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### Isoquant

1. 1. IsoquantFrom Wikipedia, the free encyclopediaIn economics, an isoquant (derived from quantity and the Greek word iso,meaning equal) is a contour line drawn through the set of points at which thesame quantity of output is produced while changing the quantities of two ormore inputs.[1][2] While an indifference curve mapping helps to solve theutility-maximizing problem of consumers, the isoquant mapping deals withthe cost-minimization problem of producers. Isoquants are typically drawnon capital-labor graphs, showing the technological tradeoff between capitaland labor in the production function, and the decreasing marginal returns ofboth inputs. Adding one input while holding the other constant eventuallyleads to decreasing marginal output, and this is reflected in the shape of theisoquant. A family of isoquants can be represented by an isoquant map, agraph combining a number of isoquants, each representing a differentquantity of output. Isoquants are also called equal product curves.An isoquant shows the extent to which the firm in question has the ability tosubstitute between the two different inputs at will in order to produce thesame level of output. An isoquant map can also indicate decreasing orincreasing returns to scale based on increasing or decreasing distancesbetween the isoquant pairs of fixed output increment, as output increases. Ifthe distance between those isoquants increases as output increases, the firmsproduction function is exhibiting decreasing returns to scale; doubling bothinputs will result in placement on an isoquant with less than double the outputof the previous isoquant. Conversely, if the distance is decreasing as outputincreases, the firm is experiencing increasing returns to scale; doubling bothinputs results in placement on an isoquant with more than twice the output ofthe original isoquant.
2. 2. As with indifference curves, two isoquants can never cross. Also, everypossible combination of inputs is on an isoquant. Finally, any combination ofinputs above or to the right of an isoquant results in more output than anypoint on the isoquant. Although the marginal product of an input decreases asyou increase the quantity of the input while holding all other inputs constant,the marginal product is never negative in the empirically observed range sincea rational firm would never increase an input to decrease output.Shapes of IsoquantsIf the two inputs are perfect substitutes, the resulting isoquant map generatedis represented in fig. A; with a given level of production Q3, input X can bereplaced by input Y at an unchanging rate. The perfect substitute inputs do notexperience decreasing marginal rates of return when they are substituted foreach other in the production function.If the two inputs are perfect complements, the isoquant map takes the form offig. B; with a level of production Q3, input X and input Y can only be combinedefficiently in the certain ratio occurring at the kink in the isoquant. The firmwill combine the two inputs in the required ratio to maximize profit.Isoquants are typically combined with isocost lines in order to solve a cost-minimization problem for given level of output. In the typical case shown inthe top figure, with smoothly curved isoquants, a firm with fixed unit costs ofthe inputs will have isocost curves that are linear and downward sloped; anypoint of tangency between an isoquant and an isocost curve represents thecost-minimizing input combination for producing the output level associatedwith that isoquant.The only relevent portion of the iso quant is the one that is convex to theorigin, part of the curve which is not convex to the origin implies negativemarginal product for factors of production. Higher ISO-Quant higher theproduction
4. 4. however, if they export cars to global markets in addition to selling to the localmarket. Economies of scale also play a role in a "natural monopoly."Natural monopolyA natural monopoly is often defined as a firm which enjoys economies of scalefor all reasonable firm sizes; because it is always more efficient for one firm toexpand than for new firms to be established, the natural monopoly has nocompetition. Because it has no competition, it is likely the monopoly hassignificant market power. Hence, some industries that have been claimed tobe characterized by natural monopoly have been regulated or publicly-owned.Economies of scale and returns to scaleEconomies of scale is related to and can easily be confused with the theoreticaleconomic notion of returns to scale. Where economies of scale refer to a firmscosts, returns to scale describe