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unobtainable with current technology, all points below are technically feasible, and all points on the function show the maximum quantity of output obtainable at the specified level of usage of the input. From point A to point C, the firm is experiencing positive but decreasing marginal returns to the variable input. As additional units of the input are employed, output increases but at a decreasing rate. Point B is the point beyond which there are diminishing average returns, as shown by the declining slope of the average physical product curve (APP) beyond point Y. Point B is just tangent to the steepest ray from the origin hence the average physical product is at a maximum. Beyond point B, mathematical necessity requires that the marginal curve must be below the average curve (See
1116:
variable input. The optimum input/output combination for the price-taking firm will be in stage 2, although a firm facing a downward-sloped demand curve might find it most profitable to operate in Stage 2. In Stage 3, too much variable input is being used relative to the available fixed inputs: variable inputs are over-utilized in the sense that their presence on the margin obstructs the production process rather than enhancing it. The output per unit of both the fixed and the variable input declines throughout this stage. At the boundary between stage 2 and stage 3, the highest possible output is being obtained from the fixed input.
20:
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theoretical construct, may be abstracting away from the secondary factors and intermediate products consumed in a production process. The production function is not a full model of the production process: it deliberately abstracts from inherent aspects of physical production processes that some would argue are essential, including error, entropy or waste, and the consumption of energy or the co-production of pollution. Moreover, production functions do not ordinarily model the
2699:
physical outputs minus the economic value of physical inputs is the income generated by the production process. By keeping the prices fixed between two periods under review we get the income change generated by a change of the production function. This is the principle how the production function is made a practical concept, i.e. measureable and understandable in practical situations.
2629:. Details related to the derivation of a specific functional form of this production function as well as empirical support for this form of the production function are discussed in more-recently published work. Note that similar arguments could be used to develop more-realistic production functions which consider other depletable natural resources beyond energy:
50:, a key focus of economics. One important purpose of the production function is to address allocative efficiency in the use of factor inputs in production and the resulting distribution of income to those factors, while abstracting away from the technological problems of achieving technical efficiency, as an engineer or professional manager might understand it.
1929:, aggregate production functions for whole nations are sometimes constructed. In theory, they are the summation of all the production functions of individual producers; however there are methodological problems associated with aggregate production functions, and economists have debated extensively whether the concept is valid.
1125:
more significant than what is required to merely balance production capacity with demand. For example, you may only need to increase production by million units per year to keep up with demand, but the production equipment upgrades that are available may involve increasing productive capacity by 2 million units per year.
1702:, a curve drawn through the set of points in say labour-capital space at which the same quantity of output is produced for varying combinations of the inputs) is homogeneous of degree zero. Due to this, along rays coming from the origin, the slopes of the isoquants will be the same. Homothetic functions are of the form
1136:
If a firm is operating at a profit-maximizing level in stage one, it might, in the long run, choose to reduce its scale of operations (by selling capital equipment). By reducing the amount of fixed capital inputs, the production function will shift down. The beginning of stage 2 shifts from B1 to B2.
2698:
The theory of the production function depicts the relation between physical outputs of a production process and physical inputs, i.e. factors of production. The practical application of production functions is obtained by valuing the physical outputs and inputs by their prices. The economic value of
1124:
By definition, in the long run the firm can change its scale of operations by adjusting the level of inputs that are fixed in the short run, thereby shifting the production function upward as plotted against the variable input. If fixed inputs are lumpy, adjustments to the scale of operations may be
127:
and may represent primary factors, which are stocks. Classically, the primary factors of production were land, labour and capital. Primary factors do not become part of the output product, nor are the primary factors, themselves, transformed in the production process. The production function, as a
110:
choice of how much of a factor input to use, or the degree to which one factor may be substituted for another. In the production function itself, the relationship of output to inputs is non-monetary; that is, a production function relates physical inputs to physical outputs, and prices and costs are
2081:
in man-hours of labor; are told something about the index-number problem in choosing a unit of output; and then are hurried on to the next question, in the hope that will forget to ask in what units K is measured. Before ever do ask, have become a professor, and so sloppy habits of thought are
1694:
If a production function is homogeneous of degree one, it is sometimes called "linearly homogeneous". A linearly homogeneous production function with inputs capital and labour has the properties that the marginal and average physical products of both capital and labour can be expressed as functions
2633:
If a geographical region runs out of the natural resources required to produce a given machine or maintain existing machines and is unable to import more or recycle, the machines in that region will eventually fall into disrepair and the machines' maximum output would be reduced to near-zero. This
2473:
This model has also been shown to predict a 28% decrease in output for a 99% decrease in energy, which further supports the revision of this model's assumptions. Note that, while inappropriate for energy, an "independent" modelling approach may be appropriate for modelling other natural resources
1107:
To simplify the interpretation of a production function, it is common to divide its range into 3 stages. In Stage 1 (from the origin to point B) the variable input is being used with increasing output per unit, the latter reaching a maximum at point B (since the average physical product is at its
1495:
means that a one percent increase in the usage levels of all inputs would result in a greater than one percent increase in output; the presence of decreasing returns means that it would result in a less than one percent increase in output. Constant returns to scale is the in-between case. In the
1115:
both decline. However, the average product of fixed inputs (not shown) is still rising, because output is rising while fixed input usage is constant. In this stage, the employment of additional variable inputs increases the output per unit of fixed input but decreases the output per unit of the
119:
for each factor. The profit-maximizing firm in perfect competition (taking output and input prices as given) will choose to add input right up to the point where the marginal cost of additional input matches the marginal product in additional output. This implies an ideal division of the income
102:
requirements needed to produce designated quantities of output. Assuming that maximum output is obtained from given inputs allows economists to abstract away from technological and managerial problems associated with realizing such a technical maximum, and to focus exclusively on the problem of
139:
The production function is central to the marginalist focus of neoclassical economics, its definition of efficiency as allocative efficiency, its analysis of how market prices can govern the achievement of allocative efficiency in a decentralized economy, and an analysis of the distribution of
101:
output obtainable from a given set of inputs. The production function, therefore, describes a boundary or frontier representing the limit of output obtainable from each feasible combination of input. Alternatively, a production function can be defined as the specification of the minimum input
114:
In the decision frame of a firm making economic choices regarding production—how much of each factor input to use to produce how much output—and facing market prices for output and inputs, the production function represents the possibilities afforded by an exogenous technology. Under certain
1094:
Any of these equations can be plotted on a graph. A typical (quadratic) production function is shown in the following diagram under the assumption of a single variable input (or fixed ratios of inputs so they can be treated as a single variable). All points above the production function are
2093:. The problem is that this independence is a precondition of constructing an isoquant. Further, the slope of the isoquant helps determine relative factor prices, but the curve cannot be constructed (and its slope measured) unless the prices are known beforehand.
