Isoquant

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In 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 the same quantity of output is produced while changing the quantities of two or more inputs.[1][2] While an indifference curve mapping helps to solve the utility-maximizing problem of consumers, the isoquant mapping deals with the cost-minimization problem of producers. Isoquants are typically drawn on capital-labor graphs, showing the technological tradeoff between capital and labor in the production function, and the decreasing marginal returns of both inputs. Adding one input while holding the other constant eventually leads to decreasing marginal output, and this is reflected in the shape of the isoquant. A family of isoquants can be represented by an isoquant map, a graph combining a number of isoquants, each representing a different quantity of output. Isoquants are also called equal product curves.

An isoquant shows the extent to which the firm in question has the ability to substitute between the two different inputs at will in order to produce the same level of output. An isoquant map can also indicate decreasing or increasing returns to scale based on increasing or decreasing distances between the isoquant pairs of fixed output increment, as output increases. If the distance between those isoquants increases as output increases, the firm's production function is exhibiting decreasing returns to scale; doubling both inputs will result in placement on an isoquant with less than double the output of the previous isoquant. Conversely, if

the distance is decreasing as output increases, the firm is experiencing increasing returns to scale; doubling both inputs results in placement on an isoquant with more than twice the output of the original isoquant.

As with indifference curves, two isoquants can never cross. Also, every possible combination of inputs is on an isoquant. Finally, any combination of inputs above or to the right of an isoquant results in more output than any point on the isoquant. Although the marginal product of an input decreases as you increase the quantity of the input while holding all other inputs constant, the marginal product is never negative in the empirically observed range since a rational firm would never increase an input to decrease output.

Shapes of Isoquants

If the two inputs are perfect substitutes, the resulting isoquant map generated is represented in fig. A; with a given level of production Q3, input X can be replaced by input Y at an unchanging rate. The perfect substitute inputs do not experience decreasing marginal rates of return when they are substituted for each other in the production function.

If the two inputs are perfect complements, the isoquant map takes the form of fig. B; with a level of production Q3, input X and input Y can only be combined efficiently in the certain ratio occurring at the kink in the isoquant. The firm will 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 in the top figure, with smoothly curved isoquants, a firm with fixed unit costs of the inputs will have isocost curves that are linear and downward sloped; any point of tangency between an isoquant and an isocost curve represents the cost-minimizing input combination for producing the output level associated with that isoquant.

The only relevent portion of the iso quant is the one that is convex to the origin, part of the curve which is not convex to the origin

implies negative marginal product for factors of production. Higher ISO-Quant higher the production

Economies of scale

Economies of scale, inmicroeconomics, refers to the cost advantages that a business obtains due to expansion. There are factors that cause a producer’s average cost per unit to fall as the scale of output is increased. "Economies of scale" is a long run concept and refers to reductions in unit cost as the size of a facility and the usage levels of other inputs increase.[1] Diseconomies of scale are the opposite. The common sources of economies of scale are purchasing (bulk buying of materials through long-term contracts), managerial (increasing the specialization of managers), financial (obtaining lower-interest charges when borrowing from banks and having access to a greater range of financial

instruments), marketing (spreading the cost of advertising over a greater range of output in media markets), and technological (taking advantage of returns to scale in the production function). Each of these factors reduces the long run average costs (LRAC) of production by shifting theshort-run average total cost (SRATC) curve down and to the right. Economies of scale are also derived partially from learning by doing.

Economies of scale is a practical concept that is important for explaining real world phenomena such as patterns of international trade, the number of firms in a market, and how firms get "too big to fail". The exploitation of economies of scale helps explain why companies grow large in some industries. It is also a justification for free trade policies, since some economies of scale may require a larger market than is possible within a particular country — for example, it would not be efficient for Liechtenstein to have its own car maker, if they would only sell to their local market. A lone car maker may be profitable, however, if they export cars to global markets in addition to selling to the local market. Economies of scale also play a role in a "natural monopoly."

Natural monopoly

A natural monopoly is often defined as a firm which enjoys economies of scale for all reasonable firm sizes; because it is always more efficient for one firm to expand than for new firms to be established, the natural monopoly has no competition. Because it has no competition, it is likely the monopoly has significant market power. Hence, some industries that have been claimed to be characterized by natural monopoly have been regulated or publicly-owned.

Economies of scale and returns to scale

Economies of scale is related to and can easily be confused with the theoretical economic notion of returns to scale. Where economies of scale refer to a firm's costs, returns to scale describe the relationship between inputs and outputs in a long-run (all inputs variable) production function. A production function hasconstant returns to scale if increasing all inputs by some proportion results in output increasing by that same proportion.

Returns are decreasing if, say, doubling inputs results in less than double the output, and increasing if more than double the output. If a mathematical function is used to represent the production function, and if that production function is homogeneous, returns to scale are represented by the degree of homogeneity of the function. Homegeneous production functions with constant returns to scale are first degree homogeneous, increasing returns to scale are represented by degrees of homogeneity greater than one, and decreasing returns to scale by degrees of homogeneity less than one.

