This is “Economies of Scale and Scope”, section 10.2 from the book Beginning Economic Analysis (v. 1.0).

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- When firms get bigger, when do average costs rise or fall?
- How does size relate to profit?

An economy of scaleSituation that exists when larger scale lowers average cost.—that larger scale lowers cost—arises when an increase in output reduces average costs. We met economies of scale and its opposite, diseconomies of scale, in the previous section, with an example where long-run average total cost initially fell and then rose, as quantity was increased.

What makes for an economy of scale? Larger volumes of productions permit the manufacture of more specialized equipment. If I am producing a million identical automotive taillights, I can spend $50,000 on an automated plastic stamping machine and only affect my costs by 5 cents each. In contrast, if I am producing 50,000 units, the stamping machine increases my costs by a dollar each and is much less economical.

Indeed, it is somewhat more of a puzzle to determine what produces a diseconomy of scale. An important source of diseconomies is managerial in nature—organizing a large, complex enterprise is a challenge, and larger organizations tend to devote a larger percentage of their revenues to management of the operation. A bookstore can be run by a couple of individuals who rarely, if ever, engage in management activities, where a giant chain of bookstores needs finance, human resource, risk management, and other “overhead” type expenses just in order to function. Informal operation of small enterprises is replaced by formal procedural rules in large organizations. This idea of managerial diseconomies of scale is reflected in the aphorism “A platypus is a duck designed by a committee.”

In his influential 1975 book *The Mythical Man-Month*, IBM software manager Fred Books describes a particularly severe diseconomy of scale. Adding software engineers to a project increases the number of conversations necessary between pairs of individuals. If there are *n* engineers, there are ½*n* (*n* – 1) pairs, so that communication costs rise at the square of the project size. This is pithily summarized in *Brooks’ Law*: “Adding manpower to a late software project makes it later.”

Another related source of diseconomies of scale involves system slack. In essence, it is easier to hide incompetence and laziness in a large organization than in a small one. There are a lot of familiar examples of this insight, starting with the Peter Principle, which states that people rise in organizations to the point of their own incompetence, meaning that eventually people cease to do the jobs that they do well.Laurence Johnston Peter (1919–1990). The notion that slack grows as an organization grows implies a diseconomy of scale.

Generally, for many types of products, economies of scale from production technology tend to reduce average cost, up to a point where the operation becomes difficult to manage. Here the diseconomies tend to prevent the firm from economically getting larger. Under this view, improvements in information technologies over the past 20 years have permitted firms to get larger and larger. While this seems logical, in fact firms aren’t getting that much larger than they used to be; and the share of output produced by the top 1,000 firms has been relatively steady; that is, the growth in the largest firms just mirrors world output growth.

Related to an economy of scale is an economy of scopeSituation that exists when producing more related goods lowers average cost.. An economy of scope is a reduction in cost associated with producing several distinct goods. For example, Boeing, which produces both commercial and military jets, can amortize some of its research and development (R&D) costs over both types of aircraft, thereby reducing the average costs of each. Scope economies work like scale economies, except that they account for advantages of producing multiple products, where scale economies involve an advantage of multiple units of the same product.

Economies of scale can operate at the level of the individual firm but can also operate at an industry level. Suppose there is an economy of scale in the production of an input. For example, there is an economy of scale in the production of disk drives for personal computers. This means that an increase in the production of PCs will tend to lower the price of disk drives, reducing the cost of PCs, which is a scale economy. In this case, it doesn’t matter to the scale economy whether one firm or many firms are responsible for the increased production. This is known as an external economy of scaleAn economy of scale that operates at the industry level, not the individual firm level., or an *industry economy of scale*, because the scale economy operates at the level of the industry rather than in the individual firm. Thus, the long-run average cost of individual firms may be flat, while the long-run average cost of the industry slopes downward.

Even in the presence of an external economy of scale, there may be diseconomies of scale at the level of the firm. In such a situation, the size of any individual firm is limited by the diseconomy of scale, but nonetheless the average cost of production is decreasing in the total output of the industry, through the entry of additional firms. Generally there is an external diseconomy of scale if a larger industry drives up input prices; for example, increasing land costs. Increasing the production of soybeans significantly requires using land that isn’t so well suited for them, tending to increase the average cost of production. Such a diseconomy is an external diseconomy rather than operating at the individual farmer level. Second, there is an external economy if an increase in output permits the creation of more specialized techniques and a greater effort in R&D is made to lower costs. Thus, if an increase in output increases the development of specialized machine tools and other production inputs, an external economy will be present.

An economy of scale arises when total average cost falls as the number of units produced rises. How does this relate to production functions? We let *y* = *f*(*x*_{1}, *x*_{2}, … , *xn*) be the output when the *n* inputs *x*_{1}, *x*_{2}, … ,*xn* are used. A rescaling of the inputs involves increasing the inputs by a fixed percentage; e.g., multiplying all of them by the constant *λ* (the Greek letter “lambda”), where *λ* > 1. What does this do to output? If output goes up by more than *λ*, we have an economy of scale (also known as increasing returns to scaleSituation that exists when increasing all inputs by the same scalar factor increases output by more than the scalar factor.): Scaling up production increases output proportionately more. If output goes up by less than *λ*, we have a diseconomy of scale, or decreasing returns to scaleSituation that exists when increasing all inputs by the same scalar factor increases output by less than the scalar factor.. And finally, if output rises by exactly *λ*, we have constant returns to scaleSituation that exists when increasing all inputs by the same scalar factor increases output by that scalar factor.. How does this relate to average cost? Formally, we have an economy of scale if
$f(\lambda {x}_{1},\lambda {x}_{2},\dots ,\lambda {x}_{n})>\lambda f({x}_{1},{x}_{2},\dots ,{x}_{n})$
if *λ* > 1.