53:
For modelling the case of many outputs and many inputs, researchers often use the so-called
Shephard's distance functions or, alternatively, directional distance functions, which are generalizations of the simple production function in economics.
2151:
Georgescu-Roegen can be understood as criticizing Solow and
Stiglitz's approach to mathematically modelling factors of production. We will use the example of energy to illustrate the strengths and weaknesses of the two approaches in question.
2365:
If workers at any stage of the production process rely on electricity to perform their jobs, a power outage would significantly reduce their maximum output, and a long-enough power outage would reduce their maximum output to zero. Therefore
2361:. However, as discussed in more-recent work, this approach does not accurately model the mechanism by which energy affects production processes. Consider the following cases which support the revision of the assumptions made by this model:
2142:, since their variant allowed man-made capital to be a complete substitute for natural resources. Neither Solow nor Stiglitz reacted to Georgescu-Roegen's criticism, despite an invitation to do so in the September 1997 issue of the journal
891:
applies to situations in which inputs must be used in fixed proportions; starting from those proportions, if usage of one input is increased without another being increased, the output will not change. This production function is given by
1402:
1695:
of the capital-labour ratio alone. Moreover, in this case, if each input is paid at a rate equal to its marginal product, the firm's revenues will be exactly exhausted and there will be no excess economic profit.
851:
693:
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production function (CES), which is a generalized form of the Cobb–Douglas function, and the quadratic production function. The best form of the equation to use and the values of the parameters (
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331:
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has demonstrated that they also have no empirical relevance, as long as the alleged good fit comes from an accounting identity, not from any underlying laws of production/distribution.
402:
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1949:). Although the criticism was directed primarily at aggregate production functions, microeconomic production functions were also put under scrutiny. The debate began in 1953 when
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Sickles, R., & Zelenyuk, V. (2019). Measurement of
Productivity and Efficiency: Theory and Practice. Cambridge: Cambridge University Press. doi:10.1017/9781139565981
2219:
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1999:
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maximum at that point). Because the output per unit of the variable input is improving throughout stage 1, a price-taking firm will always operate beyond this stage.
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1428:
881:
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2467:
2413:
1816:
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gives the technological relation between quantities of physical inputs and quantities of output of goods. The production function is one of the key concepts of
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2438:
2384:
2079:
2059:
2039:
2019:
1485:
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for further explanation and
Sickles and Zelenyuk (2019) for more extensive discussions of various production functions, their generalizations and estimations).
1073:
569:
549:
471:
451:
259:
132:, either, ignoring the role of strategic and operational business management. (For a primer on the fundamental elements of microeconomic production theory, see
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a (mathematical) function of input, because any given set of inputs can be used to produce a range of outputs. To satisfy the mathematical definition of a
1266:
453:
is a scalar, then this form does not encompass joint production, which is a production process that has multiple co-products. On the other hand, if
2779:
2101:
As a result of the criticism on their weak theoretical grounds, it has been claimed that empirical results firmly support the use of neoclassical
3554:
Sickles, R., & Zelenyuk, V. (2019). Measurement of
Productivity and Efficiency: Theory and Practice. Cambridge: Cambridge University Press.
3321:
Sickles, R., & Zelenyuk, V. (2019). Measurement of
Productivity and Efficiency: Theory and Practice. Cambridge: Cambridge University Press.
754:
580:
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are parameters that are determined empirically. Linear functions imply that inputs are perfect substitutes in production. Another is as a
2482:
The "independent" energy-dependent production function can be revised by considering energy-dependent labor and capital input functions
2420:
If there were a power outage, machines would not be able to run, and therefore their maximum output would be reduced to zero. Therefore
898:
2085:
According to the argument, it is impossible to conceive of capital in such a way that its quantity is independent of the rates of
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3064:
2134:
attempted to develop a more realistic production function by including natural resources, they did it in a manner economist
1945:
During the 1950s, '60s, and '70s there was a lively debate about the theoretical soundness of production functions (see the
2171:
Labor, capital, energy input, and technical change (omitted below for brevity) are the only relevant factors of production,
1960:"The production function has been a powerful instrument of miseducation. The student of economic theory is taught to write
154:
120:
generated from output into an income due to each input factor of production, equal to the marginal product of each input.
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264:
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1055:) vary from company to company and industry to industry. In the short run, production function at least one of the
2174:
The factors of production are independent of one another such that the production function takes the general form
3498:
2759:
1629:
3475:
Moroney, J. R. (1967). "Cobb–Douglass production functions and returns to scale in US manufacturing industry".
2144:
3592:
3275:
2729:
1920:
336:
1078:
Moysan and
Senouci (2016) provide an analytical formula for all 2-input, neoclassical production functions.
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2714:
888:
2227:
2119:
Nicholas
Georgescu-Roegen § Criticising neoclassical economics (weak versus strong sustainability)
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701:
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1705:
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There are two special classes of production functions that are often analyzed. The production function
1075:'s (inputs) is fixed. In the long run, all factor inputs are variable at the discretion of management.
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2749:
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Homothetic functions are functions whose marginal technical rate of substitution (the slope of the
1096:
133:
3111:
Stiglitz, Joseph E. (1974). "Growth with
Exhaustible Natural Resources: The Competitive Economy".
3567:
3411:
3405:
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describe an approach to modelling energy as a factor of production which assumes the following:
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2139:
94:
39:
3457:
3450:
3050:(2007). "How long can neoclassical economists ignore the contributions of Georgescu-Roegen?".
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1963:
3340:
3298:
3010:
Shaikh, A. (1974). "Laws of
Production and Laws of Algebra: The Humbug Production Function".
2734:
2061:
a rate of output of commodities. are instructed to assume all workers alike, and to measure
124:
103:
47:
3361:
1957:
was measured and how the notion of factor proportions had distracted economists. She wrote:
1444:
1407:
3496:
Pearl, D.; Enos, J. (1975). "Engineering Production Functions and Technological Progress".
