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In the cardinal utility approach, presented by Alfred Marshall, the size of the utility difference between two bundles of goods carries some significance. In other words, this approach attaches a significance to the magnitude of utility.
There is a very simple way of assigning an ordinal utility to the two bundles of goods- we just assign a higher utility to the bundle which is chosen than to the bundle which is rejected when both are available and the consumer is asked to choose any of the two.
Any assignment that follows this procedure will be a utility function. Thus according to the cardinal utility approach there is an operational criterion for determining whether one bundle gives more utility than another for our representative consumer.
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According to the cardinal approach our consumer may like one bundle twice as much as another if he is willing to pay twice as much for it. Thus the willingness to pay determines how two bundles are compared and evaluated.
For analytical purposes it is very helpful if we can summarise the consumer’s preferences by means of a utility function. A weak, though economically natural assumption (called continuity) guarantees the existence of a utility representation.
However, statements about increasing or diminishing marginal utility are meaningless because we can always find a function to represent the consumer’s preferences which contradict the statement that increases in utility tend to diminish as the consumption of a good increases.
Alfred Marshall developed the law of diminishing marginal utility. However, one undesirable feature of the Marshallian approach is that it rules out the existence of inferior goods, because an individual’s utility function is separable, increasing and strictly concave. For this and other reasons (such as irrelevance of assumption of constant marginal utility of money) modern economists have rejected the law.
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Moreover, utility is a subjective concept and cannot be objectively measured. In truth, for the purpose of constructing a meaningful theory of consumer choice, not only the measurement of utility, for the very concept itself, is unnecessary.
We can base a theory of choice on the concepts of preference and indifference and nothing more is needed for the theory than the consumer’s set of indifference curves with their assumed properties.
However, utility functions lie behind any preference ordering which is the main pillar of indifference curve approach. The utility function provides such a preference ordering.
Along an indifference curve we have:
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u(x) = u0 …. (4)
where u0 is a constant (or some given number). Different consumption bundles along an indifference curve satisfy (1). In fact, (1) shows the relation between the function u(x) and the indifference sets, which are the fundamental expressions of the consumer’s preference ordering. Since these consumption bundles yield the same value of the function they must constitute an indifference set.
It is important to note that u(x) is a strictly quasi-concave function. The assumption that the utility function is strictly quasi-concave restricts the shape of the indifference curves. Let us consider points on a given indifference curve where u0 = f(x10, x20) = f(x11, x21). Strict quasi-concavity ensures that;
u[λx10 + (1 – λ) x11, λx20 + (1 – λ) x21] > u0, for all 0 < λ < 1.
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Thus, all interior points on a line segment connecting two points on an indifference curve lie on higher indifference curves. This means that an indifference curve expresses x2 as a strictly convex function of x1, sometimes expressed by saying that indifference curves are ‘convex to the origin’ (Fig. 5.6).
Moreover, a consumption bundle which yields a higher value of the function than another will always be preferred. Therefore we can interpret the desire to choose the preferred alternative in some given set of alternatives as equivalent to maximising the function w(x) over that set.
Thus we can represent the consumer’s choice problem as one of constrained maximisation of a strictly quasi-concave function. An important property of utility functions is that they are differentiable to any required order. This assumption rules out cases in which the slope of an indifference curve makes a sudden jump as shown in Fig. 5.7.
This assumption allows us to differentiate u totally to obtain du = u1dx1 + u2dx2 = 0 where u1 and u2 are the marginal utility of x1 and x2, respectively. This equation constrains the differentials dx1 and dx2 to be such as to ensure that u = u0.
Then, by rearranging we have;
Thus the MRS at any point can be expressed as the ratio of marginal utilities at that point.
The MRS is more fundamental than u1 and u2. The preference ordering of the consumer uniquely determines the indifference sets and hence the MRS. The marginal utilities, on the other hand, depend on the particular function used to represent the consumer’s preferences, i.e., to label the indifference sets.
Properties of Marginal Utility:
Utility functions are assumed to be regular strictly quasi-concave, differentiable and increasing. If x1 increases with the quantities of x2 held constant, the consumer acquires a better bundle. So the utility number must increase, in which case marginal utility of x1 is positive: u1(x1) > 0.
(i) The sign of the marginal utility of x1 is the same for all numerical representations of the consumer’s preferences (i.e., for utility functions) but the size of the marginal utility is not. The rate of change of marginal utility of x1 with respect to x1 is the second partial derivative with respect to x1: u11 = ∂2u1/∂x12.
(ii) Neither the signs nor the magnitudes of the rate of change of u1 are the same for all representations of preferences. Hence statements about increasing or diminishing marginal utility are, therefore, meaningless. The reason is that is always possible to find a function to represent the consumer’s preferences which contradicts the statement.
Equation (2) makes the important point that ratios of marginal utilities are invariant to permissible (monotonic) transformations of the utility function since they must all equal the MRS, which is determined by the consumer’s preferences.
However, if certain additional restrictive assumptions about an individual consumer’s preferences are made it is quite sensible to talk of the rate of change of marginal utility, for example, decision making under uncertainty.
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