6 Firms and Their Production Decisions

The Policy Question
Is Introducing Technology in the Classroom the Best Way to Improve Education?

In chapter 5, we covered individual and market demand. We now turn to the supply side of the market—where the goods and services that are offered for sale in markets come from. A good is a physical product like a candy bar or a car, while a service is a task, like an accountant doing your taxes or a hairdresser cutting your hair. In economics, we say that all goods and services come from firms. Firm is a term that describes any entity that produces a good or service for sale in a market. Thus, a firm can be a mammoth conglomerate like General Electric that makes everything from kitchen appliances to advanced medical equipment to jet engines. Or a firm can be an eleven-year-old child selling lemonade on the street corner.

Our goal in this section is to understand where supply curves come from and what influences their shape and movement. We can then make reasonable predictions about how changes in production affect the price and sales of goods and services.

When we understand and are able to describe supply curves, we can match them up to the demand curves and talk about markets and market outcomes in section 6.3.

Exploring the Policy Question

Reform efforts are common in the history of public education in the United States. The Bush administration’s No Child Left Behind Act and the Obama administration’s Race to the Top initiative are examples of broad efforts to reform education. Yet despite these and many other efforts in recent decades, the United States still does relatively poorly in rankings of educational outcomes.

This suggests that policy makers need to better understand how knowledge is acquired and transmitted in a group setting. The following are important questions that need to be answered in order to make an effective policy:

• How important is the training of teachers, and what types of training are most effective?
• How important is caring for the health and well-being of students, and what are the best ways of doing so?
• How important are good textbooks, and what makes a textbook good?
• Does the quality of school infrastructure matter?
• Is it important to have computers integrated into learning in the twenty-first century?

These are all part of what an economist would describe as the education production function: the set of inputs (teachers, students, supplies, and infrastructure) that are combined by education systems to produce an output, educated children. This chapter explains production functions in economics and will give us some guidance in thinking about the education production function that we might use to guide and shape education policy.

6.1 Inputs

Learning Objective 6.1: Identify the four basic categories of inputs in production, and give examples of each.

Previously, we distilled the essence of consumers to a utility function that describes their preferences and is used to choose a consumption bundle among many possible alternatives. In this chapter, we are going to do something very similar with producers, boiling down their real-world complexity to the very essence of their function.

Another term for a producer is firm. The most basic definition of a firm is an entity that combines a set of inputs to produce a good or service for sale in a market. There are four basic categories of inputs that describe most of the potential inputs used in any production process:

1. Labor ([latex]L[/latex])
2. This category of input encompasses physical labor as well as intellectual labor. It includes less skilled or manual labor, managerial labor, and skilled labor (engineers, scientists, lawyers, etc.)—the human element that goes into the production of a good or service.
3. Capital ([latex]K[/latex])
4. This input category describes all the machines that are used in production, such as conveyor belts, robots, and computers. It also describes the buildings, such as factories, stores, and offices, and other non-human elements of production, such as delivery trucks.
5. Land ([latex]N[/latex])
6. Some goods, most notably agricultural goods, need land to produce. Fields that grow crops and forests that grow trees for lumber and pulp for paper are examples of the land input in production.
7. Materials ([latex]M[/latex])
8. This input category describes all the raw materials (trees, ore, wheat, oil, etc.) or intermediate products (lumber, rolled aluminum, flour, plastic, etc.) used in the production of the final good. Note that one firm’s final good, like aluminum, is often another firm’s input.

Not all firms use all the inputs. For example, a child who runs a lemonade stand uses labor (the time cutting, squeezing, mixing, and selling), capital (the knife to cut, the juicer to juice, and the pitcher to hold the lemonade), and materials (lemons, water, and sugar). This young entrepreneur does not use land, though the producer of the lemons does.

The providers of services use inputs as well. An accountant, for example, might use labor, a computer (capital), office supplies (materials), and so on.

6.2 Production Functions, Inputs, and Short and Long Runs

Learning Objective 6.2: Explain the concept of production functions, the difference between fixed and variable inputs, and the difference between the economic short run and long run.

The way inputs are combined to produce an output is called the firm’s technology or production process. We describe the production process with a production function: a mathematical expression of the maximum output that results from a specific amount of each input.

Let’s start with the very simple example of our lemonade stand. Suppose for each cup of lemonade the child can sell, it takes exactly one lemon, two cups of water, one tablespoon of sugar, and ten minutes of labor, including the making and the selling of the beverage. A production function that describes this process would look something like this:

[latex]\text{Cups of Lemonade}=f(\text{lemons, sugar, water, labor time})[/latex]

where [latex]f[/latex] stands for some functional form. For the moment, we are not describing the actual function but that these inputs are what go into the production of lemonade. A more generic description of a production function would look like this:

[latex]Output, Q=f(L,K,M,N)[/latex]

Similar to the way we simplified the consumer choice problem, we will generally use two input production functions to keep the problem simple and tractable. By convention, we typically use labor ([latex]L[/latex]) and capital ([latex]K[/latex]) as the two inputs, and so the generic production function is

[latex]Q=f(L,K)[/latex].

