Suppose a Technology Is Described by the Production Function $f ( \ell )$

question 16

Essay

Suppose a technology is described by the production function $f ( \ell )$
a.For a price taking producer who faces output price p and wage w,derive the first order condition and interpret it.
b.Without knowing more about the function f,is the condition you derived in (a)either necessary or sufficient for deriving the profit maximizing production plan? Explain.
c.Suppose
$f ( \ell ) = \ell ^ { \alpha }$ .Derive the first order condition you illustrated in (a)and solve for
$\ell$ .
d.For what values of
$\alpha$ is this first order condition necessary and sufficient for deriving a profit maximizing production plan? Explain.

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Definitions:

Indifference Curves

Curves that represent combinations of various goods or services that provide the same level of satisfaction to a consumer.

Normal Good

A good for which demand increases as the income of consumers increases, showing a direct relationship between income and demand.

Inferior Good

A type of good for which demand decreases when the income of consumers increases, inversely related to income changes.

Indifference Curves

Curves that represent the locus of various points indicating combinations of goods giving equal satisfaction and utility to the consumer.