1. Econ 102 SY 2008 2009 Lecture 7 Supply and Cost Minimization Sept 2, 2008

2. Econ 102 SY 2008 2009 Key concepts Profit Function of Cobb Douglas Cost function

3. Econ 102 SY 2008 2009 Illustration x w x y

4. Econ 102 SY 2008 2009 Illustration p y

5. Econ 102 SY 2008 2009 Properties of supply functions Homogeneous of degree zero in all prices, i.e., y(tp) =t 0 y(p) for all t > 0. Negative Definiteness of Substitution Matrix Let y = f(x) be a twice differentiable and strictly concave single output production function, and let x(p;w) be the input demand function. Then, the substitution matrix Is negative definite and symmetric.

6. Econ 102 SY 2008 2009

7. Properties of Profit function Non-decreasing in output prices, non-increasing in input prices. If p’ i >=p i for all outputs and w’ i =  (p;w). Homogeneous of degree 1 in p.  (tp;tw) = t 1  (p;w) for all t >= 0. Convex in p. Let p’’ = tp+(1-t)p’ for 0 <= t <= 1, Then  (p’’) <= t  (p)+(1-t)  (p’). Continuous in p. The function  (p) is continuous, at least when  (p) is well-defined and p i > 0 for all i.

8. Econ 102 SY 2008 2009 Deriving the supply function (Hotelling’s Lemma)

9. Econ 102 SY 2008 2009 Cost Minimization Cost minimization is a necessary condition for profit maximization. The cost function is the minimal cost at the factor prices w and output level y; that is: c(w; y) = wx(w; y)

10. Econ 102 SY 2008 2009 Illustration x1 x2 y=f(x1;x2)=20 c’=w 1 x’ 1 +w 2 x’ 2 =100 c’’=w 1 x’’ 1 +w 2 x’’ 2 =90=f(w 1 ;w 2 ;20)=f(w;y) c’’’=w 1 x’’’ 1 +w 2 x’’’ 2 =80

11. Econ 102 SY 2008 2009 Condition for cost minimization Ratio of input prices equal to the ratio of the corresponding marginal products or MRTS. Slope of the isocost line is equal to that of the isoquant.

12. Econ 102 SY 2008 2009 Algebra of cost minimization

13. Econ 102 SY 2008 2009 Illustration: Cobb-Douglas Cost Function Conditional input demands Cost function

14. Econ 102 SY 2008 2009 Properties of cost function Non-decreasing in w. Homogeneous of degree 1 in w. Concave in w. Continuous in w, for w > 0. For all w > 0, c(w; y) is strictly increasing y.

15. Econ 102 SY 2008 2009 Shephard’s Lemma If x(w; y) is the cost-minimizing bundle necessary to produce production level y at prices w, then x i (w; y) =  c(w;y)/  w i for all i assuming the derivative exists and that x i > 0.

16. Econ 102 SY 2008 2009 Properties of conditional input demands Negative Semi-Definite Substitution Matrix Symmetric Substitution Terms Negative Own-Substitution Terms)

17. Econ 102 SY 2008 2009 Cost function vs. profit function Re-state profit maximization as follows Max  (p,x 1,x 2 ) = py - w 1 x 1 +w 2 x 2 Max y  = py - c(w 1,w 2 ;y) First order condition

18. Econ 102 SY 2008 2009 y1 y2 y3 x1 x2

19. Econ 102 SY 2008 2009 Cost definitions Total cost = Variable cost + Fixed cost Variable cost Fixed cost Average total cost=ATC=AC=TC/y Average variable cost=AVC=VC/y Average fixed cost=AFC=FC/y Short run vs. Long run

20. Econ 102 SY 2008 2009 Variable cost Y VC

21. Econ 102 SY 2008 2009 Fixed cost Y FC

22. Econ 102 SY 2008 2009 Total cost=fixed cost +variable cost Y TC

23. Econ 102 SY 2008 2009 Average fixed cost Y AFC

24. Econ 102 SY 2008 2009 Diminishing marginal product X1 MP

25. Econ 102 SY 2008 2009 Average variable cost Y AVC

26. Econ 102 SY 2008 2009 Average total cost y atc

27. Econ 102 SY 2008 2009 Geometry of costs Y costs atc avc

28. Econ 102 SY 2008 2009 End of Lecture 6 Production and Costs