A utility function is called separable if it can be written as U(x, y) = U,(x) + U,(y), where U; >0, U" < 0, and U,, U, need not be the same function. a. What does separability assume about the cross-partial derivative U,„? Give an intuitive discussion of what word this condition means and in what situations it might be plausible. b. Show that if utility is separable then neither good can be inferior. c. Does the assumption of separability allow you to con- clude definitively whether x and y are gross substitutes or gross complements? Explain. d. Use the Cobb-Douglas utility function to show that separability is not invariant with respect to monotonic transformations. Note: Separable functions are examined in more detail in the Extensions to this chapter.

A utility function is called separable if it can be written as U(x, y) = U,(x) + U,(y), where U; >0, U" < 0, and U,, U, need not be the same function. a. What does separability assume about the cross-partial derivative U,„? Give an intuitive discussion of what word this condition means and in what situations it might be plausible. b. Show that if utility is separable then neither good can be inferior. c. Does the assumption of separability allow you to con- clude definitively whether x and y are gross substitutes or gross complements? Explain. d. Use the Cobb-Douglas utility function to show that separability is not invariant with respect to monotonic transformations. Note: Separable functions are examined in more detail in the Extensions to this chapter.

ENGR.ECONOMIC ANALYSIS

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Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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Concept explainers

Form Utility

The term utility in general refers to the satisfaction earned from usage of a particular product or service. Utility always plays an important role because it influences various important elements like demand & thereby the price of the good or service as well. Though utility is not easy to measure, various types of economic models can be used indirectly. There can be various types of utilities like economic utility, marginal utility, cardinal utility, form utility.

Question

A utility function is called separable if it can be written as
U(x, y) = U (x) + U,(y),
where U > 0, U" < 0, and U1, U, need not be the same
function.
a. What does separability assume about the cross-partial
derivative U„? Give an intuitive discussion of what word
this condition means and in what situations it might be
plausible.
b. Show that if utility is separable then neither good can be
inferior.
c. Does the assumption of separability allow you to con-
clude definitively whether x and y are gross substitutes or
gross complements? Explain.
d. Use the Cobb-Douglas utility function to show that
separability is not invariant with respect to monotonic
transformations. Note: Separable functions are examined
in more detail in the Extensions to this chapter.

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