1. 1 Chapter 7 PRODUCTION FUNCTIONS

2. Review of Chapter 5 Income and substitution effects Price changes affect quantity demanded Marshallian demand function:x=x(p,I) Hicksian (compensated) : x c = x c (p,U) Slutsky equation: Demand elasticity Consumer surplus change: 2

3. 3 Production Function The firm’s production function for a particular good (q) shows the maximum amount of the good that can be produced using alternative combinations of capital (k) and labor ( l ) q = f(k, l ) Two factors k, l : only for convenience. n factors’ case: q=f(x 1, x 2,…, x n )

4. 4 Marginal Physical Product To study variation in a single input, we define marginal physical product as the additional output that can be produced by employing one more unit of that input while holding other inputs constant.

5. 5 Diminishing Marginal Productivity The marginal physical product of an input depends on how much of that input is used In general, we assume diminishing marginal productivity.( e.g ipod production)

6. 6 Malthus’s gloomy predictions Because of diminishing marginal productivity, 19th century Malthus worried about the effect of population growth on labor productivity. But changes in the marginal productivity of labor over time also depend on changes in other inputs such as capital –we need to consider f l k which is often > 0 –Labor productivity has risen for increases in k

7. 7 Average Physical Product Labor productivity is often measured by average productivity Note that AP l also depends on the amount of capital employed

8. 8 Isoquant Maps To illustrate the possible substitution of one input for another, we use an isoquant map An isoquant shows those combinations of k and l that can produce a given level of output (q 0 ) f(k, l ) = q 0

9. 9 Isoquant Map l per period k per period Each isoquant represents a different level of output –output rises as we move northeast q = 30 q = 20

10. 10 Marginal Rate of Technical Substitution (RTS) l per period k per period q = 20 - slope = marginal rate of technical substitution (RTS) The slope of an isoquant shows the rate at which l can be substituted for k lAlA kAkA kBkB lBlB A B RTS > 0 and is diminishing for increasing inputs of labor

11. 11 Marginal Rate of Technical Substitution (RTS) The marginal rate of technical substitution (RTS) shows the rate at which labor can be substituted for capital while holding output constant along an isoquant substitute A for B=substitute B with(by) A ， 用 A 替代 B

12. 12 RTS and Marginal Productivities Take the total differential of the production function: Along an isoquant dq = 0, so

13. 13 RTS and Marginal Productivities Because MP l and MP k will both be nonnegative, RTS will be positive (or zero However, it is generally not possible to derive a diminishing RTS from the assumption of diminishing marginal productivity alone To see Why ： assume q=f(k,l),f k >0 f l >0,and f kk <0, f ll <0

14. 14 RTS and Marginal Productivities To show that isoquants are convex, we would like to show that d(RTS)/d l < 0 Since RTS = f l /f k

15. 15 RTS and Marginal Productivities Using the fact that dk/d l = -f l /f k along an isoquant and Young’s theorem (f k l = f l k ) Because we have assumed f k > 0, the denominator is positive Because f ll and f kk are both assumed to be negative, the ratio will be negative if f k l is positive

16. 16 RTS and Marginal Productivities Intuitively, it seems reasonable that f k l = f l k should be positive –if workers have more capital, they will be more productive But some production functions have f k l < 0 over some input ranges –when we assume diminishing RTS we are assuming that MP l and MP k diminish quickly enough to compensate for any possible negative cross-productivity effects

17. 17 A Diminishing RTS Suppose the production function is q = f(k, l ) = 600k 2 l 2 - k 3 l 3 For this production function MP l = f l = 1200k 2 l - 3k 3 l 2 MP k = f k = 1200k l 2 - 3k 2 l 3 –these marginal productivities will be positive for values of k and l for which k l < 400

18. 18 A Diminishing RTS Because f ll = 1200k 2 - 6k 3 l f kk = 1200 l 2 - 6k l 3 this production function exhibits diminishing marginal productivities for sufficiently large values of k and l –f ll and f kk 200

19. 19 A Diminishing RTS Cross differentiation of either of the marginal productivity functions yields f k l = f l k = 2400k l - 9k 2 l 2 which is positive only for k l < 266

20. 20 A Diminishing RTS Thus, for this production function, RTS is diminishing throughout the range of k and l where marginal productivities are positive –for higher values of k and l, the diminishing marginal productivities are sufficient to overcome the influence of a negative value for f k l to ensure convexity of the isoquants

