Chapter EighteenTechnologiesSlide 3Input BundlesProduction FunctionsSlide 6Technology SetsSlide 8Slide 9Slide 10Slide 11Technologies with Multiple InputsSlide 13Slide 14Slide 15Isoquants with Two Variable InputsSlide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Slide 32Slide 33Slide 34Slide 35Slide 36Slide 37Slide 38Cobb-Douglas TechnologiesSlide 40Slide 41Slide 42Slide 43Fixed-Proportions TechnologiesSlide 45Perfect-Substitutes TechnologiesPerfect-Substitution TechnologiesMarginal (Physical) ProductsSlide 49Slide 50Slide 51Slide 52Slide 53Slide 54Slide 55Slide 56Slide 57Slide 58Returns-to-ScaleSlide 60Slide 61Slide 62Slide 63Slide 64Slide 65Slide 66Slide 67Examples of Returns-to-ScaleSlide 69Slide 70Slide 71Slide 72Slide 73Slide 74Slide 75Slide 76Slide 77Slide 78Slide 79Slide 80Slide 81Slide 82Slide 83Slide 84Slide 85Slide 86Slide 87Slide 88Technical Rate-of-SubstitutionSlide 90Slide 91Slide 92Slide 93Slide 94Slide 95Slide 96Technical Rate-of-Substitution; A Cobb-Douglas ExampleSlide 98Slide 99Slide 100Well-Behaved TechnologiesWell-Behaved Technologies - MonotonicityWell-Behaved Technologies - ConvexitySlide 104Slide 105Slide 106Slide 107Slide 108The Long-Run and the Short-RunsSlide 110Slide 111Slide 112Slide 113Slide 114Slide 115Slide 116Slide 117Slide 118Slide 119Slide 120Slide 121Slide 122Slide 123Slide 124Slide 125Slide 126Slide 127Chapter EighteenTechnologyTechnologiesA 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.TechnologiesUsually 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?Input Bundlesxi 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).Production Functionsy denotes the output level.The technology’s production function states the maximum amount of output possible from an input bundle.y f x xn ( , , )1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 outputTechnology SetsA production plan is an input bundle and an output level; (x1, … , xn, y).A production plan is feasible ifThe collection of all feasible production plans is the technology set.y f x xn ( , , )1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.Technology SetsThe technology set is T x x y y f x x andx xn nn  {( , , , ) | ( , , ), , }.1 110 0 Technology Setsx’ xInput LevelOutput Levely’One input, one outputy”The technologysetTechnology Setsx’ xInput LevelOutput Levely’One input, one outputy”The technologysetTechnicallyinefficientplansTechnicallyefficient plansTechnologies with Multiple InputsWhat 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 isy f x x x x ( , ) .1 2 11/321/32Technologies with Multiple InputsE.g. the maximal output level possible from the input bundle(x1, x2) = (1, 8) isAnd the maximal output level possible from (x1,x2) = (8,8) isy x x       2 2 1 8 2 1 2 411/321/3 1/3 1/3.y x x       2 2 8 8 2 2 2 811/321/3 1/3 1/3.Technologies with Multiple InputsOutput, yx1x2(8,1)(8,8)Technologies with Multiple InputsThe y output unit isoquant is the set of all input bundles that yield at most the same output level y.Isoquants with Two Variable Inputsy y x1x2Isoquants with Two Variable InputsIsoquants can be graphed by adding an output level axis and displaying each isoquant at the height of the isoquant’s output level.Isoquants with Two Variable InputsOutput, yx1x2y y Isoquants with Two Variable InputsMore isoquants tell us more about the technology.Isoquants with Two Variable Inputsy y x1x2y y Isoquants with Two Variable InputsOutput, yx1x2y y y y Technologies with Multiple InputsThe complete collection of isoquants is the isoquant map.The isoquant map is equivalent to the production function -- each is the other.E.g.3/123/11212),( xxxxfy Technologies with Multiple Inputsx1x2yTechnologies with Multiple Inputsx1x2yTechnologies with Multiple Inputsx1x2yTechnologies with Multiple Inputsx1x2yTechnologies with Multiple Inputsx1x2yTechnologies with Multiple Inputsx1x2yTechnologies with Multiple Inputsx1yTechnologies with Multiple Inputsx1yTechnologies with Multiple Inputsx1yTechnologies with Multiple Inputsx1yTechnologies with Multiple Inputsx1yTechnologies with Multiple Inputsx1yTechnologies with Multiple Inputsx1yTechnologies with Multiple Inputsx1yTechnologies with Multiple Inputsx1yTechnologies with Multiple Inputsx1yCobb-Douglas TechnologiesA Cobb-Douglas production function is of the formE.g.withy A x x xa anan  1 21 2 .y x x11/321/3n A a and a   2 113131 2, , .x2x1All isoquants are hyperbolic,asymptoting to, but nevertouching any axis.Cobb-Douglas Technologiesy x xa a1 21 2x2x1All isoquants are hyperbolic,asymptoting to, but nevertouching any axis.Cobb-Douglas Technologiesx x ya a1 21 2 "y x xa a1 21 2x2x1All isoquants are hyperbolic,asymptoting to, but nevertouching any axis.Cobb-Douglas Technologiesx x ya a1 21 2 'x x ya a1 21 2 "y x xa a1 21 2x2x1All isoquants are hyperbolic,asymptoting to, but nevertouching any axis.Cobb-Douglas Technologiesx x ya a1 21 2 'x x ya a1 21 2 "y"y'>y x xa a1 21 2Fixed-Proportions TechnologiesA fixed-proportions production function is of the formE.g.withy a x a x a xn nmin{ , , , }.1 1 2 2y x xmin{ , }1 22n a and a  2 1 21 2, .Fixed-Proportions Technologiesx2x1min{x1,2x2} = 144 8 14247min{x1,2x2} = 8min{x1,2x2} = 4x1 = 2x2y x xmin{ , }1 22Perfect-Substitutes TechnologiesA perfect-substitutes