Technologies A technology is a process by which inputs are converted to an output Eg labor a computer a projector electricity and software are being combined to produce this lecture Technologies ID: 1027389
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1. Chapter EighteenTechnology
2. TechnologiesA technology is a process by which inputs are converted to an output.E.g. labor, a computer, a projector, electricity, and software are being combined to produce this lecture.
3. TechnologiesUsually several technologies will produce the same product -- a blackboard and chalk can be used instead of a computer and a projector.Which technology is “best”?How do we compare technologies?
4. Input Bundlesxi denotes the amount used of input i; i.e. the level of input i.An input bundle is a vector of the input levels; (x1, x2, … , xn).E.g. (x1, x2, x3) = (6, 0, 9×3).
5. Production Functionsy denotes the output level.The technology’s production function states the maximum amount of output possible from an input bundle.
6. Production Functionsy = f(x) is theproductionfunction.x’xInput LevelOutput Levely’y’ = f(x’) is the maximal output level obtainable from x’ input units.One input, one output
7. Technology SetsA production plan is an input bundle and an output level; (x1, … , xn, y).A production plan is feasible ifThe collection of all feasible production plans is the technology set.
8. Technology Setsy = f(x) is theproductionfunction.x’xInput LevelOutput Levely’y”y’ = f(x’) is the maximal output level obtainable from x’ input units.One input, one outputy” = f(x’) is an output level that is feasible from x’ input units.
9. Technology SetsThe technology set is
10. Technology Setsx’xInput LevelOutput Levely’One input, one outputy”The technologyset
11. Technology Setsx’xInput LevelOutput Levely’One input, one outputy”The technologysetTechnicallyinefficientplansTechnicallyefficient plans
12. Technologies with Multiple InputsWhat does a technology look like when there is more than one input?The two input case: Input levels are x1 and x2. Output level is y.Suppose the production function is
13. Technologies with Multiple InputsE.g. the maximal output level possible from the input bundle(x1, x2) = (1, 8) isAnd the maximal output level possible from (x1,x2) = (8,8) is
14. Technologies with Multiple InputsOutput, yx1x2(8,1)(8,8)
15. Technologies with Multiple InputsThe y output unit isoquant is the set of all input bundles that yield at most the same output level y.
16. Isoquants with Two Variable Inputsy º 8y º 4x1x2
17. Isoquants with Two Variable InputsIsoquants can be graphed by adding an output level axis and displaying each isoquant at the height of the isoquant’s output level.
18. Isoquants with Two Variable InputsOutput, yx1x2y º 8y º 4
19. Isoquants with Two Variable InputsMore isoquants tell us more about the technology.
20. Isoquants with Two Variable Inputsy º 8y º 4x1x2y º 6y º 2
21. Isoquants with Two Variable InputsOutput, yx1x2y º 8y º 4y º 6y º 2
22. Technologies with Multiple InputsThe complete collection of isoquants is the isoquant map.The isoquant map is equivalent to the production function -- each is the other.E.g.
23. Technologies with Multiple Inputsx1x2y
24. Technologies with Multiple Inputsx1x2y
25. Technologies with Multiple Inputsx1x2y
26. Technologies with Multiple Inputsx1x2y
27. Technologies with Multiple Inputsx1x2y
28. Technologies with Multiple Inputsx1x2y
29. Technologies with Multiple Inputsx1y
30. Technologies with Multiple Inputsx1y
31. Technologies with Multiple Inputsx1y
32. Technologies with Multiple Inputsx1y
33. Technologies with Multiple Inputsx1y
34. Technologies with Multiple Inputsx1y
35. Technologies with Multiple Inputsx1y
36. Technologies with Multiple Inputsx1y
37. Technologies with Multiple Inputsx1y
38. Technologies with Multiple Inputsx1y
39. Cobb-Douglas TechnologiesA Cobb-Douglas production function is of the formE.g.with
40. x2x1All isoquants are hyperbolic,asymptoting to, but nevertouching any axis.Cobb-Douglas Technologies
41. x2x1All isoquants are hyperbolic,asymptoting to, but nevertouching any axis.Cobb-Douglas Technologies
42. x2x1All isoquants are hyperbolic,asymptoting to, but nevertouching any axis.Cobb-Douglas Technologies
43. x2x1All isoquants are hyperbolic,asymptoting to, but nevertouching any axis.Cobb-Douglas Technologies>
44. Fixed-Proportions TechnologiesA fixed-proportions production function is of the formE.g.with
45. Fixed-Proportions Technologiesx2x1min{x1,2x2} = 144814247min{x1,2x2} = 8min{x1,2x2} = 4x1 = 2x2
46. Perfect-Substitutes TechnologiesA perfect-substitutes production function is of the formE.g.with
47. Perfect-Substitution Technologies93186248x1x2x1 + 3x2 = 18x1 + 3x2 = 36x1 + 3x2 = 48All are linear and parallel
48. Marginal (Physical) ProductsThe marginal product of input i is the rate-of-change of the output level as the level of input i changes, holding all other input levels fixed.That is,
49. Marginal (Physical) ProductsE.g. ifthen the marginal product of input 1 is
50. Marginal (Physical) ProductsE.g. ifthen the marginal product of input 1 is
51. Marginal (Physical) ProductsE.g. ifthen the marginal product of input 1 isand the marginal product of input 2 is
52. Marginal (Physical) ProductsE.g. ifthen the marginal product of input 1 isand the marginal product of input 2 is
53. Marginal (Physical) ProductsTypically the marginal product of oneinput depends upon the amount used of other inputs. E.g. if then,and if x2 = 27 thenif x2 = 8,
54. Marginal (Physical) ProductsThe marginal product of input i is diminishing if it becomes smaller as the level of input i increases. That is, if
55. Marginal (Physical) ProductsandE.g. ifthen
56. Marginal (Physical) ProductsandsoE.g. ifthen
57. Marginal (Physical) ProductsandsoandE.g. ifthen
58. Marginal (Physical) ProductsandsoandBoth marginal products are diminishing.E.g. ifthen
59. Returns-to-ScaleMarginal products describe the change in output level as a single input level changes.Returns-to-scale describes how the output level changes as all input levels change in direct proportion (e.g. all input levels doubled, or halved).
