Economics D10-1: Lecture 12

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) = minzV(q)wz • Or, in the single output case,C(q,w) = minz≥0{wz: f(z) ≥ q} • C is linearly homogeneous in factor prices:Pf: C(q,tw) = minzV(q)twz = tminzV(q)wz = tC(q,w)

4. Solutions to this problem are conditional factor demands: • z(q,w) = argminz≥0{wz: f(z) ≥ q} • Law of Downward-Sloping Conditional Factor Demand Let z0z(q,w0) and z1z(q,w1). Then, zw = (z1-z0)(w1-w0) = (w1z1 - w1z0)+ (w0z0 - w0z1) ≤ 0 • Concavity of costs in factor prices Let wt = tw0 + (1-t)w1 and ztz(q,wt) C(q,wt)=wtzt=tw0zt+(1-t)w1zt≥tw0z0+(1-t)w1z1=tC(q,w0)+(1-t)C(q,w1) • Concavity of f (convexity of Y) implies C convex in q Let z0z(q0,w), z1z(q1,w), qt = tq0 + (1-t)q1, ztz(qt,w), and z* = tz0+(1-t)z1.Note that, by concavity, f(z*) ≥ f(zt). Therefore, C(qt,w) = wzt ≤ wz* = twz0 + (1-t)wz1 = tC(q0,w) + (1-t)C(q1,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 z0 = z(q,w0) and define the function g(w)=C(q,w)-w·z0 Clearly, g(w) ≤ 0 and g(w0) = 0, so that g is maximized at w0. Therefore, if g is differentiable at w0 Dg(w0) = DwC(q,w0) - z0 = 0 and DwC(q,w0) = z0 = z(q,w0) • Law of Downward-sloping conditional factor demand If g is twice differentiable at w0, its Hessian matrix D2g(w0)=D2wC(q,w0)=Dwz(q,w0) is negative semi-definite there. Thus, all diagonal elements zi/wi are non positive.

6. Cost minimization: the Neoclassical approach Assume that fC2. The LaGrangian expression for the cost minimization problem is L = w·z - (f(z)-q) with FONCs of Li= wi - fi ≥ 0; zi≥ 0; ziLi= 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. Let f: m++ +. variable cost function: Cv(q,k,w) = minz{wz: f(z,k)≥q} zv(q,k,w)=argminz{wz: f(z,k)≥q} short-run cost function: Cs(q,k,w,r)= Cv(q,k,w)+rk. long-run cost function: C(q,w,r)=mink≥0Cs(q,k,w,r) k(q,w,r)=argmink≥0Cs(q,k,w,r) z(q,w,r)=zv(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 The restricted (“short-run”) cost function

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 a0 • 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