Economics
Production Function
A production function in economics represents the relationship between inputs and outputs in the production process. It shows how much output can be produced from a given combination of inputs, such as labor and capital. The function is a key concept in understanding the efficiency and productivity of production processes.
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12 Key excerpts on "Production Function"
eBook - PDF- Sharma, Arti(Authors)
- 2021(Publication Date)
- Scholars World(Publisher)
Chapter 2 Theory of Production Function and Economic Efficiency A Production Function relates to physical output of a production process to physical inputs or factors of production. The Production Function is one of the key concepts of neoclassical theories, used to define marginal product and to distinguish allocative efficiency, the defining focus of economics. The primary purpose of the Production Function is to address technical efficiency, resource use efficiency and 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, aggregate Production Functions are estimated to create a framework in which to distinguish how much of economic growth to attribute to changes in factor allocation and how much to attribute to advancing technology. 2.1 Concept of Production Functions The theory of production is mainly concerned with maximizing profit or in some instances minimizing cost. Both are indicative of economic efficiency. 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 generated from output into an income due to each input factor of production, equal to the marginal product of each input. To satisfy the mathematical definition of a function, a Production Function is customarily assumed to specify the maximum output This ebook is exclusively for this university only. Cannot be resold/distributed. 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.
eBook - PDF- Kumar, K Nirmal Ravi(Authors)
- 2021(Publication Date)
- Daya Publishing House(Publisher)
From the above discussion, it is clear that, a Production Function is an expression of quantitative (physical) relation between change in inputs and the resulting change in output. So, it is expressed as Y = f(X 1 , X 2 , X 3 , —, Xn). In microeconomic analysis, conventionally, we study the following two aspects of relation between inputs and output. ‘In what manner the output changes, if only one of the inputs is varied and other inputs are kept constant ’. This falls under Short run Production Function explained by LDR. ‘In what manner the output changes, if all the inputs all varied simultaneously and in the same proportion’ . This This ebook is exclusively for this university only. Cannot be resold/distributed. falls under long run Production Function and is explained by the concept of RTS. Technical unit: It refers to a convenient unit of studying about production programme. It also refers to a single convenient unit in production for which output, costs and returns will be worked. e.g .: Cost of cultivation of paddy per hectare. Economic unit : It is also called as ‘Plant’. It refers to aggregation of all the individual units for which output, costs, returns will be worked as whole. e.g .: Cost of cultivation of paddy in a college farm say, in 17.55 hectares. Production period: It is also called as Transformation period. It is defined as the period taken by the resources and resource services to be completely transformed into final product in a production programme. For example, production period of paddy is 120 days. Farm productivity: Farm productivity implies output per unit of land. For example, productivity of paddy is 2.5 tonnes/ha. Choice indicator: Choice indicator is an yardstick or criterion indicating, which of the available alternatives will optimize or maximize the given objective or given end. That is, the farmer has to opt for one of the available alternatives in a given production programme, so as to minimize the cost or maximize the profits.
- Humberto Barreto(Author)
- 2009(Publication Date)
- Cambridge University Press(Publisher)
2.1 Production Function 2.1.1 Production Function Let us choose that function P = bL k C k − 1 and find such numerical values of b and k that P will “best” approximate P [product or output] in the sense of the Theory of Least Squares. Then relative to the indices and the period we have the norm P = 1 . 01 L 3 / 4 C 1 / 4 . Charles W. Cobb and Paul H. Douglas The Production Function is the backbone of the Theory of the Firm. It describes the current state of technology and how input can be transformed into output. The Production Function can be displayed in a variety of ways, including product curves and isoquants. In every optimization problem faced by the firm, the Production Function is included. Key Definitions and Assumptions Inputs , or factors of production , are used to make output , or product . As shown in Figure 2.1.1.1 , the firm is a highly abstract entity – a black box – that simply transforms inputs into output. Inputs are often broken down into large categories: land, labor, raw mate-rials, and capital. Capital can be confusing. Capital, K , as a factor of produc-tion means physical capital goods such as machinery, tools, or equipment. That is different from financial or venture capital that is a fund of money. Like labor, capital is rented. The firm does not own any of its machines. This is extremely unrealistic but allows us to avoid complicated issues in-volving depreciation, financing of machinery purchases (debt versus equity, for example), and so on. Like the consumer, the firm exists only for a nanosecond. It makes deci-sions about how much to produce to maximize profits with no time horizon. It produces the output in an instant. Another simplifying assumption is that the firm produces only one prod-uct. That makes revenues simply price times quantity sold. 267 268 Production Function Inputs Outpu t Figure 2.1.1.1.