When we put in actual amounts for labor and capital, this function tells us the maximum output that can be achieved with the two amounts. Producing less than that output is possible, but we will soon find that firms that are trying to maximize profits or minimize costs would never produce less than the full amount of something that they can sell for a positive price. Said another way, if a firm has a specific output in mind, it has no interest in using more than the minimum amounts of inputs.

Producing a good or service for the market requires firms to make choices about the type and number of inputs to use. As we will see, this choice depends critically on the contribution the input makes to the final output relative to its cost. But a firm can only make choices about inputs that are variable: inputs whose quantities can be adjusted by the firm. In shorter time periods, some inputs are not variable but fixed: the firm cannot adjust these quantities.

Consider a firm that produces tablet computers for use in classrooms using an assembly line. A conveyor belt (capital) carries the computers in various stages of construction past stations where the next part is added by a worker (labor). The firm might decide that it wants to expand production and add a second conveyor belt, but it takes time to expand the factory, build the belt, and get it operational. In this case, capital (the conveyor belt) is a fixed input—an input than cannot be adjusted by the firm in a given time period. If the firm wants to increase production for the time being, it will have to adjust labor, a variable input—an input that can be adjusted by the firm in a given time period.

Given enough time, this firm could add the second conveyor belt and increase production on its new level of capital. This concept is how we define short run and long run in economics: the short run is a period of time in which some inputs are fixed, and the long run is a period of time long enough that all inputs can be adjusted. It is important to note that these are not defined by any objective period of time (day, month, year, etc.) but are specific to the particular firm and its particular inputs.

Suppose our tablet computer manufacturer needs three months to install a second assembly line—that is, its short run is less than three months, and its long run is three months or more. For our lemonade salesperson, it might take an hour-long trip to the store to buy a new juicer or pitcher. In this case, the short run is less than an hour, and the long run is an hour or more.

6.3 Production Functions and Characteristics in the Short Run

Learning Objective 6.3: Explain the concepts of the marginal product of labor, the total product of labor, the average product of labor, and the law of diminishing marginal returns.

The short run, as was just described, is a period short enough that at least one input is fixed. We will follow the convention that the capital ([latex]K[/latex]) input is fixed in the short run and labor is variable. This assumption is not always true, but labor hours often can be adjusted quickly—for example, by having workers put in a little overtime—and capital inputs generally take some time to adjust. To describe the short run using our production function, we write it like this:

[latex]Q=f(L,\overline{K})[/latex],

where [latex]Q[/latex] is the quantity of the output, [latex]L[/latex] is the quantity of the labor (generally measured in labor hours), and [latex]K[/latex] is the quantity of capital (generally measured in capital hours). The bar over the capital variable indicates that it is fixed: [latex]\overline{K}[/latex].

Let’s go back to our computer manufacturer. In the short run, the firm cannot add capital in the form of a new assembly line. But it can add workers. It could, for example, add a second shift to production, using the same assembly line through the night. Or if already running twenty-four hours, the firm could add more workers to the same assembly line and squeeze more production out of it.

The relationship between the amount of the variable input used and the amount of output produced (given a level of the fixed input) is the total product, or in this case the total product of labor ([latex]Q[/latex]), as labor is the variable input. Table 6.1 summarizes this information for our tablet computer manufacturer along with some additional information, which we will discuss.

As the number of workers (labor) increases incrementally by one in table 6.1, the output also increases. Note that at first, increases in production are pretty big. This is due to the efficiencies from increased specialization made possible by more workers. After a while, the increases in output from increased labor get smaller and smaller as the assembly line gets crowded and the additional workers have difficulty being as productive. The maximum number of workers is reached at ten due to the firm being unable to fit more workers on the assembly line.

The extra contribution to output of each worker is critically important to the firm’s decision about how many workers to employ. The extra output achieved from the addition of a single unit of labor is the marginal product of labor ([latex]MP_L[/latex]). In our case, the extra unit is an additional worker, but in general, we might measure units as worker hours. [latex]MP_L[/latex] is a key measure listed in the fifth column of table 6.1. Formally, it is

[latex]MP_L=\frac{\Delta Q}{\Delta L}[/latex]

Table 6.1 shows that this marginal product increases at first as we add more workers, peaks at five workers, but then starts to decline and even goes negative as we reach ten workers.

Also important to keep track of is the average product of labor ([latex]AP_L[/latex]), or how much output per worker is being produced at each level of employment:

[latex]AP_L=\frac{Q}{L}[/latex]

In table 6.1, we see that the average product of labor increases at first, peaks at five workers, but then starts to decline.

These measures of labor productivity can also be represented as curves. We will draw them as smooth continuous curves because we can hire fractional workers in the sense that we could hire one full-time employee and another who works 60 percent of the time, and so on.

In figure 6.1, we represent the total product of labor curve in the upper panel and the marginal and average product of labor curves in the lower panel.

