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

3. Production Technology: y = f (x 1, x 2, x 3,... x n ) x i ’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)

4. Last year depicted the relationship between inputs as an isoquant y = f (K, L) K L yoyo y1y1

5. L K y=output y0y0 L0L0 K0K0 y=f(K,L) An alternative representation is:

6. 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 MP K MP L And along a given isoquant dy = 0

7. If dy = 0 then MP k dK+MP L dL=0 MP K dk = - MP L 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) = MRTS KL

8. Usually assume that MRTS KL is diminishing Follows from the fact that MP of capital and labour is decreasing. Thus, K L = MP L, gets smaller as we increase L when we substitute L for K, while = MP k gets bigger as K gets smaller.

9. 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’ x2x2 x1x1

10. 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 xixi

11. 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. x2x2 x1x1

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

13. Returns to Scale A function is homogenous if degree k iff f (t K, t L) = t k f(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) = 4 2 f (K,L)=16f (K,L) if DRS e.g. k = ½ then f (4K,4L) = 4 ½ f (K,L)=2f (K,L)

14. Ch 18 Varian Problem 1. The Profit maximisation problem Where x i are inputs w i 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 f (K, L) - rK - wL

15. 1). 2). So profit maximisation requires that = 0 First Order Conditions: MP L =

16. Similarly Or P.MP K = r Or finally i.e. Ratio of the marginal products = Ratio of the Marginal Costs

17. 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

18. y L MP L = w/p y K MP K = r/p L K y0y0 L0L0 K0K0 y=f(K,L)

19. L OR Tells us the slope of the isoquant, but not which isoquant K yoyo y1y1

20. So if in the short-run the capital stock is fixed at some amount then we can solve for ideal L and hence y y L MP L = w/p L0L0 y0y0 but what about the long run? We need something more

21. 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. P x x+ P y y=M So why can’t we do the same thing here in production

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

23. Isoquant Map of f (K, L) Suppose now have a constraint on output e.g. venture capitalist will only lend you £10m K yoyo y1y1 L y2y2 L0L0 K0K0

24. Isoquant Map of f(K, L) =Ratio of factor prices constraint K yoyo y1y1 L

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

26. 3. Alternative to the alternative representation of the problem! General cost function: C = wL + rK K L c1c1 Iso-cost Lines Intercept will be C/r and slope – w/r For different levels of C we can draw iso-cost lines c3c3

27. 3. Alternative to the alternative representation of the problem! K L c1c1 Iso-cost Lines Now for a given output target, say, 10,000 units of output (a specific order for Sainsbury’s) we want to minimise costs. c3c3 yoyo Pick K & L to Minimise costs C = wL + rK subject to producing Output y 0 =f(K.L)

28. So have 3 Distinct Problems 1) Maximise profits Max x1, x2  = P f (x 1, x 2 ) – w 1 x 1 – w 2 x 2 Gives factor demand functions X 1 = x 1 (w 1, w 2 ) X 2 = x 2 (w 1, w 2 ) May not be well defined if there are constant returns to scale

29. 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.

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

31. From 1 + 2

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

33. Next Consider Problem 3 Minimise cost : wL + w 2 K s.t. f(x 1, x 2 ) =

34. 3 EQNS – 3 unknowns x 1, x 2, So solve for ‘Quantity Constrained’ Conditional factor demands X 1 = x 1 (w 1, w 2, ) X 2 = x 2 (w 1, w 2, )

35. Cobb-Douglas Example (? = w 1 x 1 + w 2 x 2 + [ - x 1 a x 2 b ] FOC

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

37. Similarly we can solve for x 2 Re-arranging the bottom line

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

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

40. Note can now formalise the cost function for the item C = w 1 x 1 + w 2 x 2