the relationship between inputs and outputs ina long-run (all inputs variable) production function. A production functionhasconstant returns to scale if increasing all inputs by some proportion resultsin output increasing by that same proportion. Returns are decreasing if, say,doubling inputs results in less than double the output, and increasing if morethan double the output. If a mathematical function is used to represent theproduction function, and if that production function is homogeneous, returnsto scale are represented by the degree of homogeneity of the function.Homegeneous production functions with constant returns to scale are firstdegree homogeneous, increasing returns to scale are represented by degrees ofhomogeneity greater than one, and decreasing returns to scale by degrees ofhomogeneity less than one.If the firm is a perfect competitor in all input markets, and thus the per-unitprices of all its inputs are unaffected by how much of the inputs the firmpurchases, then it can be shown[2][3][4] that at a particular level of output, thefirm has economies of scale if and only if it has increasing returns to scale, hasdiseconomies of scale if and only if it has decreasing returns to scale, and hasneither economies nor diseconomies of scale if it has constant returns to scale.In this case, with perfect competition in the output market the long-runequilibrium will involve all firms operating at the minimum point of theirlong-run average cost curves (i.e., at the borderline between economies anddiseconomies of scale).If, however, the firm is not a perfect competitor in the input markets, then theabove conclusions are modified. For example, if there are increasing returns to
5. 5. scale in some range of output levels, but the firm is so big in one or more inputmarkets that increasing its purchases of an input drives up the inputs per-unitcost, then the firm could have diseconomies of scale in that range of outputlevels. Conversely, if the firm is able to get bulk discounts of an input, then itcould have economies of scale in some range of output levels even if it hasdecreasing returns in production in that output range.The literature assumed that due to the competitive nature of Reverse Auction,and in order to compensate for lower prices and lower margins, suppliers seekhigher volumes to maintain or increase the total revenue. Buyers, in turn,benefit from the lower transaction costs and economies of scale that resultfrom larger volumes. In part as a result, numerous studies have indicated thatthe procurement volume must be sufficiently high to provide sufficient profitsto attract enough suppliers, and provide buyers with enough savings to covertheir additional costs[5].However, surprisingly enough, Shalev and Asbjornsen found, in their researchbased on 139 reverse auctions conducted in the public sector by public sectorbuyers, that the higher auction volume, or economies of scale, did not lead tobetter success of the auction!. They found that Auction volume did notcorrelate with competition, nor with the number of bidder, suggesting thatauction volume does not promote additional competition. They noted,however, that their data included a wide range of products, and the degree ofcompetition in each market varied significantly, and offer that furtherresearch on this issue should be conducted to determine whether thesefindings remain the same when purchasing the same product for both smalland high volumes. Keeping competitive factors constant, increasing auctionvolume may further increase competition[6].Diseconomy of scale
6. 6. Diseconomies of scale are the forces that cause larger firms andgovernments to produce goods andservices at increased per-unit costs. Theyare less well known than whateconomists have long understood as "economiesof scale", the forces which enable larger firms to produce goods and services atreduced per-unit costs.[citation needed]However the political philosophy ofconservatism has long recognized the concept when applied to government.CausesSome of the forces which cause a diseconomy of scale are listed below:Cost of communicationIdeally, all employees of a firm would have one-on-one communication witheach other so they know exactly what the other workers are doing.