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Chapter 7

Chapter 7. PRODUCTION FUNCTIONS. 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:. Production Function.

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Chapter 7

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  1. Chapter 7 PRODUCTION FUNCTIONS

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

  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(x1, x2,…, xn )

  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. 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. 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 flk which is often > 0 • Labor productivity has risen for increases in k

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

  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 (q0) f(k,l) = q0

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

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

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

  13. RTS and Marginal Productivities • Because MPl and MPk 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),fk>0 fl>0,and fkk<0, fll<0

  14. RTS and Marginal Productivities • To show that isoquants are convex, we would like to show that d(RTS)/dl < 0 • Since RTS = fl/fk

  15. RTS and Marginal Productivities • Using the fact that dk/dl = -fl/fk along an isoquant and Young’s theorem (fkl = flk) • Because we have assumed fk > 0, the denominator is positive • Because fll and fkk are both assumed to be negative, the ratio will be negative if fkl is positive

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

  17. A Diminishing RTS • Suppose the production function is q = f(k,l) = 600k2l2 - k3l3 • For this production function MPl = fl = 1200k2l - 3k3l2 MPk = fk = 1200kl2 - 3k2l3 • these marginal productivities will be positive for values of k and l for which kl < 400

  18. A Diminishing RTS • Because fll = 1200k2 - 6k3l fkk = 1200l2 - 6kl3 this production function exhibits diminishing marginal productivities for sufficiently large values of k and l • fll and fkk < 0 if kl > 200

  19. A Diminishing RTS • Cross differentiation of either of the marginal productivity functions yields fkl = flk = 2400kl - 9k2l2 which is positive only for kl < 266

  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 fkl to ensure convexity of the isoquants

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

  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. Constant Returns to Scale • Constant returns-to-scale production functions are homogeneous of degree one in inputs f(tk,tl) = t1f(k,l) = tq • This implies that the marginal productivity functions are homogeneous of degree zero • Differentiating the above equation,

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

  29. Returns to Scale • Returns to scale can be generalized to a production function with n inputs q = f(x1,x2,…,xn) • If all inputs are multiplied by a positive constant t, we have f(tx1,tx2,…,txn) = tkf(x1,x2,…,xn)=tkq • 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. 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. Both RTS and k/l will change as we move from point A to point B RTSA A RTSB (k/l)A B (k/l)B Elasticity of Substitution  is the ratio of these proportional changes k per period  measures the curvature(曲率) of the isoquant q = q0 l per period

  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. 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. 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(tk,tl) = atk + btl = t(ak + bl) = tf(k,l) • All isoquants are straight lines • RTS is constant •  = 

  35. RTS is constant as k/l changes slope = -b/a q2 q3 q1 The Linear Production Function Capital and labor are perfect substitutes k per period  =  l per period

  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. k/l is fixed at b/a q3 q3/a q2 q1 q3/b Fixed Proportions No substitution between labor and capital is possible k per period  = 0 l per period

  38. Cobb-Douglas Production Function • Suppose that the production function is q = f(k,l) = Akalb A,a,b > 0 • This production function can exhibit any returns to scale f(tk,tl) = A(tk)a(tl)b = Ata+b kalb = ta+bf(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 • The Cobb-Douglas production function is linear in logarithms ln q = ln A + a ln k + b lnl • a is the elasticity of output with respect to k • b is the elasticity of output with respect to l

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

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

  45. Technical Progress • Dividing by q gives us

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

  47. Technical Progress • Since

  48. Technical Progress in the Cobb-Douglas Function • Suppose that the production function is q = A(t)f(k,l) =A(t)kl1- • If we assume that technical progress occurs at a constant exponential () then A(t) = Aet q = Aetkl1-

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

  50. Technical Progress in the Cobb-Douglas Function

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