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1、Chapter EighteenTechnologyTechnologiesuA technology is a process by which inputs are converted to an output.uE.g. labor, a computer, a projector, electricity, and software are being combined to produce this lecture.TechnologiesuUsually several technologies will produce the same product - a blackboar
2、d and chalk can be used instead of a computer and a projector.uWhich technology is “best”?uHow do we compare technologies?Input Bundlesuxi denotes the amount used of input i; i.e. the level of input i.uAn input bundle is a vector of the input levels; (x1, x2, , xn).uE.g. (x1, x2, x3) = (6, 0, 93).Pr
3、oduction Functionsuy denotes the output level.uThe technologys production function states the maximum amount of output possible from an input bundle.yf xxn (,)1Production Functionsy = f(x) is theproductionfunction.xxInput LevelOutput Levelyy = f(x) is the maximal output level obtainable from x input
4、 units.One input, one outputTechnology SetsuA production plan is an input bundle and an output level; (x1, , xn, y).uA production plan is feasible ifuThe collection of all feasible production plans is the technology set.yf xxn (,)1Technology Setsy = f(x) is theproductionfunction.xxInput LevelOutput
5、Levelyy”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 Txxyyf xxandxxnnn (, ) |(,),.11100Technology SetsxxInput LevelOutput LevelyOne input, one outputy”The
6、technologysetTechnology SetsxxInput LevelOutput LevelyOne input, one outputy”The technologysetTechnicallyinefficientplansTechnicallyefficient plansTechnologies with Multiple InputsuWhat does a technology look like when there is more than one input?uThe two input case: Input levels are x1 and x2. Out
7、put level is y.uSuppose the production function isyf xxxx (,).1211/321/32Technologies with Multiple InputsuE.g. the maximal output level possible from the input bundle(x1, x2) = (1, 8) isuAnd the maximal output level possible from (x1,x2) = (8,8) isyxx 2218212411/321/31/31/3.yxx 2288222811/321/31/31
8、/3.Technologies with Multiple InputsOutput, yx1x2(8,1)(8,8)Technologies with Multiple InputsuThe 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 8 8y 4 4x1x2Isoquants with Two Variable InputsuIsoquants can be graph
9、ed by adding an output level axis and displaying each isoquant at the height of the isoquants output level. Isoquants with Two Variable InputsOutput, yx1x2y 8 8y 4 4Isoquants with Two Variable InputsuMore isoquants tell us more about the technology.Isoquants with Two Variable Inputsy 8 8y 4 4x1x2y 6
10、 6y 2 2Isoquants with Two Variable InputsOutput, yx1x2y 8 8y 4 4y 6 6y 2 2Technologies with Multiple InputsuThe complete collection of isoquants is the isoquant map.uThe isoquant map is equivalent to the production function - each is the other.uE.g.3/123/11212),(xxxxfy Technologies with Multiple Inp
11、utsx1x2yTechnologies with Multiple Inputsx1x2yTechnologies with Multiple Inputsx1x2yTechnologies with Multiple Inputsx1x2yTechnologies with Multiple Inputsx1x2yTechnologies with Multiple Inputsx1x2yTechnologies with Multiple Inputsx1yTechnologies with Multiple Inputsx1yTechnologies with Multiple Inp
12、utsx1yTechnologies with Multiple Inputsx1yTechnologies with Multiple Inputsx1yTechnologies with Multiple Inputsx1yTechnologies with Multiple Inputsx1yTechnologies with Multiple Inputsx1yTechnologies with Multiple Inputsx1yTechnologies with Multiple Inputsx1yCobb-Douglas TechnologiesuA Cobb-Douglas p
13、roduction function is of the formuE.g.withyAxxxaanan 1212.yxx 11/321/3nAaand a 21131312,.x2x1All isoquants are hyperbolic,asymptoting to, but nevertouching any axis.Cobb-Douglas Technologiesyxxaa 1212x2x1All isoquants are hyperbolic,asymptoting to, but nevertouching any axis.Cobb-Douglas Technologie