[citationneeded] A firm with a single worker does not require any communicationbetween employees. A firm with two workers requires one communicationchannel, directly between those two workers. A firm with three workersrequires three communication channels (between employees A & B, B & C,and A & C). Here is a chart of one-on-one communication channels required:The one-on-one channels of communication grow more rapidly than thenumber of workers, thus increasing the time, and therefore costs, ofcommunication. At some point one-on-one communications between allworkers becomes impractical; therefore only certain groups of employees willcommunicate with one another (salespeople with salespeople, productionworkers with production workers, etc.). This reduced communication slows,but doesnt stop, the increase in time and money with firm growth, but alsocosts additional money, due to duplication of effort, owing to this reducedlevel of communication.Duplication of effortA firm with only one employee cant have any duplication of effort betweenemployees. A firm with two employees could have duplication of efforts, butthis is improbable, as the two are likely to know what each other is working on
7. 7. at all times. When firms grow to thousands of workers, it is inevitable thatsomeone, or even a team, will take on a project that is already being handledby another person or team. General Motors, for example, developed two in-house CAD/CAMsystems: CADANCE was designed by the GM Design Staff,while Fisher Graphics was created by the former Fisher Body division. Thesesimilar systems later needed to be combined into a single Corporate GraphicsSystem, CGS, at great expense. A smaller firm would neither have had themoney to allow such expensive parallel developments, or the lack ofcommunication and cooperation which precipitated this event. In addition toCGS, GM also used CADAM, UNIGRAPHICS, CATIA and other off-the-shelfCAD/CAM systems, thus increasing the cost of translating designs from onesystem to another. This endeavor eventually became so unmanageable thatthey acquired Electronic Data Systems (EDS) in an effort to control thesituation.Office politics"Office politics" is management behavior which a manager knows is counter tothe best interest of the company, but is in her/his personal best interest. Forexample, a manager might intentionally promote an incompetent workerknowing that that worker will never be able to compete for the managers job.This type of behavior only makes sense in a company with multiple levels ofmanagement. The more levels there are, the more opportunity for thisbehavior. At a small company, such behavior would likely cause the companyto go bankrupt, and thus cost the manager his job, so he would not make sucha decision. At a large company, one bad manager would not have much effecton the overall health of the company, so such "office politics" are in theinterest of individual managers.Isolation of decision makers from results of their decisionsIf a single person makes and sells donuts and decides to try jalapeño flavoring,they would likely know that day whether their decision was good or not, basedon the reaction of customers. A decision maker at a huge company that makesdonuts may not know for many months if such a decision worked out or not.
8. 8. By that time they may very well have moved on to another division orcompany and thus see no consequences from their decision. This lack ofconsequences can lead to poor decisions and cause an upward sloping averagecost curve.Slow response timeIn a reverse example, the single worker donut firm will know immediately ifpeople begin to request healthier offerings, like whole grain bagels, and beable to respond the next day. A large company would need to do research,create an assembly line, determine which distribution chains to use, plan anadvertising campaign, etc., before any change could be made. By this timesmaller competitors may well have grabbed that market niche.Inertia (unwillingness to change)This will be defined as the "weve always done it that way, so theres no need toever change" attitude (see appeal to tradition). An old, successful company isfar more likely to have this attitude than a new, struggling one. While "changefor changes sake" is counter-productive, refusal to consider change, evenwhen indicated, is toxic to a company, as changes in the industry and marketconditions will inevitably demand changes in the firm, in order to remainsuccessful. A recent example is Polaroid Corporations refusal to move intodigital imaging until after this lag adversely affected the company, ultimatelyleading to bankruptcy.[citation needed]CannibalizationA small firm only competes with other firms, but larger firms frequently findtheir own products are competing with each other. A Buick was just as likely tosteal customers from another GM make, such as an Oldsmobile, as it was tosteal customers from other companies. This may help to explain whyOldsmobiles were discontinued after 2004. This self-competition wastesresources that should be used to compete with other firms.
