Chapter 9: Profit Maximization Flashcards
Profit Maximization Problem
maximize p\cdot z subject to z\in Z.
Z*(p)
argmax{p\cdotz:z\in Z}
pi(p)
sup{p\cdot z:z\in Z}
Properties of the profit function? (2)
homogenous of degree 1 and convex.
When is Z*(p) convex for each p?
When Z is convex.
When is Z*(p) a singleton or empty?
If between any two points in Z, there exists some small perturbation between the two points that is also within the set.
RCP
For all z in Z, if {z_n} is a sequence that diverges to infinity (in norm), then eveyr accumulation point z_n/||z_n|| lies in R_{-}^k (may include zero).
Berge’s for Profit Maximization (Conditions?)
Z is closed, nonempty, RCP
Berge’s for Profit Maximization (Properties?)
Z*(p) is nonempty valued, locally bounded, and upper semi continuous.
\pi(p) is continuous.
Afriat’s Theorem (Condition)
No revealed choice results in higher profit.
Recovering Production Possibility
Convex combination of all points (and everything below by Free disposal).
Do Giffen Goods exist for Profit Maximizing firms?
No, since there is no income effect. Strict increases in prices lead to weakly increasing income.
Good z_j will be weakly sold more in response.
Given a profit function, what is the production possibility set which is consistent?
Z*=\cap_p {z:p\cdot z\leq \pi(p)}
(takes all the possible productions which produce equal to or less profit than the profit function for each price, since it must be consistent with profit at every point p)
When do two sets have the same profit function?
If they have the same closure of the free disposal convex hull. The idea is that these replicate the same production capabilities.
What if closure of FDCH is different?
Profit function cannot be identical: that is, there exists some price vector for which they differ.