If the firm is a perfect competitor in all input markets, and thus the per-unit prices of all its inputs are unaffected by how much of the inputs the firm purchases, then it can be shown[2][3][4] that at a particular level of output, the firm has economies of scale if and only if it has increasing returns to scale, has diseconomies of scale if and only if it has decreasing returns to scale, and has neither economies nor diseconomies of scale if it has constant returns to scale. In this case, with perfect competition in the output market the long-run equilibrium will involve all firms operating at the minimum point of their long-run average cost curves (i.e., at the borderline between economies and diseconomies of scale).

If, however, the firm is not a perfect competitor in the input markets, then the above conclusions are modified. For example, if there are increasing returns to scale in some range of output levels, but the firm is so big in one or more input markets that increasing its purchases of an input drives up the input's per-unit cost, then the firm could have diseconomies of scale in that range of output levels. Conversely, if the firm is able to get bulk discounts of an input, then it could have economies of scale in some range of output levels even if it has decreasing 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 seek higher volumes to maintain or increase the total revenue. Buyers, in turn, benefit from the lower transaction costs and economies of scale that result from larger volumes. In part as a result, numerous studies have indicated that the procurement volume must be sufficiently high to provide

sufficient profits to attract enough suppliers, and provide buyers with enough savings to cover their additional costs[5].

However, surprisingly enough, Shalev and Asbjornsen found, in their research based on 139 reverse auctions conducted in the public sector by public sector buyers, that the higher auction volume, or economies of scale, did not lead to better success of the auction!. They found that Auction volume did not correlate with competition, nor with the number of bidder, suggesting that auction volume does not promote additional competition. They noted, however, that their data included a wide range of products, and the degree of competition in each market varied significantly, and offer that further research on this issue should be conducted to determine whether these findings remain the same when purchasing the same product for both small and high volumes. Keeping competitive factors constant, increasing auction volume may further increase competition[6].

Diseconomy of scale

Diseconomies of scale are the forces that cause larger firms and governments to produce goods andservices at increased per-unit costs. They are less well known than whateconomists have long understood as "economies of scale", the forces which enable larger firms to produce goods and services at reduced per-unit costs.[citation

needed]However the political philosophy of conservatism has long recognized the concept when applied to government.

Causes

Some of the forces which cause a diseconomy of scale are listed below:

Cost of communication

Ideally, all employees of a firm would have one-on-one communication with each other so they know exactly what the other workers are doing.[citation needed] A firm with a single worker does not require any communication between employees. A firm with two workers requires one communication channel, directly between those two workers. A firm with three workers requires 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 the number of workers, thus increasing the time, and therefore costs, of communication. At some point one-on-one communications between all workers becomes impractical; therefore only certain groups of employees will communicate with one another (salespeople with salespeople, production workers with production workers, etc.). This reduced communication slows, but doesn't stop, the increase in time and money with firm growth, but also costs additional money, due to duplication of effort, owing to this reduced level of communication.

Duplication of effort

A firm with only one employee can't have any duplication of effort between employees. A firm with two employees could have duplication of efforts, but this is improbable, as the two are likely to know what each other is working on at all times. When firms grow to thousands of workers, it is inevitable that someone, or even a team, will take on a project that is already being handled by

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. These similar systems later needed to be combined into a single Corporate Graphics System, CGS, at great expense. A smaller firm would neither have had the money to allow such expensive parallel developments, or the lack of communication and cooperation which precipitated this event. In addition to CGS, GM also used CADAM, UNIGRAPHICS, CATIA and other off-the-shelf CAD/CAM systems, thus increasing the cost of translating designs from one system to another. This endeavor eventually became so unmanageable that they acquired Electronic Data Systems (EDS) in an effort to control the situation.

Office politics

"Office politics" is management behavior which a manager knows is counter to the best interest of the company, but is in her/his personal best interest. For example, a manager might intentionally promote an incompetent worker knowing that that worker will never be able to compete for the manager's job. This type of behavior only makes sense in a company with multiple levels of management. The more levels there are, the more opportunity for this behavior. At a small company, such behavior would likely cause the company to go bankrupt, and thus cost the manager his job, so he would not make such a decision. At a large company, one bad manager would not have much effect on the overall health of the company, so such "office politics" are in the interest of individual managers.

Isolation of decision makers from results of their decisions

If 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, based on the reaction of customers.

A decision maker at a huge company that makes donuts may not know for many months if such a decision worked out or not. By that time they may very well have moved on to another division or company and thus see no consequences from their decision. This lack of consequences can lead to poor decisions and cause an upward sloping average cost curve.

Slow response time

In a reverse example, the single worker donut firm will know immediately if people begin to request healthier offerings, like whole grain bagels, and be able to respond the next day. A large company would need to do research, create an assembly line, determine which distribution chains to use, plan an advertising campaign, etc., before any change could be made. By this time smaller competitors may well have grabbed that market niche.