eBook - PDFMicroeconomics
Equilibrium and Efficiency
- Thijs ten Raa(Author)
- 2017(Publication Date)
- Red Globe Press(Publisher)
R. Solow, “Technical Change and the Aggregate Production Function,” Review of Economics and Statistics 39, 3, 312–320 (1957). 121 7 Production Functions 7.1 Introduction A Production Function helps to determine how much output can be produced with given inputs. For instance, if you have one million workers, 100,000 tools, and 100 square miles of arable land, then you can grow a certain quantity of rice. This is not to say that these inputs are actually available. The popula-tion figure may be too low, or the workers may not be willing to supply their labor services in full if wages are too low (or too high – so that they can earn enough working part time). Thus the Production Function is an important tool that economists use to understand the supply of goods and services in an economy and, ultimately, the standard of living. Some economies have strong Production Functions that yield a lot of output per input, while others have weaker produc-tion functions. It is not always obvious why such differences exist. After all, if a third-world country had the same inputs as a developed country – land, fertiliz-ers, knowledge, and skills – then it should be able to produce the same quantity of output. Many economists argue that this is the case, since Production Functions are the same across countries; but we disagree. Why do Production Functions vary? First it is important to remember that, as a rule, models are incomplete. There are always unaccounted inputs, such as cli-mate conditions – for example, in an arid country fewer crops (output) will be harvested with the same labor, capital, and produced inputs as in a temperate country. Second, and more subtly, some countries are more successful than oth-ers in extracting the maximum possible output from given inputs. For instance, their labor may be more used to discipline, their products more reliable (this is especially important when they are used as inputs), and so on.
eBook - PDFEnergy and Economic Myths
Institutional and Analytical Economic Essays
- Nicholas Georgescu-Roegen(Author)
- 2014(Publication Date)
- Pergamon(Publisher)
Indeed, search as one may, even in the works of the most respected modern authorities one finds hardly any discussion of the empirical scaffold of the Production Function. All seem satisfied with some purely formal definition, some variation on Wicksteed's theme. For a representative sample, we may cite, first, Boulding's defini-tion : the basic transformation function of an enterprise is its production junction, which shows what quantities of inputs (factors) can be transformed into what quantities of output (product) ; 2 1 Wicksteed, op, cit. p. 4. This cavalier treatment is characteristic of most text-books nowadays. Cf. A. W. Stonier and D. C. Hague, A Textbook of Economics (2nd edn., London, 1958), p. 119, or Richard H. Leftwich, The Price System and Resource Allocation (rev. edn., New York, 1960), p. 108. 2 Kenneth E. Boulding, Economic Analysis (3rd edn., New York, 1955), p. 585. This definition represents the most popular variant. Cf. A. L. Bowley, The 74 Energy and Economic Myths second, Stigler's : A Production Function may be defined as the relationship between inputs of productive services per unit of time and outputs of products per unit of time. 1 and, finally, Carlson's, according to which the Production Function is the relationship between the quantity of input bought at the beginning of the production period and the quantity of output turned out by the process at the period's end. 2 We find no attempt to supplement such formal definitions with some operational instructions about how to determine a Production Function at least under hypothetically ideal conditions. Boulding likens a production process to a recipe from a cook-book. Just as one such recipe says that c if we put 4 eggs, 2 cups of milk .