The shape of the total product of labor curve confirms that, at first, adding workers allows more specialization and more output per worker. This can be seen in the initial part of the curve where the slope is getting steeper. Then given the fact that the capital is fixed, adding more workers continues to increase output but at a decreasing rate. We see this past point A on the graph, where the slope is still positive but decreasing. Finally, more workers actually cause output to fall.

From the total product of labor graph, we can actually measure the average and marginal product of labor. The average product of labor is the same as the slope of the ray from the origin that intersects the total product curve (note that the slope’s rise/run is output/labor). This reaches its highest slope right at point B, where the ray is just touching the total product curve. The marginal product of labor curve is the change in output over the change in labor, which is the same as the slope of the total product curve itself. Notice that the maximum slope is right at point A so that the average and the marginal are exactly the same at point B and that the slope turns negative at point C—the point of maximum output. These observations can help us draw the average and marginal product curves in the lower panel. We know marginal product reaches a maximum at A, crosses the average product curve at B, and turns negative at C. We also know that when the marginal product is above the average, the average is increasing, and when the marginal product is below the average, the average is decreasing. Thus the two sets of curves relate directly to each other.

There is one more important characteristic of production to note before we move on: the law of diminishing marginal returns. This “law” states that if a firm increases one input while holding all others constant, the marginal product of the input will start to get smaller, just like in our example at three labor units. It is important to remember that we are talking about the marginal returns here—that the additional output will be positive but smaller and smaller. We can also talk about diminishing returns—what happens to total output as labor is increased, which in our case does eventually diminish (at nine labor units), but this is not a general rule.

6.4 Production Functions in the Long Run

Learning Objective 6.4: Describe and illustrate isoquants generally and for Cobb-Douglas, perfect complements (fixed proportions), and perfect substitutes.

In the long run, all production inputs are variable. Because of this, firms have more flexibility in how they choose to produce, expand, and contract their output in the long run than in the short run. But this situation also complicates firms’ decision-making. They have to weigh the marginal contribution to output an extra unit of each input would provide and the cost of that extra unit. We will now focus on this decision process.

Since we have simplified our firm to one that uses only two inputs, capital ([latex]K[/latex]) and labor ([latex]L[/latex]), we have a choice problem that shares many similarities with the consumers’ problem we studied in section 6.1. It should come as no surprise then that the way we solve the problem is similar as well. Let’s start with the production function.

[latex]Q=f(L,K)[/latex]

Note that there is no bar above the [latex]K[/latex], meaning that both labor and capital are now variable. This makes the function mathematically similar to the utility function from chapter 2. The graph of these two variable functions is three-dimensional.

From table 6.2, we can see the diminishing marginal productivity of one variable input. Holding one input constant—for example, holding capital at three—we can see that the increase in the total output falls as we increase labor. The extra output from going from one unit of labor to two is 130 (380 − 250), from two to three is 120, from three to four is 60, and from four to five is 40.

Like indifference curves for utility functions, we can visualize this output function in two dimensions by drawing contour lines: lines that connect all combinations of labor and capital that lead to an identical output. To draw one, we can start by arbitrarily choosing an output level and then finding all the combinations of labor and capital the firm could possibly use to produce this quantity:

[latex]\overline{Q}=f(L,K)[/latex]

Drawing this on a graph with capital on the vertical axis and labor on the horizontal axis produces an isoquant: a curve that shows all the possible combinations of inputs that produce the same output. Iso means “the same” and quant means “quantity.” So an isoquant is a curve that for every point on it, the output quantity is the same.

Figure 6.2 shows three isoquants for table 6.2, corresponding to the outputs 180, 300, and 500. In the figure, the combinations of inputs that yield the same output are connected by a smooth curve. We draw them as smooth curves to illustrate that firms can use any fraction of a unit of input they desire.

You can, of course, do this for any output level you choose so the entire space is filled with potential isoquants; figure 6.2 simply sketches three. Note that we do not have a specific functional form for the production function, so we are drawing hypothetical isoquants. We will derive an isoquant from a specific production function momentarily.

Properties of Isoquants

Just like indifference curves, isoquants have properties that we can characterize.

Three Types of Production Functions and Their Related Isoquants

As with preferences, there are three basic types of production functions. Though they are essentially the same as the three indifference curves that represent the three basic types of preferences, it is good to keep in mind that the context is very different. Depending on the production process, inputs into the production process could be easily substitutable, might need to be used in fixed proportions, or might be mixed together but have flexibility in terms of proportions.

Perfect Substitute Production Functions

Some production processes can substitute one input for another in a fixed ratio. For example, our lemonade-selling child might have an electric juicer that will transform whole lemons into lemonade without any human input. This child might also be able to make lemonade entirely by hand.