3227:
3167:
1220:
859:
36:
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2443:
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criticized as a "conjuring trick": Solow and Stiglitz had failed to take into account the
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are the quantities of factor inputs (such as capital, labour, land or raw materials). For
8:
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62:
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2004:
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Cobb–Douglas production function referred to above, returns to scale are increasing if
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534:
456:
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244:
19:
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different types of output based on the joint usage of the specified quantities of the
3556:
https://assets.cambridge.org/97811070/36161/frontmatter/9781107036161_frontmatter.pdf
3541:
3461:
3434:
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https://assets.cambridge.org/97811070/36161/frontmatter/9781107036161_frontmatter.pdf
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should be modeled as significantly affecting the total output. Therefore, therefore
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1434:
1431:
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116:
70:
43:
3352:
Craig, C.; Harris, R. (1973). "Total Productivity Measurement at the Firm Level".
2914:
2863:"Retrospectives: Whatever Happened to the Cambridge Capital Theory Controversies?"
2570:. This approach yields an energy-dependent production function given generally as
2739:
2654:
should be modeled as depending directly on time-dependent natural resource input
2164:
2131:
1397:{\displaystyle f(kX_{1},kX_{2},\dotsc ,kX_{n})=k^{m}f(X_{1},X_{2},\dotsc ,X_{n})}
148:
A production function can be expressed in a functional form as the right side of
66:
3053:
Ecological Economics and Sustainable Development. Selected Essays of Herman Daly
1937:
There are two major criticisms of the standard form of the production function.
140:
income, which attributes factor income to the marginal product of factor input.
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2862:
1926:
129:
58:
3586:
3247:
3187:
3132:
2709:
1950:
1086:
3179:
2973:
Robinson, Joan (1953). "The Production Function and the Theory of Capital".
2774:
2160:
2127:
3156:"Thermodynamic laws, economic methods and the productive power of energy"
3047:
81:, however, reject the very concept of an aggregate production function.
16:
Used to define marginal product and to distinguish allocative efficiency
3519:
3336:
3140:
3085:
The Economic Growth Engine: How Useful Work Creates Material Prosperity
3033:
2996:
2440:
should be modeled as depending directly on time-dependent energy input
2386:
should be modeled as depending directly on time-dependent energy input
1137:
The (unchanged) profit-maximizing output level will now be in stage 2.
1111:
In Stage 2, output increases at a decreasing rate, and the average and
531:
then it is a joint production function expressing the determination of
74:
69:
to attribute to changes in factor allocation (e.g. the accumulation of
3154:
KĂĽmmel, Reiner; Ayres, Robert U.; Lindenberger, Dietmar (2010-07-01).
2305:
This approach yields an energy-dependent production function given as
846:{\displaystyle Q=a_{0}X_{1}^{a_{1}}X_{2}^{a_{2}}\cdots X_{n}^{a_{n}}}
688:{\displaystyle Q=a_{1}X_{1}+a_{2}X_{2}+a_{3}X_{3}+\dotsb +a_{n}X_{n}}
28:
3511:
3362:"Putting an end to the aggregate function of production... forever?"
3124:
3025:
2988:
2829:
Daly, H (1997). "Forum on Georgescu-Roegen versus Solow/Stiglitz".
2086:
1699:
3386:
3373:
1140:
2950:
Mishra, S. K. (2007). "A Brief History of Production Functions".
995:{\displaystyle Q=\min(a_{1}X_{1},a_{2}X_{2},\dotsc ,a_{n}X_{n}).}
3448:
Maurice, S. Charles; Phillips, Owen R.; Ferguson, C. E. (1982).
3214:
Keen, Steve; Ayres, Robert U.; Standish, Russell (2019-03-01).
2224:
Labor, capital, and energy input only depend on time such that
1932:
3273:
97:, a production function is customarily assumed to specify the
2090:
115:
assumptions, the production function can be used to derive a
3399:. Macmillan Studies in Economics. New York: Macmillan Press.
3433:(Second ed.). Oxford: Philip Allan. pp. 124–137.
1789:
is a monotonically increasing function (the derivative of
123:
The inputs to the production function are commonly termed
65:
to create a framework in which to distinguish how much of
2477:
3573:
Anatomy of Cobb–Douglas Type Production Functions in 3D
3447:
3153:
2687:
2899:"A note on 2-input neoclassical production functions"
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2640:
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2532:
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2067:
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2007:
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339:
267:
247:
157:
143:
84:
3314:
Production and Productivity as Sources of Well-being
2155:
231:{\displaystyle Q=f(X_{1},X_{2},X_{3},\dotsc ,X_{n})}
3306:. Espoo, Finland: European Productivity Conference.
3277:
Long-term Estimates of U.S. Productivity and Growth
3449:
3290:Operatiivinen laskentatoimi (Operative accounting)
3287:
2675:
2646:
2621:
2562:
2518:
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2432:
2407:
2378:
2353:
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2013:
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875:
845:
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687:
563:
543:
523:
494:
465:
445:
422:
396:
325:
253:
230:
3540:(3rd ed.). Englewood Cliffs: Prentice Hall.
3274:Jorgenson, D.W.; Ho, M.S.; Samuels, J.D. (2014).
3213:
2896:
2354:{\displaystyle Q=AL^{\beta }K^{\alpha }E^{\chi }}
2126:are usually absent in production functions. When
1914:
1119:
1081:
430:since we cannot produce anything without inputs.
3584:
3300:Productivity. Theory and Measurement in Business
2780:Productivity improving technologies (historical)
1855:{\displaystyle \mathrm {d} F/\mathrm {d} y>0}
908:
3407:Mathematical Optimalization and Economic Theory
1141:Homogeneous and homothetic production functions
326:{\displaystyle X_{1},X_{2},X_{3},\dotsc ,X_{n}}
3578:Anatomy of CES Type Production Functions in 3D
3374:"Total Factor Productivity: A Short Biography"
2105:aggregate production functions. Nevertheless,
1619:{\displaystyle a_{1}+a_{2}+\dotsb +a_{n}<1}
1554:{\displaystyle a_{1}+a_{2}+\dotsb +a_{n}>1}
1212:{\displaystyle Q=f(X_{1},X_{2},\dotsc ,X_{n})}
3568:A further description of production functions
3456:(Fourth ed.). Homewood: Irwin. pp.
