1. Economics D10-1: Lecture 12 Cost minimization and the cost function

2. The minimum cost function is the workhorse of the theory of the firm The cost function expresses the minimized costs of the firm as a function of output level(s) and input prices. The cost function is exactly analogous to the expenditure function of consumer theory. The cost function can also be viewed as a restricted profit function, with the level of output taken as parametric. The properties of the cost function -e.g., linear homogeneity and concavity in factor prices- can be developed using the algebraic approach, duality, or the Neoclassical approach.

3. Cost minimization: the algebraic approach The cost minimization problem: produce a specified output level with the least expenditure on inputs. C(q,w) = min z  V(q) w  z Or, in the single output case, C(q,w) = min z≥0 {w  z: f(z) ≥ q} C is linearly homogeneous in factor prices: Pf: C(q,tw) = min z  V(q) tw  z = tmin z  V(q) w  z = tC(q,w)

4. Solutions to this problem are conditional factor demands: z(q,w) = argmin z≥0 {w  z: f(z) ≥ q} Law of Downward-Sloping Conditional Factor Demand Let z 0  z(q,w 0 ) and z 1  z(q,w 1 ). Then,  z  w = (z 1 -z 0 )(w 1 -w 0 ) = (w 1 z 1 - w 1 z 0 ) + (w 0 z 0 - w 0 z 1 ) ≤ 0 Concavity of costs in factor prices Let w t = tw 0 + (1-t)w 1 and z t  z(q,w t ) C(q,w t )=w t z t =tw 0 z t +(1-t)w 1 z t ≥tw 0 z 0 +(1-t)w 1 z 1 =tC(q,w 0 )+(1-t)C(q,w 1 ) Concavity of f (convexity of Y) implies C convex in q Let z 0  z(q 0,w), z 1  z(q 1,w), q t = tq 0 + (1-t)q 1, z t  z(q t,w), and z* = tz 0 +(1-t)z 1. Note that, by concavity, f(z*) ≥ f(z t ). Therefore, C(q t,w) = w  z t ≤ w  z* = tw  z 0 + (1-t)w  z 1 = tC(q 0,w) + (1-t)C(q 1,w)

5. Cost minimization: the dual approach The dual approach to cost minimization derives differential comparative statics results using a Mirrlees construction. The derivative property (Shephard’s Lemma): Let z 0 = z(q,w 0 ) and define the function g(w)=C(q,w)-w·z 0 Clearly, g(w) ≤ 0 and g(w 0 ) = 0, so that g is maximized at w 0. Therefore, if g is differentiable at w 0 Dg(w 0 ) = D w C(q,w 0 ) - z 0 = 0 and D w C(q,w 0 ) = z 0 = z(q,w 0 ) Law of Downward-sloping conditional factor demand If g is twice differentiable at w 0, its Hessian matrix D 2 g(w 0 )=D 2 w C(q,w 0 )=D w z(q,w 0 ) is negative semi-definite there. Thus, all diagonal elements  z i /  w i are non positive.

6. Cost minimization: the Neoclassical approach Assume that f  C 2. The LaGrangian expression for the cost minimization problem is L = w·z - (f(z)-q) with FONCs of L i = w i - f i ≥ 0; z i ≥ 0; z i L i = 0 L = f(z) - q ≥ 0; ≥ 0; L = 0 Assume z(q,w) is differentiable and strictly positive. By ET  L/  q =  C/  q = (q,w) Total differentiation of the FONCs yields:

7. The Neoclassical approach: two input example

8. The restricted (“short-run”) cost function Let f:  m +  +  +. variable cost function: C v (q,k,w) = min z {wz: f(z,k)≥q} z v (q,k,w)=argmin z {wz: f(z,k)≥q} short-run cost function: C s (q,k,w,r)= C v (q,k,w)+rk. long-run cost function: C(q,w,r)=min k≥0 C s (q,k,w,r) k(q,w,r)=argmin k≥0 C s (q,k,w,r) z(q,w,r)=z v (q,k(q,w,r),w) Breaking down an optimization problem into 2 or more stages Makes it possible to focus on variables of interest Short-run - long-run distinction traditional in economics, but “time” really plays no role Short-run and variable cost formulations can be used to focus on any decision variable

9. SRMC = LRMC (Jacob Viner’s Envelope Theorem) Long-run cost equals short-run at the plant size efficient for that output level. By the Envelope Theorem, SRMC=LRMC there. Intermediate Micro diagrams also show SRMC curves to be steeper than LRMC where they intersect This fact --that value functions are less steep in the long-run-- is an example of the Le Chatlier Principle

10. The Le Chatlier Principle I: Value Functions Partition the decision variables into: “fixed” k and “variable” x Define the value function for the sub optimization problem Set fixed variables to a level optimal for some arbitrary (interior) parameter value a 0 Construct a Mirrlees’ function equal to the difference between short run and long-run value functions Le Chatlier Principle follows from SONC

11. Le Chatlier Principle II: Decision Variables Vector of parameters and the objective functions takes the special linear form. Again, use Mirrlees’ construction. FONC establish equality of SR and LR value functions Envelope Theorem and linearity establish equality between SR and LR decision variables SONCs establish responsiveness result