Chapter 19. Technology

Transcription

1 Chpter 9 Technolog

2 Technologies A technolog is process b which inputs re converted to n output. E.g. lbor, computer, projector, electricit, nd softwre re being combined to produce this lecture.

3 Technologies Usull severl technologies will produce the sme product -- blckbord nd chlk cn be used insted of computer nd projector. Which technolog is best? How do we compre technologies?

4 Input Bundles i denotes the mount used of input i; i.e. the level of input i. An input bundle is vector of the input levels; (,,, n ). E.g. (,, 3 ) = (6, 0, 9 3).

5 Production Functions denotes the output level. The technolog s production function sttes the mimum mount of output possible from n input bundle. f(,, n )

6 Production Functions One input, one output Output Level = f() is the production function. = f( ) is the miml output level obtinble from input units. Input Level

7 Technolog Sets A production pln is n input bundle nd n output level; (,, n, ). A production pln is fesible if f(,, n ) The collection of ll fesible production plns is the technolog set.

8 Technolog Sets One input, one output Output Level = f() is the production function. = f( ) is the miml output level obtinble from input units. = f( ) is n output level tht is fesible from input units. Input Level

9 Technolog Sets The technolog set is T {(,,, ) f(,, ) nd n n 0,, 0}. n

10 Output Level Technolog Sets One input, one output The technolog set Input Level

11 Output Level Technolog Sets One input, one output Technicll inefficient plns Technicll efficient plns The technolog set Input Level

12 Technologies with Multiple Inputs Wht does technolog look like when there is more thn one input? The two input cse: Input levels re nd. Output level is. Suppose the production function is f (, ) /3 /3.

13 Technologies with Multiple Inputs E.g. the miml output level possible from the input bundle (, ) = (, 8) is 8 /3 /3 /3 /3 4. And the miml output level possible from (, ) = (8,8) is /3 /3 /3 /

14 Technologies with Multiple Output, Inputs (8,) (8,8)

15 Technologies with Multiple Inputs The output unit isoqunt is the set of ll input bundles tht ield t most the sme output level.

16 Isoqunts with Two Vrible Inputs

17 Isoqunts with Two Vrible Inputs Isoqunts cn be grphed b dding n output level is nd displing ech isoqunt t the height of the isoqunt s output level.

18 Isoqunts with Two Vrible Output, Inputs

19 Isoqunts with Two Vrible Inputs More isoqunts tell us more bout the technolog.

20 Isoqunts with Two Vrible Inputs

21 Isoqunts with Two Vrible Output, Inputs

22 Technologies with Multiple Inputs The complete collection of isoqunts is the isoqunt mp. The isoqunt mp is equivlent to the production function -- ech is the other. E.g. f (, ) /3 / 3

23 Technologies with Multiple Inputs

24 Technologies with Multiple Inputs

25 Technologies with Multiple Inputs

26 Technologies with Multiple Inputs

27 Technologies with Multiple Inputs

28 Technologies with Multiple Inputs

29 Technologies with Multiple Inputs

30 Technologies with Multiple Inputs

31 Technologies with Multiple Inputs

32 Technologies with Multiple Inputs

33 Technologies with Multiple Inputs

34 Technologies with Multiple Inputs

35 Technologies with Multiple Inputs

36 Technologies with Multiple Inputs

37 Technologies with Multiple Inputs

38 Technologies with Multiple Inputs

39 Cobb-Dougls Technologies A Cobb-Dougls production function is of the form A n n. E.g. /3 /3 with n, A, nd 3 3.

40 Cobb-Dougls Technologies All isoqunts re hperbolic, smptoting to, but never touching n is.

41 Cobb-Dougls Technologies All isoqunts re hperbolic, smptoting to, but never touching n is. "

42 Cobb-Dougls Technologies All isoqunts re hperbolic, smptoting to, but never touching n is. " '

43 Cobb-Dougls Technologies All isoqunts re hperbolic, smptoting to, but never touching n is. " ' > " '

44 Fied-Proportions Technologies A fied-proportions production function is of the form E.g. min{,,, n n }. min{, } With n, nd. Q: Fied-proportions is nlogous to wht other concept?