2937:Fundamental Methods of Mathematical Economics
2860:
2096:
23:Graph of total, average, and marginal product
3531:. Princeton, NJ: Princeton University Press.
3410:. Englewood Cliffs: Prentice-Hall. pp.
3403:
3216:"A Note on the Role of Energy in Production"
2948:On the history of production functions, see
2082:handed on from one generation to the next".
1940:
1933:Criticisms of the production function theory
3351:
1684:{\displaystyle a_{1}+a_{2}+\dotsb +a_{n}=1}
3538:Economics of the Firm: Theory and Practice
3394:
3495:
3452:Economic Analysis: Theory and Application
3385:
3160:Journal of Non-Equilibrium Thermodynamics
3079:
2880:
1911:is a homogeneous function of any degree.
574:One formulation is as a linear function:
511:
482:
73:) and how much to attribute to advancing
3535:
3526:
3110:
2972:
1127:
1085:
18:
3529:Theory of Cost and Production Functions
3474:
3428:
2856:
2854:
2852:
2828:
1102:
397:{\displaystyle X_{1}=X_{2}=...=X_{n}=0}
3585:
3371:
3283:. Tokyo: Third World KLEMS Conference.
3009:
2949:
2861:Cohen, A. J.; Harcourt, G. C. (2003).
2824:
2822:
3335:
3310:
3296:
3209:
3207:
3205:
3106:
3104:
3003:
2478:Inter-dependent factors of production
61:, aggregate production functions are
3046:
3040:
2897:Moysan and, G.; Senouci, M. (2016).
2849:
2688:The practice of production functions
2294:{\displaystyle K=K(t),L=L(t),E=E(t)}
2112:
1953:criticized the way the factor input
2819:
1007:constant elasticity of substitution
13:
3489:10.1111/j.1465-7295.1967.tb01174.x
3347:. New York: Wiley. pp. 62–74.
3329:
3202:
3101:
3013:Review of Economics and Statistics
2770:Productive and unproductive labour
1839:
1826:
1491:returns to scale. The presence of
1048:{\displaystyle a_{0},\dots ,a_{n}}
737:{\displaystyle a_{1},\dots ,a_{n}}
144:Specifying the production function
85:The theory of production functions
14:
3604:
3561:
3404:Intriligator, Michael D. (1971).
3288:Riistama, K.; Jyrkkiö E. (1971).
2903:Journal of Mathematical Economics
2156:Independent factors of production
1753:{\displaystyle F(h(X_{1},X_{2}))}
1467:. If it is homogeneous of degree
1243:, if given any positive constant
2868:Journal of Economic Perspectives
524:{\displaystyle \mathbb {R} ^{k}}
495:{\displaystyle \mathbb {R} ^{n}}
3499:Journal of Industrial Economics
3147:
3073:
2760:Production possibility frontier
111:not reflected in the function.
89:In general, economic output is
3431:Introduction to Microeconomics
3372:Hulten, C. R. (January 2000).
3360:Guerrien B. and O. Gun (2015)
3240:10.1016/j.ecolecon.2018.11.002
3113:The Review of Economic Studies
2966:
2942:
2929:
2889:
2808:
2755:Production, costs, and pricing
2670:
2664:
2622:{\displaystyle Q=f(L(E),K(E))}
2616:
2613:
2607:
2598:
2592:
2586:
2557:
2554:
2548:
2542:
2513:
2510:
2504:
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2456:
2450:
2402:
2396:
2288:
2282:
2267:
2261:
2246:
2240:
2208:
2190:
1988:
1976:
1915:Aggregate production functions
1904:{\displaystyle h(X_{1},X_{2})}
1898:
1872:
1805:
1799:
1776:
1770:
1747:
1744:
1718:
1712:
1391:
1346:
1327:
1273:
1206:
1161:
1132:Shifting a production function
1120:Shifting a production function
1082:Production function as a graph
986:
911:
261:is the quantity of output and
225:
167:
1:
3292:. Weilin + Göös. p. 335.
3069:– via United Diversity.
2939:, third edition, McGraw-Hill.
2915:10.1016/j.jmateco.2016.09.011
2843:10.1016/S0921-8009(97)00080-3
2796:
2730:Economic region of production
1921:Cambridge capital controversy
1090:Quadratic production function
3345:Quantitative Economic Theory
3059:. Cheltenham: Edward Elgar.
2801:
2790:Second Industrial Revolution
2715:Computer-aided manufacturing
889:Leontief production function
7:
3378:NBER Working Paper No. 7471
2702:
10:
3609:
3395:Heathfield, D. F. (1971).
3366:Real World Economic Review
3266:
2976:Review of Economic Studies
2882:10.1257/089533003321165010
2691:
2214:{\displaystyle Q=f(L,K,E)}
2116:
2097:On the empirical relevance
2041:a quantity of capital and
1918:
3341:"The Production Function"
3083:; Warr, Benjamin (2009).
2563:{\displaystyle K=K(E(t))}
2519:{\displaystyle L=L(E(t))}
2136:Nicholas Georgescu-Roegen
1941:On the concept of capital
1113:marginal physical product
885:total factor productivity
79:non-mainstream economists
42:theories, used to define
3477:Western Economic Journal
2935:Chiang, Alpha C. (1984)
2750:Production theory basics
2720:Distribution (economics)
2021:is a quantity of labor,
1994:{\displaystyle Q=f(L,K)}
1430:, the function exhibits
1097:production theory basics
1005:Other forms include the
134:production theory basics
3429:Laidler, David (1981).
3354:Sloan Management Review
3180:10.1515/jnetdy.2010.009
3317:. MIDO OY. p. 25.
2694:Production (economics)
2677:
2648:
2623:
2564:
2520:
2463:
2434:
2409:
2380:
2355:
2295:
2215:
2140:laws of thermodynamics
2075:
2055:
2035:
2015:
1995:
1905:
1856:
1812:
1783:
1754:
1685:
1620:
1555:
1481:
1461:
1460:{\displaystyle m<1}
1424:
1423:{\displaystyle m>1}
1398:
1257:
1237:
1213:
1133:
1091:
1069:
1049:
996:
877:
847:
738:
689:
565:
545:
525:
496:
467:
447:
424:
398:
327:
255:
232:
106:, associated with the
24:
3536:Thompson, A. (1981).
