Consider the problem of maximizing profit (pi symbol, but i'll use &)  
&=py-wl  subject to the production function y=l^a  (l to the power of
alpha, alpha is an element of (0,1)).  Derive the profit function and
confirm that Hotelling?s lemma is satisfied.  What happens if a=1 ?
(alpha is equal to 1)

thanks

Hi dime365!
First we'll check that Hotelling's Lemma is satisfied. You can find
Hotelling's Lemma at the following page:

The Profit Function
http://cepa.newschool.edu/het/essays/product/profit.htm
(search for "Hotelling" inside the page)

We have:

& = p*y - w*l

so

(d&/dp) = 1*y = y , and
(d&/dw) = -1*l = -l

Therefore Hotelling's Lemma is satisfied.

Finding the profit function can be done by simply replacing y=l^a into
the profit formula py-wl. Therefore we have that the profit function
is:

& = p*l^a - w*l

If "a" belongs to (0,1) (not including either 0 or 1) then we can use
calculus in order to find the "l" that maximizes profits. This is
because when "a" 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if a=1, this is no longer valid, because the profit function is not
concave in this case. When a=1, we have:

& = p*l - w*l
& = (p-w)*l

We have thus 3 possible cases:

1) p > w : In this case, (p-w)>0, therefore the firm must hire
infinite "l", because as l rises, so do profits.

2) p < w : In this case, (p-w)<0, therefore the firm must hire 0 "l".
Any positive quantity 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.


Google search strategy
"hotelling's lemma"
://www.google.com.ar/search?q=%22hotelling%27s+lemma%22&ie=UTF-8&oe=UTF-8&hl=es&meta=


I hope this helps! If you have any doubts regarding my answer, please
don't hesitate to request clarification. Otherwise I await your rating
and final comments.

Best wishes!
elmarto

Elmarto always provides excellent answers!  Thanks!