1. Chapter 11 PRODUCTION FUNCTIONS Copyright ©2002 by South-Western, a division of Thomson Learning. All rights reserved. MICROECONOMIC THEORY BASIC PRINCIPLES AND EXTENSIONS EIGHTH EDITION WALTER NICHOLSON

2. 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)

3. 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

4. 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

5. Diminishing Marginal Productivity Because of diminishing marginal productivity, 19th century economist Thomas 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 LK which is often > 0

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

7. A Two-Input Production Function Suppose the production function for flyswatters can be represented by q = f(K,L) = 600K 2 L 2 - K 3 L 3 To construct MP L and AP L, we must assume a value for K –Let K = 10 The production function becomes q = 60,000L 2 - 1000L 3

8. A Two-Input Production Function The marginal productivity function is MP L =  q/  L = 120,000L - 3000L 2 which diminishes as L increases This implies that q has a maximum value: 120,000L - 3000L 2 = 0 40L = L 2 L = 40 Labor input beyond L=40 reduces output

9. A Two-Input Production Function To find average productivity, we hold K=10 and solve AP L = q/L = 60,000L - 1000L 2 AP L reaches its maximum where  AP L /  L = 60,000 - 2000L = 0 L = 30

10. A Two-Input Production Function In fact, when L=30, both AP L and MP L are equal to 900,000 Thus, when AP L is at its maximum, AP L and MP L are equal

11. 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

12. 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

13. 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

14. 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

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

16. RTS and Marginal Productivities Because MP L and MP K will both be nonnegative, RTS will also be nonnegative However, it is not possible to derive a diminishing RTS from the assumption of diminishing marginal productivity alone

17. RTS and Marginal Productivities To show that isoquants are convex, we would like to show that d(RTS)/dL < 0 Since RTS = f L /f K

18. RTS and Marginal Productivities Using the fact that dK/dL = -f L /f K along an isoquant and Young’s theorem (f KL = f LK ) 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 KL is positive

19. RTS and Marginal Productivities Intuitively, it seems reasonable that f KL =f LK should be positive –if workers have more capital, they will be more productive But some production functions have f KL < 0 over some input ranges –Thus, 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

20. 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 = 1200KL 2 - 3K 2 L 3 These marginal productivities will be positive for values of K and L for which KL < 400

21. A Diminishing RTS Because f LL = 1200K 2 - 6K 3 L f KK = 1200L 2 - 6KL 3 this production function exhibits diminishing marginal productivities for sufficiently large values of K and L –f LL and f KK 200

22. A Diminishing RTS Cross differentiation of either of the marginal productivity functions yields f KL = f LK = 2400KL - 9K 2 L 2 which is positive only for KL < 266

23. 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 KL to ensure convexity of the isoquants

24. 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

25. 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

26. Returns to Scale If the production function is given by q = f(K,L) and all inputs are multiplied by the same positive constant (m > 1), then

27. 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

28. Constant Returns to Scale Constant returns-to-scale production functions have the useful theoretical property that that the RTS between K and L depends only on the ratio of K to L, not the scale of operation Geometrically, all of the isoquants are “radial blowups” of the unit isoquant

29. 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

30. 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 m, we have f(mX 1,mX 2,…,mX n ) = m k f(X 1,X 2,…,X n )=m 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

31. 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

32. 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 of the isoquant

33. 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

34. The Linear Production Function Suppose that the production function is q = f(K,L) = aK + bL This production function exhibits constant returns to scale f(mK,mL) = amK + bmL = m(aK + bL) = mf(K,L) All isoquants are straight lines –RTS is constant –  = 

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. Fixed Proportions Suppose that the production function is q = min (aK,bL) 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. 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. 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(mK,mL) = A(mK) a (mL) b = Am a+b K a L b = m 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. Cobb-Douglas Production Function Suppose that hamburgers are produced according to the Cobb-Douglas function q = 10K 0.5 L 0.5 Since a+b=1  constant returns to scale The isoquant map can be derived q = 50 = 10K 0.5 L 0.5  KL = 25 q = 100 = 10K 0.5 L 0.5  KL = 100 –The isoquants are rectangular hyperbolas

40. Cobb-Douglas Production Function The RTS can easily be calculated The RTS declines as L rises and K falls The RTS depends only on the ratio of K and L Because the RTS changes exactly in proportion to changes in K/L,  = 1

41. 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

42. 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

43. 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

44. 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

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

46. Technical Progress Dividing by q gives us

47. 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

48. Technical Progress Since

49. Technical Progress in the Cobb-Douglas Function Suppose that the production function is q = 10e 0.05t K 0.5 L 0.5 Taking logarithms yields ln q = ln 10 + 0.05t + 0.5 ln K + 0.5 ln L Differentiating with respect to t gives the growth equation

50. Technical Progress in the Cobb-Douglas Function We can put this in terms of growth rates G q = 0.05 + 0.5G K + 0.5G L When K and L are constant, output grows at 5 percent per period –G K = G L = 0 –G q = 0.05

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 –the marginal physical productivity of an input is assumed to decline as use of the input increases

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 measures how one input can be substituted for another while holding output constant RTS is the ratio of the marginal physical productivities of the two inputs –Isoquants are assumed to be convex they obey the assumption of a diminishing RTS

53. 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 –if there are greater than proportionate increases in output, there are increasing returns to scale –if there are less than proportionate increases in output, there are decreasing returns to scale

54. Important Points to Note: The elasticity of substitution (  ) provides a measure of how easy it is to substitute on input for another in production –a high  implies nearly straight isoquants –a low  implies that isoquants are nearly L- shaped Technical progress shifts the entire production function and isoquant map –may arise from the use of more productive inputs or better economic organization