The following utility function is known as CES (constant elasticity of substitution) function: U (x, y) = (ax° + By')'/º, where a > 0, B > 0 a) Is this function homothetic? b) How does the M RSy depend on the ratio x/y? Specifically, show that the MRSy is strictly decreasing in the ratio x/y for all values S < 1, increasing in the ratio x/y for all values &>1 and constant for 8 = 1. (Hint: take the derivative of the MRS with respect to the ratio x/y as z = x/y and take the derivative with respect to z) c) Show that if x = y, the MRS of this function depends only on the ratio a/6.
The following utility function is known as CES (constant elasticity of substitution) function: U (x, y) = (ax° + By')'/º, where a > 0, B > 0 a) Is this function homothetic? b) How does the M RSy depend on the ratio x/y? Specifically, show that the MRSy is strictly decreasing in the ratio x/y for all values S < 1, increasing in the ratio x/y for all values &>1 and constant for 8 = 1. (Hint: take the derivative of the MRS with respect to the ratio x/y as z = x/y and take the derivative with respect to z) c) Show that if x = y, the MRS of this function depends only on the ratio a/6.
Chapter4: Utility Maximization And Choice
Section: Chapter Questions
Problem 4.13P
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The following utility function is known as CES (constant elasticity of substitution) function:
U (x, y) = (ax° + By')'/º, where a > 0, B > 0
%3D
a) Is this function homothetic?
b) How does the MRSY depend on the ratio x/y? Specifically, show that the MRSxy is strictly
decreasing in the ratio x/y for all values d < 1, increasing in the ratio x/y for all values &> 1 and
constant for 8 = 1. (Hint: take the derivative of the MRS with respect to the ratio x/y as z =
and take the derivative with respect to z)
x,y
x/y
c) Show that if x = y, the MRS of this function depends only on the ratio /B.
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