9. 9. Large market portfolioA small investment fund can potentially return a larger percentage because itcan concentrate its investments in a small number of good opportunitieswithout driving up the price of the investment securities.[1] Conversely, a largeinvestment fund like Fidelity Magellan must spread its investments among somany securities that its results tend to track those of the market as a whole.[2]Inelasticity of SupplyA company which is heavily dependent on its resource supply will have troubleincreasing production. For instance a timber company can not increaseproduction above the sustainable harvest rate of its land. Similarly servicecompanies are limited by available labor, STEM (Science TechnologyEngineering and Mathematics professions) being the most cited example.Public and government oppositionSuch opposition is largely a function of the size of the firm. BehaviorfromMicrosoft, which would have been ignored from a smaller firm, was seenas an anti-competitive and monopolistic threat, due to Microsofts size, thusbringing about public opposition and government lawsuits.SolutionsSolutions to the diseconomy of scale for large firms involve changing thecompany into one or more small firms. This can either happen by defaultwhen the company, in bankruptcy, sells off its profitable divisions and shutsdown the rest, or can happen proactively, if the management is willing.Returning to the example of the large donut firm, each retail location could beallowed to operate relatively autonomously from the company headquarters,with employee decisions (hiring, firing, promotions, wage scales, etc.) made bylocal management, not dictated by the corporation. Purchasing decisionscould also be made independently, with each location allowed to choose itsown suppliers, which may or may not be owned by the corporation (whereverthey find the best quality and prices). Each locale would also have the option
10. 10. of either choosing their own recipes and doing their own marketing, or theymay continue to rely on the corporation for those services. If the employeesown a portion of the local business, they will also have more invested in itssuccess. Note that all these changes will likely result in a substantial reductionin corporate headquarters staff and other support staff. For this reason, manybusinesses delay such a reorganization until it is too late to be effective.Cobb-Douglas Production Function1 IntroductionIn economics, the Cobb-Douglas functional form of productionfunctions is widely used to represent the relationshipof an output to inputs. It was proposed by KnutWicksell (1851 - 1926), and tested against statistical evidenceby Charles Cobb and Paul Douglas in 1928.In 1928 Charles Cobb and Paul Douglas published astudy in which they modeled the growth of the Americaneconomy during the period 1899 - 1922. They considereda simplified view of the economy in which productionoutput is determined by the amount of labor involvedand the amount of capital invested. While thereare many other factors affecting economic performance,their model proved to be remarkably accurate.The function they used to model production was of the form:P(L,K) = bL_K_where:• P = total production (the monetary value of all goods produced in a year)• L = labor input (the total number of person-hours worked in a year)• K = capital input (the monetary worth of all machinery, equipment, and buildings)• b = total factor productivity• _ and _ are the output elasticities of labor and capital, respectively. These values areconstantsdetermined by available technology.1Output elasticity measures the responsiveness of output to a change in levels of eitherlabor or
11. 11. capital used in production, ceteris paribus. For example if _ = 0.15, a 1% increase inlabor wouldlead to approximately a 0.15% increase in output.Further, if:_ + _ = 1,the production function has constant returns to scale. That is, if L and K are eachincreased by20%, then P increases by 20%.Returns to scale refers to a technical property of production that examines changesin output subsequent to a proportional change in all inputs (where all inputs increaseby a constant factor). If output increases by that same proportional change then thereare constant returns to scale (CRTS), sometimes referred to simply as returns to scale.If output increases by less than that proportional change, there are decreasing returnsto scale (DRS). If output increases by more than that proportion, there are increasingreturns to scale (IRS)However, if_ + _ < 1,returns to scale are decreasing, and if_ + _ > 1,returns to scale are increasing. Assuming perfect competition, _ and _ can be shown tobe laborand capital’s share of output.22 DiscoveryThis section will discuss the discovery of the production formula and how partialderivatives areused in the Cobb-Douglas model.2.1 Assumptions MadeIf the production function is denoted by P = P(L,K), then the partial derivative@P@Lis therate at which production changes with respect to the amount of labor. Economists call itthemarginal production with respect to labor or the marginal productivity of labor. Likewise,thepartial derivative@P@Kis the rate of change of production with respect to capital and is called themarginal productivity of capital.In these terms, the assumptions made by Cobb and Douglas can be stated as follows:1. If either labor or capital vanishes, then so will production.2. The marginal productivity of labor is proportional to the amount of production per unitoflabor.