eBook - PDF- Dominic J. Brewer, Patrick J. McEwan(Authors)
- 2010(Publication Date)
- Elsevier(Publisher)
III. THE PRODUCTION, COSTS AND FINANCING OF EDUCATION This page intentionally left blank Education Production Functions: Concepts D N Harris, University of Wisconsin–Madison, Madison, WI, USA ã 2010 Elsevier Ltd. All rights reserved. Glossary Education Production Function – This constitutes all combinations of inputs that produce a given set of outputs. In education, outputs are typically measured by achievement scores and graduation rates. Inputs include both school and family factors that influence outcomes. Value-added model – A way of estimating the impact of school programs in inputs that uses data on individual students inked over time so that each individual’s achievement growth can be calculated. Value-added models can be used to estimate effects of programs and inputs (e.g., teacher experience) as well as the effects of individual teachers and schools. The concept of value-added is rooted in economics and refers to the difference in the economic value of outputs and inputs. Introduction The Production Function is one of the core concepts of microeconomics and its application to education has been one of the central research activities of quantitative education researchers, especially economists of educa-tion, for many decades. The education production func-tion (EPF) is implicitly part of any research that attempts to establish a statistical relationship between education resources (e.g., class size) and measures of student out-comes (e.g., scores on standardized achievement tests). Results from these studies have had a significant impact on policy debates about how, and how much, public funding should be provided for education. The quasi-experimental research methods associated with EPFs have also become an increasingly common approach to education program evaluation, influencing a wide range of policies beyond education funding.
eBook - PDFMicroeconomic Theory
Basic Principles and Extensions
- Walter Nicholson, Christopher Snyder(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
Mas-Collell, A., M. D. Whinston, and J. R. Green. Microeconomic Theory . New York: Oxford University Press, 1995. Chapter 5 provides a sophisticated, if somewhat spare, review of production theory. The use of the profit function (see Chapter 11) is sophisticated and illuminating. Shephard, R. W. Theory of Cost and Production Functions . Princeton, NJ: Princeton University Press, 1978. Extended analysis of the dual relationship between production and cost functions. Silberberg, E., and W. Suen. The Structure of Economics: A Mathematical Analysis , 3rd ed. Boston: Irwin/McGraw-Hill, 2001. Thorough analysis of the duality between Production Functions and cost curves. Provides a proof that the elasticity of substitution can be derived as shown in footnote 6 of this chapter. Stigler, G. J. “The Division of Labor Is Limited by the Extent of the Market.” Journal of Political Economy 59 (June 1951): 185–93. Careful tracing of the evolution of Smith’s ideas about economies of scale. Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300 322 EXTENSIONS MANY-INPUT Production FunctionS Most of the Production Functions illustrated in Chapter 9 can be easily generalized to many-input cases. Here we show this for the Cobb–Douglas and CES cases and then examine two flexible forms that such Production Functions might take. In all these examples, the α’s are non-negative parameters and the n inputs are represented by x 1 , c , x n . E9.1 Cobb–Douglas The many-input Cobb–Douglas Production Function is given by q 5 q n i 5 1 x α i i . (i) a. This function exhibits constant returns to scale if a n i 5 1 α i 5 1 . (ii) b. In the constant-returns-to-scale Cobb–Douglas func-tion, α i is the elasticity of q with respect to input x i . Because 0 # α i , 1, each input exhibits diminishing marginal productivity.