45 Fied-Proportions Technologies min{,} = min{, } = 4 min{, } = 8 min{, } = 4

46 Perfect-Substitutes Technologies A perfect-substitutes production function is of the form E.g. n n 3. with n, nd 3.

47 Perfect-Substitution Technologies = = = 48 All re liner nd prllel 9 8 4

48 Mrginl (Phsicl) Products f (,, n ) The mrginl product of input i is the rte-of-chnge of the output level s the level of input i chnges, holding ll other input levels fied. Tht is, MP i i

49 Mrginl (Phsicl) Products E.g. if /3 / 3 f (, ) then the mrginl product of input is obtined b the prtil derivtive of output WRT good : MP 3 /3 /3

50 Mrginl (Phsicl) Products E.g. if f(, ) / /3 3 then the mrginl product of input is MP / 3 / 3 3 nd the mrginl product of input is

51 Mrginl (Phsicl) Products E.g. if f(, ) / /3 3 then the mrginl product of input is MP / 3 / 3 3 nd the mrginl product of input is MP /3 / 3 3.

52 Mrginl (Phsicl) Products Tpicll the mrginl product of one input depends upon the mount used of other inputs. E.g. if MP 3 if = 8, /3 /3 MP nd if = 7 then then, /3 /3 4 / MP /3 7 /3 3 / 3 3.

53 Mrginl (Phsicl) Products The mrginl product of input i is diminishing if it becomes smller s the level of input i increses. Tht is, if the second derivtive of output WRT good I is negtive: 0. i i i i i MP

54 Mrginl (Phsicl) Products E.g. if /3 /3 then MP 3 /3 /3 nd MP /3 /3 3

55 so Mrginl (Phsicl) Products /3 / 3 E.g. if MP 3 /3 /3 MP nd then MP /3 /3 3 5/3 / 3 9 0

56 Mrginl (Phsicl) Products E.g. if then MP / 3 / 3 nd MP 3 3 so MP 5 / 3 / nd MP /3 4 / /3 / 3 /3 /3

57 Mrginl (Phsicl) Products E.g. if MP so nd 3 nd /3 /3 /3 / 3 MP MP MP /3 /3 5/3 / 3 9 /3 4/ 3 9 then Both mrginl products re diminishing

58 Returns-to-Scle Mrginl products describe the chnge in output level s single input level chnges. Returns-to-scle describes how the output level chnges s ll input levels chnge in direct proportion (e.g. ll input levels doubled, or hlved).

59 Returns-to-Scle If, for n input bundle (,, n ), f ( k, k,, kn) kf (,,, n then the technolog described b the production function f ehibits constnt returns-to-scle. E.g. (k = ) doubling ll input levels doubles the output level. )

60 Output Level Returns-to-Scle One input, one output = f() Constnt returns-to-scle Input Level

61 Returns-to-Scle If, for n input bundle (,, n ), f ( k, k,, kn) kf (,,, n then the technolog ehibits diminishing returns-to-scle. E.g. (k = ) doubling ll input levels less thn doubles the output level. )

62 Output Level Returns-to-Scle One input, one output f( ) f( ) f( ) = f() Decresing returns-to-scle Input Level

63 Returns-to-Scle If, for n input bundle (,, n ), f( k, k,, k ) kf(,,, ) n then the technolog ehibits incresing returns-to-scle. E.g. (k = ) doubling ll input levels more thn doubles the output level. n

64 Returns-to-Scle One input, one output Output Level f( ) Incresing returns-to-scle = f() f( ) f( ) Input Level

65 Returns-to-Scle A single technolog cn locll ehibit different returns-to-scle.

66 Output Level Returns-to-Scle One input, one output Incresing returns-to-scle = f() Decresing returns-to-scle Input Level

67 Emples of Returns-to-Scle The perfect-substitutes production function is n n Epnd ll input levels proportiontel b k. The output level becomes ( k ) ( k) n ( k n ).

