Definition of Increasing Returns in Economics

Cost Function Estimates

K. Carey , in Encyclopedia of Health Economics, 2014

Economies of Scale

Economies of scale refer to the notion that average cost falls as the firm expands. Conversely, diseconomies of scale occur when expansion incurs increasing average costs. From a technical standpoint, a measure of economies of scale is equivalent to the ratio of marginal to average costs. This is because if cost at the margin is lower than average cost, then average cost will fall with increased output.

In the multiproduct context, there are two distinct economies of scale concepts. Product specific economies of scale characterize the cost effects of expanding each output separately while holding production levels of other outputs constant. The alternative adaptation is ray scale economies, which assumes a proportional increase in cost resulting from a simultaneous proportional increase in all outputs. Either construct may be appropriate; the choice depends on the context involved in the specific analysis.

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The Ideal Neoclassical Market and General Equilibrium*

Wolfram Elsner , ... Henning Schwardt , in The Microeconomics of Complex Economies, 2015

Returns to scale is a term that refers to the proportionality of changes in output after the amounts of all inputs in production have been changed by the same factor. Technology exhibits increasing, decreasing, or constant returns to scale. Constant returns to scale prevail, i.e., by doubling all inputs we get twice as much output; formally, a function that is homogeneous of degree one, or, F(cx)=cF(x) for all $c\ge 0$ . If we multiply all inputs by two but get more than twice the output, our production function exhibits increasing returns to scale. Formally, we use a function with a degree of homogeneity greater than one to depict this, F(cx)>cF(x) for c>1. Vice versa, decreasing returns to scale are defined by F(cx)<cF(x) for c>1. Increasing returns to scale might prevail if a technology becomes feasible only if a certain minimum level of output is produced. On the other hand, limited availability of scarce resources (natural resources or managerial talent) might be limiting firm size in which case decreasing returns to scale are more likely. Also, it is possible that a technology exhibits increasing returns at low levels of production and decreasing returns at high levels.

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The economic role of the state in education

D.N. Plank , T.E. Davis , in The Economics of Education (Second Edition), 2020

Economies of scale

Economies of scale arise when producers' average total cost falls as output increases (Mankiw, 1998). In education, this suggests that larger schools and districts may face a lower per-pupil cost. For example, larger schools have a greater ability to provide science laboratories and libraries by spreading the cost over more tax-paying households. There are also potential scale economies in information gathering, organization, and in the development of a curriculum (Belfield, 2000). To the extent that there are economies of scale in the delivery of education services, market forces may result in a monopoly, as smaller schools are driven out of business by established state schools. Faced with this tendency for the market to drive out small schools, the state has two options: either run large schools as state monopolies, or actively encourage competition by leveling the playing field for smaller schools.

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Spatial Externalities

Dean M. Hanink , in Encyclopedia of Social Measurement, 2005

Positive Externalities and Increasing Returns

Economies of scale are a decrease in the average cost of production with an increase in output. That is, increasing size leads to increasing efficiency, at least to a point. At an extreme, if economies of scale did not exist, then everyone would produce everything for her- or himself. Without economies of scale, there would be no need for firms. In a spatial sense, without externalities, there would be no need for cities; the landscape could consist of uniformly distributed households instead of the various levels of concentrated settlements that actually occur.

Economies of scale are often called increasing returns. Conventional economic analysis often characterizes production in a constant returns model that indicates that there are no economies of scale (and in relation, perfect competition). The Cobb-Douglas production function is a constant returns model that takes the following form:

(1) $Q=\left(K^{\text{β}},L^{\text{α}}\right),\text{α}+\text{β}=1,$

where Q is output and K and L are capital and labor inputs, respectively. Increasing returns can be similarly modeled:

(2) $Q=\left(K^{\text{β}},L^{\text{α}}\right),\text{α}+\text{β}>1,$

so that output increases in observed, internal factors of production.

Externalities that contribute to output are often associated with concentrations of people and producers. For analytical purposes, whether the externalities are spatial or geographical is largely a matter of scale of interest (although that is not the case with respect to policy). Given that externality effects seem to vary with concentration, however, most of them that arise without policy inducement can be considered spatial in the sense that they will spill over from place of origin to neighboring places.