14、sxxyaa1212 yxxaa 1212x2x1All isoquants are hyperbolic,asymptoting to, but nevertouching any axis.Cobb-Douglas Technologiesxxyaa1212 xxyaa1212 yxxaa 1212x2x1All isoquants are hyperbolic,asymptoting to, but nevertouching any axis.Cobb-Douglas Technologiesxxyaa1212 xxyaa1212 yyyxxaa 1212Fixed-Proportio
15、ns TechnologiesuA fixed-proportions production function is of the formuE.g.withya xa xa xnn min,.112 2yxx min,122naand a 21212,.Fixed-Proportions Technologiesx2x1minx1,2x2 = 144814247minx1,2x2 = 8minx1,2x2 = 4x1 = 2x2yxx min,122Perfect-Substitutes TechnologiesuA perfect-substitutes production functi
16、on is of the formuE.g.withya xa xa xnn 1122.yxx 123naand a 21312,.Perfect-Substitution Technologies93186248x1x2x1 + 3x2 = 18x1 + 3x2 = 36x1 + 3x2 = 48All are linear and parallelyxx 123Marginal (Physical) ProductsuThe marginal product of input i is the rate-of-change of the output level as the level
17、of input i changes, holding all other input levels fixed.uThat is,yf xxn (,)1iixyMP Marginal (Physical) ProductsE.g. ifyf xxxx (,)/1211/322 3then the marginal product of input 1 isMarginal (Physical) ProductsE.g. ifyf xxxx (,)/1211/322 3then the marginal product of input 1 isMPyxxx1112 322 313 /Marg
18、inal (Physical) ProductsE.g. ifyf xxxx (,)/1211/322 3then the marginal product of input 1 isMPyxxx1112 322 313 /and the marginal product of input 2 isMarginal (Physical) ProductsE.g. ifyf xxxx (,)/1211/322 3then the marginal product of input 1 isMPyxxx1112 322 313 /and the marginal product of input
19、2 isMPyxxx2211/321/323 .Marginal (Physical) ProductsTypically the marginal product of oneinput depends upon the amount used of other inputs. E.g. if MPxx112 322 313 /then,MPxx112 32 312 313843 /and if x2 = 27 thenif x2 = 8,MPxx112 32 312 313273 /.Marginal (Physical) ProductsuThe marginal product of
20、input i is diminishing if it becomes smaller as the level of input i increases. That is, if. 022 iiiiixyxyxxMP Marginal (Physical) ProductsMPxx112 322 313 /MPxx211/321/323 andE.g. ifyxx 11/322 3/thenMarginal (Physical) ProductsMPxx112 322 313 /MPxx211/321/323 andso MPxxx1115 322 3290 /E.g. ifyxx 11/
21、322 3/thenMarginal (Physical) ProductsMPxx112 322 313 /MPxx211/321/323 andso MPxxx1115 322 3290 / MPxxx2211/324 3290 /.andE.g. ifyxx 11/322 3/thenMarginal (Physical) ProductsMPxx112 322 313 /MPxx211/321/323 andso MPxxx1115 322 3290 / MPxxx2211/324 3290 /.andBoth marginal products are diminishing.E.g
22、. ifyxx 11/322 3/thenReturns-to-ScaleuMarginal products describe the change in output level as a single input level changes.uReturns-to-scale describes how the output level changes as all input levels change in direct proportion (e.g. all input levels doubled, or halved).Returns-to-ScaleIf, for any
23、input bundle (x1,xn),f kxkxkxkf xxxnn(,)(,)1212 then the technology described by theproduction function f exhibits constantreturns-to-scale.E.g. (k = 2) doubling all input levelsdoubles the output level.Returns-to-Scaley = f(x)xxInput LevelOutput LevelyOne input, one output2x2yConstantreturns-to-sca
24、leReturns-to-ScaleIf, for any input bundle (x1,xn),f kxkxkxkf xxxnn(,)(,)1212 then the technology exhibits diminishingreturns-to-scale.E.g. (k = 2) doubling all input levels less than doubles the output level.Returns-to-Scaley = f(x)xxInput LevelOutput Levelf(x)One input, one output2xf(2x)2f(x)Decre
25、asingreturns-to-scaleReturns-to-ScaleIf, for any input bundle (x1,xn),f kxkxkxkf xxxnn(,)(,)1212 then the technology exhibits increasingreturns-to-scale.E.g. (k = 2) doubling all input levelsmore than doubles the output level.Returns-to-Scaley = f(x)xxInput LevelOutput Levelf(x)One input, one output