Suppose the juicer can always make one gallon of lemonade in half an hour and the child can always make one gallon in one hour. Thus the production function for the quantity of lemonade ([latex]Q[/latex]) given labor ([latex]L[/latex]) and capital ([latex]K[/latex]) looks like this:

[latex]Q = L + 2K[/latex]

To see this, think in terms of an isoquant: producing exactly one gallon of lemonade ([latex]Q = 1[/latex]) requires either one hour of labor ([latex]L[/latex]) or [latex]\frac{1}{2}[/latex] hour of capital, the juicer ([latex]K[/latex]). You can also mix inputs—for example, you could use [latex]\frac{1}{2}[/latex] hour of labor and [latex]\frac{1}{4}[/latex] hour of capital. To produce two gallons ([latex]Q = 2[/latex]), you could use two hours of labor or one hour of capital. You could also mix and use one hour of labor and [latex]\frac{1}{2}[/latex] hour of capital or any linear combination of the two inputs.

Isoquants for a perfect substitute production function are therefore straight lines. Figure 6.3 presents three isoquants for a production function, [latex]Q = L + K[/latex].

Perfect substitute production functions generally have the form

[latex]Q=aL+\beta K[/latex],

where α and β are positive constants. A more general form of this production function that incorporates a measure of overall productivity is this:

[latex]Q=A(aL+\beta K)[/latex],

where [latex]A[/latex] is also a positive constant. Careful observers will notice that the [latex]A[/latex] is unnecessary because it can be incorporated into [latex]\alpha[/latex] and [latex]\beta[/latex], but expressing the production function this way provides an easy method of adjusting for (and empirically estimating) productivity. We will examine [latex]A[/latex] and productivity in more detail in section 6.6.

Perfect Complement (Fixed-Proportions) Production Functions

If inputs have to be used in fixed proportions, then we have a fixed-proportions production function where the inputs are perfect complements, also known as a fixed-proportions production function. Imagine a metal-stamping business where the business takes metal sheets from its clients and stamps them into shapes, such as auto body parts. Suppose the stamping machines require exactly one operator. Without a full-time operator, the machine will not work. Without a machine, a worker cannot stamp the metal sheet. Thus the inputs are perfect complements—they have to be applied in fixed proportions to get any extra output. Suppose also that one machine and one worker can produce one hundred stamped panels in a day.

A production function that describes the daily output of the metal-stamping business looks like this:

[latex]Q = 100^*min[L, K][/latex],

where the min function simply takes the smaller of the two values of the arguments [latex]L[/latex] and [latex]K[/latex]. For example, if the business has ten workers ([latex]L[/latex]) and eight machines ([latex]K[/latex]), the value of the function, [latex]Q[/latex], would be eight times one hundred, or eight hundred. Similarly, if the business has twelve machines but only seven workers, the daily output would be seven hundred.

We can draw the isoquants for this firm starting with equal inputs. At point A, there are five machines and five workers, and the daily output is five hundred. Now suppose we move to point B, where there are still five machines and seven workers. Does this yield any more output? No. So this point is on the same indifference curve. This is similar to point C, where there are seven machines and five workers.

Perfect complement production functions have the general functional form of

[latex]Q=A ^*min[\alpha L,\beta K][/latex],

where [latex]A[/latex], [latex]\alpha[/latex], and [latex]\beta[/latex] are positive constants. As in the case of perfect substitutes, the [latex]A[/latex] is unnecessary because it can be incorporated into [latex]\alpha[/latex] and [latex]\beta[/latex], but we will keep it for simplicity and clarity.

Figure 6.4 presents three isoquants for the production function [latex]Q = min[L,K][/latex].

Cobb-Douglas Production Functions

Perfect substitute and fixed-proportion production functions are special cases of a more general production function that describes inputs as imperfect substitutes for each other. In other words, we can get rid of some machines (capital) in exchange for more workers (labor) but at a ratio that changes depending on the current mix of workers and machines. The Cobb-Douglas function that we used to describe consumer choice with the preference for variety assumption is used for just such a production process.

An example of a Cobb-Douglas production function is

[latex]Q=AL^\alpha K^\beta[/latex],

where [latex]A[/latex], [latex]\alpha[/latex], and [latex]\beta[/latex] are positive constants. Notice that [latex]A[/latex] represents overall productivity (as [latex]A[/latex] increases, the same number of inputs yields more output), and the [latex]\alpha[/latex] and [latex]\beta[/latex] parameters represent the contribution to the output of each input. In other words, the higher the level of α, the more one unit of capital will produce in extra output, and likewise for [latex]\beta[/latex], labor. The essential aspect of this production function is always having the ability to substitute one input for the other and maintain the same level of output.

From this production function, we can study the substitutability of inputs into production. In order to do this, we need to define the marginal rate of technical substitution, which is similar to the marginal rate of substitution from consumer theory. The marginal rate of technical substitution (MRTS) describes how much you must increase one input if you decrease the other input by one unit in order to produce the same output. The MRTS is simply the ratio of the marginal product of labor to the marginal product of capital:

[latex]MRTS =-\frac{MP_L}{MP_K}[/latex]

To see where this ratio comes from, remember that the marginal product of an input tells us how much extra output will result from a one-unit increase in the input. Earlier we defined the marginal product of labor as

[latex]MP_L=\Delta Q/\Delta L[/latex]

Similarly for capital, we have

[latex]MP_K=\Delta Q/\Delta K[/latex]

Note that we can rearrange the marginal product of labor and express it as

[latex]\Delta Q=MP_L^*\Delta L[/latex]

So if the [latex]MP_L[/latex] is four and the firm increases labor by one unit, then the extra output is four.