eBook - PDFManagerial Economics
Problem-Solving in a Digital World
- Nick Wilkinson(Author)
- 2022(Publication Date)
- Cambridge University Press(Publisher)
We can now examine these effects as they relate to the various forms of Production Function described in (6.2) to (6.7). First we need to explain more precisely the economic interpretation of marginal effects in the context of production theory. A marginal effect is given mathematically by a derivative, or, more precisely, in the case of the two-input Production Function, a partial derivative (obtained by differentiating the Production Function with respect to one variable, while keeping other variables constant). The economic interpretation of this is a marginal product. The marginal product of labour is the additional output resulting from using one more unit of labour, while holding the capital input constant. Likewise, the marginal product of capital is the additional output resulting from using one more unit of capital, while holding the labour input constant. These marginal products can thus be expressed mathematically in terms of the following partial derivatives: Table 6.1 Input–output table for cubic function – Viking Shoes Capital input (machines) K 1 2 3 4 5 6 7 8 Labour input (workers) L 1 4 9 13 18 23 27 31 35 2 8 17 27 36 45 54 62 70 3 12 26 39 53 67 80 B 92 102 4 16 33 50 68 85 102 117 131 5 18 38 59 80 C 100 A 119 137 153 6 20 42 64 87 110 131 150 167 7 20 43 66 90 113 135 155 171 8 19 41 64 87 110 131 149 164 282 6 Production Theory MP L ¼ ∂Q=∂L and MP K ¼ ∂Q=∂K: Expressions for marginal product can now be derived for each of the mathematical forms (6.2) to (6.7). These are shown in Table 6.2, in terms of the marginal product of labour. The marginal product of capital will have the same general form because of the symmetry of the functions. The linear Production Function has constant marginal product, meaning that the marginal product is not affected by the level of either the labour or the capital input. This is not normally a realistic situation, and such functions, in spite of their simplicity, are not frequently used.
eBook - PDFMacro-Econophysics
New Studies on Economic Networks and Synchronization
- Hideaki Aoyama, Yoshi Fujiwara, Yuichi Ikeda, Hiroshi Iyetomi, Wataru Souma, Hiroshi Yoshikawa(Authors)
- 2017(Publication Date)
- Cambridge University Press(Publisher)
4.1 (d) is the CCDF ¯ F (c). In these figures, we fit the data by GB2 defined by Eq. 3.47. A selected example of the distribution of labor productivity for non-manufacturing sectors is shown in Fig. 4.2 The context of Fig. 4.2 is similar to that of Fig. 4.1. The distribution of labor productivity follows the GB2. However, there is another peak at the low side of productivity. This peak is observed for the entire non-manufacturing sector (Ikeda et al., 2010). (a) (b) –1 –2 –3 –4 –5 (c) –1 –2 –3 –4 –5 (d) Fig. 4.2 The distributions of labor productivity for the service sector in 2006 are shown as an example of distributions for the non-manufacturing sector. The solid curve represents the results of fitting by GB2 (from Ikeda et al. (2010)). 4.1.2 Production Function The Production Function is one of basic ingredients in microeconomic theory (Wicksteed, 1894; Varian, 1992). It specifies output (value added) Y of a firm for given input factors such as labor L and capital K: Y = Φ(L, K), (4.4) 100 Macro Econophysics where it is assumed that each firm produces goods optimally using its own Production Function. Figure 4.3 shows scatter plots for all pairs of the three financial quantities, i.e., Y, L, K, in the manufacturing sector in 2006. We see that these variables are mutually correlated. Spearman's rank correlation coefficient ρ S for the three pairs, K-Y , L-Y , and L-K, are 0.86, 0.95, and 0.83, respectively. Thus the L-Y pair has stronger correlation than the remaining two pairs K-Y and L-K have. Moreover the remaining ones have correlations of similar strength. The detailed correlation structure for each pair is described in terms of copulas later, with the implications of these results for the overall correlations.
- Berkeley Hill(Author)
- 2013(Publication Date)
- Pergamon(Publisher)
CHAPTER 5 Production Economics (Theory of the Firm) Note on Chapter 5: this chapter contains essential material for students whose prime interest is business management. However, those with major interests in other areas may prefer to pass on first reading to the last section on Time and Scale of Production. The omission of the first part of this chapter will not undermine their understanding of subsequent chapters. The Scope of Production Economics Production Economics studies how one sector of the economic system, firms, allocate their resources in the pursuit of given objectives. Parallels exist between this area of economic theory and the Theory of Consumer Choice already encountered in Chapter 2. When the Theory of Consumer Choice attempted to explain how individual consumers allocated their purchasing power, it was assumed that the goal of any consumer was to maximise his satisfaction. Similarly, in Production Economics the assumption is made that the goal of any firm is the maximisation of profit. However, the two areas of theory differ in that profit, unlike personal satisfaction, is fairly easily measured. (Note that in this chapter the term profit is used rather loosely and in an accountancy sense to refer to the difference between revenues and whatever costs are being considered at the time.) Production Economics recognises that in practice firms do have other objectives (or goals) besides profit. These were discussed in relation to the Theory of Supply (Chapter 3) and included the avoidance of risk, the prestige of the business and the personal preferences of the entre-preneur. For the sake of simplicity, however, Production Economics 120 Production Economics 121 assumes that the sole objective of production is the maximising of profits.
