between x and y at the point . This definition of the MRS is based only on the ordinal preference relation – it does not depend on a numeric utility function. If the preference relation is represented by a utility function and the function is differentiable, then the MRS can be calculated from the derivatives of that function: For example, if the preference relation is represented by then . The MRS is the same for the function . This is not a coincidence as these two functions represent the same preference relation – each one is an increasing monotone transformation of the other. In general, the MRS may be different at different points . For example, it is possible that at the MRS is low because the person has a lot of x and only one y, but at or the MRS is higher. Some special cases are described below. Linearity When the MRS of a certain preference relation does not depend on the bundle, i.e., the MRS is the same for all , the indifference curves are linear and of the form: and the preference relation can be represented by a linear function: (Of course, the same relation can be represented by many other non-linear functions, such as or , but the linear function is simplest.) Quasilinearity When the MRS depends on but not on , the preference relation can be represented by a quasilinear utility function, of the form where is a certain monotonically increasing function. Because the MRS is a function , a possible function can be calculated as an integral of : In this case, all the indifference curves are parallel – they are horizontal transfers of each other. Additivity with two goods A more general type of utility function is an additive function: 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 and implies so this "double-cancellation" property is a necessary condition for additivity. 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. Corresponding tradeoffs property If the preferences are represented by an additive function, then a simple arithmetic calculation shows that so this "corresponding tradeoffs" property is a necessary condition for additivity. This