Similarly, if the [latex]MP_K[/latex] is two and the firm decreases capital by one unit, then the decline in output is two, or

[latex]\Delta Q=MP_K^*\Delta K[/latex].

Notice that in order to stay on the same isoquant, the net change in output has to be zero, or

[latex](MP_L^*\Delta L)+(MP_K^*\Delta K)=0[/latex].

Rearranging the steps, we get the following:

[latex]-(MP_L^*\Delta L)=(MP_K^*\Delta K)[/latex]

which we can simplify to give us:

[latex]-\frac{MP_L}{MP_K}=\frac{\Delta K}{\Delta L}=MRTS[/latex]

Note that the MRTS describes the slope of the isoquant—how many units of capital you would need to add (rise) if the labor decreases by one unit (run) to maintain the same output (stay on the isoquant).

Now let’s return to the Cobb-Douglas utility function, [latex]Q=AL^\alpha K^\beta[/latex], and derive the MRTS. Since the parameters [latex]\alpha[/latex] and [latex]\beta[/latex] are positive constants that describe the contribution of each input to the final output, they are related to the marginal products. It turns out that the marginal product of labor is just α times the average product of labor:

[latex]MP_L=aAP_L=aQ/L[/latex]

And the marginal product of capital is just β times the marginal product of capital:

[latex]MP_K=aAP_K=aQ/K[/latex]

Thus for the MRTS, we have

[latex]-\frac{MP_L}{MP_K}=-\frac{aQ/L}{\beta Q/K}=-\frac{\alpha }{\beta }\frac{K}{L}[/latex].

Notice the absence of [latex]A[/latex] in this ratio. Since we are talking about marginal returns to inputs and [latex]A[/latex] only affects total return, it does not appear in this calculation.

Let’s look at an example of a specific production function. Suppose a firm’s production function is estimated to be

[latex]Q=3.7L^{.45}K^3[/latex].

In this case, [latex]A[/latex]=3.7, [latex]\alpha[/latex]= .45, and [latex]\beta[/latex]= .3. Thus the MRTS for this production function is

[latex]MRTS=-\frac{MP_L}{MP_K}=-\frac{\alpha }{\beta }\frac{K}{L}=-\frac{.45}{.3}\frac{K}{L}=−1.5\frac{K}{L}[/latex].

So the slope of this firm’s isoquant depends on the specific mix of capital ([latex]K[/latex]) and labor ([latex]L[/latex]) but is [latex]−1.5(K/L)[/latex]. Note that when [latex]K[/latex] is relatively large and [latex]L[/latex] is relatively small (as in point [1, 16] in figure 6.5), the slope is negative and steep. Alternatively, when [latex]K[/latex] is relatively small and [latex]L[/latex] is relatively large, the slope is negative and shallow. The slope is also always changing along the isoquant, and thus we have the “bowed-in” shape (concave to the origin) typical to the Cobb-Douglas production function.

6.5 Returns to Scale

Learning Objective 6.5: Define the three categories of returns to scale, and describe how to identify them.

Up to now, our focus on marginal returns has centered on the change of one input, holding the other input constant. In this section, we consider what happens to output when we increase or decrease both inputs. Increasing or decreasing both inputs can be described as scaling the firm up or down, and we want to know how it affects output. In particular, we would like to know about proportional changes to output. For example, if the firm doubles both inputs, does output double as well? Does it increase more or less than double the previous amount? The answer to these questions determines the returns to scale of the firm: the rate at which the output increases when all inputs are increased proportionally.

Constant Returns to Scale (CRS)

If a firm’s output changes in exact proportion to changes in the inputs, we say the firm exhibits constant returns to scale (CRS). Let’s consider this situation using the doubling example.

Suppose a firm has a production function of

[latex]Q=f(L,K)[/latex].

Now the firm doubles its inputs. We can show this by multiplying the current inputs by two:

[latex]f(2L,2K)[/latex]

If this yields exactly double the output, we can write this expression as

[latex]f(2L,2K)=2f(L,K)=2Q[/latex].

This expression says that if both inputs double, output doubles. It is an example of a CRS production function. The CRS condition can be expressed more generally as

[latex]f(\Phi L,\Phi K)=\Phi f(L,K)=\Phi Q[/latex],

where [latex]\Phi[/latex] is a strictly positive constant, like two in our example. In the case of scaling down, [latex]\Phi[/latex] would be a fraction, such as [latex]\frac{1}{2}[/latex].

Increasing Returns to Scale (IRS)

Increasing returns to scale (IRS) production functions are those for which the output increases or decreases by a greater percentage than the percentage increase or decrease in inputs. In the case of doubling inputs, the IRS expression would look like this:

[latex]f(2L,2K)>2f(L,K)=2Q[/latex]

Look carefully at what this says. The first term on the left is the output that results from the doubling of the two inputs. We compare this to the doubling of the output, represented by the two expressions on the right of the inequality sign. This expression says that you get more output from doubling the inputs than you get from doubling the output—or that if you double the inputs, you get more than double the output. In general terms,

[latex]f(\Phi L, \Phi K)> \Phi f (L,K)=\Phi Q[/latex],

again where [latex]\Phi[/latex] is a strictly positive constant.

