. The relation is called the utility function. The range of the function is a set of real numbers. The actual values of the function have no importance. Only the ranking of those values has content for the theory. More precisely, if , then the bundle is described as at least as good as the bundle . If , the bundle is described as strictly preferred to the bundle . Consider a particular bundle and take the total derivative of about this point: or, without loss of generality, (Eq. 1) where is the partial derivative of with respect to its first argument, evaluated at . (Likewise for ) The indifference curve through must deliver at each bundle on the curve the same utility level as bundle . That is, when preferences are represented by a utility function, the indifference curves are the level curves of the utility function. Therefore, if one is to change the quantity of by , without moving off the indifference curve, one must also change the quantity of by an amount such that, in the end, there is no change in U: , or, substituting 0 into (Eq. 1) above to solve for dy/dx: . Thus, the ratio of marginal utilities gives the absolute value of the slope of the indifference curve at point . This ratio is called the marginal rate of substitution between and . Examples Linear utility If the utility function is of the form then the marginal utility of is and the marginal utility of is . The slope of the indifference curve is, therefore, Observe that the slope does not depend on or : the indifference curves are straight lines. Cobb–Douglas utility A class of utility functions known as Cobb-Douglas utility functions are very commonly used in economics for two reasons: 1. They represent ‘well-behaved’ preferences, such as more is better and preference for variety. 2. They are very flexible and can be adjusted to fit real-world data very easily. If the utility function is of the form the marginal utility of is and the marginal utility of is .Where . The slope of the indifference curve, and therefore the negative of the marginal rate of substitution, is then CES utility A general CES (Constant Elasticity of Substitution) form is where and . (The Cobb–Douglas is a special case of the CES utility, with .) The marginal utilities are given by