60. Returns-to-ScaleIf, for any input bundle (x1,…,xn),then the technology described by theproduction function f exhibits constantreturns-to-scale.E.g. (k = 2) doubling all input levelsdoubles the output level.
61. Returns-to-Scaley = f(x)x’xInput LevelOutput Levely’One input, one output2x’2y’Constantreturns-to-scale
62. Returns-to-ScaleIf, for any input bundle (x1,…,xn),then the technology exhibits diminishingreturns-to-scale.E.g. (k = 2) doubling all input levels less than doubles the output level.
63. Returns-to-Scaley = f(x)x’xInput LevelOutput Levelf(x’)One input, one output2x’f(2x’)2f(x’)Decreasingreturns-to-scale
64. Returns-to-ScaleIf, for any input bundle (x1,…,xn),then the technology exhibits increasingreturns-to-scale.E.g. (k = 2) doubling all input levelsmore than doubles the output level.
65. Returns-to-Scaley = f(x)x’xInput LevelOutput Levelf(x’)One input, one output2x’f(2x’)2f(x’)Increasingreturns-to-scale
66. Returns-to-ScaleA single technology can ‘locally’ exhibit different returns-to-scale.
67. Returns-to-Scaley = f(x)xInput LevelOutput LevelOne input, one outputDecreasingreturns-to-scaleIncreasingreturns-to-scale
68. Examples of Returns-to-ScaleThe perfect-substitutes productionfunction isExpand all input levels proportionatelyby k. The output level becomes
69. Examples of Returns-to-ScaleThe perfect-substitutes productionfunction isExpand all input levels proportionatelyby k. The output level becomes
70. Examples of Returns-to-ScaleThe perfect-substitutes productionfunction isExpand all input levels proportionatelyby k. The output level becomesThe perfect-substitutes productionfunction exhibits constant returns-to-scale.
71. Examples of Returns-to-ScaleThe perfect-complements productionfunction isExpand all input levels proportionatelyby k. The output level becomes
72. Examples of Returns-to-ScaleThe perfect-complements productionfunction isExpand all input levels proportionatelyby k. The output level becomes
73. Examples of Returns-to-ScaleThe perfect-complements productionfunction isExpand all input levels proportionatelyby k. The output level becomesThe perfect-complements productionfunction exhibits constant returns-to-scale.
74. Examples of Returns-to-ScaleThe Cobb-Douglas production function isExpand all input levels proportionatelyby k. The output level becomes
75. Examples of Returns-to-ScaleThe Cobb-Douglas production function isExpand all input levels proportionatelyby k. The output level becomes
76. Examples of Returns-to-ScaleThe Cobb-Douglas production function isExpand all input levels proportionatelyby k. The output level becomes
77. Examples of Returns-to-ScaleThe Cobb-Douglas production function isExpand all input levels proportionatelyby k. The output level becomes
78. Examples of Returns-to-ScaleThe Cobb-Douglas production function isThe Cobb-Douglas technology’s returns-to-scale isconstant if a1+ … + an = 1
79. Examples of Returns-to-ScaleThe Cobb-Douglas production function isThe Cobb-Douglas technology’s returns-to-scale isconstant if a1+ … + an = 1increasing if a1+ … + an > 1
80. Examples of Returns-to-ScaleThe Cobb-Douglas production function isThe Cobb-Douglas technology’s returns-to-scale isconstant if a1+ … + an = 1increasing if a1+ … + an > 1decreasing if a1+ … + an < 1.
81. Returns-to-ScaleQ: Can a technology exhibit increasing returns-to-scale even though all of its marginal products are diminishing?
82. Returns-to-ScaleQ: Can a technology exhibit increasing returns-to-scale even if all of its marginal products are diminishing?A: Yes.E.g.