Decreasing Returns to Scale (DRS)

Finally, decreasing returns to scale (DRS) production functions are those for which the output increases or decreases by a smaller percentage than the increase or decrease in inputs. In the case of doubling the inputs, the DRS expression would look like this:

[latex]f(2L,2K)<2f(L,K)=2Q[/latex]

This expression says that you get less output from doubling the inputs than you get from doubling the output—or that if you double the inputs, you get less than double the output. In general terms,

[latex]f(\Phi L, \Phi K)< \Phi f(L,K)=\Phi Q[/latex],

again where [latex]\Phi[/latex] is a strictly positive constant.

Varying Returns to Scale

Some production processes have varying returns to scale. The most common is where there are increasing returns to scale for small output levels, constant returns for medium output levels, and decreasing for very large output levels.

Returns to Scale and Production Functions

Let’s think about the three types of production functions we have studied—perfect substitutes, perfect complements, and Cobb-Douglas—and their returns to scale. The question we want to answer can be expressed as, If the amounts of inputs are doubled, by how much does the output increase? The answer will tell us if the production function exhibits constant, increasing, or decreasing returns to scale.

Perfect Substitutes

Recall the expression for perfect substitutes:

[latex]Q=\alpha L+\beta K[/latex]

Here we double both inputs, so we replace L and K with 2L and 2K:

[latex]\alpha 2L+\beta 2K=2Q[/latex]

By simple algebra, we can factor out the common two and come up with

[latex]2(\alpha L+\beta K)=2Q[/latex].

By the above equation, we know that the term in the parentheses is equal to [latex]Q[/latex]. So we know that by doubling both [latex]L[/latex] and [latex]K[/latex], we exactly double [latex]Q[/latex], or we have constant returns to scale. This means that production functions for perfect substitutes are CRS.

Perfect Complements

The perfect complement function is

[latex]Q=A^*min[\alpha L,\beta K][/latex].

Again, we want to begin by replacing [latex]L[/latex] and [latex]K[/latex] with [latex]2L[/latex] and [latex]2K[/latex] and note that since we are doubling both inputs, the minimum of the two numbers must also be doubled so we can pull the two out in front of the min:

[latex]Amin[\alpha 2L,\beta 2K]=A2min[\alpha L,\beta K][/latex]

Since the order of multiplication does not matter, we know

[latex]A2min[\alpha L,\beta K]=2A^*[\alpha L,\beta K][/latex],

which is the same as doubling output:

[latex]2A^*min[\alpha L,\beta K]=2Q[/latex]

The first step is not immediately obvious, but think about the way the min function works: it just picks the smaller of the two numbers in the square brackets. Doubling them both does not change which one is smaller, and in the end, the value is just double the smaller one, so it is the same as doubling after we choose the min of the two numbers. From there, since order doesn’t matter in multiplication, we can move the two out in front of the [latex]A[/latex], and now we have two multiplied by the original production functions (in parentheses), which is the same as [latex]Q[/latex]. Again, we know that by doubling both [latex]L[/latex] and [latex]K[/latex], we exactly double [latex]Q[/latex], or we have constant returns to scale. Thus production functions for perfect complements are also CRS.

Cobb-Douglas

Recall the Cobb-Douglas function:

[latex]Q=AL^\alpha K^\beta[/latex]

We will start the same way, replacing [latex]L[/latex] and [latex]K[/latex] with [latex]2L[/latex] and [latex]2K[/latex]:

[latex]A(2L)^\alpha (2K)^\beta[/latex]

You have to be careful here because the exponents α and β are on [latex]L[/latex] and [latex]K[/latex], respectively, and when we replace [latex]L[/latex] with [latex]2L[/latex], for example, the α is now the exponent for [latex]2L[/latex]. By the rules of exponents, we can separate the 2s and the [latex]L[/latex] and [latex]K[/latex]:

[latex]A2^\alpha L^\alpha 2^\beta K^\beta[/latex]

Next, we can rearrange terms at will, as the order of operations does not matter in multiplication:

[latex]2^\alpha 2^\beta AL^\alpha K^\beta[/latex]

We can also, by the rules of exponents, express [latex]2^\alpha 2^\beta[/latex] as [latex]2^{\alpha +\beta }[/latex]:

[latex]2^{\alpha +\beta }(AL^\alpha K^\beta) = 2^{\alpha +\beta }Q[/latex]

Now we have an expression with [latex]Q[/latex] and can evaluate the returns to scale. We do so by considering the three possibilities for the exponent [latex]\alpha + \beta[/latex]:

1. [latex]\alpha + \beta =1[/latex].
2. Any number to the power of one is itself, so [latex]2^1=2[/latex]. So in this case, we have exactly double the output, and the production function is CRS.
3. [latex]\alpha + \beta \lt 1[/latex].
4. In this case, we have some number less than two multiplied by [latex]Q[/latex], and so we will have something less than double the output. That is, the production function exhibits decreasing returns to scale (DRS).
5. [latex]\alpha + \beta \gt 1[/latex].
6. In this case, [latex]Q[/latex] is multiplied by a number larger than two, so the output is more than double. That is, the production function exhibits increasing returns to scale (IRS).