12. 12. 3. The marginal productivity of capital is proportional to the amount of production perunit ofcapital.2.2 SolvingBecause the production per unit of labor isPL, assumption 2 says that@P@L=_PLfor some constant _. If we keep K constant(K = K0) , then this partial differentialequationbecomes an ordinary differential equation:dPdL=_PLThis separable differential equation can be solved by re-arranging the terms andintegrating bothsides: Z1PdP = _Z1LdLln(P) = _ ln(cL)ln(P) = ln(cL_)3And finally,P(L,K0) = C1(K0)L_ (1)where C1(K0) is the constant of integration and we write it as a function of K0 since itcoulddepend on the value of K0.Similarly, assumption 3 says that@P@K=_PKKeeping L constant(L = L0), this differential equation can be solved to get:
13. 13. P(L0,K) = C2(L0)K_ (2)And finally, combining equations (1) and (2):P(L,K) = bL_K_ (3)where b is a constant that is independent of both L and K.Assumption 1 shows that _ > 0 and _ > 0.Notice from equation (3) that if labor and capital are both increased by a factor m, thenP(mL,mK) = b(mL)_(mK)_= m_+_bL_K_= m_+_P(L,K)If _ + _ = 1, then P(mL,mK) = mP(L,K), which means that production is also increasedbya factor of m, as discussed earlier in Section 1.43 UsageThis section will demonstrate the usage of the production formula using real world data.3.1 An ExampleYear 1899 1900 1901 1902 1903 1904 1905 ... 1917 1918 1919 1920P 100 101 112 122 124 122 143 ... 227 223 218 231L 100 105 110 117 122 121 125 ... 198 201 196 194K 100 107 114 122 131 138 149 ... 335 366 387 407Table 1: Economic data of the American economy during the period 1899 - 1920 [1].Portionsnot shown for the sake of brevityUsing the economic data published by the government , Cobb and Douglas took theyear 1899 asa baseline, and P, L, and K for 1899 were each assigned the value 100. The values forother yearswere expressed as percentages of the 1899 figures. The result is Table 1.Next, Cobb and Douglas used the method of least squares to fit the data of Table 1 tothe function:P(L,K) = 1.01(L0.75)(K0.25) (4)For example, if the values for the years 1904 and 1920 were plugged in:P(121, 138) = 1.01(1210.75)(1380.25) _ 126.3P(194, 407) = 1.01(1940.75)(4070.25) _ 235.8which are quite close to the actual values, 122 and 231 respectively.The production function P(L,K) = bL_K_ has subsequently been used in many settings,rangingfrom individual firms to global economic questions. It has become known as the Cobb-Douglasproduction function. Its domain is {(L,K) : L _ 0,K _ 0} because L and K represent laborand capital and are therefore never negative.3.2 DifficultiesEven though the equation (4) derived earlier works for the period 1899 - 1922, there arecurrentlyvarious concerns over its accuracy in different industries and time periods.
