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1、Chapter EighteenTechnology技术技术StructureuDescribing technologieslProduction set or technology setlProduction functionlIsoquantuMarginal productuReturns to scaleuTechnical rate of substitutionuWell-behaved technologiesuLong run and short runTechnologiesuA technology is a process by which inputs are co
2、nverted 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 blackboard and chalk can be used instead of a computer and a projector.uWhich technology is“best”?uHow
3、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).Production Functions(生产函数）uy denotes the output level.uThe technologys production function states the maximum amount of output possible
4、from an input bundle.Production Functionsy=f(x)is theproductionfunction.xxInput LevelOutput Levelyy=f(x)is the maximal output level obtainable from x input units.One input,one outputTechnology SetsuA production plan is an input bundle and an output level;(x1,xn,y).uA production plan is feasible（可行）可
5、行）ifuThe collection of all feasible production plans is the production set(生产集生产集)or technology set（技术集）技术集）.Technology Setsy=f(x)is theproductionfunction.xxInput LevelOutput 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
6、 feasible from x input units.Technology SetsThe production set or technology set is Technology SetsxxInput LevelOutput LevelyOne input,one outputy”The technologysetTechnology SetsxxInput LevelOutput LevelyOne input,one outputy”The technologysetTechnicallyinefficientplansTechnicallyefficient plansTec
7、hnologies 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.Output level is y.uSuppose the production function isTechnologies with Multiple InputsuE.g.the maximal output level possible from the input bundle(x1,x2)=(
8、1,8)isuAnd the maximal output level possible from(x1,x2)=(8,8)isTechnologies with Multiple InputsuAn isoquant(等产量线等产量线)is the set of all possible combinations of inputs 1 and 2 that are just sufficient to produce a given amount of output.Isoquants with Two Variable Inputsy 8 8y 4 4x1x2Technologies w
9、ith 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.Isoquants with Two Variable Inputsy 8 8y 4 4x1x2y 6 6y 2 2Examples of TechnologiesuCobb-DouglasuFixed-Proportions TechnologiesuPerfect-Substi
10、tution TechnologiesCobb-Douglas TechnologiesuA Cobb-Douglas production function is of the formuE.g.withx2x1All isoquants are hyperbolic(双曲线双曲线),asymptoting(渐进渐进)to,but never touching any axis.Cobb-Douglas Technologiesx2x1All isoquants are hyperbolic,asymptoting to,but nevertouching any axis.Cobb-Dou
11、glas TechnologiesFixed-Proportions Technologiesx2x1minx1,2x2=144814247minx1,2x2=8minx1,2x2=4x1=2x2Perfect-Substitution Technologies93186248x1x2x1+3x2=18x1+3x2=36x1+3x2=48All are linear and parallelMarginal(Physical)ProductsuThe marginal product(边际产量边际产量)of input i is the rate-of-change of the output
12、 level as the level of input i changes,holding all other input levels fixed.uThat is,Marginal(Physical)ProductsE.g.ifthen the marginal product of input 1 isand the marginal product of input 2 isMarginal(Physical)ProductsTypically the marginal product of oneinput depends upon the amount used of other
13、 inputs.E.g.if then,and if x2=27 thenif x2=8,Marginal(Physical)ProductsuThe marginal product of input i is diminishing if it becomes smaller as the level of input i increases.That is,ifMarginal(Physical)ProductsandsoandBoth marginal products are diminishing.E.g.ifthenReturns-to-Scale（规模收益）uMarginal
14、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 input bundle(x1,xn),then the technology described b
15、y theproduction function f exhibits constantreturns-to-scale（规模报酬不变）规模报酬不变）.E.g.(k=2)doubling all input levelsdoubles the output level.Returns-to-Scaley=f(x)xxInput LevelOutput LevelyOne input,one output2x2yConstantreturns-to-scaleReturns-to-ScaleIf,for any input bundle(x1,xn),then the technology ex
16、hibits diminishingreturns-to-scale（规模报酬递减）规模报酬递减）.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)Decreasingreturns-to-scaleReturns-to-ScaleIf,for any input bundle(x1,xn),then the technology exhi
