Chapter 19 Technology

1. Chapter 19 Technology

2. Technologies • A technology is a process by which inputs are converted to an output. • E.g. labor, a computer, a projector, electricity, and software are being combined to produce this lecture.

3. Technologies • Usually several technologies will produce the same product -- a blackboard and chalk can be used instead of a computer and a projector. • Which technology is “best”? • How do we compare technologies?

4. Input Bundles • xi denotes the amount used of input i; i.e. the level of input i. • An input bundle is a vector of the input levels; (x1, x2, … , xn). • E.g. (x1, x2, x3) = (6, 0, 9×3).

5. Production Functions • y denotes the output level. • The technology’s production function states the maximum amount of output possible from an input bundle.

6. Production Functions One input, one output Output Level y = f(x) is the production function. y’ y’ = f(x’) is the maximal output level obtainable from x’ input units. x’ x Input Level

7. Technology Sets • A production plan is an input bundle and an output level; (x1, … , xn, y). • A production plan is feasible if • The collection of all feasible production plans is the technology set.

8. Technology Sets One input, one output Output Level y = f(x) is the production function. y’ y’ = f(x’) is the maximal output level obtainable from x’ input units. y” y” = f(x’) is an output level that is feasible from x’ input units. x’ x Input Level

9. Technology Sets The technology set is

10. Technology Sets One input, one output Output Level y’ The technologyset y” x’ x Input Level

11. Technology Sets One input, one output Output Level Technicallyefficient plans y’ The technologyset Technicallyinefficientplans y” x’ x Input Level

12. Technologies with Multiple Inputs • What does a technology look like when there is more than one input? • The two input case: Input levels are x1 and x2. Output level is y. • Suppose the production function is

13. Technologies with Multiple Inputs • E.g. the maximal output level possible from the input bundle(x1, x2) = (1, 8) is • And the maximal output level possible from (x1,x2) = (8,8) is

14. Technologies with Multiple Inputs • The y output unit isoquant is the set of all input bundles that yield (at most) the same output level y.

15. Isoquants with Two Variable Inputs x2 y º 8 y º 4 x1

16. Isoquants with Two Variable Inputs • More isoquants tell us more about the technology.

17. Isoquants with Two Variable Inputs x2 y º 8 y º 6 y º 4 y º 2 x1

18. Isoquants with Two Variable Inputs Output, y y º 8 y º 6 y º 4 x2 y º 2 x1

19. Cobb-Douglas Technologies x2 All isoquants are hyperbolic,asymptoting to, but nevertouching any axis. x1

20. Fixed-Proportions Technologies • A fixed-proportions production function is of the form • E.g.with

21. Fixed-Proportions Technologies x2 x1 = 2x2 min{x1,2x2} = 14 7 min{x1,2x2} = 8 4 2 min{x1,2x2} = 4 4 8 14 x1

22. Perfect-Substitutes Technologies • A perfect-substitutes production function is of the form • E.g.with

23. Perfect-Substitution Technologies x2 x1 + 3x2 = 18 x1 + 3x2 = 36 x1 + 3x2 = 48 8 6 All are linear and parallel 3 x1 9 18 24

24. Marginal (Physical) Products • The marginal product of input i is the rate-of-change of the output level as the level of input i changes, holding all other input levels fixed. • That is,

25. Marginal (Physical) Products • The marginal product of input i is diminishing if it becomes smaller as the level of input i increases. That is, if

26. Marginal (Physical) Products E.g. if then and

27. Marginal (Physical) Products E.g. if then and so

28. Marginal (Physical) Products E.g. if then and so and

29. Marginal (Physical) Products E.g. if then and so and Both marginal products are diminishing.

30. Returns-to-Scale • Marginal products describe the change in output level as a single input level changes. • Returns-to-scale describes how the output level changes as all input levels change in direct proportion (e.g. all input levels doubled, or halved).

31. Returns-to-Scale If, for any input bundle (x1,…,xn), then the technology described by theproduction function f exhibits constantreturns-to-scale.E.g. (k = 2) doubling all input levelsdoubles the output level.

32. Returns-to-Scale One input, one output Output Level y = f(x) 2y’ Constantreturns-to-scale y’ x’ 2x’ x Input Level

33. Returns-to-Scale If, for any input bundle (x1,…,xn), then the technology exhibits diminishingreturns-to-scale.E.g. (k = 2) doubling all input levels less than doubles the output level.

34. Returns-to-Scale One input, one output Output Level 2f(x’) y = f(x) f(2x’) Decreasingreturns-to-scale f(x’) x’ 2x’ x Input Level

35. Returns-to-Scale If, for any input bundle (x1,…,xn), then the technology exhibits increasingreturns-to-scale.E.g. (k = 2) doubling all input levelsmore than doubles the output level.

36. Returns-to-Scale One input, one output Output Level Increasingreturns-to-scale y = f(x) f(2x’) 2f(x’) f(x’) x’ 2x’ x Input Level

37. Returns-to-Scale • A single technology can ‘locally’ exhibit different returns-to-scale.

38. Returns-to-Scale One input, one output Output Level y = f(x) Increasingreturns-to-scale Decreasingreturns-to-scale x Input Level

39. Examples of Returns-to-Scale The perfect-substitutes productionfunction is Expand all input levels proportionatelyby k. The output level becomes

40. Examples of Returns-to-Scale The perfect-substitutes productionfunction is Expand all input levels proportionatelyby k. The output level becomes

41. Examples of Returns-to-Scale The perfect-substitutes productionfunction is Expand all input levels proportionatelyby k. The output level becomes The perfect-substitutes productionfunction exhibits constant returns-to-scale.

42. Examples of Returns-to-Scale The perfect-complements productionfunction is Expand all input levels proportionatelyby k. The output level becomes

43. Examples of Returns-to-Scale The perfect-complements productionfunction is Expand all input levels proportionatelyby k. The output level becomes

44. Examples of Returns-to-Scale The perfect-complements productionfunction is Expand all input levels proportionatelyby k. The output level becomes The perfect-complements productionfunction exhibits constant returns-to-scale.

45. Examples of Returns-to-Scale The Cobb-Douglas production function is Expand all input levels proportionatelyby k. The output level becomes

46. Examples of Returns-to-Scale The Cobb-Douglas production function is Expand all input levels proportionatelyby k. The output level becomes

47. Examples of Returns-to-Scale The Cobb-Douglas production function is Expand all input levels proportionatelyby k. The output level becomes

48. Examples of Returns-to-Scale The Cobb-Douglas production function is Expand all input levels proportionatelyby k. The output level becomes

49. Examples of Returns-to-Scale The Cobb-Douglas production function is The Cobb-Douglas technology’s returns-to-scale isconstant if a1+ … + an = 1

50. Examples of Returns-to-Scale The Cobb-Douglas production function is The Cobb-Douglas technology’s returns-to-scale isconstant if a1+ … + an = 1increasing if a1+ … + an > 1