Utility. M. Utku Ünver Micro Theory. M. Utku Ünver Micro Theory Utility 1 / 15

Transcription

1 Utility M. Utku Ünver Micro Theory M. Utku Ünver Micro Theory Utility 1 / 15

2 Utility Function The preferences are the fundamental description useful for analyzing choice and utility is simply a way of describing preferences. A utility function is a way of assigning a number to every possible consumption bundle such that more-preferred bundles get assigned bigger numbers than less-preferred bundles. That is; (x 1, x 2 ) (y 1, y 2 ) u(x 1, x 2 ) > u(y 1, y 2 ) The only property of a utility assignment that is important is how it orders the bundles of goods. The magnitude is important only to an extent it effects the ranking of different bundles, the size of the utility difference does not matter. M. Utku Ünver Micro Theory Utility 2 / 15

3 Example: Suppose (w 1, w 2 ) (x 1, x 2 ) (y 1, y 2 ) (z 1, z 2 ) Utility function representing these preferences u(w 1, w 2 ) = 20 > u(x 1, x 2 ) = 10 > u(y 1, y 2 ) = 4 > u(z 1, z 2 ) = Since only the ranking of the bundles matters, there can be no unique way to assign utilities to bundles of goods. Example continued: We can also use the following utility function representing the above preferences v(w 1, w 2 ) = 12 > v(x 1, x 2 ) = 11 > v(y 1, y 2 ) = 10 > v(z 1, z 2 ) = 0 A monotonic transformation is a way of transforming one set of numbers into another set of numbers in a way that preserves the order of the numbers. Example continued There is a monotonic transformation f that transforms u to v, that is v = f (u) : 0 = f (0.001), 10 = f (4), 11 = f (10), 12 = f (20) As u goes up f goes up. So f is monotonic. M. Utku Ünver Micro Theory Utility 3 / 15

4 Examples of monotonic transformations: v = 2u v = u 2 (if u 0) v = ln u (if u > 0) v = 1 u v = u + 5 M. Utku Ünver Micro Theory Utility 4 / 15

5 A monotonic transformation M. Utku Ünver Micro Theory Utility 5 / 15

6 We typically represent a monotonic transformation by a function f (u) that transforms each u into another f (u) in a way that preserves the order of the numbers; u 1 > u 2 f (u 1 ) > f (u 2 ). Here f (u(x 1, x 2 )) is also a utility function that represents those same preferences. The reason is the following; 1 u(x 1, x 2 ) represents some preferences means; u(x 1, x 2 ) > u(y 1, y 2 ) (x 1, x 2 ) (y 1, y 2 ) 2 But f (u) is a monotonic transformation, therefore 3 1 and 2 imply; u(x 1, x 2 ) > u(y 1, y 2 ) f (u(x 1, x 2 )) > f (u(y 1, y 2 )) f (u(x 1, x 2 )) > f (u(y 1, y 2 )) (x 1, x 2 ) (y 1, y 2 ) 4 But that means f (u(x 1, x 2 )) represents these preferences as well. Geometrically, a utility function is a way to label indifference curves. A monotonic transformation is a relabeling of the indifference curves. M. Utku Ünver Micro Theory Utility 6 / 15

7 Indifference curves from Utility functions Given a utility function u(x 1, x 2 ), we should plot all the points (x 1, x 2 ) such that u(x 1, x 2 ) equals a constant. Different constants will correspond to different utility levels (i. e. different indifference curves). M. Utku Ünver Micro Theory Utility 7 / 15

8 Example: u(x 1, x 2 ) = x 1 x 2 u(x 1, x 2 ) = k x 2 = k x 1 Remark: If we use any monotonic transformation it will give us the same curves. M. Utku Ünver Micro Theory Utility 8 / 15

9 Perfect Substitutes: u(x 1, x 2 ) = ax 1 + bx 2 M. Utku Ünver Micro Theory Utility 9 / 15

10 Perfect Complements: Suppose a consumer always uses 2 teaspoons of sugar with each cup of tea. If x 1 is the cups of tea and x 2 is the teaspoons of sugar available, then the number of correctly sweetened cups of tea will be min{x 1, 1 2 x 2}. In general u(x 1, x 2 ) = min{ax 1, bx 2 } M. Utku Ünver Micro Theory Utility 10 / 15

11 Quasilinear Preferences: u(x 1, x 2 ) = v(x 1 ) + x 2 Ex: u = ln x 1 + x 2, u = x 1 + x 2 Each indifference curve is a vertical shifted version of a single indifference curve. M. Utku Ünver Micro Theory Utility 11 / 15

12 Other preferences with kinks: Example: u = min{x 1 + 2x 2, 2x 1 + x 2 } M. Utku Ünver Micro Theory Utility 12 / 15

13 Cobb-Douglas Preferences: u(x 1, x 2 ) = x c 1 x d 2 These are nice convex and monotonic preferences. If we take the natural logarithm then; v(x 1, x 2 ) = ln(u(x 1 x 2 )) = c ln x 1 + d ln x 2. Since ln is a monotonic transformation this represents the same utility function. If we raise the utility function to 1 c+d power, then c+d v(x 1 x 2 ) = x c c+d 1 x d 2 and if we name a = c c+d then v(x 1, x 2 ) = x a 1 x 1 a 2 That means we can always take a monotonic transformation of the Cobb-Douglas utility function that make the exponents sum to 1. This will turn out to have a useful interpretation later on. M. Utku Ünver Micro Theory Utility 13 / 15

14 Marginal Utility Consider a consumer who is consuming some bundle of goods (x 1, x 2 ). How does his utility change as we give him a little more of good 1? This rate of change is called marginal utility with respect to good 1. Formally it is: MU 1 = u x 1 = u(x 1 + x 1, x 2 ) u(x 1, x 2 ) x 1 As x 1 0, MU 1 = u(x 1,x 2 ) x 1. Marginal Utility depends on the particular utility function. Note that U = MU 1 x 1 measures the change in utility, when consumption of good 1 changes by x 1. M. Utku Ünver Micro Theory Utility 14 / 15

15 Marginal Utility and MRS A utility function u(x 1, x 2 ) can be used to measure the MRS defined in Ch3. Recall that the MRS measures the slope of the indifference curve at the given bundle of goods; it can be interpreted as the rate at which a consumer is just willing to substitute a small amount of good 2 for good 1. Consider a change in the consumption of each good ( x 1, x 2 ) that keeps utility constant. In other words this change in consumption moves us along the indifference curve. Then Therefore; MU 1 x 1 + MU 2 x 2 = U = 0 MRS = x 2 x 1 = MU 1 MU 2 = u(x 1,x 2 ) x 1 u(x 1,x 2 ) x 2 While the marginal utilities depend on the particular utility function, their ratio (i. e. the MRS) does not. M. Utku Ünver Micro Theory Utility 15 / 15