Chapter 7 PRODUCTION FUNCTIONS. Copyright 2005 by South-Western, a division of Thomson Learning. All rights reserved.

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2 Production Function The firm s production function for a particuar good (q) shows the maximum amount of the good that can be produced using aternative combinations of capita () and abor () q = f(,) 2

3 Margina Physica Product To study variation in a singe input, we define margina physica product as the additiona output that can be produced by empoying one more unit of that input whie hoding other inputs constant margina margina physica physica product product of of capita abor q = MP = = f q = MP = = f 3

4 Diminishing Margina Productivity ( 边际生产力递减 ) The margina physica product of an input depends on how much of that input is used In genera, we assume diminishing margina productivity 2 2 MP f MP f = = f = f11 < 0 2 = = f = 22 < 0 2 f 4

5 Diminishing Margina Productivity Because of diminishing margina productivity, 19th century economist Thomas Mathus worried about the effect of popuation growth on abor productivity But changes in the margina productivity of abor over time aso depend on changes in other inputs such as capita we need to consider f which is often > 0 5

6 Average Physica Product Labor productivity is often measured by average productivity output q AP = = = abor input f (, ) Note that AP aso depends on the amount of capita empoyed 6

7 A Two-Input Production Function Suppose the production function for fyswatters can be represented by q = f(,) = To construct MP and AP, we must assume a vaue for et = 10 The production function becomes q = 60,

8 A Two-Input Production Function The margina productivity function is MP = q/ = 120, which diminishes as increases This impies that q has a maximum vaue: 120, = 0 40 = 2 = 40 Labor input beyond = 40 reduces output 8

9 A Two-Input Production Function To find average productivity, we hod =10 and sove AP = q/ = 60, AP reaches its maximum where AP / = 60, = 0 = 30 9

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

11 Isoquant Maps To iustrate the possibe substitution of one input for another, we use an isoquant map ( 等产量线图 ) An isoquant shows those combinations of and that can produce a given eve of output (q 0 ) f(,) = q 0 11

12 Isoquant Map Each isoquant represents a different eve of output output rises as we move northeast per period q = 30 q = 20 per period 12

13 Margina Rate of Technica Substitution (RTS) The sope of an isoquant shows the rate at which can be substituted for per period A A - sope = margina rate of technica substitution (RTS) B RTS > 0 and is diminishing for increasing inputs of abor B q = 20 A B per period 13

14 Margina Rate of Technica Substitution (RTS) 边际技术替代率 The margina rate of technica substitution (RTS) shows the rate at which abor can be substituted for capita whie hoding output constant aong an isoquant RTS ( for ) = d d q= q 0 14

15 RTS and Margina Productivities Tae the tota differentia of the production function: dq f f = d + d = MP d Aong an isoquant dq = 0, so + MP d MP d = MP d d RTS ( for ) = = d q= q 0 MP MP 15

16 RTS and Margina Productivities Because MP and MP wi both be nonnegative, RTS wi be positive (or zero) However, it is generay not possibe to derive a diminishing RTS from the assumption of diminishing margina productivity aone 16

17 RTS and Margina Productivities To show that isoquants are convex, we woud ie to show that d(rts)/d < 0 Since RTS = f /f drts d f f = ( / ) d d drts d = [ f ( f + f d / d) f ( f ) 2 ( f + f d / d)] 17

18 RTS and Margina Productivities Using the fact that d/d = -f /f aong an isoquant and Young s theorem (f = f ) drts d = ( f 2 f 2fff ( f ) Because we have assumed f > 0, the denominator is positive Because f and f are both assumed to be negative, the ratio wi be negative if f is positive 3 + f 2 f ) 18

19 RTS and Margina Productivities Intuitivey, it seems reasonabe that f = f shoud be positive if worers have more capita, they wi be more productive But some production functions have f < 0 over some input ranges when we assume diminishing RTS we are assuming that MP and MP diminish quicy enough to compensate for any possibe negative cross-productivity effects 19

20 A Diminishing RTS Suppose the production function is q = f(,) = For this production function MP = f = MP = f = these margina productivities wi be positive for vaues of and for which <

21 Because A Diminishing RTS f = f = this production function exhibits diminishing margina productivities for sufficienty arge vaues of and f and f < 0 if >