17、bits increasingreturns-to-scale（规模报酬递增）规模报酬递增）.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 output2xf(2x)2f(x)Increasingreturns-to-scaleReturns-to-ScaleuA single technology can locally exhibit different returns-
18、to-scale.Returns-to-Scaley=f(x)xInput LevelOutput LevelOne input,one outputDecreasingreturns-to-scaleIncreasingreturns-to-scaleuPerfect substitutesuPerfect complementsuCobb-DouglasExamples of Returns-to-ScaleExamples of Returns-to-ScaleThe perfect-substitutes productionfunction isExpand all input le
19、vels proportionatelyby k.The output level becomesThe perfect-substitutes productionfunction exhibits constant returns-to-scale.Examples of Returns-to-ScaleThe perfect-complements productionfunction isExpand all input levels proportionatelyby k.The output level becomesThe perfect-complements producti
20、onfunction exhibits constant returns-to-scale.Examples of Returns-to-ScaleThe Cobb-Douglas production function isExpand all input levels proportionatelyby k.The output level becomesExamples of Returns-to-ScaleThe Cobb-Douglas production function isThe Cobb-Douglas technologys returns-to-scale iscons
21、tant if a1+an =1increasing if a1+an 1decreasing 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
22、 diminishing?uA:Yes.uE.g.Returns-to-Scaleso this technology exhibitsincreasing returns-to-scale.But diminishes as x1increases anddiminishes as x1increases.Returns-to-ScaleuA marginal product is the rate-of-change of output as one input level increases,holding all other input levels fixed.uMarginal p
23、roduct 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 input levels are increased proportionately,there need be no diminution of marginal products since each input will always have
24、 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-Substitution(技术替代率)uAt what rate can a firm substitute one input for another without changing its output level?Technical Rate-of-Substitution
25、x2x1y100100Technical 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.The slope of an isoquant is its technical rate-of-substitution.Technical Rate-of-SubstitutionuHow is a technical rate-of-sub
26、stitution computed?uThe production function isuA small change(dx1,dx2)in the input bundle causes a change to the output level ofTechnical 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 satisfyTechnical Rate-of-Substit
27、utionrearranges tosoTechnical 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.Technical Rate-of-Substitution;A Cobb-Douglas ExamplesoandThe technical rate-of-substitution isx2x1Technical Rate-
28、of-Substitution;A Cobb-Douglas Examplex2x1Technical Rate-of-Substitution;A Cobb-Douglas Example84x2x1Technical Rate-of-Substitution;A Cobb-Douglas Example612Well-Behaved TechnologiesuA well-behaved technology islmonotonic,andlconvex.Well-Behaved Technologies-MonotonicityuMonotonicity:More of any inp
29、ut 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-Convexityx2x1y100100Well-Behaved Technol
30、ogies-Convexityx2x1y100100Well-Behaved Technologies-Convexityx2x1y100100y120120Well-Behaved Technologies-Convexityx2x1Convexity implies that the TRSincreases(becomes lessnegative)as x1 increases.Well-Behaved Technologies Monotonicity and convexityx2x1y100100y5050y200200higher outputThe Long-Run and
31、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 restricted in some way in its choice of at least one input level.The Long-Run and the Short-RunsuExa
32、mples 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 regulations.The Long-Run and the Short-RunsuWhat do short-run restrictions imply for a firms techn
33、ology?uSuppose the short-run restriction is fixing the level of input 2.uInput 2 is thus a fixed input in the short-run.Input 1 remains variable.The Long-Run and the Short-Runsx1yFour short-run production functions.The Long-Run and the Short-Runs is the long-run productionfunction(both x1 and x2 are variable).The short-run production function whenx2 1 isThe short-run production function when x2 10 isThe Long-Run and the Short-Runsx1yFour short-run production functions.