So for Cobb-Douglas production functions, the returns to scale depend on the sum of the exponents: if they equal one, they are CRS; if they are less than one, they are DRS; and if they are more than one, they are IRS.

Returns to Scale and Isoquants

We can see returns to scale in isoquants by examining how much they increase with input increases for a particular production function. For example, in figure 6.6(a), when we double inputs from ten to twenty, we double output from one hundred to two hundred. Since output increases at the same rate as inputs, we know it is CRS.

The isoquants in figure 6.6(b) illustrate increasing returns to scale. When we double inputs from 10 to 20, we increase output by more than double the original quantity, from 100 to 235.

The isoquants in figure 6.6(c) illustrate decreasing returns to scale. When we double inputs from 10 to 20, we increase output by less than double the original quantity, from 100 to 160.

6.6 Technological Change and Productivity Growth

Learning Objective 6.6: Discuss technological change and productivity increases and how they affect production functions.

A firm’s production function need not be static over time. Many firms adopt new technologies or new ways of organizing and doing things in order to become more productive. Technological change refers to new production technology or knowledge that changes firms’ production functions so that more output is produced by the same amount of inputs. Technological change is also known as total factor productivity growth.

We can express changes in overall productivity with production functions. So far we have assumed that firms are efficient, meaning that they do not waste inputs—they use the minimum of inputs necessary to produce a particular output level. You can see this by the way we assume that certain inputs yield specific amounts of output. There is no waste in this scenario. Productivity, on the other hand, refers to how much overall output a firm gets from a set amount of inputs. The more a firm produces with the same inputs, the more productive it is.

As was noted in section 6.4, the constant [latex]A[/latex] is a measure of overall productivity in the types of production functions we have been studying. All else equal, increasing [latex]A[/latex] increases the output that a firm produces from a set amount of inputs.

Graphically adjusting [latex]A[/latex] in any of our three production function types relabels our isoquants. As figure 6.7 shows, when [latex]A[/latex] is increased from one to three in the case of a simple Cobb-Douglas production function, all the outputs associated with the individual isoquants increase by three as well.

We measure productivity increases by comparing output before and after the productivity change. In the case above, our production function

[latex]Q=(1)L^\alpha K^\beta[/latex]

changed to

[latex]Q=(3)L^\alpha K^\beta[/latex].

The output of the firm thus increased by [latex]\Delta Q/Q[/latex], or [latex][(3)L^\alpha K^\beta −(1)L^\alpha K^\beta]/(1)L^\alpha K^\beta[/latex], or two. In percentage terms, output increased by 200 percent.

6.7 the Policy Example
Is Introducing Technology in the Classroom the Best Way to Improve Education?

Learning Objective 6.7: Use a production function to model the process of learning in education.

Let’s return now to the policy question introduced earlier. One way to think of education is similar to a firm: you take a mix of inputs—students, teachers, books, buildings, pedagogy, and technology—and mix them together and get an output. The output in this case is the education itself—how much the students learn. If learning is the output and we can identify the inputs, then it is possible to think of education as a type of production that is governed by a production function.

Modeling education as a production function is mechanical and has limitations for sure. The true test of this endeavor is whether we can gain any insight into how student learning is accomplished. We would have to start by coming up with a plausible production function. By far the most flexible—and perhaps the most relevant to education—is the Cobb-Douglas production function, where inputs are imperfectly substitutable.

For example, you don’t get any learning without a teacher, but you might be able to substitute some technology-based instruction for some live teaching. This is the essence of the Cobb-Douglas production function. We can express an education production function, apply it to real-world data, and estimate its parameters to find out, for example, how important good teachers are and how substitutable they could be using technology. Doing so will give policy makers a better idea of the most effective ways to spend scarce resources if the ultimate goal of education funding is learning.

For example, suppose we posit that learning is a function of teacher quality, the technology used in the classroom, the quality of the class materials, and the size of the class (number of students per teacher). So we imagine that the learning production function looks something like this:

[latex]learning=Teacher^\alpha Technology^\beta Materials^\gamma Class\,Size^\delta[/latex]

With the appropriate data, the parameters alpha, beta, gamma, and delta can be measured, giving policy makers an idea about where the highest return on investment is found. In fact, economists have been doing this for quite some time. See Measuring Investment in Education.

Exploring the Policy Question

1. What do you think are the relevant inputs into the learning production function?
2. Which type of production function best represents the grade school learning environment?
3. If you believe that teachers are easily replaced by online videos, where one hour of online video is exactly equivalent to one hour of live teaching, what type of production function would you use to describe this?
4. If there were a magic “learning pill” that students could take and that would, regardless of the current level and mix of inputs, double their learning, how would you express this in a production function?