22 A Diminishing RTS Cross differentiation of either of the margina productivity functions yieds f = f = which is positive ony for <

23 A Diminishing RTS Thus, for this production function, RTS is diminishing throughout the range of and where margina productivities are positive for higher vaues of and, the diminishing margina productivities are sufficient to overcome the infuence of a negative vaue for f to ensure convexity of the isoquants 23

24 Returns to Scae 规模回报 How does output respond to increases in a inputs together? suppose that a inputs are doubed, woud output doube? Returns to scae have been of interest to economists since the days of Adam Smith 24

25 Returns to Scae Smith identified two forces that come into operation as inputs are doubed greater division of abor and speciaization of function oss in efficiency because management may become more difficut given the arger scae of the firm 25

26 Returns to Scae If the production function is given by q = f(,) and a inputs are mutipied by the same positive constant (t >1), then Effect on Output Returns to Scae f(t,t) = tf(,) f(t,t) < tf(,) f(t,t) > tf(,) Constant Decreasing Increasing 26

27 Returns to Scae It is possibe for a production function to exhibit constant returns to scae for some eves of input usage and increasing or decreasing returns for other eves economists refer to the degree of returns to scae with the impicit notion that ony a fairy narrow range of variation in input usage and the reated eve of output is being considered 27

28 Constant Returns to Scae Constant returns-to-scae production functions are homogeneous of degree one in inputs f(t,t) = t 1 f(,) = tq This impies that the margina productivity functions are homogeneous of degree zero if a function is homogeneous of degree, its derivatives are homogeneous of degree -1 28

29 Constant Returns to Scae The margina productivity of any input depends on the ratio of capita and abor (not on the absoute eves of these inputs) The RTS between and depends ony on the ratio of to, not the scae of operation 29

30 Constant Returns to Scae The production function wi be homothetic Geometricay, a of the isoquants are radia expansions of one another 30

31 Constant Returns to Scae Aong a ray from the origin (constant /), the RTS wi be the same on a isoquants per period The isoquants are equay spaced as output expands q = 1 q = 2 q = 3 per period 31

32 Returns to Scae Returns to scae can be generaized to a production function with n inputs q = f(x 1,x 2,,x n ) If a inputs are mutipied by a positive constant t, we have f(tx 1,tx 2,,tx n ) = t f(x 1,x 2,,x n )=t q If = 1, we have constant returns to scae If < 1, we have decreasing returns to scae If > 1, we have increasing returns to scae 32

33 Easticity of Substitution The easticity of substitution (σ) measures the proportionate change in / reative to the proportionate change in the RTS aong an isoquant σ = % ( / ) % RTS = d( / ) drts RTS / = n( / ) nrts The vaue of σ wi aways be positive because / and RTS move in the same direction 33

34 Easticity of Substitution Both RTS and / wi change as we move from point A to point B per period σ is the ratio of these proportiona changes A RTS A RTS B σ measures the curvature of the isoquant (/) A B q = q 0 (/) B per period 34

35 Easticity of Substitution If σ is high, the RTS wi not change much reative to / the isoquant wi be reativey fat If σ is ow, the RTS wi change by a substantia amount as / changes the isoquant wi be sharpy curved It is possibe for σ to change aong an isoquant or as the scae of production changes 35

36 Easticity of Substitution Generaizing the easticity of substitution to the many-input case raises severa compications if we define the easticity 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 hod output and the eves of other inputs constant 36

37 The Linear Production Function Suppose that the production function is q = f(,) = a + b This production function exhibits constant returns to scae f(t,t) = at + bt = t(a + b) = tf(,) A isoquants are straight ines RTS is constant σ = 37

38 The Linear Production Function Capita and abor are perfect substitutes per period RTS is constant as / changes sope = -b/a σ = q 1 q 2 q 3 per period 38

39 Fixed Proportions Suppose that the production function is q = min (a,b) a,b > 0 Capita and abor must aways be used in a fixed ratio the firm wi aways operate aong a ray where / is constant Because / is constant, σ = 0 39

40 Fixed Proportions No substitution between abor and capita is possibe per period / is fixed at b/a q 3 /a q 3 σ = 0 q 2 q 1 q 3 /b per period 40