3527:Shephard, R. (1970).
3356:(Spring 1973): 13–28.
2735:Industrial Revolution
2678:
2649:
2624:
2565:
2521:
2464:
2435:
2410:
2381:
2356:
2296:
2216:
2076:
2056:
2036:
2016:
1996:
1906:
1862:)), and the function
1857:
1813:
1784:
1755:
1686:
1621:
1556:
1482:
1462:
1425:
1399:
1258:
1238:
1214:
1131:
1089:
1070:
1050:
997:
878:
876:{\displaystyle a_{0}}
848:
748:production function:
739:
690:
566:
546:
526:
497:
468:
448:
425:
399:
328:
256:
233:
125:factors of production
104:allocative efficiency
48:allocative efficiency
22:
3593:Production economics
3397:Production Functions
3220:Ecological Economics
2831:Ecological Economics
2676:{\displaystyle N(t)}
2658:
2638:
2574:
2530:
2486:
2462:{\displaystyle E(t)}
2444:
2424:
2408:{\displaystyle E(t)}
2390:
2370:
2309:
2228:
2178:
2145:Ecological Economics
2065:
2045:
2025:
2005:
1964:
1866:
1822:
1811:{\displaystyle F(y)}
1793:
1782:{\displaystyle F(y)}
1764:
1706:
1630:
1565:
1500:
1471:
1445:
1441:returns to scale if
1408:
1267:
1247:
1227:
1149:
1103:Stages of production
1059:
1013:
899:
860:
755:
702:
581:
555:
535:
506:
477:
457:
437:
408:
337:
265:
245:
155:
3232:2019EcoEc.157...40K
3172:2010JNET...35..145K
1947:Capital controversy
842:
817:
795:
423:{\displaystyle Q=0}
46:and to distinguish
33:production function
3311:Saari, S. (2011).
3297:Saari, S. (2006).
2785:Productivity model
2725:Division of labour
2673:
2644:
2619:
2560:
2516:
2459:
2430:
2405:
2376:
2351:
2291:
2211:
2071:
2051:
2031:
2011:
1991:
1901:
1852:
1808:
1779:
1750:
1681:
1626:, and constant if
1616:
1551:
1493:increasing returns
1477:
1457:
1437:, and it exhibits
1420:
1394:
1253:
1233:
1209:
1134:
1092:
1065:
1045:
992:
873:
843:
821:
796:
774:
734:
685:
561:
541:
521:
492:
463:
443:
420:
394:
323:
251:
228:
130:business processes
25:
3094:978-1-84844-182-8
3066:978-1-84720-101-0
2765:Productive forces
2647:{\displaystyle K}
2433:{\displaystyle K}
2379:{\displaystyle L}
2124:Natural resources
2113:Natural resources
2074:{\displaystyle L}
2054:{\displaystyle Q}
2034:{\displaystyle K}
2014:{\displaystyle L}
1480:{\displaystyle 1}
1256:{\displaystyle k}
1236:{\displaystyle m}
1068:{\displaystyle X}
883:is the so-called
564:{\displaystyle n}
544:{\displaystyle k}
466:{\displaystyle f}
446:{\displaystyle Q}
254:{\displaystyle Q}
3600:
3551:
3532:
3523:
3492:
3471:
3455:
3444:
3425:
3400:
3391:
3389:
3357:
3348:
3318:
3307:
3305:
3293:
3284:
3282:
3260:
3259:
3211:
3200:
3199:
3151:
3145:
3144:
3108:
3099:
3098:
3087:. Edward Elgar.
3081:Ayres, Robert U.
3077:
3071:
3070:
3058:
3044:
3038:
3037:
3007:
3001:
3000:
2970:
2964:
2963:
2946:
2940:
2933:
2927:
2926:
2893:
2887:
2886:
2884:
2858:
2847:
2846:
2826:
2817:
2812:
2682:
2680:
2679:
2674:
2653:
2651:
2650:
2645:
2628:
2626:
2625:
2620:
2569:
2567:
2566:
2561:
2525:
2523:
2522:
2517:
2468:
2466:
2465:
2460:
2439:
2437:
2436:
2431:
2414:
2412:
2411:
2406:
2385:
2383:
2382:
2377:
2360:
2358:
2357:
2352:
2350:
2349:
2340:
2339:
2330:
2329:
2300:
2298:
2297:
2292:
2220:
2218:
2217:
2212:
2080:
2078:
2077:
2072:
2060:
2058:
2057:
2052:
2040:
2038:
2037:
2032:
2020:
2018:
2017:
2012:
2000:
1998:
1997:
1992:
1910:
1908:
1907:
1902:
1897:
1896:
1884:
1883:
1861:
1859:
1858:
1853:
1842:
1837:
1829:
1817:
1815:
1814:
1809:
1788:
1786:
1785:
1780:
1759:
1757:
1756:
1751:
1743:
1742:
1730:
1729:
1690:
1688:
1687:
1682:
1674:
1673:
1655:
1654:
1642:
1641:
1625:
1623:
1622:
1617:
1609:
1608:
1590:
1589:
1577:
1576:
1561:, decreasing if
1560:
1558:
1557:
1552:
1544:
1543:
1525:
1524:
1512:
1511:
1486:
1484:
1483:
1478:
1466:
1464:
1463:
1458:
1435:returns to scale
1429:
1427:
1426:
1421:
1403:
1401:
1400:
1395:
1390:
1389:
1371:
1370:
1358:
1357:
1342:
1341:
1326:
1325:
1304:
1303:
1288:
1287:
1262:
1260:
1259:
1254:
1242:
1240:
1239:
1234:
1218:
1216:
1215:
1210:
1205:
1204:
1186:
1185:
1173:
1172:
1074:
1072:
1071:
1066:
1054:
1052:
1051:
1046:
1044:
1043:
1025:
1024:
1001:
999:
998:
993:
985:
984:
975:
974:
956:
955:
946:
945:
933:
932:
923:
922:
882:
880:
879:
874:
872:
871:
852:
850:
849:
844:
841:
840:
839:
829:
816:
815:
814:
804:
794:
793:
792:
782:
773:
772:
743:
741:
740:
735:
733:
732:
714:
713:
694:
692:
691:
686:
684:
683:
674:
673:
655:
654:
645:
644:
632:
631:
622:
621:
609:
608:
599:
598:
570:
568:
567:
562:
550:
548:
547:
542:
530:
528:
527:
522:
520:
519:
514:
501:
499:
498:
493:
491:
490:
485:
472:
470:
469:
464:
452:
450:
449:
444:
429:
427:
426:
421:
403:
401:
400:
395:
387:
386:
362:
361:
349:
348:
332:
330:
329:
324:
322:
321:
303:
302:
290:
289:
277:
276:
260:
258:
257:
252:
237:
235:
234:
229:
224:
223:
205:
204:
192:
191:
179:
178:
117:marginal product
71:physical capital
44:marginal product
3608:
3607:
3603:
3602:
3601:
3599:
3598:
3597:
3583:
3582:
3564:
3548:
3512:10.2307/2098099
3468:
3441:
3422:
3332:
3330:Further reading
3327:
3303:
3280:
3269:
3264:
3263:
3212:
3203:
3152:
3148:
3125:10.2307/2296378
3109:
3102:
3095:
3078:
3074:
3067:
3056:
3048:Daly, Herman E.
