Let u(r) be a utility function that represents and let f(.) be a continuous monotonic function. f(z) is monotonic when z > y = f(z) > f(y) (a) Show that any monotonic transformation of the utility function (f o u) can also represent the same preferences. (b) Can you explain why taking a monotonic transformation of a utility function doesn't change the marginal rate of substitution? (Hint: MRS = ) (c) What kind of preferences are represented by a utility function of the form u(z1, 12) = Vn+n? What about the function e(1,72) = 13r, + 13r2? (d) Consider the function u(z1,72) = Van. What kind of preferences does it represent? Is the function v(I1,I2) = r¡12 a monotonie trans- formation of u(11,z2)? Is the function w(I1,72) = r7 monotonic transformation of u(1,12)?
Let u(r) be a utility function that represents and let f(.) be a continuous monotonic function. f(z) is monotonic when z > y = f(z) > f(y) (a) Show that any monotonic transformation of the utility function (f o u) can also represent the same preferences. (b) Can you explain why taking a monotonic transformation of a utility function doesn't change the marginal rate of substitution? (Hint: MRS = ) (c) What kind of preferences are represented by a utility function of the form u(z1, 12) = Vn+n? What about the function e(1,72) = 13r, + 13r2? (d) Consider the function u(z1,72) = Van. What kind of preferences does it represent? Is the function v(I1,I2) = r¡12 a monotonie trans- formation of u(11,z2)? Is the function w(I1,72) = r7 monotonic transformation of u(1,12)?
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Mainly looking for how to work out b-d, as I asked about part a in a seperate question on here. Thanks so much!
Transcribed Image Text:Let u(r) be a utility function that represents and let f(.) be a continuous
monotonic function. f(z) is monotonic when z > y = f(z) > f(y)
(a) Show that any monotonic transformation of the utility function (f o
u) can also represent the same preferences.
(b) Can you explain why taking a monotonic transformation of a utility
function doesn't change the marginal rate of substitution? (Hint:
MRS = )
(c) What kind of preferences are represented by a utility function of the
form u(z1, 12) = Vn+n? What about the function e(1,72) =
13r, + 13r2?
(d) Consider the function u(z1,72) = Van. What kind of preferences
does it represent? Is the function v(I1,I2) = r¡12 a monotonie trans-
formation of u(11,z2)? Is the function w(I1,72) = r7 monotonic
transformation of u(1,12)?
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