meaning. 04.46 ← 2. Thus, given any one utility function, any monotonic transformation of it will represent the same preferences. e-books-MICRO-INTERMDE... 70 UTILITY (Ch. 4) 3. The marginal rate of substitution, MRS, can be calculated from the utility function via the formula MRS = Ax2/Ax₁ = -MU₁/MU2. REVIEW QUESTIONS 1. The text said that raising a number to an odd power was a monotonic transformation. What about raising a number to an even power? Is this a monotonic transformation? (Hint: consider the case f(u) = u².) 2. Which of the following are monotonic transformations? (1) u = 2v-13; (2) u = 1/v²; (3) u = 1/v²; (4) u = lnv; (5) u = −ev; (6) u = v²; (7) uv² for v> 0; (8) u = v² for v<0. 3. We claimed in the text that if preferences were monotonic, then a diag- onal line through the origin would intersect each indifference curve exactly once. Can you prove this rigorously? (Hint: what would happen if it intersected some indifference curve twice?) 4. What kind of preferences are represented by a utility function of the form u(x1, x2) = √√x1 + x2? What about the utility function v(x1, x2) = 13x1 + 13x2? 5. What kind of preferences are represented by a utility function of the form u(x1, x2)=x1+√√x2? Is the utility function v(x1, x2) = x²+2x1√x2+x2 a monotonic transformation of u(x1, x2)? 6. Consider the utility function u(x1, x2) = √x1x2. What kind of pref- erences does it represent? Is the function v(x1, x2) = xx2 a monotonic transformation of u(x1, x2)? Is the function w(x1, x2) = x²x² a monotonic transformation of u(x1, x2)? 7. Can you explain why taking a monotonic transformation of a utility function doesn't change the marginal rate of substitution? APPENDIX First, let us clarify what is meant by "marginal utility." As elsewhere in eco- nomics, "marginal" just means a derivative. So the marginal utility of good 1 is just MU₁ = lim Δε1+0 u(x1+x1, x2)- u(x1, x2) Δει Ju(x1, x2) Əx1 Note that we have used the partial derivative here, since the marginal utility of good 1 is computed holding good 2 fixed. 4G..Il
meaning. 04.46 ← 2. Thus, given any one utility function, any monotonic transformation of it will represent the same preferences. e-books-MICRO-INTERMDE... 70 UTILITY (Ch. 4) 3. The marginal rate of substitution, MRS, can be calculated from the utility function via the formula MRS = Ax2/Ax₁ = -MU₁/MU2. REVIEW QUESTIONS 1. The text said that raising a number to an odd power was a monotonic transformation. What about raising a number to an even power? Is this a monotonic transformation? (Hint: consider the case f(u) = u².) 2. Which of the following are monotonic transformations? (1) u = 2v-13; (2) u = 1/v²; (3) u = 1/v²; (4) u = lnv; (5) u = −ev; (6) u = v²; (7) uv² for v> 0; (8) u = v² for v<0. 3. We claimed in the text that if preferences were monotonic, then a diag- onal line through the origin would intersect each indifference curve exactly once. Can you prove this rigorously? (Hint: what would happen if it intersected some indifference curve twice?) 4. What kind of preferences are represented by a utility function of the form u(x1, x2) = √√x1 + x2? What about the utility function v(x1, x2) = 13x1 + 13x2? 5. What kind of preferences are represented by a utility function of the form u(x1, x2)=x1+√√x2? Is the utility function v(x1, x2) = x²+2x1√x2+x2 a monotonic transformation of u(x1, x2)? 6. Consider the utility function u(x1, x2) = √x1x2. What kind of pref- erences does it represent? Is the function v(x1, x2) = xx2 a monotonic transformation of u(x1, x2)? Is the function w(x1, x2) = x²x² a monotonic transformation of u(x1, x2)? 7. Can you explain why taking a monotonic transformation of a utility function doesn't change the marginal rate of substitution? APPENDIX First, let us clarify what is meant by "marginal utility." As elsewhere in eco- nomics, "marginal" just means a derivative. So the marginal utility of good 1 is just MU₁ = lim Δε1+0 u(x1+x1, x2)- u(x1, x2) Δει Ju(x1, x2) Əx1 Note that we have used the partial derivative here, since the marginal utility of good 1 is computed holding good 2 fixed. 4G..Il
Chapter21: Demand: Consumer Choic
Section: Chapter Questions
Problem 1E
Related questions
Question
Transcribed Image Text:meaning.
04.46
←
2. Thus, given any one utility function, any monotonic transformation of
it will represent the same preferences.
e-books-MICRO-INTERMDE...
70 UTILITY (Ch. 4)
3. The marginal rate of substitution, MRS, can be calculated from the
utility function via the formula MRS = Ax2/Ax₁ = -MU₁/MU2.
REVIEW QUESTIONS
1. The text said that raising a number to an odd power was a monotonic
transformation. What about raising a number to an even power? Is this a
monotonic transformation? (Hint: consider the case f(u) = u².)
2. Which of the following are monotonic transformations? (1) u = 2v-13;
(2) u = 1/v²; (3) u = 1/v²; (4) u = lnv; (5) u = −ev; (6) u = v²;
(7) uv² for v> 0; (8) u = v² for v<0.
3. We claimed in the text that if preferences were monotonic, then a diag-
onal line through the origin would intersect each indifference curve exactly
once. Can you prove this rigorously? (Hint: what would happen if it
intersected some indifference curve twice?)
4. What kind of preferences are represented by a utility function of the
form u(x1, x2) = √√x1 + x2? What about the utility function v(x1, x2) =
13x1 + 13x2?
5. What kind of preferences are represented by a utility function of the form
u(x1, x2)=x1+√√x2? Is the utility function v(x1, x2) = x²+2x1√x2+x2
a monotonic transformation of u(x1, x2)?
6. Consider the utility function u(x1, x2) = √x1x2. What kind of pref-
erences does it represent? Is the function v(x1, x2) = xx2 a monotonic
transformation of u(x1, x2)? Is the function w(x1, x2) = x²x² a monotonic
transformation of u(x1, x2)?
7. Can you explain why taking a monotonic transformation of a utility
function doesn't change the marginal rate of substitution?
APPENDIX
First, let us clarify what is meant by "marginal utility." As elsewhere in eco-
nomics, "marginal" just means a derivative. So the marginal utility of good 1 is
just
MU₁ = lim
Δε1+0
u(x1+x1, x2)- u(x1, x2)
Δει
Ju(x1, x2)
Əx1
Note that we have used the partial derivative here, since the marginal utility
of good 1 is computed holding good 2 fixed.
4G..Il
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