2.2 Utility Functions A utility function is a real-valued function u defined on the consumption set X such that preference rankings are preserved by the magnitude of u. That is, a utility function u has the property that given any two elements x, y in X, u(x)>u(y)if and only if x2y. But not all preference relations can be represented by utility functions. A rather general result is that any continuous preference ordering can be represented by a continuous utility function. This is a very difficult result to prove(Debreu(1959).(Moreover, while any continuous ordering is al ways representable, continuity is not necessary. The necessary and sufficient conditions fo representation is rather technical; see Ng 1979/83, App. IB )We will focus on a somewhat simpler result-the one that can be proved constructively. The main ideas are e select arbitrary fixed line that cuts all of the indifference curves(or surfaces) Once utility is defined along this line, the utility of any other point is found by tracing the appropriate indifference curve to the line and using the utility value there The assumption of strong monotonicity guarantees that the indifference curves exit and that any line of the form ae, a>0 and e>0, cuts them all Existence of Utility functions Suppose that a reference relation on X=R+ is complete, reflexive, transitive, continuous, and strongly monotone. Then there exists a continuous utility function u:R"+->R which represents the preference relation all ones. Then given any vector r, let g of Let e be the vector in r+ consist u (x)= a such that xae We now need to show that u(x)is well-defined, 1. e, it exists and unique Define the following two sets A={a:a≥0,ae2x} B={a:a≥0,xae} Then by strong monotonicity, A is nonempty. B is certainly nonempty since 0 E B. Both A and B are closed by the continuity assumption. On the other hand, by the completeness assumption, we know that every a(e 0)must belong to AUB, that is, AUB=R Note that if a E AnB, then ae -x so that we can let u(x)=a. Therefore, we need to prove that AnB is nonempty By monotonicity, it follows that a E A implies that a'E A for all a'>a. Since A is closed subset of R+, it must be in a form of closed interval [ a, +oo ) which implies that B=[0, a]since B is a nonempty closed set such that AUB=R+ We now have to prove that the value a must be unique. Let aie- x and aze -x. Then it is clear that aie- aie(transitivity property of " -") By strong monotonicity, it must be the case that al Let us prove that the above-defined utility function actually represents the preference relation Consider two bundles x and y, and their associated utility levels u(x)and dy), which by definition satisfy u(x)e-x and u(y)e-y. Now,4 2.2 Utility Functions A utility function is a real-valued function u defined on the consumption set X such that preference rankings are preserved by the magnitude of u. That is, a utility function u has the property that given any two elements x, y in X, u(x)  u(y) if and only if x y. But not all preference relations can be represented by utility functions. A rather general result is that any continuous preference ordering can be represented by a continuous utility function. This is a very difficult result to prove (Debreu (1959)). (Moreover, while any continuous ordering is always representable, continuity is not necessary. The necessary and sufficient conditions for representation is rather technical; see Ng 1979/83, App. 1B.) We will focus on a somewhat simpler result - the one that can be proved constructively. The main ideas are: • We select arbitrary fixed line that cuts all of the indifference curves (or surfaces). • Once utility is defined along this line, the utility of any other point is found by tracing the appropriate indifference curve to the line and using the utility value there. • The assumption of strong monotonicity guarantees that the indifference curves exit and that any line of the form  e,  > 0 and e > 0, cuts them all. Existence of Utility Functions • Suppose that a reference relation on X = R m + is complete, reflexive, transitive, continuous, and strongly monotone. Then there exists a continuous utility function u: R m + → R which represents the preference relation. Proof Let e be the vector in R m + consisting of all ones. Then given any vector x, let u(x) =  such that x ~ e. We now need to show that u(x) is well-defined, i.e., it exists and unique. Define the following two sets: A = {:   0, e x} B = {:   0, x e} Then by strong monotonicity, A is nonempty. B is certainly nonempty since 0  B. Both A and B are closed by the continuity assumption. On the other hand, by the completeness assumption, we know that every  ( 0) must belong to AB, that is, AB = R+. Note that if  *  AB, then  * e ~ x so that we can let u(x) =  * . Therefore, we need to prove that AB is nonempty. By monotonicity, it follows that   A implies that '  A for all '  . Since A is closed subset of R+, it must be in a form of closed interval [ * , +), which implies that B = [0,  * ] since B is a nonempty closed set such that AB = R+. We now have to prove that the value  * must be unique. Let 1e ~ x and 2e ~ x. Then it is clear that 1e ~ 1e (transitivity property of "~"). By strong monotonicity, it must be the case that 1 = 2. Let us prove that the above-defined utility function actually represents the preference relation. Consider two bundles x and y, and their associated utility levels u(x) and u(y), which by definition satisfy u(x)e ~ x and u(y)e ~ y. Now, x2 e x1