Production theory

Question 1

Efficient frontier

Answer 1

Y^e={y in Y|y’>y then y’ not in Y}

Question 2

Existence of supply correspondence

Answer 2

Y bounded above p>>0 then there exists a solution

Question 3

Supply correspondence and profit function properties

Answer 3

Y closed and with free disposal

• profit function is homogeneous of degree one
• y(p) is homogeneous of degree 0
• profits are convex
• net supply law (p’-p’’)(y(p’)-y(p’’))
• Y convex, then y(p) convex set, for all p. Y strictly convex then there is a unique solution.
• Hotelling lemma: y(p) is unique in p, then profit function is differentiable in p, and the gradient is the supply correspondence.
• y(p) is diff in p then D_py(p) is the hessian of the profit function and it is symmetric, positive semidefinite with Dy(p)p=0
  *

Question 4

Conditional factor demand and cost function properties

Answer 4

Y closed and satisfies the free disposal property

• c(w,q) is non decreasing in (w,q)
• z(w,q) is homogeneous of degree 0 in w.
• c(w,q) is homogeneous of degree 1 in w
• c(w,q) is concave in w
• (w’-w’’)(z(w’,q)-z(w’‘,q))<=0
• f(z) is quasiconcave, z(w,q) is a convex set

Question 5

Conditional factor demand and cost function properties 2

Answer 5

Y closed and satisfies the free disposal property

• f(z) is strictly quasiconcave, z(w,q) is a function.
• z(w,q) unique, c(w,q) is diff in w and the gradient is the conditional factor demand.
• z diff in w, then Dwz(w,q) is the hessian of the cost function is a negative semidefinite matrix, Dz w=0
• f homogeneous of degree 1 then c and z are homogeneous of degree one.
• f concave, c is convex q (marginal costs are non decreasing).