83. Returns-to-Scaleso this technology exhibitsincreasing returns-to-scale.
84. Returns-to-Scaleso this technology exhibitsincreasing returns-to-scale.But diminishes as x1increases
85. Returns-to-Scaleso this technology exhibitsincreasing returns-to-scale.But diminishes as x1increases anddiminishes as x1increases.
86. Returns-to-ScaleSo a technology can exhibit increasing returns-to-scale even if all of its marginal products are diminishing. Why?
87. Returns-to-ScaleA marginal product is the rate-of-change of output as one input level increases, holding all other input levels fixed.Marginal product diminishes because the other input levels are fixed, so the increasing input’s units have each less and less of other inputs with which to work.
88. Returns-to-ScaleWhen all input levels are increased proportionately, there need be no diminution of marginal products since each input will always have the same amount of other inputs with which to work. Input productivities need not fall and so returns-to-scale can be constant or increasing.
89. Technical Rate-of-SubstitutionAt what rate can a firm substitute one input for another without changing its output level?
90. Technical Rate-of-Substitutionx2x1yº100
91. Technical Rate-of-Substitutionx2x1yº100The slope is the rate at which input 2 must be given up as input 1’s level is increased so as not to change the output level. The slope of an isoquant is its technical rate-of-substitution.
92. Technical Rate-of-SubstitutionHow is a technical rate-of-substitution computed?
93. Technical Rate-of-SubstitutionHow is a technical rate-of-substitution computed?The production function isA small change (dx1, dx2) in the input bundle causes a change to the output level of
94. Technical Rate-of-SubstitutionBut dy = 0 since there is to be no changeto the output level, so the changes dx1and dx2 to the input levels must satisfy
95. Technical Rate-of-Substitutionrearranges toso
96. Technical Rate-of-Substitutionis the rate at which input 2 must be givenup as input 1 increases so as to keepthe output level constant. It is the slopeof the isoquant.
97. Technical Rate-of-Substitution; A Cobb-Douglas ExamplesoandThe technical rate-of-substitution is
98. x2x1Technical Rate-of-Substitution; A Cobb-Douglas Example
99. x2x1Technical Rate-of-Substitution; A Cobb-Douglas Example84
100. x2x1Technical Rate-of-Substitution; A Cobb-Douglas Example612
101. Well-Behaved TechnologiesA well-behaved technology ismonotonic, andconvex.
102. Well-Behaved Technologies - MonotonicityMonotonicity: More of any input generates more output.yxyxmonotonic notmonotonic
103. Well-Behaved Technologies - ConvexityConvexity: If the input bundles x’ and x” both provide y units of output then the mixture tx’ + (1-t)x” provides at least y units of output, for any 0 < t < 1.
104. Well-Behaved Technologies - Convexityx2x1yº100
105. Well-Behaved Technologies - Convexityx2x1yº100
106. Well-Behaved Technologies - Convexityx2x1yº100yº120
107. Well-Behaved Technologies - Convexityx2x1Convexity implies that the TRSincreases (becomes lessnegative) as x1 increases.
108. Well-Behaved Technologiesx2x1yº100yº50yº200higher output
109. The Long-Run and the Short-RunsThe long-run is the circumstance in which a firm is unrestricted in its choice of all input levels.There are many possible short-runs.A short-run is a circumstance in which a firm is restricted in some way in its choice of at least one input level.
110. The Long-Run and the Short-RunsExamples of restrictions that place a firm into a short-run:temporarily being unable to install, or remove, machinerybeing required by law to meet affirmative action quotashaving to meet domestic content regulations.
111. The Long-Run and the Short-RunsA useful way to think of the long-run is that the firm can choose as it pleases in which short-run circumstance to be.
112. The Long-Run and the Short-RunsWhat do short-run restrictions imply for a firm’s technology?Suppose the short-run restriction is fixing the level of input 2.Input 2 is thus a fixed input in the short-run. Input 1 remains variable.
113. The Long-Run and the Short-Runsx2x1y
114. The Long-Run and the Short-Runsx2x1y
115. The Long-Run and the Short-Runsx2x1y
116. The Long-Run and the Short-Runsx2x1y
117. The Long-Run and the Short-Runsx2x1y
118. The Long-Run and the Short-Runsx2x1y
119. The Long-Run and the Short-Runsx2x1y
120. The Long-Run and the Short-Runsx2x1y
121. The Long-Run and the Short-Runsx2x1y
122. The Long-Run and the Short-Runsx2x1y
123. The Long-Run and the Short-Runsx1y
124. The Long-Run and the Short-Runsx1y
125. The Long-Run and the Short-Runsx1yFour short-run production functions.
126. The Long-Run and the Short-Runs is the long-run productionfunction (both x1 and x2 are variable).The short-run production function whenx2 º 1 isThe short-run production function when x2 º 10 is
127. The Long-Run and the Short-Runsx1yFour short-run production functions.