Consumption- Preferences Ut A utility function assigns a number to each bundle of goods, r Early economists thought of utility as being a measure of happiness or satisfaction. In this case, the numbers attached to various bundles of goods matter. It means something for a bundle to have twice the utility of another. most economists think of utility as being ordinal. The numbers themselves only matter insofar as the rank the different bundles. If u(r)>u(y) then r >y. Therefore preferences can be represented by many different utility functions. All of the below utility functions are equivalent u2 298 mic transformation of a utility function yields another utility function which represents the preferences. A monotonic transformation is simply one which leaves order unchanged- like multiplying by 10. Utility functions can be constructed from indifference curves and vice-versa. In fact, indifference curves are like cross sections of the utility function. Some examples will clarify this point. Consumption-Preferenees Some Example 1. Perfect Substitutes: Consider u(I1, I2)=I1+I2. To draw indifference curves from such a function set utility equal to a constant, c. It must be the case that the consumer is indifferent between any combination of goods which sults in the same level of utility. c. I1+I2= c defines a straight line(I2 intercept of this line is at c. The below graph can be drawn. Perfect Substitutes ts:The same operation can be done for u(r1, I2)= min(I1, I21. See abowe. 3. Quasilinear Preferences: u(rn, I2)=v(z1)+I2 where u(- )is an increasing function. In this case each identical shape, simply 1.Cobb-Douglas Preferences: u(r1, r2)=r r2. Another useful way to write this is to take logarithms(a monotonic transformation) yielding the equivalent v(rn, I2)=aln r+8lnz2Consumption — Preferences 9 Utility Functions • A utility function assigns a number to each bundle of goods, x. • Early economists thought of utility as being a measure of happiness or satisfaction. In this case, the numbers attached to various bundles of goods matter. It means something for a bundle to have twice the utility of another. This is called cardinal utility. • Nowadays most economists think of utility as being ordinal. The numbers themselves only matter insofar as they rank the different bundles. If u(x) > u(y) then x  y. Therefore preferences can be represented by many different utility functions. All of the below utility functions are equivalent. Bundle u1 (·) u2 (·) u3 (·) x 1 0.01 -10 y 2 98 0 z 3 101 65 • Any monotonic transformation of a utility function yields another utility function which represents the same preferences. A monotonic transformation is simply one which leaves order unchanged — like multiplying by 10. • Utility functions can be constructed from indifference curves and vice-versa. In fact, indifference curves are like cross sections of the utility function. Some examples will clarify this point. Consumption — Preferences 10 Some Examples 1. Perfect Substitutes: Consider u(x1, x2) = x1 + x2. To draw indifference curves from such a function set utility equal to a constant, c. It must be the case that the consumer is indifferent between any combination of goods which results in the same level of utility, c. x1 + x2 = c defines a straight line (x2 = c − x1) with a slope of −1. The intercept of this line is at c. The below graph can be drawn. . x2 x1 . . . ................................................................................................................................................................................................................................................................................ Perfect Substitutes u = 2 u = 3 u = 4 x2 x1 u = 1 u = 2 u = 3 .............................................................................................................................................................................................. ........................................................................................................................................ .................................................................................. .................................................................................................................................................................................................................................................................................. Perfect Complements 0 0 2. Perfect Complements: The same operation can be done for u(x1, x2) = min{x1, x2}. See above. 3. Quasilinear Preferences: u(x1, x2) = v(x1) + x2 where v(·) is an increasing function. In this case each indifference curve is an identical shape, simply shifted upward. 4. Cobb-Douglas Preferences: u(x1, x2) = x1 αx2 β . Another useful way to write this is to take logarithms (a monotonic transformation) yielding the equivalent v(x1, x2) = α ln x1 + β ln x2