3045:
3041:
3026:10.2307/1927538
3008:
3004:
2989:10.2307/2296002
2971:
2967:
2947:
2943:
2934:
2930:
2894:
2890:
2859:
2850:
2827:
2820:
2813:
2809:
2804:
2799:
2794:
2740:Mass production
2705:
2696:
2690:
2659:
2656:
2655:
2639:
2636:
2635:
2575:
2572:
2571:
2531:
2528:
2527:
2487:
2484:
2483:
2480:
2445:
2442:
2441:
2425:
2422:
2421:
2391:
2388:
2387:
2371:
2368:
2367:
2345:
2341:
2335:
2331:
2325:
2321:
2310:
2307:
2306:
2229:
2226:
2225:
2179:
2176:
2175:
2165:Joseph Stiglitz
2158:
2132:Joseph Stiglitz
2121:
2115:
2099:
2066:
2063:
2062:
2046:
2043:
2042:
2026:
2023:
2022:
2006:
2003:
2002:
1965:
1962:
1961:
1943:
1935:
1923:
1917:
1892:
1888:
1879:
1875:
1867:
1864:
1863:
1838:
1833:
1825:
1823:
1820:
1819:
1794:
1791:
1790:
1765:
1762:
1761:
1738:
1734:
1725:
1721:
1707:
1704:
1703:
1669:
1665:
1650:
1646:
1637:
1633:
1631:
1628:
1627:
1604:
1600:
1585:
1581:
1572:
1568:
1566:
1563:
1562:
1539:
1535:
1520:
1516:
1507:
1503:
1501:
1498:
1497:
1472:
1469:
1468:
1446:
1443:
1442:
1409:
1406:
1405:
1385:
1381:
1366:
1362:
1353:
1349:
1337:
1333:
1321:
1317:
1299:
1295:
1283:
1279:
1268:
1265:
1264:
1248:
1245:
1244:
1228:
1225:
1224:
1200:
1196:
1181:
1177:
1168:
1164:
1150:
1147:
1146:
1143:
1122:
1105:
1084:
1060:
1057:
1056:
1039:
1035:
1020:
1016:
1014:
1011:
1010:
980:
976:
970:
966:
951:
947:
941:
937:
928:
924:
918:
914:
900:
897:
896:
867:
863:
861:
858:
857:
835:
831:
830:
825:
810:
806:
805:
800:
788:
784:
783:
778:
768:
764:
756:
753:
752:
728:
724:
709:
705:
703:
700:
699:
679:
675:
669:
665:
650:
646:
640:
636:
627:
623:
617:
613:
604:
600:
594:
590:
582:
579:
578:
556:
553:
552:
536:
533:
532:
515:
510:
509:
507:
504:
503:
486:
481:
480:
478:
475:
474:
458:
455:
454:
438:
435:
434:
409:
406:
405:
382:
378:
357:
353:
344:
340:
338:
335:
334:
317:
313:
298:
294:
285:
281:
272:
268:
266:
263:
262:
246:
243:
242:
219:
215:
200:
196:
187:
183:
174:
170:
156:
153:
152:
146:
87:
67:economic growth
17:
12:
11:
5:
3606:
3596:
3595:
3581:
3580:
3575:
3570:
3563:
3562:External links
3560:
3559:
3558:
3552:
3546:
3533:
3524:
3493:
3472:
3466:
3445:
3439:
3426:
3420:
3401:
3392:
3369:
3358:
3349:
3331:
3328:
3326:
3325:
3319:
3308:
3294:
3285:
3270:
3268:
3265:
3262:
3261:
3201:
3166:(2): 145–179.
3146:
3100:
3093:
3072:
3065:
3039:
3020:(1): 115–120.
3002:
2965:
2941:
2928:
2888:
2875:(1): 199–214.
2848:
2837:(3): 261–306.
2818:
2806:
2805:
2803:
2800:
2798:
2795:
2793:
2792:
2787:
2782:
2777:
2772:
2767:
2762:
2757:
2752:
2747:
2742:
2737:
2732:
2727:
2722:
2717:
2712:
2706:
2704:
2701:
2692:Main article:
2689:
2686:
2685:
2684:
2672:
2669:
2666:
2663:
2643:
2618:
2615:
2612:
2609:
2606:
2603:
2600:
2597:
2594:
2591:
2588:
2585:
2582:
2579:
2559:
2556:
2553:
2550:
2547:
2544:
2541:
2538:
2535:
2515:
2512:
2509:
2506:
2503:
2500:
2497:
2494:
2491:
2479:
2476:
2474:such as land.