41 Cobb-Dougas Production Function Suppose that the production function is q = f(,) = A a b A,a,b > 0 This production function can exhibit any returns to scae f(t,t) = A(t) a (t) b = At a+b a b = t a+b f(,) if a + b = 1 constant returns to scae if a + b > 1 increasing returns to scae if a + b < 1 decreasing returns to scae 41

42 Cobb-Dougas Production Function The Cobb-Dougas production function is inear in ogarithms n q = n A + a n + b n a is the easticity of output with respect to b is the easticity of output with respect to 42

43 CES Production Function Suppose that the production function is q = f(,) = [ ρ + ρ ] γ/ρ ρ 1, ρ 0, γ > 0 γ > 1 increasing returns to scae γ < 1 decreasing returns to scae For this production function σ = 1/(1-ρ) ρ = 1 inear production function ρ = - fixed proportions production function ρ = 0 Cobb-Dougas production function 43

44 A Generaized Leontief Production Function Suppose that the production function is q = f(,) = + + 2() 0.5 Margina productivities are f = 1 + (/) -0.5 Thus, RTS = f = 1 + (/) 0.5 f f = 1+ ( / ) 1+ ( / )

45 Technica Progress Methods of production change over time Foowing the deveopment of superior production techniques, the same eve of output can be produced with fewer inputs the isoquant shifts in 45

46 Technica Progress Suppose that the production function is q = A(t)f(,) where A(t) represents a infuences that go into determining q other than and changes in A over time represent technica progress A is shown as a function of time (t) da/dt > 0 46

47 Technica Progress Differentiating the production function with respect to time we get dq da df (, ) = f (, ) + A dt dt dt dq dt = da dt q A + q f (,) f d dt + f d dt 47

48 48 Technica Progress Dividing by q gives us dt d f f dt d f f A dt da q dt dq + + = ), ( / ), ( / / / dt d f f dt d f f A dt da q dt dq / ), ( / ), ( / / + + =

49 49 Technica Progress For any variabe x, [(dx/dt)/x] is the proportiona growth rate in x denote this by G x Then, we can write the equation in terms of growth rates G f f G f f G G A q + + = ), ( ), (

50 50 Technica Progress Since G e G e G G q q A q,, + + = e q q q f f, ), ( = =, ), ( e q q q f f = =

51 Technica Progress in the Cobb-Dougas Function Suppose that the production function is q = A(t)f(,) = A(t) α 1-α If we assume that technica progress occurs at a constant exponentia (θ) then A(t) = Ae θ-t q = Ae θ-t α 1-α 51

52 Technica Progress in the Cobb-Dougas Function Taing ogarithms and differentiating with respect to t gives the growth equation nq nq q q = = / t q t q t = G q 52

53 Technica Progress in the Cobb-Dougas Function G q (n A = = θ + α + θt + αn + (1 α)n) t n n + (1 α) = θ + αg t t + (1 α) G 53

54 Important Points to Note: If a but one of the inputs are hed constant, a reationship between the singe variabe input and output can be derived the margina physica productivity is the change in output resuting from a one-unit increase in the use of the input assumed to decine as use of the input increases 54

55 Important Points to Note: The entire production function can be iustrated by an isoquant map the sope of an isoquant is the margina rate of technica substitution (RTS) it shows how one input can be substituted for another whie hoding output constant it is the ratio of the margina physica productivities of the two inputs 55

56 Important Points to Note: Isoquants are usuay assumed to be convex they obey the assumption of a diminishing RTS this assumption cannot be derived excusivey from the assumption of diminishing margina productivity one must be concerned with the effect of changes in one input on the margina productivity of other inputs 56

57 Important Points to Note: The returns to scae exhibited by a production function record how output responds to proportionate increases in a inputs if output increases proportionatey with input use, there are constant returns to scae 57

58 Important Points to Note: The easticity of substitution (σ) provides a measure of how easy it is to substitute one input for another in production a high σ impies neary straight isoquants a ow σ impies that isoquants are neary L-shaped 58

59 Important Points to Note: Technica progress shifts the entire production function and isoquant map technica improvements may arise from the use of more productive inputs or better methods of economic organization 59

( ) is just a function of x, with

( ) is just a function of x, with II. MULTIVARIATE CALCULUS The first ecture covered functions where a singe input goes in, and a singe output comes out. Most economic appications aren t so simpe. In most cases, a number of variabes infuence