Consider the problem of maximizing the profit function (pi)= pY -wL subject to the production function Y= L to the alpha (as the exponent) where alpha E (epsilon) (0,1)

First we'll check that Hotelling's Lemma is satisfied. The profit function relates factor prices to the maximum profit levels achievable at those output prices and factor prices. It is characterized by Hotteling's Lemma, which states that differentiating the profit function with respect to output price yields output quantity, while differentiating it with respect to price of a particular factor yields (the negative of) the corresponding factor quantity.

To verify this for your problem, first we differentiate:
Π; = pY - wL
so (dΠ;/dp) = 1Y= Y , and
(dΠ;/dw) = -1L = -L

Therefore Hotelling's Lemma is satisfied.

Finding the profit function can be done by substituting Y=L^? into the profit function. Therefore we have

Π = pL^? - wL

If ? belongs to (0,1) (not including either 0 or 1) then we can use calculus in order to find th eL that maximizes profits. This is because when ? belongs to (0,1), the profit function is concave; so if we take the derivative of the profit function with respect to L and equate it to 0, we'll get the L that maximizes profits. However, if ?=1, this is no longer valid, because the profit function is not concave in this case. When ?=1, we have:

Π; = pL - wL
Π; = (p-w)L

Thus we have 3 possible cases:

1) p > w : In this case, (p-w)>0, therefore the firm will want infinite L because as L rises, so do profits.

2) p < w : In this case, (p-w)<0, therefore the firm will want zero L. Any positive amount of L will cause the firm to have negative profits. Clearly, the firm manager would prefer to produce nothing.

3) p = w : In this case, (p-w)=0, therefore the firm is indifferent among any quantity of L. It doesn't matter how much L the firm hires, the profits will always be 0.