2471:
2470:
2458:
2455:
2452:
2449:
2429:
2417:
2416:
2404:
2401:
2398:
2395:
2375:
2348:
2344:
2338:
2334:
2328:
2324:
2320:
2317:
2314:
2303:
2302:
2290:
2287:
2284:
2281:
2278:
2275:
2272:
2269:
2266:
2263:
2260:
2257:
2254:
2251:
2248:
2245:
2242:
2239:
2236:
2233:
2222:
2210:
2207:
2204:
2201:
2198:
2195:
2192:
2189:
2186:
2183:
2172:
2157:
2154:
2114:
2111:
2098:
2095:
2070:
2050:
2030:
2010:
1990:
1987:
1984:
1981:
1978:
1975:
1972:
1969:
1942:
1939:
1934:
1931:
1927:macroeconomics
1916:
1913:
1900:
1895:
1891:
1887:
1882:
1878:
1874:
1871:
1851:
1848:
1845:
1841:
1836:
1832:
1828:
1807:
1804:
1801:
1798:
1778:
1775:
1772:
1769:
1749:
1746:
1741:
1737:
1733:
1728:
1724:
1720:
1717:
1714:
1711:
1680:
1677:
1672:
1668:
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1661:
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1649:
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1636:
1615:
1612:
1607:
1603:
1599:
1596:
1593:
1588:
1584:
1580:
1575:
1571:
1550:
1547:
1542:
1538:
1534:
1531:
1528:
1523:
1519:
1515:
1510:
1506:
1487:, it exhibits
1476:
1456:
1453:
1450:
1419:
1416:
1413:
1393:
1388:
1384:
1380:
1377:
1374:
1369:
1365:
1361:
1356:
1352:
1348:
1345:
1340:
1336:
1332:
1329:
1324:
1320:
1316:
1313:
1310:
1307:
1302:
1298:
1294:
1291:
1286:
1282:
1278:
1275:
1272:
1252:
1232:
1219:is said to be
1208:
1203:
1199:
1195:
1192:
1189:
1184:
1180:
1176:
1171:
1167:
1163:
1160:
1157:
1154:
1142:
1139:
1121:
1118:
1104:
1101:
1083:
1080:
1064:
1042:
1038:
1034:
1031:
1028:
1023:
1019:
1003:
1002:
991:
988:
983:
979:
973:
969:
965:
962:
959:
954:
950:
944:
940:
936:
931:
927:
921:
917:
913:
910:
907:
904:
870:
866:
854:
853:
838:
834:
828:
824:
820:
813:
809:
803:
799:
791:
787:
781:
777:
771:
767:
763:
760:
731:
727:
723:
720:
717:
712:
708:
696:
695:
682:
678:
672:
668:
664:
661:
658:
653:
649:
643:
639:
635:
630:
626:
620:
616:
612:
607:
603:
597:
593:
589:
586:
560:
540:
518:
513:
489:
484:
462:
442:
419:
416:
413:
393:
390:
385:
381:
377:
374:
371:
368:
365:
360:
356:
352:
347:
343:
320:
316:
312:
309:
306:
301:
297:
293:
288:
284:
280:
275:
271:
250:
239:
238:
227:
222:
218:
214:
211:
208:
203:
199:
195:
190:
186:
182:
177:
173:
169:
166:
163:
160:
145:
142:
86:
83:
59:macroeconomics
15:
9:
6:
4:
3:
2:
3605:
3594:
3591:
3590:
3588:
3579:
3576:
3574:
3571:
3569:
3566:
3565:
3557:
3553:
3549:
3547:0-13-231423-1
3543:
3539:
3534:
3530:
3525:
3521:
3517:
3513:
3509:
3505:
3501:
3500:
3494:
3490:
3486:
3482:
3478:
3473:
3469:
3467:0-256-02614-9
3463:
3459:
3454:
3453:
3446:
3442:
3440:0-86003-131-4
3436:
3432:
3427:
3423:
3421:0-13-561753-7
3417:
3413:
3409:
3408:
3402:
3398:
3393:
3388:
3387:10.3386/w7471
3383:
3379:
3375:
3370:
3367:
3363:
3359:
3355:
3350:
3346:
3342:
3338:
3334:
3333:
3324:
3320:
3316:
3315:
3309:
3302:
3301:
3295:
3291:
3286:
3279:
3278:
3272:
3271:
3257:
3253:
3249:
3245:
3241:
3237:
3233:
3229:
3225:
3221:
3217:
3210:
3208:
3206:
3197:
3193:
3189:
3185:
3181:
3177:
3173:
3169:
3165:
3161:
3157:
3150:
3142:
3138:
3134:
3130:
3126:
3122:
3118:
3114:
3107:
3105:
3096:
3090:
3086:
3082:
3076:
3068:
3062:
3055:
3054:
3049:
3043:
3035:
3031:
3027:
3023:
3019:
3015:
3014:
3006:
2998:
2994:
2990:
2986:
2983:(2): 81–106.