68 Emples of Returns-to-Scle The perfect-substitutes production function is n n Epnd ll input levels proportiontel b k. The output level becomes ( k ) ( k ) ( k ) k( ) n n n. n

69 Emples of Returns-to-Scle. n n The perfect-substitutes production function is Epnd ll input levels proportiontel b k. The output level becomes. ) ( ) ( ) ( ) ( k k k k k n n n n The perfect-substitutes production function ehibits constnt returns-to-scle.

70 Emples of Returns-to-Scle The perfect-complements production function is }. min{,,, n n Epnd ll input levels proportiontel b k. The output level becomes min{ ( k ), ( k),, n ( k n )}

71 Emples of Returns-to-Scle The perfect-complements production function is min{,,, }. Epnd ll input levels proportiontel b k. The output level becomes min{ ( k k(min{ ),, ( k ),,,, n n n }) n ( k n n )}

72 Emples of Returns-to-Scle The perfect-complements production function is min{,,, n n Epnd ll input levels proportiontel b k. The output level becomes min{ ( k k(min{ k. ),, ( k ),,,, n n n ( k The perfect-complements production function ehibits constnt returns-to-scle. }) n )} }.

73 Emples of Returns-to-Scle The Cobb-Dougls production function is Epnd ll input levels proportiontel b k. The output level becomes n n k ( k ) ( ) ( k ). n n

74 Emples of Returns-to-Scle. n n The Cobb-Dougls production function is Epnd ll input levels proportiontel b k. The output level becomes n n n n k k k k k k ) ( ) ( ) (

75 Emples of Returns-to-Scle The Cobb-Dougls production function is. n n Epnd ll input levels proportiontel b k. The output level becomes n ( k ) ( k ) ( k ) k k k n k n n n n n

76 Emples of Returns-to-Scle. n n The Cobb-Dougls production function is Epnd ll input levels proportiontel b k. The output level becomes. ) ( ) ( ) ( k k k k k k k k n n n n n n n n

77 Emples of Returns-to-Scle The Cobb-Dougls production function is ) ( k ) n ( k n n ( k n n The Cobb-Dougls technolog s returnsto-scle is constnt if + + n =. ) k.

78 Emples of Returns-to-Scle The Cobb-Dougls production function is ( k ) ( k ) ( k ) n k n. n n n. The Cobb-Dougls technolog s returnsto-scle is constnt if + + n = incresing if + + n >

79 Emples of Returns-to-Scle The Cobb-Dougls production function is. n n n ( k ) ( k ) ( k ) n k n. The Cobb-Dougls technolog s returnsto-scle is constnt if + + n = incresing if + + n > decresing if + + n <.

80 Returns-to-Scle Q: Cn technolog ehibit incresing returns-to-scle even though ll of its mrginl products re diminishing?

81 Returns-to-Scle Q: Cn technolog ehibit incresing returns-to-scle even if ll of its mrginl products re diminishing? A: Yes. E.g. /3 /3.

82 Returns-to-Scle /3 /3 4 3 so even though ech individul eponent is <, this technolog ehibits incresing returns-to-scle.

83 Returns-to-Scle /3 /3 But 4 3 MP 3 /3 so this technolog ehibits incresing returns-to-scle. /3 diminishes s increses

84 Returns-to-Scle / / so this technolog ehibits 3 incresing returns-to-scle. 3 /3 / 3 But MP diminishes s increses nd increses. /3 /3 MP diminishes s 3

85 Returns-to-Scle So technolog cn ehibit incresing returns-to-scle even if ll of its mrginl products re diminishing. Wh?

86 Returns-to-Scle A mrginl product is the rte-ofchnge of output s one input level increses, holding ll other input levels fied. Mrginl product diminishes becuse the other input levels re fied, so the incresing input s units hve ech less nd less of other inputs with which to work.

87 Returns-to-Scle When ll input levels re incresed proportiontel, there need be no diminution of mrginl products since ech input will lws hve the sme mount of other inputs with which to work. Input productivities need not fll nd so returns-to-scle cn be constnt or incresing.

88 Technicl Rte-of-Substitution TRS = the rte t which firm substitute one input for nother without chnging its output level (nlogous to MRS).

