Consider the Utility Function $u \left( x _ { 1 } , x _ { 2 } \right) = \ln x _ { 1 } + x _ { 2 } ^ { 1 / 2 }$

question 13

Essay

Consider the utility function $u \left( x _ { 1 } , x _ { 2 } \right) = \ln x _ { 1 } + x _ { 2 } ^ { 1 / 2 }$ .(Explain all your answers.)
a.Derive the function for the marginal rate of substitution.
b.Do the tastes represented by this utility function satisfy diminishing MRS?
c.The marginal utility of a good is defined as the change in utility from additional consumption of that good (holding all else constant).Derive the marginal utility of
$x _ { 1 }$ and
$x _ { 2 }$ .
d.Why does an ordinal approach to utility theory not any attention to what you derived in (c)?
e.Why does an ordinal approach to utility not treat the marginal rate of substitution the way it treats the marginal utility concept?

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Definitions:

Caffeine Intake

The consumption of the stimulant caffeine, found in coffee, tea, and many soft drinks, which affects the central nervous system.

Dependent Variable

The variable in an experiment that is expected to change in response to manipulations of the independent variable.

Independent Variable

"A manipulated experimental factor; the variable that the experimenter changes to see what its effects are."

Longitudinal Design

A special kind of systematic observation, used by correlational researchers, that involves obtaining measures of the variables of interest in multiple waves over time.