Concentrations of producers may generate so-called localization externalities that result, for example, from the ability to share a trained labor force. Costs of training are externalized, either to competitors or perhaps to public educational institutions. Other externality benefits arise from the availability of specialized suppliers (input–output relationships) that develop in industrial concentrations. Adam Smith noted that the division of labor "is limited by the extent of the market." The division of labor is efficiency-inducing specialization, which increases with the size of the localized market.

Other positive localization externalities arise from formal and informal producer networks. Formal networks with political goals are often strengthened when they represent geographical concentrations rather than dispersed ones, especially when it comes to guiding the development and implementation of local industrial regulations and policies. Other local networks facilitate knowledge spillovers so that production technology and managerial expertise are effectively reduced in cost for producers found within concentrations.

Agglomeration externalities are associated simply with concentrations of people as opposed to a specific set of producers. High levels of market potential, concentrations of general infrastructure, and a dynamic, knowledge-generating milieu have all been cited as sources of agglomeration-based efficiencies.

Linear regression analysis is often used to detect the externality effects in concentrations or agglomerations. For example, a cross-sectional production function could be estimated that takes the general operational form

(3) $\ln \left(Q_j\right)=\text{α}+\text{β}\ln \left(K_j\right)+\text{γ}\ln \left(L_j\right)+\text{δ}\ln \left(E_j\right)+\text{ɛ},$

where Q is output at the jth place, K and L are capital and labor quantities, and E is a measure of externality-inducing characteristics.

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The theory of production decisions

Leonardo Becchetti , ... Stefano Zamagni , in The Microeconomics of Wellbeing and Sustainability, 2020

4.5 Economies of scale

A distinctive characteristic of production in modern economic systems is the use of processes that are only implemented at high output volumes; their salient characteristic is that they are more efficient than small scale operations. The expression economies of scale designates all the factors that make unit costs at high production levels less than unit costs at lower output levels. What brings about economies of scale? Two distinct types of economies of scale are usually noted: real and financial.

For example, financial economies benefit companies that are able to pay lower prices for the inputs it uses, due to the fact that increasing the scale of production also increases the quantities demanded for inputs. Examples of financial economies of scale are lower prices for raw materials, lower costs for outside financing, lower costs to transport and distribute the output, and so on. As we can see, these are cost savings outside the company, because they have to do with bargaining power in the input markets.

Real economies of scale are associated with reductions in input quantities used when a given company increases output levels. Consider the following examples.

•

Starting any production process always requires a certain monetary outlay; the higher the production level, the less that outlay impacts each unit of output. In other words, this is a typical fixed cost effect. The initial fixed costs include the cost of installing machinery, fine tuning the equipment, product line costs, and so on.

•

Another example of real economies of scale is economies of reserve capacity. A company always maintains a certain reserve capacity so output is not interrupted in the event of a machine failure. A small company that uses, say, only one machine will need to maintain another to ensure against the risk of interruptions; a company that uses four machines certainly does not need eight machines to cover itself; six or so should be sufficient. The same applies to the repair workforce. The situation of economies of reserves is similar. Raw materials reserves increase when output increases, but in general the amount of reserves – which cost the company to maintain – vary less than proportionally with respect to output.

•

Increasing returns to scale , which we have already discussed, are ultimately another important cause of real economies of scale.

A final remark. Economies of scale should not be confused with economies of scope. When a company is able to produce more than one good – that is, it can implement a multi-product production – it can happen that by producing several products together the overall costs can be lower at the same output levels than the sum of the production costs for each output considered by itself.

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Biodiesel Fuels

Leon G. Schumacher , ... Brian Adams , in Encyclopedia of Energy, 2004

8.2 Cost of Biodiesel

Economies of scale are a factor when buying biodiesel. Since biodiesel is not yet available on the pipeline network, most deliveries are made by truck or rail. Transportation costs can be a significant portion of the product cost.

The most significant contributor to the cost of the fuel is the cost of the oil itself. As noted in the feedstock section, the price of the oil has varied from as little as 3 to 4 cents/pound to a high of 25.8 cents/pound. If soybeans were used as the feedstock, the cost could range from $1.73 to $3.10/gallon. If a less expensive feedstock were used, the price range would even be greater anywhere from $0.50 to $3.10/gallon. (See Fig. 6.)

Figure 6. Production cost per gallon. Reprinted the from National Renewable Energy Laboratory, Kansas Cooperative Development Center/KSU.

