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:

Capacity

The maximum level of output that a company can sustain to make a product or provide a service.

Production Possibilities Frontier

A curve depicting all maximum output possibilities for two goods, given a set of inputs, assuming all resources are fully and efficiently utilized.

Unemployment Rate

The slice of the labor pool actively seeking employment but currently unemployed.

Full Production

The state where all available resources are being used efficiently to produce goods and services.