26、2xf(2x)2f(x)Increasingreturns-to-scaleReturns-to-ScaleuA single technology can locally exhibit different returns-to-scale.Returns-to-Scaley = f(x)xInput LevelOutput LevelOne input, one outputDecreasingreturns-to-scaleIncreasingreturns-to-scaleExamples of Returns-to-Scaleya xa xa xnn 1122.The perfect
27、-substitutes productionfunction isExpand all input levels proportionatelyby k. The output level becomesakxakxakxnn1122()()() Examples of Returns-to-Scaleya xa xa xnn 1122.The perfect-substitutes productionfunction isExpand all input levels proportionatelyby k. The output level becomesakxakxakxk a xa
28、 xa xnnn n11221 122()()()() Examples of Returns-to-Scaleya xa xa xnn 1122.The perfect-substitutes productionfunction isExpand all input levels proportionatelyby k. The output level becomesakxakxakxk a xa xa xkynnn n11221 122()()()(). The perfect-substitutes productionfunction exhibits constant retur
29、ns-to-scale.Examples of Returns-to-Scaleya xa xa xnn min,.1122The perfect-complements productionfunction isExpand all input levels proportionatelyby k. The output level becomesmin(),(),()akxakxakxnn1122Examples of Returns-to-Scaleya xa xa xnn min,.1122The perfect-complements productionfunction isExp
30、and all input levels proportionatelyby k. The output level becomesmin(),(),()(min,)akxakxakxka xa xa xnnn n11221 122 Examples of Returns-to-Scaleya xa xa xnn min,.1122The perfect-complements productionfunction isExpand all input levels proportionatelyby k. The output level becomesmin(),(),()(min,).a
31、kxakxakxka xa xa xkynnn n11221 122 The perfect-complements productionfunction exhibits constant returns-to-scale.Examples of Returns-to-Scaleyxxxaanan 1212.The Cobb-Douglas production function isExpand all input levels proportionatelyby k. The output level becomes()()()kxkxkxaanan1212Examples of Ret
32、urns-to-Scaleyxxxaanan 1212.The Cobb-Douglas production function isExpand all input levels proportionatelyby k. The output level becomes()()()kxkxkxkkkxxxaanaaaaaaannn12121212 Examples of Returns-to-Scaleyxxxaanan 1212.The Cobb-Douglas production function isExpand all input levels proportionatelyby
33、k. The output level becomes()()()kxkxkxkkkxxxkxxxaanaaaaaaaaaaaanannnnn12121212121212 Examples of Returns-to-Scaleyxxxaanan 1212.The Cobb-Douglas production function isExpand all input levels proportionatelyby k. The output level becomes()()().kxkxkxkkkxxxkxxxkyaanaaaaaaaaaaaanaaannnnnn1212121212121
34、21 Examples of Returns-to-Scaleyxxxaanan 1212.The Cobb-Douglas production function is()()().kxkxkxkyaanaaann12121 The Cobb-Douglas technologys returns-to-scale isconstant if a1+ + an = 1Examples of Returns-to-Scaleyxxxaanan 1212.The Cobb-Douglas production function is()()().kxkxkxkyaanaaann12121 The
35、 Cobb-Douglas technologys returns-to-scale isconstant if a1+ + an = 1increasing if a1+ + an 1Examples of Returns-to-Scaleyxxxaanan 1212.The Cobb-Douglas production function is()()().kxkxkxkyaanaaann12121 The Cobb-Douglas technologys returns-to-scale isconstant if a1+ + an = 1increasing if a1+ + an 1
36、decreasing if a1+ + an 1.Returns-to-ScaleuQ: Can a technology exhibit increasing returns-to-scale even though all of its marginal products are diminishing?Returns-to-ScaleuQ: Can a technology exhibit increasing returns-to-scale even if all of its marginal products are diminishing?uA: Yes.uE.g.yxx 12
37、 322 3/.Returns-to-Scaleyxxxxaa 12 322 31212/aa12431 so this technology exhibitsincreasing returns-to-scale.Returns-to-Scaleyxxxxaa 12 322 31212/aa12431 so this technology exhibitsincreasing returns-to-scale.But MPxx111/322 323 /diminishes as x1increasesReturns-to-Scaleyxxxxaa 12 322 31212/aa12431 s