This corresponds to decreasing average cost. Let *w*_{1} be the price of input one, *w*_{2} the price of input two, and so on. Then the average cost of producing *y* = *f*(*x*_{1}, *x*_{2}, … , *xn*) is
*AVC* = $\frac{{w}_{1}{x}_{1}+{w}_{2}{x}_{2}+\mathrm{\dots}+{w}_{n}{x}_{n}}{f({x}_{1},{x}_{2},\dots ,{x}_{n})}\text{.}$

What happens to average cost as we scale up production by *λ* > 1? Call this *AVC*(*λ*).

Thus, average cost falls if there is an economy of scale and rises if there is a diseconomy of scale.

Another insight about the returns to scale concerns the value of the marginal product of inputs. Note that if there are constant returns to scale, then

$${x}_{1}\frac{\partial f}{\partial {x}_{1}}+{x}_{2}\frac{\partial f}{\partial {x}_{2}}+\mathrm{\dots}+{x}_{n}\frac{\partial f}{\partial {x}_{n}}={\frac{d}{d\lambda}f(\lambda {x}_{1},\lambda {x}_{2},\dots ,\lambda {x}_{n})|}_{\lambda \to 1}$$ $$=\underset{\lambda \to 1}{\mathrm{lim}}\text{\hspace{0.17em}}\frac{f(\lambda {x}_{1},\lambda {x}_{2},\dots ,\lambda {x}_{n})-f({x}_{1},{x}_{2},\dots ,{x}_{n})}{\lambda -1}=f({x}_{1},{x}_{2},\dots ,{x}_{n})\text{.}$$The value $\frac{\partial f}{\partial {x}_{1}}$
is the marginal product of input *x*_{1}, and similarly $\frac{\partial f}{\partial {x}_{2}}$
is the marginal product of the second input, and so on. Consequently, if the production function exhibits constant returns to scale, it is possible to divide up output in such a way that each input receives the value of the marginal product. That is, we can give ${x}_{1}\frac{\partial f}{\partial {x}_{1}}$
to the suppliers of input one, ${x}_{2}\frac{\partial f}{\partial {x}_{2}}$
to the suppliers of input two, and so on; and this exactly uses up all of the output. This is known as “paying the marginal product,” because each supplier is paid the marginal product associated with the input.

If there is a diseconomy of scale, then paying the marginal product is feasible; but there is generally something left over, too. If there are increasing returns to scale (an economy of scale), then it is not possible to pay all the inputs their marginal product; that is, ${x}_{1}\frac{\partial f}{\partial {x}_{1}}+{x}_{2}\frac{\partial f}{\partial {x}_{2}}+\dots +{x}_{n}\frac{\partial f}{\partial {x}_{n}}>f({x}_{1},{x}_{2},\dots ,{x}_{n}).$

- An economy of scale arises when an increase in output reduces average costs.
- Specialization may produce economies of scale.
- An important source of diseconomies is managerial in nature—organizing a large, complex enterprise is a challenge, and larger organizations tend to devote a larger percentage of their revenues to management of the operation.
- An economy of scope is a reduction in cost associated with producing several related goods.
- Economies of scale can operate at the level of the individual firm but can also operate at an industry level. At the industry level, scale economies are known as an external economies of scale or an industry economies of scale.
- The long-run average cost of individual firms may be flat, while the long-run average cost of the industry slopes downward.
- Generally there is an external diseconomy of scale if a larger industry drives up input prices. There is an external economy if an increase in output permits the creation of more specialized techniques and a greater effort in R&D is made to lower costs.
- A production function has increasing returns to scale if an increase in all inputs by a constant factor
*λ*increases output by more than*λ*. - A production function has decreasing returns to scale if an increase in all inputs by a constant factor
*λ*increases output by less than*λ*. - The production function exhibits increasing returns to scale if and only if the cost function has an economy of scale.
- When there is an economy of scale, the sum of the values of the marginal product exceeds the total output. Consequently, it is not possible to pay all inputs their marginal product.
- When there is a diseconomy of scale, the sum of the values of the marginal product is less than the total output. Consequently, it is possible to pay all inputs their marginal product and have something left over for the entrepreneur.

- Given the Cobb-Douglas production function $f({x}_{1},{x}_{2},\mathrm{\dots},{x}_{n})={x}_{1}^{{a}_{1}}\text{\hspace{0.17em}}{x}_{2}^{{a}_{2}}\text{\hspace{0.17em}}\mathrm{\dots}\text{\hspace{0.17em}}{x}_{n}^{{a}_{n}}\text{,}$ show that there is constant returns to scale if ${a}_{1}+{a}_{2}+\dots +{a}_{n}=1\text{,}$ increasing returns to scale if ${a}_{1}+{a}_{2}+\dots +{a}_{n}>1$ , and decreasing returns to scale if ${a}_{1}+{a}_{2}+\dots +{a}_{n}<1\text{.}$
- Suppose a company has total cost given by $rK+\frac{{q}^{2}}{2K}\text{,}$
where capital
*K*can be adjusted in the long run. Does this company have an economy of scale, diseconomy of scale, or constant returns to scale in the long run? - A production function
*f*is*homogeneous of degree r*if $f(\lambda {x}_{1},\lambda {x}_{2},\dots ,\lambda {x}_{n})={\lambda}^{r}f({x}_{1},{x}_{2},\dots ,{x}_{n})\text{.}$ Consider a firm with a production function that is homogeneous of degree*r*. Suppose further that the firm pays the value of marginal product for all of its inputs. Show that the portion of revenue left over is 1 –*r*.