21. 21 Returns to Scale How does output respond to increases in all inputs together? –suppose that all inputs are doubled, would output double? Returns to scale have been of interest to economists since the days of Adam Smith

22. 22 Returns to Scale Smith identified two forces that come into operation as inputs are doubled –greater division of labor and specialization of function –loss in efficiency because management may become more difficult given the larger scale of the firm

23. 23 Returns to Scale If the production function is given by q = f(k, l ) and all inputs are multiplied by the same positive constant (t >1), then

24. 24 Returns to Scale It is possible for a production function to exhibit constant returns to scale for some levels of input usage and increasing or decreasing returns for other levels –economists refer to the degree of returns to scale with the implicit notion that only a fairly narrow range of variation in input usage and the related level of output is being considered

25. 25 Constant Returns to Scale Constant returns-to-scale production functions are homogeneous of degree one in inputs f(tk,t l ) = t 1 f(k, l ) = tq This implies that the marginal productivity functions are homogeneous of degree zero –Differentiating the above equation,

26. 26 Constant Returns to Scale The marginal productivity of any input depends on the ratio of capital and labor (not on the absolute levels of these inputs) The RTS between k and l depends only on the ratio of k to l, not the scale of operation => homothetic prod.func.(chat.2 p55) t=1/l

27. 27 Constant Returns to Scale The production function will be homothetic Geometrically, all of the isoquants are radial expansions of one another CRS is a reasonably good approximation to use for an aggregate production function in the empirical researches.

28. 28 Constant Returns to Scale l per period k per period Along a ray from the origin (constant k/ l ), the RTS will be the same on all isoquants q = 3 q = 2 q = 1 The isoquants are equally spaced as output expands

29. 29 Returns to Scale Returns to scale can be generalized to a production function with n inputs q = f(x 1,x 2,…,x n ) If all inputs are multiplied by a positive constant t, we have f(tx 1,tx 2,…,tx n ) = t k f(x 1,x 2,…,x n )=t k q –If k = 1, we have constant returns to scale –If k < 1, we have decreasing returns to scale –If k > 1, we have increasing returns to scale

30. 30 Elasticity of Substitution The elasticity of substitution (  ) measures the proportionate change in k/ l relative to the proportionate change in the RTS along an isoquant The value of  will always be positive because k/ l and RTS move in the same direction

31. 31 Elasticity of Substitution l per period k per period Both RTS and k/ l will change as we move from point A to point B A B q = q 0 RTS A RTS B (k/ l ) A (k/ l ) B  is the ratio of these proportional changes  measures the curvature( 曲率 ) of the isoquant

32. 32 Elasticity of Substitution If  is high, the RTS will not change much relative to k/ l –the isoquant will be relatively flat If  is low, the RTS will change by a substantial amount as k/ l changes –the isoquant will be sharply curved It is possible for  to change along an isoquant or as the scale of production changes

33. 33 Elasticity of Substitution Generalizing the elasticity of substitution to the many-input case raises several complications –if we define the elasticity of substitution between two inputs to be the proportionate change in the ratio of the two inputs to the proportionate change in RTS, we need to hold output and the levels of other inputs constant

34. 34 The Linear Production Function Suppose that the production function is q = f(k, l ) = ak + b l This production function exhibits constant returns to scale f(tk,t l ) = atk + bt l = t(ak + b l ) = tf(k, l ) All isoquants are straight lines –RTS is constant –  = 

35. 35 The Linear Production Function l per period k per period q1q1 q2q2 q3q3 Capital and labor are perfect substitutes RTS is constant as k/ l changes slope = -b/a  = 

36. 36 Fixed Proportions Suppose that the production function is q = min (ak,b l ) a,b > 0 Capital and labor must always be used in a fixed ratio –the firm will always operate along a ray where k/ l is constant Because k/ l is constant,  = 0

37. 37 Fixed Proportions l per period k per period q1q1 q2q2 q3q3 No substitution between labor and capital is possible  = 0 k/ l is fixed at b/a q 3 /b q 3 /a

38. 38 Cobb-Douglas Production Function Suppose that the production function is q = f(k, l ) = Ak a l b A,a,b > 0 This production function can exhibit any returns to scale f(tk,t l ) = A(tk) a (t l ) b = At a+b k a l b = t a+b f(k, l ) –if a + b = 1  constant returns to scale –if a + b > 1  increasing returns to scale –if a + b < 1  decreasing returns to scale