Learn: key topics

Terms

Labor ([latex]L[/latex])

Input that encompasses physical labor as well as intellectual labor. Includes less skilled or manual labor, managerial labor, and skilled labor (engineers, scientists, lawyers, etc,) i.e., the human element that goes into the production of a good or service.

Capital ([latex]K[/latex])

Input category which describes all the machines that are used in production: conveyor belts, robots, and computers. It also describes the buildings, such as factories, stores, offices, and other non-human elements of production, such as delivery trucks.

Land ([latex]N[/latex])

Input category that encompasses land used for production, i.e., fields that grow crops and forests that grow trees for lumber and pulp for paper.

Materials ([latex]M[/latex])

All the raw materials (trees, ore, wheat, oil, etc.) or intermediate products (lumber, rolled aluminum, flour, plastic, etc.) used in the production of the final good.

Production function

A mathematical expression of the maximum output that results from a specific amount of each input.

Variable input

An input that can be adjusted by the firm in a given time period.

Fixed input

An input that cannot be adjusted by the firm in a given time period.

Short run

A period of time in which some inputs are fixed; not defined by any objective period of time (day, month, year, etc.) but are specific to the particular firm and its particular inputs.

Long run

A period of time long enough that all inputs can be adjusted; not defined by any objective period of time (day, month, year, etc.) but are specific to the particular firm and its particular inputs.

Total product of labor ([latex]Q[/latex])

The relationship between the amount of the variable input used and the amount of output produced (given a level of the fixed input).

Average product of labor ([latex]AP_L[/latex])

How much output per worker is being produced at each level of employment.

Marginal product of labor ([latex]MP_L[/latex])

The extra output achieved from the addition of a single unit of labor.

Law of diminishing marginal returns

If a firm increases one input while holding all others constant, the marginal product of the input will start to get smaller. It will still remain positive, just smaller and smaller.

Isoquant

From iso meaning “same” and quant meaning quantity. A curve that shows all the possible combinations of inputs that produce the same output.

Cobb-Douglas production functions

An example:

[latex]Q=AL^\alpha K^\beta[/latex]

A production function that describes a ratio that changes depending on the current mix of workers (labor) and capital (machines, etc). A variant of the Cobb-Douglas function that describes consumer choice with the preference for variety assumption.

Perfect substitute production functions

A production function used when two means of production can substitute for one another in a fixed ratio. Generally have the form:

[latex]Q=aL+\beta K[/latex]

where α and β are positive constants

Perfect complement (fixed-proportions) production functions

A fixed-proportions production function where the inputs are perfect complements:

[latex]Q=A ^*min[\alpha L,\beta K][/latex]

where [latex]A[/latex], [latex]\alpha[/latex], and [latex]\beta[/latex] are positive constants

Marginal rate of technical substitution (MRTS)

A ration of the marginal product of labor ([latex]MP_L[/latex]) and the marginal product of capital ([latex]MP_K[/latex])

[latex]MRTS =-\frac{MP_L}{MP_K}[/latex]

Returns to scale

The rate at which the output increases when all inputs are increased proportionally.

Constant returns to scale (CRS)

Exhibited when a firm’s output changes in exact proportion to changes in the inputs

[latex]f(\Phi L,\Phi K)=\Phi f(L,K)=\Phi Q[/latex]

where [latex]\Phi[/latex] is a strictly positive constant. [latex]\Phi[/latex] becomes fractional in the case of scaling down.

Increasing returns to scale (IRS)

When the output increases or decreases by a greater percentage than the percentage increase or decrease in inputs

[latex]f(\Phi L, \Phi K)> \Phi f (L,K)=\Phi Q[/latex]

where [latex]\Phi[/latex] is a strictly positive constant.

Decreasing returns to scale (DRS)

When the outputs increases or decreases by a smaller percentage than the percentage increase or decrease in inputs.

[latex]f(\Phi L, \Phi K)< \Phi f(L,K)=\Phi Q[/latex]

where [latex]\Phi[/latex] is a strictly positive constant.

Graphs

Short-run total product curve and average and marginal product curves

Examples of isoquants

Isoquants for a production function with inputs that are perfect complements

Isoquants for a production function with inputs that are perfect substitutes

Isoquant with MRTS

Isoquants and returns to scale

Isoquant Cobb-Douglas with outputs increasing

Equations

Average product of labor ([latex]AP_L[/latex])

[latex]AP_L=\frac{Q}{L}[/latex]

Marginal product of labor ([latex]MP_L[/latex])

[latex]MP_L=\frac{\Delta Q}{\Delta L}[/latex]

See also:

[latex]MP_L=\frac{\partial Q}{\partial L}[/latex]

The partial derivative is used when working with [latex]MP_L[/latex], regardless if it is univariate in presentation. In general, it is a bivariate function.

Marginal rate of technical substitution (MRTS)

[latex]MRTS =-\frac{MP_L}{MP_K}[/latex]