Definition
Why we need cost minimization problem besides profit maximization problem:
(1) It leads to useful results and constructs.
(2) When a firm is not a price taker in output market, we can no longer use the profit function. However, as long as the firm is a price taker in the input market, results from cost minimization problem would be valid.
(3) when the production set exhibits non decreasing return to scale (for example, when the profit function and only take 0 or 
), the value function and optimizing vectors of cost minimization problem, which keep the levels of output fixed, are better behaved than the profit function and supply correspondence of PMP.
The cost minimization problem (assume single output, free disposal of output):
s.t. 
(f(z) is production function)
Where w is the price vector of z
z is the input vector.
The optimized value of CMP is given by the cost function c(w, q), and the corresponding optimizing set of input (for factor) is the conditional factor demand z(w, q)
If 
is optimal in the CMP and production function 
is differentiable, then for some 
, the first order condition is for every 
, with equality if 
CMP is the production analogy of expenditure minimization problem with consumption theory.
Proposition 5.C.2
Suppose that c(w, q) is the cost function of a single-output technology Y with production function 
and that z(w, q) is the associated conditional factor demand correspondence.
Assume also that Y is closed and satisfies the free disposal property, then:
(1) 
is homogeneous of degree one in w and non decreasing in q.
(2) is a concave function of w.
(3) If the sets 
are convex for every q, then 
(4) 
is homogeneous of degree zero in w.
(5) If the set 
is convex, then z(w, q) is a convex set. Moreover, if is a strictly convex set, then z(w, q) is single-valued.
(6) (Shepard's lemma) If 
consists of a single point, then 
is differentiable with respect to w at 
and 
, or 
(7) If is differentiable at 
, then 
is a symmetric and negative semi-definite matrix with 
(8) If 
is homogeneous of degree one (constant returns to scale), then and are homogeneous of degree one in q.
(9) If is concave, then is a convex function of q (marginal costs are non-decreasing in q).
At an interior optimum (
) price equals marginal cost. If c(w, q) is convex in q, the first order condition is also sufficient for 
to be the firm's optimal output level.