Suppose a technology is described by the production function 
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 
. Derive the first order condition you illustrated in (a) and solve for 
.
d. For what values of 
is this first order condition necessary and sufficient for deriving a profit maximizing production plan? Explain.
What will be an ideal response?
which gives us the first order condition 
. Since the 
, this first order condition can be written as 
In words, this means that the marginal revenue product -- i.e. the increase in revenue from the last worker hired -- is equal to the wage.b. No. The condition is necessary and sufficient only if we know that the solution is an interior solution and that there are no local minima that have the same tangency between isoprofits and the production frontier.
c.
-- so the first order condition is 
This solves to
d. The answer above is correct for
In that case, the producer choice set is fully convex and a single interior solution exists. For 
, we have increasing marginal product of labor throughout -- implying that the profit maximizing solution would be to produce an infinite amount.You might also like to view...
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Borem is a big fan of wine from Trader Moe's. Moe's sells a high-quality expensive wine and a cheap low-quality wine. Borem buys both types but tends to buy more bottles of the cheaper wine
Borem later moves to a new city where he must drive a long distance to get his wine at Trader Moe's. Assuming the income effect is small, and that Borem still buys his wine from Moe's, how is he going to change his relative consumption of the expensive and cheap wines?