Consider a utility function of two goods x and y: U (x,y) = A (ax' +by') where A >0, a>0, b>0, r € (-0,0)U(0, 1) are constants. This utility function is called a "constant elasticity of substitution (CES)" function and is frequently used in Macroeconom- ics. (a) Prove that when a+b = 1, this utility function converges to a Cobb-Douglas utility function as r→0. Hint: apply l'Hopital's rule to lim In (x9) – lim Im(ax'+by') (b) Calculate the slope of the indifference curves of U. Based on your answer, are good x and y perfect/imperfect substitutes/complements when r → 1? When r → -? (c) Is U a homogeneous function? If so, what's its degree? If not, please explain. (d) Is U a homothetic function? Please explain.
Consider a utility function of two goods x and y: U (x,y) = A (ax' +by') where A >0, a>0, b>0, r € (-0,0)U(0, 1) are constants. This utility function is called a "constant elasticity of substitution (CES)" function and is frequently used in Macroeconom- ics. (a) Prove that when a+b = 1, this utility function converges to a Cobb-Douglas utility function as r→0. Hint: apply l'Hopital's rule to lim In (x9) – lim Im(ax'+by') (b) Calculate the slope of the indifference curves of U. Based on your answer, are good x and y perfect/imperfect substitutes/complements when r → 1? When r → -? (c) Is U a homogeneous function? If so, what's its degree? If not, please explain. (d) Is U a homothetic function? Please explain.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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11.) Answer only part A
Transcribed Image Text:Consider a utility function of two goodsx and y:
U (x, y) = A (ax' +by')
where A > 0, a> 0, b>0, r € (-∞,0)U (0,1) are constants. This utility function is called a
"constant elasticity of substitution (CES)" function and is frequently used in Macroeconom-
ics.
(a) Prove that when a+b = 1, this utility function converges to a Cobb-Douglas utility
function as r→ 0. Hint: apply l'Hopital's rule to lim In U(x.y) = lim In(ax' +by')
(b) Calculate the slope of the indifference curves of U. Based on your answer, are good x
and y perfect/imperfect substitutes/complements when r → 1? When r → -00?
(c) Is U a homogeneous function? If so, what's its degree? If not, please explain.
(d) Is U a homothetic function? Please explain.
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