The next highest cost when producing biodiesel is the cost to convert the feedstock from a pure oil to a transesterified fuel. Some researchers report that this amounts to approximately 30% of the total cost (as nearly 70% of the cost is tied up in the raw materials [soybean oil feedstock, methanol, catalyst]).

The conversion costs for biodiesel can range from $0.30 to $0.60/gallon. One study reported that it cost $0.58/gallon for transesterification and $0.33/gallon for overhead. A credit of $0.39/gallon was applied for the resale of the glycerol, bringing the total for transesterification to $0.52/gallon.

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Mergers and Alliances in the Biopharmaceuticals Industry

H. Grabowski , M. Kyle , in Encyclopedia of Health Economics, 2014

Access to New Technologies and Therapeutic Areas

Beyond economies of scale, biopharmaceutical firms may engage in mergers to gain a presence in an emerging therapeutic category that represents significant future growth opportunities. For example, the oncology class has been characterized by several new 'first-in-class' drugs in recent years (DiMasi and Grabowski, 2007). The oncology class is now the fastest growing therapeutic category among all major drug classes. The novel entities in this class have emerged primarily from the biotech sector, utilizing molecular biology techniques (e.g., new monoclonal antibody products and other targeted agents). Mergers provide a more expeditious way to enter such high opportunity fields relative to internal expansion. It can take several years or even decades to build the internal scientific capability to enter a new therapeutic area or implement a new research platform in an emerging scientific field. This appears to be an important motivation underlying both acquisitions and alliances of developing biotechnology firms by established pharmaceutical firms.

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The Theory of Decreasing Cost Production

Richard W. Tresch , in Public Finance (Third Edition), 2015

Appendix: Returns to Scale, Homogeneity, and Decreasing Cost

Since increasing returns to scale implies decreasing average cost, the two terms are used interchangeably in the chapter. To see that the former implies the latter, consider the homogeneous production function:

(9A.1) $Y=f(X_1,\dots ,X_N)=f\left(X_i\right)$

where X _i   =   input i, i  =   1,…, N, and Y  =   output. By the definition of homogeneous functions,

(9A.2) ${\lambda }^{\mathrm{B}}Y=f(\lambda ·X_i)$

Increasing returns to scale implies that β  >   1, or a scalar increase (decrease) in each of the factors generates a more-than-proportionate increase (decrease) in output. Furthermore, if the production function is homogeneous of degree β, then the marginal product functions, ∂ Y/∂ X _k   ≡ f _K (X _i ) are homogeneous of degree β  −   1. This follows immediately by differentiating λ ^β Y  = λ ^β f(X _i )   = f(λX _i ) with respect to X _k :

(9A.3) ${\lambda }^{\mathrm{B}}{f}_k\left(X_i\right)=\frac{\partial f\left(\lambda X_i\right)}{\partial X_K}=\lambda f_K\left(\lambda X_i\right)$

Hence,

(9A.4) ${\lambda }^{\beta -1}{f}_K=f_K\left(\lambda X_i\right)k=1,\dots ,N$

To minimize production costs for any given output, the firm solves the following problem:

$\begin{array}{l}\underset{\left(Xi\right)}{\mathrm{m}\mathrm{i}\mathrm{n}}\sum P_i{X}_i\\ \mathrm{s}.\mathrm{t}.Y=f\left(X_i\right)\end{array}$

The first-order conditions imply

(9A.5) $\frac{P_i}{P_l}=\frac{f_i\left(X_i\right)}{f_l\left(X_i\right)}i=2,\dots ,N$

The ratio of factor prices equals the marginal rate of technical substitution of the factors in production. Suppose the firm increases (decreases) its use of all factors X _i by the scalar λ. Since f _i (λX _i )   =   λ^β−1 f _i (X _i ), this scalar increase (decrease) continues to satisfy the first-order conditions:

(9A.6) $\frac{f_i\left(\lambda X_i\right)}{f_i\left(\lambda X_i\right)}=\frac{{\lambda }^{\beta –1}{f}_i\left(X_i\right)}{{\lambda }^{\beta –1}{f}_1\left(X_i\right)}=\frac{f_i\left(X_i\right)}{f_i\left(X_i\right)}=\frac{P_i}{P_1}$