89 Technicl Rte-of-Substitution ' '

90 Technicl Rte-of-Substitution ' The slope is the rte t which input must be given up s input s level is incresed so s not to chnge the output level. The slope of n isoqunt is its technicl rte-ofsubstitution. '

91 Technicl Rte-of-Substitution How is technicl rte-of-substitution computed?

92 Technicl Rte-of-Substitution How is technicl rte-of-substitution computed? The production function is (, A smll chnge (d, d ) in the input bundle cuses chnge to the output level of f ). d d d.

93 Technicl Rte-of-Substitution d d d. But d = 0 since there is to be no chnge to the output level, so the chnges d nd d to the input levels must stisf 0 d d.

94 Technicl Rte-of-Substitution 0 d d rerrnges to d d so. / / d d

95 Technicl Rte-of-Substitution d d / / is the rte t which input must be given up s input increses so s to keep the output level constnt. It is the slope of the isoqunt.

96 so Technicl Rte-of-Substitution; A Cobb-Dougls Emple f (, ) b b nd b b The technicl rte-of-substitution is. d d / / b b b b.

97 Technicl Rte-of-Substitution; A Cobb-Dougls Emple ; ndb 3 /3 /3 3 TRS b (/ 3) ( / 3)

98 Technicl Rte-of-Substitution; A Cobb-Dougls Emple /3 /3 ; ndb TRS TRS b (/ 3) ( / 3) 8 4 4

99 Technicl Rte-of-Substitution; 6 A Cobb-Dougls Emple / ; nd b 3 3 TRS ( / 3) b ( / 3) TRS 6 4 /3 3

100 Well-Behved Technologies A well-behved technolog is monotonic, nd conve.

101 Well-Behved Technologies - Monotonicit Monotonicit: More of n input genertes more output. monotonic not monotonic

102 Well-Behved Technologies - Conveit Conveit: If the input bundles nd both provide units of output then the miture t + (-t) provides t lest units of output, for n 0 < t <.

103 Well-Behved Technologies - Conveit ' " ' "

104 Well-Behved Technologies - Conveit ' " ' ' " ' " t ( t), t ( t) "

105 Well-Behved Technologies - Conveit ' " ' ' " ' " t ( t), t ( t) "

106 Well-Behved Technologies - ' Conveit Conveit implies tht the TRS increses (becomes less negtive) s increses. " ' "

107 Well-Behved Technologies higher output

108 The Long-Run nd the Short- Runs The long-run is the circumstnce in which firm is unrestricted in its choice of ll input levels. There re mn possible short-runs. A short-run is circumstnce in which firm is restricted in some w in its choice of t lest one input level.

109 The Long-Run nd the Short- Runs Emples of restrictions tht plce firm into short-run: temporril being unble to instll, or remove, mchiner being required b lw to meet ffirmtive ction quots hving to meet domestic content regultions.

110 The Long-Run nd the Short- Runs A useful w to think of the long-run is tht the firm cn choose s it pleses in which short-run circumstnce to be.

111 The Long-Run nd the Short- Runs Wht do short-run restrictions impl for firm s technolog? Suppose the short-run restriction is fiing the level of input. Input is thus fied input in the short-run. Input remins vrible.

112 The Long-Run nd the Short- Runs

113 The Long-Run nd the Short- Runs

114 The Long-Run nd the Short- Runs

115 The Long-Run nd the Short- Runs

116 The Long-Run nd the Short- Runs

117 The Long-Run nd the Short- Runs

118 The Long-Run nd the Short- Runs

119 The Long-Run nd the Short- Runs

120 The Long-Run nd the Short- Runs

121 The Long-Run nd the Short- Runs

122 The Long-Run nd the Short- Runs

123 The Long-Run nd the Short- Runs

124 The Long-Run nd the Short- Runs Four short-run production functions.

125 The Long-Run nd the Short- /3 /3 Runs is the long-run production function (both nd re vrible). The short-run production function when is /3 /3 /3. The short-run production function when 0 is /3 /3 /3 0 5.

126 The Long-Run nd the Short- Runs / = 3 0 / 3 / = 3 5 / 3 = / 3 / 3 = / 3 / 3 Four short-run production functions.