39. 39 Cobb-Douglas Production Function The Cobb-Douglas production function is linear in logarithms ln q = ln A + a ln k + b ln l –a is the elasticity of output with respect to k –b is the elasticity of output with respect to l

40. 40 CES CES Production Function Suppose that the production function is q = f(k, l ) = [k  + l  ]  /    1,   0,  > 0 –  > 1  increasing returns to scale –  < 1  decreasing returns to scale For this production function  = 1/(1-  ) –  = 1  linear production function –  = -   fixed proportions production function –  = 0  Cobb-Douglas production function

41. 41 A Generalized Leontief Production Function Suppose that the production function is q = f(k, l ) = k + l + 2(k l ) 0.5 Marginal productivities are f k = 1 + (k/ l ) -0.5 f l = 1 + (k/ l ) 0.5 Thus,

42. 42 Technical Progress Methods of production change over time Following the development of superior production techniques, the same level of output can be produced with fewer inputs –the isoquant shifts in. many empirical literatures about technical progress

43. 43 Technical Progress Suppose that the production function is q = A(t)f(k, l ) where A(t) represents all influences that go into determining q other than k and l –changes in A over time represent technical progress A is shown as a function of time (t) dA/dt > 0

44. 44 Technical Progress Differentiating the production function with respect to time we get

45. 45 Technical Progress Dividing by q gives us

46. 46 Technical Progress For any variable x, [(dx/dt)/x] is the proportional growth rate in x –denote this by G x Then, we can write the equation in terms of growth rates

47. 47 Technical Progress Since

48. 48 Technical Progress in the Cobb-Douglas Function Suppose that the production function is q = A(t)f(k, l ) = A(t)k  l 1-  If we assume that technical progress occurs at a constant exponential (  ) then A(t) = Ae  t q = Ae  t k  l 1- 

49. 49 Technical Progress in the Cobb-Douglas Function Taking logarithms and differentiating with respect to t gives the growth equation

50. 50 Technical Progress in the Cobb-Douglas Function

51. 51 Important Points to Note: If all but one of the inputs are held constant, a relationship between the single variable input and output can be derived –the marginal physical productivity is the change in output resulting from a one-unit increase in the use of the input assumed to decline as use of the input increases

52. 52 Important Points to Note: The entire production function can be illustrated by an isoquant map –the slope of an isoquant is the marginal rate of technical substitution (RTS) it shows how one input can be substituted for another while holding output constant it is the ratio of the marginal physical productivities of the two inputs

53. 53 Important Points to Note: Isoquants are usually assumed to be convex –they obey the assumption of a diminishing RTS this assumption cannot be derived exclusively from the assumption of diminishing marginal productivity one must be concerned with the effect of changes in one input on the marginal productivity of other inputs

54. 54 Important Points to Note: The returns to scale exhibited by a production function record how output responds to proportionate increases in all inputs –if output increases proportionately with input use, there are constant returns to scale

55. 55 Important Points to Note: The elasticity of substitution (  ) provides a measure of how easy it is to substitute one input for another in production –a high  implies nearly straight isoquants –a low  implies that isoquants are nearly L-shaped

56. 56 Important Points to Note: Technical progress shifts the entire production function and isoquant map –technical improvements may arise from the use of more productive inputs or better methods of economic organization

57. CES function CES utility(chat.3 p85) Arrow et. al.(1961): a general form of CES function 57

58. 变替代弹性（ VES ）生产函数模型 Variable Elasticity Substitution 考虑到要素替代弹性与要素投入比例有关，当 K/L 较大时，资 本替代劳动比较困难，而当 K/L 较小时，资本替代劳动就要容易 得多，因此假设要素替代弹性为投入要素比例的线性函数： 从而生产函数的一般形式为： 其中： Z ＝ Y/L ， k ＝ K/L

59. 变替代弹性（ VES ）生产函数模型 当 b=0 时， VES 等同于 CES 当 b=0 、 a=1 时， VES 等同于 C-D 模型 当 a=1 时， VES 的形式为： 这是最一般而常用的 VES 的理论形式，其替代弹 性为：

60. 超越对数 translog 生产函数 可以根据估计估计结果差数要素的替代性质： 如果 ，表现为 C-D 生产函数 如果 ，表现为 CES 生产函数