Chap 2 (Varian) - Profit max Flashcards
If the profit is maximized, then the technology is…
is efficient
What is the implication of Hotelling’s lemma?
The partial derivative of the profit function w.r.t the input, yields the negative of the profit maximizing demand for that input.
dPI/dw_i = -w*_i
The partial derivative of the profit function w.r.t the output price, yields the supply function.
dPI/dP_i = Y*
What is the implication of Shepards Lemma
The partial derivative of the cost function w.r.t the input, yields the conditional demand for that input.
dC/dw_i = w*_i(w_i, y^0)
How do we get the cost function from the profit function?
C(w,y) = y = p1w1 + p2w2. So we need those two. Therefore we utilize both of Hotelling’s lemma’s:
dPI/dw_i = -w_i
dPI/dP_i = Y
We then plug this in to the cost function and then gets an expression for the minimum cost of producing a specific output.
Show that the profit function is H.M 1 in price
See notion.
PI(ap) = max_y apy y in Y
Move out a
aPi(p) = a[max_y py] y in Y
Show that the profit function is convex in prices
This is kind of the opposite as with the expenditure function and cost function.
For a graphical approach, show a diagram with profits on the $y$-axis and p on the $x$-axis. If the price increases and the firm is lazy and continues to use the same production plan, then profits will increase linear, as a 45 degree straight line. Then pre profits from pursuing an optimal production plan can not be worse then whenbeingg lazy,thuse this choice must be above the 45 degree line. That is, show with a convex curve how profits rise exponentially with price when firms chooses the optimal production plan.
Show it formally:
See notion.
How do we get the supply function from the profit function?
By Hotellings lemma
dPI/p = y*
How do we get the factor demands from the profit function?
By Hotellings lemma:
dPI/dwi = -zi.
Can er have giffen behaviour for firms?
No, since firms do not have any income effect.
That is. The law of supply holds for any price changes. Because, in contrast with demand theory, there is no budget constraint, there is no compensation requirement of any sort.
In essence, we have no wealth effect here, only substitution effect.
The law of supply is given by
(P-p’)(y-y’) ≥ 0
Show that the Profit function is h.m = 0 in p,w
See notion
If the profit function exists, then show that the new supply function is H.M 0 in P
- Show that the profit function is H.M = 1.
- Use Hoteling’s lemma (dPI/dP) to get the net supply function.
- By definition, if f(.) is H.M = K in p, then df(.)/dp will be H.M = k-1 in p.
