) 2, where g(x, y) = 1 ( 2π exp x2 + y 2 ) 2 Then T 1 (x) = 1 1 g(x, 1) = f (x)f (1) 2π 2π and T 2 (y) = 1 1 g(1, y) = f (1)f (y), 2π 2π where f (x) = 1 ) exp ( x2 2π 2

22 EXAMPLES II. A Gaussian utility function (2) The copula C(u, v) develops like this. u = T 1 (x) T 1 (1) = g(1, 1) g(x, 1) = f (1) ( f (1) f (x) ) v = T 2 (y) T 2 (1) = g(1, 1) g(1, y) = f (1) ( f (1) f (y) ) In this example k = (2π) 1 g(1, 1), and the utility copula (6) C(u, v) = uv g(1, 1) = 2πe uv, on [0, (2πe) 1 ] 2 = [0, g(1, 1)] 2 by Equation (4). This is an independent copula, discussed below.

23 EXAMPLES Gaussian Utility Copula (Independent) Perspective and Level Curves Gaussian Utility Copula (Independent) Gaussian Utility Copula (Independent) Level Curves C(u,v) v u v C(u,v) u

24 EXAMPLES II. A Gaussian utility function (3) Note that the domain of this copula is [0, g(1, 1)] 2, at which outer bounds the margins are uniform. The finite domain owes to the fact that the utility function is bounded. In general the copular bounds are these, which may be +. ( α := lim inf T1 (x) T 1 (a) ) x (7) ( β := lim inf T2 (y) T 2 (b) ) y Further, to preserve uniformity in the margins of the copula α = C(α, β) = β

25 EXAMPLES II. A Gaussian utility function (4) One may extend this copula to a one-parameter family in many ways. Here are two. C (1) θ (u, v) = uv g(1, 1) ( 1 θ(1 u)(1 v) ) C (2) θ (u, v) = uv g(1, 1) ( 1 (1 u θ )(1 v θ ) ) 1 θ on [0, g(1, 1)] 2, where θ [0, 1], and C (2) 0 := uv/g(1, 1).

26 EXAMPLES Inversion from chosen utility margins Equation (4) provides the means to invert the process of finding a utility copula by starting with a utility copula and choosing margins. Making the necessary substitutions, (8) T(x, y) + T(a, b) = T 1 (x) + T 2 (y) C ( T 1 (x) T 1 (a), T 2 (y) T 2 (b) ) Other than choosing T 1 and T 2 to be utility functions on the requisite domains, one must only ensure that T 1 (a) = T 2 (b), because their value becomes T(a, b) in the generated utility function, and that the conditions of Equations (7) are observed so that the margins fit the copula.

27 EXAMPLES III. Independence (1) The copula of the Gaussian utility function, as in Equation (4), closely resembles the independent probability copula, and therefore inspires the thought of the class of utility functions having a copula of the form (9) C(u, v) = 1 α uv Such a utility function T(x, y) has this measure, by the inversion formula, Equation (8). (10) 2 T x y = 1 dt 1 dt 2 α dx dy

28 EXAMPLES III. Independence (2) Conversely, any bounded utility function satisfying Equation (10) has the copula of Equation (9). Application of the Fundamental Theorem of the Calculus reveals that the point (u, v) = ( T 1 (x) T 1 (a), T 2 (y) T 2 (b) ) has the value C(u, v) = 1 α( T1 (x) T 1 (a) )( T 2 (y) T 2 (b) ) = 1 α uv

29 EXAMPLES III. Independence (3) Definition A utility copula C(u, v) = (1/α)uv is an independent copula 1 of density 1 α, or 2 of [linear] size α, or 3 of measure α.

30 EXAMPLES IV. Complementarity All complementary utility functions, e.g., Guns and Butter, have separable variables (making the mixed second partial derivative zero.) By extending the definition of the independent copula to unbounded utility functions by letting α, an unbounded complementary utility function becomes independent. (This is the copula which is zero everywhere, save for the lines at infinity, on which is it uniform.) A bounded complementary utility function cannot be independent, for it violates Equation (10). In fact it cannot even have a copula, for by the definition the margins would be zero, violating the ground condition. This is an uninteresting case.