Inertia (unwillingness to change)

This will be defined as the "we've always done it that way, so there's no need to ever change" attitude (see appeal to tradition). An old, successful company is far more likely to have this attitude than a new, struggling one. While "change for change's sake" is counter-productive, refusal to consider change, even when indicated, is toxic to a company, as changes in the industry and market conditions will inevitably demand changes in the firm, in order to remain successful. A recent example is Polaroid Corporation's refusal to move into digital imaging until after this lag adversely affected the company, ultimately leading to bankruptcy.[citation needed]

Cannibalization

A small firm only competes with other firms, but larger firms frequently find their own products are competing with each other. A Buick was just as likely to steal customers from

another GM make, such as an Oldsmobile, as it was to steal customers from other companies. This may help to explain why Oldsmobiles were discontinued after 2004. This self-competition wastes resources that should be used to compete with other firms.

Large market portfolio

A small investment fund can potentially return a larger percentage because it can concentrate its investments in a small number of good opportunities without driving up the price of the investment securities.[1] Conversely, a large investment fund like Fidelity Magellan must spread its investments among so many securities that its results tend to track those of the market as a whole.[2]

Inelasticity of Supply

A company which is heavily dependent on its resource supply will have trouble increasing production. For instance a timber company can not increase production above the sustainable harvest rate of its land. Similarly service companies are limited by available labor, STEM (Science Technology Engineering and Mathematics professions) being the most cited example.

Public and government opposition

Such opposition is largely a function of the size of the firm. Behavior fromMicrosoft, which would have been ignored from a smaller firm, was seen as an anti-competitive and monopolistic threat, due to Microsoft's size, thus bringing about public opposition and government lawsuits.

Solutions

Solutions to the diseconomy of scale for large firms involve changing the company into one or more small firms. This can either

happen by default when the company, in bankruptcy, sells off its profitable divisions and shuts down the rest, or can happen proactively, if the management is willing. Returning to the example of the large donut firm, each retail location could be allowed to operate relatively autonomously from the company headquarters, with employee decisions (hiring, firing, promotions, wage scales, etc.) made by local management, not dictated by the corporation. Purchasing decisions could also be made independently, with each location allowed to choose its own suppliers, which may or may not be owned by the corporation (wherever they find the best quality and prices). Each locale would also have the option of either choosing their own recipes and doing their own marketing, or they may continue to rely on the corporation for those services. If the employees own a portion of the local business, they will also have more invested in its success. Note that all these changes will likely result in a substantial reduction in corporate headquarters staff and other support staff. For this reason, many businesses 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 production

output 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 are constantsdetermined by available technology.1Output elasticity measures the responsiveness of output to a change in levels of either labor orcapital used in production, ceteris paribus. For example if _ = 0.15, a 1% increase in labor 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 each increased 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 to be laborand capital’s share of output.

2

2 DiscoveryThis section will discuss the discovery of the production formula and how partial derivatives 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 it themarginal 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 unit oflabor.3. The marginal productivity of capital is proportional to the amount of production per unit 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 differential equationbecomes an ordinary differential equation:dPdL= _P

LThis separable differential equation can be solved by re-arranging the terms and integrating 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 it coulddepend 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: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 increased bya factor of m, as discussed earlier in Section 1.4

3 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 1920

P 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 the year 1899 asa baseline, and P, L, and K for 1899 were each assigned the value 100. The values for other 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 to the 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 are currentlyvarious concerns over its accuracy in different industries and time periods.Cobb and Douglas were influenced by statistical evidence that appeared to show that labor andcapital shares of total output were constant over time in developed countries; they explained this5by statistical fitting least-squares regression of their production function. However, there is 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 the nature ofthe machinery and other capital goods (the K) differs between time-periods and according to whatis being produced. So do the skills of labor (the L).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 becauseit had attractive mathematical characteristics, such as diminishing marginal returns to either factorof production.Crucially, there are no microfoundations for it. In the modern era, economists have insisted thatthe micro-logic of any larger-scale process should be explained. The C-D production function 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 ifL1

L2

=K1

K2

and _ + _ = 1, i.e. for constant returns to scale technology.It is thus a mathematical mistake to assume that just because the Cobb-Douglas function appliesat the micro-level, it also applies at the macro-level. Similarly, there is no reason that a macroCobb-Douglas applies at the disaggregated level.

Cobb–Douglas

A two-input Cobb–Douglas production function

In economics, the Cobb–Douglas functional form of production functions is widely used to represent the

relationship of an output to inputs. It was proposed by Knut Wicksell (1851–1926), and tested against statistical

evidence 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

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 capital

used in production, ceteris paribus. For example if α = 0.15, a 1% increase in labor would lead to

approximately 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 capital's 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)

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 = AL1αK1

β

and, for sector 2,

Y2 = AL2αK2

β,

that, in general, does not imply that

Y1 + 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

macro-level. Similarly, there is no reason that a macro Cobb–Douglas

applies at the disaggregated level.