![]() ![]() The slope of the indifference curve is 1/r (a x r + b y r) 1/r - 1 (r a x r - 1 + r b y r - 1 dy/dx) = 0. The most elementary technique for finding the slope of the indifference curve through (x, y) is to differentiate the equation u(x, y) = c implicitly. ![]() The first step is to determine the slope of the indifference curve through a given point (x, y), and then set the slope of the indifference curve through (x, y) equal to the slope of the budget equation. The technique for determining demand functions is similar to the technique that was used above to determine the demand for the Cobb-Douglas utility function. The Cobb-Douglas utility function has the form u(x, y) = x a y 1 - a for 0 0, b >0, and r < 1 and r 0. Diminishing MRS is both an intuitive condition on preferences, and also a mild assumption, but in almost all cases it is enough to guarantee that the demand for a commodity decreases as its price increases. This reduction in the marginal rate of substitution is one of the main assumptions about preferences or utility beyond the regularity assumptions that were described in earlier sections. The MRS going from (x 2, y 2) to (x 3, y 3) is only 1.5, since the consumer is only willing to give up 3 units of Y to get 2 more units of X. The marginal rate of substitution (MRS) going from (x 1, y 1) to (x 2, y 2) is 3, since the consumer is willing to give up 3 units of Y to get one additional unit of X. Combinations of X and Y that produce the same utility level are called an "indifference curve." In figure 10, the combinations (x 1, y 1) = (3, 12), (x 2, y 2) = (4, 9), and (x 3, y 3) = (6, 6) all lead to the same level of utility. The curve in the figure shows a many combinations of commodities X and Y that all result in the same utility. Each of these utility functions has specific properties and uses which are discussed below after the demand function for each is derived.ĭiminishing Marginal Rate of SubstitutionĮach of the utility functions examined below exhibits a property called "diminishing marginal rate of substitution." This means that as consumption of commodity X increases, the amount of another commodity that the consumer would give up to aquire an additional unit of X decreases.įigure 10 below demonstrates this idea. A third common utility function is quadratic, which has the form u(x, y) = 2 a x - (b - y) 2. This function has the form u(x, y) = (a x r + b y r) 1/r. Another common form for utility is the Constant Elasticity of Substitution (CES) utility function. One of the most common is the Cobb-Douglas utility function, which has the form u(x, y) = x a y 1 - a. There are several classes of utility functions that are frequently used to generate demand functions. Once demand is represented by a function, it can be used to develop a model of exchange, and it can be combined with the supply functions of firms to model trade in a market. In this section, we assume that the consumer has preferences that are represented by a utility function, and we then carry out this derivation of demand. If preferences are represented by a utility function, then demand can be derived from maximization of utility for various prices and income. But when economists evaluate markets, they would like to have a representation of demand. ![]() ![]() Faced with choice alternatives, it is reasonable to expect that a consumer will be able to rank the alternatives. Preferences are a natural psychological concept. Much of the preceeding material in the consumer theory section is focused on the relationship between a consumer's preferences and a utility function that represents these preferences. ![]()
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