31 EXAMPLES V. Mean Variance copulas The only concern is the monotone-decreasing desirability of the variance. A suggestion is simply to transform the variance into another variable which is monotone increasing, like its reciprocal. (As an aside, the reciprocal is frequently misidentified as the inverse. Insofar as the variance is a function of a random variable, its inverse must be a random variable, which clearly is not unique.) One then proceeds as in other cases with monotone increasing variables, recovering the variance at a later step, if desired.

32 EXAMPLES VI. An unbounded utility function Power margins with a logarithmic substitution copula Let T 1 (x) = x T 2 (y) = y Then by Equations (5) and (8) T(x, y) = log [ exp ( x ) + exp ( y ) 1 ]

33 EXAMPLES Substitution Utility Function with Logarithm and Power Margins Substitution Utility Function Substitution Utility Function with Power Margins T(x,y) 1.50 T(x,y) x y x y

34 EXAMPLES VII. A bounded utility function Exponential margins with an independent copula On [0, ) let T 1 (x) = 1 ( 1 e x ) 2πe T 2 (y) = 1 ( 1 e y ) 2πe Then T 1 (0) = T 2 (0) = 0, and by Equations (6) and (8) T(x, y) = 1 (1 e (x+y)) 2πe

35 EXAMPLES Gaussian Utility Function with Gaussian and Exponential Margins Gaussian Utility Function Gaussian Utility Function (Independent) with Exponential Margins T(x,y) x y T(x,y) x y

36 EXAMPLES Research plan The research plan on this subject of utility copulas involves three aspects, the first of which is under way. 1. Theory a. Properties b. Stochastic evolution c. Constrained optimization 2. Estimation a. Bias b. Consistency c. Efficiency 3. Empirical study a. Microeconomic models b. Macroeconomic models c. Public policy

37 EXAMPLES General reading For general reading on probability and Lévy copulas, and applications, look to these references, listed in detail below: Genest and Rivest (1993) Shih and Louis (1995) Nelsen (1998) Cherubini, Luciano, and Vecchiato (2004) Cont and Tankov (2004) Barndorff-Nielsen and Lindner (2006) Kallsen and Tankov (2006)

38 REFERENCES References, Page 1 Barndorff-Nielsen, O. E. and A. M. Lindner (2006). Lévy copulas: dynamics and transforms of upsilon-type. To appear, Scand. J. Statist. Cherubini, U., E. Luciano, and W. Vecchiato (2004). Copula Methods in Finance. Chichester: Wiley. Cont, R. and P. Tankov (2004). Financial Modelling with Jump Processes. Boca Raton: Chapman & Hall/CRC. Genest, C. and L.-P. Rivest (1993, Sep.). Statistical inference procedures for bivariate Archimedean copulas. J. Am. Stat. Assoc. 88(423),

39 REFERENCES References, Page 2 Kallsen, J. and P. Tankov (2006, Aug.). Characterization of dependence of multidimensional Lévy processes using Lévy copulas. J. Multivariate Anal. 97(7), Nelsen, R. B. (1998). An Introduction to Copulas. New York: Springer-Verlag. Shih, J. H. and T. A. Louis (1995, Dec.). Inferences on the association parameter in copula models for bivariate survival data. Biometrics 51(4),

40 EPILOGUE Epilogue Endlich kann kein Ding Wert sein, ohne Gebrauchsgegenstand zu sein. Ist es nutzlos, so ist auch die in ihm enthaltene Arbeit nutzlos, zählt nicht als Arbeit und bildet daher keinen Wert. Lastly nothing can have value, without being an object of utility. If the thing is useless, so is the labor contained in it; the labor does not count as labor, and therefore creates no value. Karl Marx, Das Kapital, 1867 [Part 1, Chapter 1, Section 1, at end]

41 CONTACT INFORMATION To reach me Paul C. Kettler paulck/ Telephone:

42 CONTACT INFORMATION