Production

1. Production • Reading Varian 17-20 • But particularly, All Ch 17 and the Appendices to Chapters 18 & 19. • We start with Chapter 17.

2. Production • Technology: y = f (x1, x2, x3, ... xn) • xi’s = inputs into the production process • For simplicity consider the case of 2 inputs e.g. labour and capital, L and K • y = f (K, L)

3. Last year depicted the relationship • between inputs as an isoquant • y = f (K, L) K y1 yo L

4. An alternative representation is: y=output y=f(K,L) y0 K K0 L L0

5. This year want to analyse isoquants and the firm’s production problem in the same fashion as utility. • y = y(K,L) • Taking the total derivative dy = dK + dL MPK MPL And along a given isoquant dy = 0

6. If dy = 0 then MPkdK+MPLdL=0 • MPK dk = - MPL dL Or slope of the isoquant Marginal Rate of Technical Substitution of K per unit of L (Amount of K that must be substituted per unit of L in order to keep output constant) = MRTSKL

7. K L • Usually assume that MRTSKL is diminishing • Follows from the fact that MP of capital and labour is decreasing. Thus, = MPL, gets smaller as we increase L when we substitute L for K, while = MPk gets bigger as K gets smaller.

8. x2 x1 So as L gets bigger and K gets smaller, the top of the line goes down while the bottom goes up, so dK/dL gets smaller as L gets bigger That is, Isoquants are Quasi ‘convex’

9. Note MRTS different from diminishing marginal product • As we noted above, ‘Law’ of diminishing marginal product says df/dL gets smaller as L gets bigger holding all other inputs constant y xi

10. x2 x1 But in this exercise we are reducing K as we increase L, so all other things are not constant So MRTS is not the same as Diminishing Marginal Product, though they are related.

11. So Distinct Concepts • 1. Diminishing Marginal Product • 2. Diminishing Marginal Rate of ……Technical Substitution • 3. Returns to Scale

12. Returns to Scale A function is homogenous if degree k iff f (t K, t L) = tkf(K,L,) e.g. if k = 1, i.e. there are C.R.S. then f (4K, 4L) = 4f (K,L) if IRS, e.g. k = 2 then f (4K, 4L) = 42f (K,L)=16f (K,L) if DRS e.g. k = ½ then f (4K,4L) =4½f (K,L)=2f (K,L)

13. Where xi are inputs wi are the prices of inputs Now we usually know what y is because unlike utility we can get this from engineering studies etc. y = f (K, L) Max w.r.t.K,L p = P f (K, L) - rK - wL Ch 18 Varian Problem 1. The Profit maximisation problem

14. First Order Conditions: 1). = 0 2). = 0 So profit maximisation requires that MPL=

15. Or P.MPK = r Similarly Or finally i.e. Ratio of the marginal products = Ratio of the Marginal Costs

16. So first order conditions (1) + (2) gives us Or in other words it tells us how much K to use given L, and how much L to use given K But not how much k and L to use

17. MPL= w/p MPK= r/p y=f(K,L) y0 y y K K0 L K L L0

18. OR K Tells us the slope of the isoquant, but not which isoquant y1 yo L

19. MPL= w/p y0 L0 y L So if in the short-run the capital stock is fixed at some amount then we can solve for ideal L and hence y but what about the long run? We need something more

20. 2. Alternative View • Recall in consumer demand, we derived a demand curve for x without any great problems? • E.G.for a Cobb-Douglas utility function: • Max U(x,y) s.t. Pxx+ Pyy=M So why can’t we do the same thing here in production

21. Profit Maximisation Problem 2 • Appendix to Ch 18 • An alternative to first problem • Maximising output subject to a cost constraint

22. Suppose now have a constraint on output e.g. venture capitalist will only lend you £10m K Isoquant Map of f (K, L) K0 y2 y1 yo L L0

23. constraint K Isoquant Map of f(K, L) y1 =Ratio of factor prices yo L

24. Profit Maximisation Problem 3 • Varian Appendix Ch19 • Alternative to the Alternative in problem 2 • Minimising Cost subject to an Output constraint

25. K c1 L 3. Alternative to the alternative representation of the problem! General cost function: C = wL + rK Iso-cost Lines c3 Intercept will be C/r and slope – w/r For different levels of C we can draw iso-cost lines

26. K c1 L 3. Alternative to the alternative representation of the problem! Pick K & L to Minimise costs C = wL + rK subject to producing Output y0=f(K.L) Iso-cost Lines yo c3 Now for a given output target, say, 10,000 units of output (a specific order for Sainsbury’s) we want to minimise costs.

27. So have 3 Distinct Problems 1) Maximise profits Maxx1, x2 p = P f (x1, x2) – w1x1 – w2x2 Gives factor demand functions X1 = x1 (w1, w2) X2 = x2 (w1, w2) May not be well defined if there are constant returns to scale

28. 2) Maximise subject to a constraint 3) Minimise subject to a constraint Problem 3) is the Dual of 2) Called Duality Theory Essentially allows us to look at problems in reverse and can often give very important insights.

29. Take Problem 2: e.g (1) (2) (3)

30. From 1 + 2

31. Substitute into (3) This is a Cost constrained factor demand function

32. Next Consider Problem 3 Minimise cost : wL + w2 K s.t. f(x1, x2) =

33. 3 EQNS – 3 unknowns x1, x2,  So solve for ‘Quantity Constrained’ Conditional factor demands X1 = x1 (w1, w2, ) X2 = x2 (w1, w2, )

34. Cobb-Douglas Example (? = w1 x1+ w2x2 +  [ - x1a x2b] FOC

35. If we have CRS a + b = 1 [and notes we invert brackets when we bring it to other side] Conditional demand function for x1

36. Similarly we can solve for x2 Re-arranging the bottom line

37. [If a+b = 1] Getting rid of the Power [b] on the RHS 

38. Now note So conditional factor demand functions always slope down Constant – so no ‘income’ type effect

39. Note can now formalise the cost function for the item C = w1 x1 + w2x2