38、o this technology exhibitsincreasing returns-to-scale.But MPxx111/322 323 /diminishes as x1increases andMPxx212 321/323 /diminishes as x1increases.Returns-to-ScaleuSo a technology can exhibit increasing returns-to-scale even if all of its marginal products are diminishing. Why?Returns-to-ScaleuA mar
39、ginal product is the rate-of-change of output as one input level increases, holding all other input levels fixed.uMarginal product diminishes because the other input levels are fixed, so the increasing inputs units have each less and less of other inputs with which to work.Returns-to-ScaleuWhen all
40、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.Technical Rate-of-SubstitutionuA
41、t what rate can a firm substitute one input for another without changing its output level?Technical Rate-of-Substitutionx2x1y100100 x2x1Technical Rate-of-Substitutionx2x1y100100The slope is the rate at which input 2 must be given up as input 1s level is increased so as not to change the output level
42、. The slope of an isoquant is its technical rate-of-substitution.x2x1Technical Rate-of-SubstitutionuHow is a technical rate-of-substitution computed?Technical Rate-of-SubstitutionuHow is a technical rate-of-substitution computed?uThe production function isuA small change (dx1, dx2) in the input bund
43、le causes a change to the output level ofyf xx (,).12dyyxdxyxdx 1122.Technical Rate-of-Substitutiondyyxdxyxdx 1122.But dy = 0 since there is to be no changeto the output level, so the changes dx1and dx2 to the input levels must satisfy01122 yxdxyxdx .Technical Rate-of-Substitution01122 yxdxyxdxrearr
44、anges to yxdxyxdx2211 sodxdxyxyx2112 /.Technical Rate-of-Substitutiondxdxyxyx2112 /is 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.Technical Rate-of-Substitution; A Cobb-Douglas Exampleyf xxx xa b (,)1212so yxax
45、xab1112 yxbx xa b2121 .andThe technical rate-of-substitution isdxdxyxyxaxxbx xaxbxaba b211211212121 /.x2x1Technical Rate-of-Substitution; A Cobb-Douglas ExampleTRSaxbxxxxx 2121211 3232( /)(/)yxxaand b 11/322 31323/;x2x1Technical Rate-of-Substitution; A Cobb-Douglas ExampleTRSaxbxxxxx 2121211 3232( /
46、)(/)yxxaand b 11/322 31323/;84TRSxx 2128241x2x1Technical Rate-of-Substitution; A Cobb-Douglas ExampleTRSaxbxxxxx 2121211 3232( /)(/)yxxaand b 11/322 31323/;612TRSxx 212621214Well-Behaved TechnologiesuA well-behaved technology islmonotonic, andlconvex.Well-Behaved Technologies - MonotonicityuMonotoni
47、city: More of any input generates more output.yxyxmonotonic notmonotonicWell-Behaved Technologies - ConvexityuConvexity: 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. Well-Behaved Technologies - Convexit
48、yx2x1x2x1x2x1y100100Well-Behaved Technologies - Convexityx2x1x2x1x2x1 txt xtxt x112211(),() y100100Well-Behaved Technologies - Convexityx2x1x2x1x2x1 txt xtxt x112211(),() y100100y120120Well-Behaved Technologies - Convexityx2x1x2x1x2x1Convexity implies that the TRSincreases (becomes lessnegative) as
49、x1 increases.Well-Behaved Technologiesx2x1y100100y5050y200200higher outputThe Long-Run and the Short-RunsuThe long-run is the circumstance in which a firm is unrestricted in its choice of all input levels.uThere are many possible short-runs.uA short-run is a circumstance in which a firm is restricte
50、d in some way in its choice of at least one input level.The Long-Run and the Short-RunsuExamples of restrictions that place a firm into a short-run:ltemporarily being unable to install, or remove, machinerylbeing required by law to meet affirmative action quotaslhaving to meet domestic content regul