eBook - PDF- David Besanko, Ronald Braeutigam(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
15 The fixed-proportions Production Function is also called the Leontief Production Function, after the econo- mist Wassily Leontief, who used it to model relationships between sectors in a national economy. CHAPTER 6 INPUTS AND Production FunctionS 244 When inputs are combined in fixed proportions, the elasticity of substitution is zero (i.e., σ = 0), because the marginal rate of technical substitution along the isoquant of a fixed-proportions Production Function changes from ∞ to 0 when we pass through the corner of an isoquant (e.g., point A, B, or C). A firm facing a fixed-proportions pro- duction function has no flexibility in its ability to substitute among inputs. We can see this in Figure 6.15: To produce a single molecule of water, there is only one sensible input combination—two atoms of hydrogen and one atom of oxygen. We often observe production processes with fixed proportions. The production of certain chemicals requires the combination of other chemicals, and sometimes heat, in fixed proportions. Every bicycle must always have two tires and one frame. An auto- mobile requires one engine, one chassis, and four tires, and these inputs cannot be substituted for one another. Cobb–Douglas Production Function Figure 6.16 illustrates isoquants for the Cobb–Douglas Production Function, which is intermediate between a linear Production Function and a fixed-proportions produc- tion function. The Cobb–Douglas Production Function is given by the formula Q = AL α K β , where A, α, and β are positive constants (in Figure 6.16, their values are 100, 0.4, and 0.6, respectively). With the Cobb–Douglas Production Function, capital and labor can be substituted for each other. Unlike a fixed-proportions production func- tion, capital and labor can be used in variable proportions. Unlike a linear Production Function, though, the rate at which labor can be substituted for capital is not constant as you move along an isoquant.
eBook - PDFIntermediate Microeconomics
An Intuitive Approach with Calculus
- Thomas Nechyba(Author)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
To provide an illustration of these methods in a setting where calculations are more manageable, we will now conclude the chapter by offering an example in which we do not have to worry about corner solutions or multiple potential solutions, because we assume from the outset a producer choice set that is strictly convex. Suppose the producer choice set and the production frontier are defined by the Production Function: f ( ℓ ) 5 Aℓ a . (11.25) What has to be true about a in order for this Production Function to exhibit diminishing marginal product of labour? Exercise 11B.11 Suppose 0 , a , 1 and A . 0. Holding price fixed, is the labour demand function downward sloping in the wage? Holding wage fixed, is it upward or downward sloping in price? Can you graphically illustrate why your answers hold? Exercise 11B.12 Setting up the profit-maximization problem, our first way of calculating input demands and output supplies, we get: max x , / p 5 px 2 w / subject to x 5 A / a , (11.26) or, with the constraint placed into the objective function: max / p 5 pA / a 2 w / . (11.27) Taking the first derivative with respect to ℓ , we get the first-order condition: a Apℓ a 2 1 2 w 5 0, (11.28) and solving this for ℓ , we find the labour demand function: / ( p , w ) 5 S w a Ap D 1 y ( a 2 1) . (11.29) Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. THE MATHEMATICS OF THE SHORT-RUN MODEL 277 The output supply function is the Production Function evaluated at ℓ ( p, w ): x ( p , w ) 5 f ( / ( p , w )) 5 A S w a Ap D a y ( a 2 1) .
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