Editing Ordinal utility (section)

== Additivity with two goods ==
A more general type of utility function is an [[additive function]]:
:<math>v(x,y)=v_X(x)+v_Y(y)</math>

There are several ways to check whether given preferences are representable by an additive utility function.

=== Double cancellation property ===
If the preferences are additive then a simple arithmetic calculation shows that
:<math>(x_1,y_1)\succeq (x_2,y_2)</math> and
:<math>(x_2,y_3)\succeq(x_3,y_1) </math> implies 
:<math>(x_1,y_3)\succeq(x_3,y_2)</math>
so this "double-cancellation" property is a necessary condition for additivity.

[[Debreu theorems|Debreu (1960)]] showed that this property is also sufficient: i.e., if a preference relation satisfies the double-cancellation property then it can be represented by an additive utility function.<ref name="Ted">{{cite web | url=http://www.econ.ucsb.edu/~tedb/Courses/GraduateTheoryUCSB/separabilitynotes.pdf | title=Lecture Notes on Separable Preferences | publisher=UCSB Econ | access-date=18 August 2015 | author=Bergstrom, Ted}}</ref>

=== Corresponding tradeoffs property ===
If the preferences are represented by an additive function, then a simple arithmetic calculation shows that
:<math>MRS(x_2,y_2)=\frac{MRS(x_1,y_2)\cdot MRS(x_2,y_1)}{MRS(x_1,y_1)}</math> 
so this "corresponding tradeoffs" property is a necessary condition for additivity. 
This condition is also sufficient.<ref name=LuceTukey1964>{{cite journal|doi=10.1016/0022-2496(64)90015-x|title=Simultaneous conjoint measurement: A new type of fundamental measurement|journal=Journal of Mathematical Psychology|volume=1|pages=1–27|year=1964|last1=Luce|first1=R.Duncan|last2=Tukey|first2=John W.|citeseerx=10.1.1.334.5018}}</ref><ref name=KeeneyRaiffa1993/>{{rp|91}}