2982:
2978:
2977:
2969:
2961:
2957:
2953:
2952:Working Paper
2945:
2938:
2932:
2924:
2920:
2916:
2912:
2908:
2904:
2900:
2892:
2883:
2878:
2874:
2870:
2869:
2864:
2857:
2855:
2853:
2844:
2840:
2836:
2832:
2825:
2823:
2816:
2811:
2807:
2791:
2788:
2786:
2783:
2781:
2778:
2776:
2773:
2771:
2768:
2766:
2763:
2761:
2758:
2756:
2753:
2751:
2748:
2746:
2743:
2741:
2738:
2736:
2733:
2731:
2728:
2726:
2723:
2721:
2718:
2716:
2713:
2711:
2710:Assembly line
2708:
2707:
2700:
2695:
2667:
2661:
2641:
2632:
2631:
2630:
2610:
2604:
2601:
2595:
2589:
2583:
2580:
2577:
2551:
2545:
2539:
2536:
2533:
2507:
2501:
2495:
2492:
2489:
2475:
2453:
2447:
2427:
2419:
2418:
2399:
2393:
2373:
2364:
2363:
2362:
2346:
2342:
2336:
2332:
2326:
2322:
2318:
2315:
2312:
2285:
2279:
2276:
2273:
2270:
2264:
2258:
2255:
2252:
2249:
2243:
2237:
2234:
2231:
2223:
2205:
2202:
2199:
2196:
2193:
2187:
2184:
2181:
2173:
2170:
2169:
2168:
2166:
2162:
2153:
2149:
2147:
2146:
2141:
2137:
2133:
2129:
2125:
2120:
2110:
2108:
2104:
2094:
2092:
2088:
2083:
2068:
2048:
2028:
2008:
1985:
1982:
1979:
1973:
1970:
1967:
1958:
1956:
1952:
1951:Joan Robinson
1948:
1938:
1930:
1928:
1922:
1912:
1893:
1889:
1885:
1880:
1876:
1869:
1849:
1846:
1843:
1834:
1830:
1818:is positive (
1802:
1796:
1773:
1767:
1739:
1735:
1731:
1726:
1722:
1715:
1709:
1701:
1696:
1692:
1678:
1675:
1670:
1666:
1662:
1659:
1656:
1651:
1647:
1643:
1638:
1634:
1613:
1610:
1605:
1601:
1597:
1594:
1591:
1586:
1582:
1578:
1573:
1569:
1548:
1545:
1540:
1536:
1532:
1529:
1526:
1521:
1517:
1513:
1508:
1504:
1494:
1490:
1474:
1454:
1451:
1448:
1440:
1436:
1433:
1417:
1414:
1411:
1386:
1382:
1378:
1375:
1372:
1367:
1363:
1359:
1354:
1350:
1343:
1338:
1334:
1330:
1322:
1318:
1314:
1311:
1308:
1305:
1300:
1296:
1292:
1289:
1284:
1280:
1276:
1270:
1250:
1230:
1222:
1201:
1197:
1193:
1190:
1187:
1182:
1178:
1174:
1169:
1165:
1158:
1155:
1152:
1138:
1130:
1126:
1117:
1114:
1109:
1100:
1098:
1088:
1079:
1076:
1062:
1040:
1036:
1032:
1029:
1026:
1021:
1017:
1008:
989:
981:
977:
971:
967:
963:
960:
957:
952:
948:
942:
938:
934:
929:
925:
919:
915:
905:
902:
895:
894:
893:
890:
886:
868:
864:
836:
832:
826:
822:
818:
811:
807:
801:
797:
789:
785:
779:
775:
769:
765:
761:
758:
751:
750:
749:
747:
729:
725:
721:
718:
715:
710:
706:
680:
676:
670:
666:
662:
659:
656:
651:
647:
641:
637:
633:
628:
624:
618:
614:
610:
605:
601:
595:
591:
587:
584:
577:
576:
575:
572:
558:
538:
516:
487:
460:
440:
431:
417:
414:
411:
391:
388:
383:
379:
375:
372:
369:
366:
363:
358:
354:
350:
345:
341:
318:
314:
310:
307:
304:
299:
295:
291:
286:
282:
278:
273:
269:
248:
220:
216:
212:
209:
206:
201:
197:
193:
188:
184:
180:
175:
171:
164:
161:
158:
151:
150:
149:
141:
137:
135:
131:
126:
121:
118:
112:
109:
105:
100:
96:
92:
82:
80:
76:
72:
68:
64:
60:
55:
51:
49:
45:
41:
38:
34:
30:
21:
3537:
3528:
3506:(1): 55–72.
3503:
3497:
3483:(1): 39–51.
3480:
3476:
3451:
3430:
3406:
3396:
3377:
3365:
3353:
3344:
3313:
3299:
3289:
3276:
3223:
3219:
3163:
3159:
3149:
3116:
3112:
3084:
3075:
3052:
3042:
3017:
3011:
3005:
2980:
2974:
2968:
2951:
2944:
2936:
2931:
2906:
2902:
2891:
2872:
2866:
2834:
2830:
2810:
2775:Productivity
2697:
2481:
2472:
2304:
2161:Robert Solow
2159:
2150:
2143:
2128:Robert Solow
2122:
2107:Anwar Shaikh
2103:well behaved
2102:
2100:
2084:
1959:
1944:
1936:
1924:
1697:
1693:
1488:
1144:
1135:
1123:
1110:
1106:
1093:
1077:
1004:
855:
746:Cobb–Douglas
697:
573:
432:
240:
147:
138:
122:
113:
107:
98:
90:
88:
56:
52:
40:neoclassical
32:
26:
3337:Brems, Hans
3119:: 139–152.
1221:homogeneous
404:it must be
2797:References
2745:Production
2117:See also:
1919:See also:
1439:decreasing
1432:increasing
1223:of degree
473:maps from
75:technology
37:mainstream
3256:158863011
3248:0921-8009
3226:: 40–46.
3188:1437-4358
3133:0034-6527
2909:: 80–86.
2802:Citations
2347:χ
2337:α
2327:β
1660:⋯
1595:⋯
1530:⋯
1376:…
1309:…
1191:…
1030:…
961:…
819:⋯
719:…
660:⋯
308:…
210:…
63:estimated
29:economics
3587:Category
3339:(1968).
3196:73538957
2703:See also
2087:interest
1700:isoquant
1489:constant
571:inputs.
108:economic
95:function
77:. Some
3520:2098099
3458:169–222
3412:178–189
3267:Sources
3228:Bibcode
3168:Bibcode
3141:2296378
3034:1927538
2997:2296002
2960:1020577
2923:3581910
1955:capital
99:maximum
3544:
3518:
3464:
3437:
3418:
3254:
3246:
3194:
3186:
3139:
3131:
3091:
3063:
3032:
2995:
2958:
2921:
2001:where
1760:where
887:. The
856:where
698:where
241:where
3516:JSTOR
3304:(PDF)
3281:(PDF)
3252:S2CID
3192:S2CID
3137:JSTOR
3057:(PDF)
3030:JSTOR
2993:JSTOR
2919:S2CID
2091:wages
1404:. If
3542:ISBN
3462:ISBN
3435:ISBN
3416:ISBN
3368:N°73
3244:ISSN
3184:ISSN
3129:ISSN
3089:ISBN
3061:ISBN
2956:SSRN
2895:see
2163:and
2130:and
2089:and
1847:>
1611:<
1546:>
1452:<
1415:>
31:, a
3508:doi
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3236:doi
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3176:doi
3121:doi
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2911:doi
2877:doi
2839:doi
1925:In
909:min
502:to
433:If
136:).
91:not
57:In
27:In
3589::
3514:.
3504:24
3502:.
3479:.
3460:.
3414:.
3380:.
3376:.
3364:,
3343:.
3250:.
3242:.
3234:.
3222:.
3218:.
3204:^
3190:.
3182:.
3174:.
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3162:.
3158:.
3135:.
3127:.
3117:41
3115:.
3103:^
3028:.
3018:56
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2917:.
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2905:.
2901:.
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2821:^
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1691:.
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3510::
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3487::
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3123::
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