14. 14. Cobb and Douglas were influenced by statistical evidence that appeared to show thatlabor andcapital shares of total output were constant over time in developed countries; theyexplained this5by statistical fitting least-squares regression of their production function. However, thereis nowdoubt over whether constancy over time exists.Neither Cobb nor Douglas provided any theoretical reason why the coefficients _ and _should beconstant over time or be the same between sectors of the economy. Remember that thenature ofthe machinery and other capital goods (the K) differs between time-periods andaccording to whatis being produced. So do the skills of labor (the L).The Cobb-Douglas production function was not developed on the basis of anyknowledge of engineering,technology, or management of the production process. It was instead developedbecauseit had attractive mathematical characteristics, such as diminishing marginal returns toeither factorof production.Crucially, there are no microfoundations for it. In the modern era, economists haveinsisted thatthe micro-logic of any larger-scale process should be explained. The C-D productionfunction failsthis test.For example, consider the example of two sectors which have the exactly same Cobb-Douglastechnologies:if, for sector 1,P1 = b(L_1 )(K_1)and, for sector 2,P2 = b(L_2 )(K_2 ),that, in general, does not imply thatP1 + P2 = b(L1 + L2)_(K1 + K2)_This holds only ifL1L2=K1K2
15. 15. and _ + _ = 1, i.e. for constant returns to scale technology.It is thus a mathematical mistake to assume that just because the Cobb-Douglasfunction appliesat the micro-level, it also applies at the macro-level. Similarly, there is no reason that amacroCobb-Douglas applies at the disaggregated level.Cobb–DouglasA two-input Cobb–Douglas production functionIn economics, the Cobb–Douglas functional form of production functions is widely used to represent therelationship of an output to inputs. It was proposed by Knut Wicksell (1851–1926), and tested against statisticalevidence by Charles Cobb and Paul Douglas in 1900–1928.For production, the function is Y = ALαKβ, where:  Y = total production (the monetary value of all goods produced in a year)  L = labor input  K = capital input  A = total factor productivity
16. 16.  α and β are the output elasticities of labor and capital, respectively. These values are constants determined by available technology.Output elasticity measures the responsiveness of output to a change in levels of either labor or capitalused in production, ceteris paribus. For example if α = 0.15, a 1% increase in labor would lead toapproximately a 0.15% increase in output.Further, if: α + β = 1, the production function has constant returns to scale. That is, if L and K are each increased by 20%, Y increases by 20%. If α + β < 1, returns to scale are decreasing, and if α+β>1 returns to scale are increasing. Assuming perfect competition and α + β = 1, α and β can be shown to be labor and capitals share of output. Cobb and Douglas were influenced by statistical evidence that appeared to show that labor and capital shares of total output were constant over time in developed countries; they explained this by statistical fitting least-squares regression of their production function. There is now doubt over whether constancy over time exists. Difficulties and criticisms ]Lack of constancy over time Neither Cobb nor Douglas provided any theoretical reason why the coefficients α and β should be constant over time or be the same between sectors of the economy. Remember that the nature of the machinery and other capital goods (the K) differs between time-periods and according to what is being produced. So do the skills of labor (the L). Dimensional analysis The Cobb–Douglas model is criticized on the basis of dimensional analysis of not having meaningful or economically reasonable units of measurement.[1] The units of the quantities are:  Y: widgets/year (wid/yr)
17. 17.  L: man-hours/year (manhr/yr)  K: capital-hours/year (caphr/yr; this raises issues of heterogeneous capital)  α, β: pure numbers (non-dimensional), due to being exponents  A: (widgets * yearα + β – 1)/(caphrα * manhrβ), a balancing quantity. The model is accordingly criticized because the quantities Lα and Kβ have economically meaningless units unless α=β=1 (which is economically unreasonable, as there are then no decreasing returns to scale). For instance, if α=1/2, Lα has units of "square root of man-hours over square root of years", neither of which is meaningful. Total factor productivity A is yet harder to interpret economically. Lack of microfoundations The Cobb–Douglas production function was not developed on the basis of any knowledge of engineering, technology, or management of the production process. It was instead developed because it had attractive mathematical characteristics, such as diminishing marginal returns to either factor of production and the property that expenditure on any given input is a constant fraction of total cost. Crucially, there are no microfoundations for it. In the modern era, economists have insisted that the micro-logic of any larger-scale process should be explained. The C–D production function fails this test. For example, consider two sectors which have exactly the same Cobb–Douglas technologies: if, for sector 1,Y1 = AL1αK1β and, for sector 2,Y2 = AL2αK2β, that, in general, does not imply thatY1 + Y2 = A(L1 + L2)α(K1 + K2)β This holds only if L1 / L2 = K1 / K2 and α+β = 1, i.e. for constant returns to scale technology. It is thus a mathematical mistake to assume that just because the Cobb–Douglas function applies at the micro-level, it also applies at the
18. 18. macro-level. Similarly, there is no reason that a macro Cobb–Douglasapplies at the disaggregated level.