Hence, if a vector of inputs $\overrightarrow{\mathbf{P}}{\text{∗}}_i$ minimizes cost, so too will any vector $\lambda \overrightarrow{\mathbf{P}}{\text{∗}}_i$ . But, if all inputs are increased by the scalar λ, total costs increase by λ and output increases by a factor λ^β. Thus, the total cost function must be of the form:

(9A.7) $\mathrm{TC}=k{Y}^{1/\beta }$

$\mathrm{since}k·{\left({\lambda }^{\beta }Y\right)}^{1/\beta }=\lambda ·k·Y^{1/\beta }=\lambda ·\mathrm{TC}.\mathrm{Finally}\text{,}$

(9A.8) $\mathrm{AC}=\mathrm{TC}/\mathrm{Y}={kY}^{(1/\beta -1)}={kY}^{(1-\beta )/\beta }$

Differentiating,

(9A.9) $\frac{\partial \mathrm{AC}}{\partial \mathrm{Y}}=\frac{1-\beta }{\beta }k·Y^{\left(\right(1–\beta )/\beta –1)}<0,\mathrm{for}\beta >1$

Hence, average cost declines continuously as output increases with increasing returns to scale.

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Introduction to the Microeconomics of Complex Economies

Wolfram Elsner , ... Henning Schwardt , in The Microeconomics of Complex Economies, 2015

1.8.3 Network Externalities and Increasing Returns to Scale of Institutions

Related concepts are economies of scale or increasing returns to scale . Economies of scale is the term used for describing falling average costs as a result of increasing production volumes or numbers. The more a firm produces of a good, the cheaper every single unit becomes. To expand production is thus a generally advantageous strategy to pursue in this case because it purports a potential cost advantage over other producers in a similar market segment. Returns to scale are derived from production technologies and refer to the proportionality of output changes following changes in all input factors (the latter changed in a fixed relation with each other). If output increases over-proportionally following an equal increase in all inputs, we refer to this as increasing returns (with constant and decreasing returns as the equivalent categories for proportional and under-proportional increases; we will explain this in more detail in Chapter 5). Economies of scale is a consequence of increasing returns to scale.

The application of a rule shows a similar characteristic to that of economies of scale in the number of both agents that use it and times it has been used by agents. The more often agents apply a rule, the less they have to invest in the taking of decisions in similar situations any longer, as they can simply apply a rule in a certain situation without further consideration. Thereby they also reduce the necessary effort level for other agents, because they increasingly know what to expect from their peers. The more agents apply a rule and the more often a rule has been applied, the less costly its application becomes.

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Theories of Statistical Discrimination and Affirmative Action: A Survey*

Hanming Fang , Andrea Moro , in Handbook of Social Economics, 2011

4.1.3 The effect of group size

Constant returns to scale imply that only relative group size matters. In general, analyzing group size effects would mean comparing different sets of equilibria. Not only the analysis becomes more complicated, but also as one parameter such as relative group size changes, some equilibria may disappear and new ones may appear. Therefore, results depend on the details of the equilibrium selection. Intuitively, as the relative size of one group increases and approaches 1, equilibrium investment for this group will approach the values corresponding to the symmetric equilibria of the model (which are equivalent to the equilibrium of a model with only one group). As for the smaller group, depending on the parameterization either lower or higher investment could be consistent with equilibrium.

Nevertheless, we can rely on the simple corner solution constructed in example at the end of the previous section to understand the importance of group size. Because both factors are essential, as discriminated group becomes larger, it becomes more difficult to sustain the extreme type of task segregation implied by the discriminatory equilibrium constructed in the previous section. To see this, note that as the discriminated group becomes larger, the mass of workers employed in the simple task gets larger, and therefore the ratio or marginal products x_u /x_q gets smaller; eventually, the inequality (20) cannot be satisfied and some group-B workers have to be employed in the complex task. Then the incentives to invest in human capital for B workers become strictly positive.

Hence, in a sense, sustaining extreme segregation in equilibrium against large groups may be difficult, rationalizing the existence of institutionalized segregation, such as apartheid in South Africa, where the larger group was segregated into lower paying tasks before the collapse of apartheid. It can also be shown that the incentives for the small group workers to keep the larger group into the segregation-type of equilibrium gets larger the bigger the large group is. The reason is that the larger the mass of workers employed in the simple task is, the higher is the marginal product in the complex job. This increases the incentives to invest for the small group and their benefits from investment.

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Definition of Increasing Returns in Economics

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