For more information on these topics, see Varian Chapter 18: Technology.
This chapter begins our study of producer theory. If you understand consumer theory, understanding producer theory will be much easier, because, as you’ll see, there will be a lot of important symmetries.
A production set describes all the technologically feasible combinations of inputs and outputs for a firm. It shows all the possible ways a firm can transform inputs into outputs given its available technology.
Any input to a production process is called a factor of production. There are four main categories of inputs: land, labor, capital goods, and raw materials. Capital goods refer to machinery: they are inputs to the production process which are themselves produced goods. Think tractors, buildings, computers, etc.
For example: suppose it only takes flour to make bread (ignoring other ingredients, labor hours, ovens, electricity, and land), and it takes 1 cup of flour to make 1 loaf of bread. Then how many loaves of bread could you make if you had 4 cups of flour? You could make 0 loaves, 1 loaf, 2 loaves, 3 loaves, or (at the most) 4 loaves. Your production set would look like this, with cups of flour on the x-axis and loaves of bread on the y-axis:
The boundary of the production set represents the maximum output that can be produced for a given level of inputs. This boundary of the production set is known as the production function. In the bread example, the production function would be \(f(x) = x\), where \(x\) refers to cups of flour and \(f(x)\) refers to the output, which is bread.
The marginal product of an input is the additional output produced by using one more unit of that input, holding all other inputs constant. It’s a measure of how much extra output you get from a small increase in one input.
It’s also the derivative of the production function. Take our bread example. If \(f(x) = x\), then we have \(f'(x) = 1\): if you add another cup of flour, you can always make another loaf of bread.
In many production processes, we observe the principle of diminishing marginal product. This means that as we increase the amount of one input while holding others constant, the marginal product of that input tends to decrease.
Suppose a data entry clerk is tasked with processing records. The number of records processed (output) can be considered as a function of labor hours (input). Assuming the worker’s skill level and tools used remain constant, the amount of data entered is primarily dependent on the time spent working.
Here’s how the output might look over an 8-hour workday:
In this example, we can see that the marginal product (the number of additional records processed each hour) is decreasing. This illustrates diminishing marginal product. In this case, the clerk is getting tired and slows down. But you can imagine more reasons that increasing one input while holding others constant will lead to smaller and smaller increases in output:
Graphically, here’s what a production function will look like when it has diminishing marginal product:
Returns to scale is a slightly different idea from what we’ve discussed so far. Instead of holding other inputs constant and increasing just one input, returns to scale asks what happens if you double (or triple, or whatever) ALL inputs.
For example, suppose you have a pizza shop. What happens if you double everything - twice as much dough, twice as many ovens, twice as many workers, twice as much space? Will you make exactly twice as many pizzas? More than twice as many? Or less than twice as many?
The answer to this question determines the type of returns to scale your pizza shop exhibits:
Constant Returns to Scale: When all inputs are increased by a factor of \(k\), output also increases by a factor of \(k\). Mathematically: \(f(kx, ky) = kf(x, y)\).
Increasing Returns to Scale: When all inputs are increased by a factor of \(k\), output increases by more than \(k\). Mathematically: \(f(kx, ky) > kf(x, y)\).
Decreasing Returns to Scale: When all inputs are increased by a factor of \(k\), output increases by less than \(k\). Mathematically: \(f(kx, ky) < kf(x, y)\).
It’s important to note that a firm can experience different types of returns to scale at different levels of production. For instance, a firm might experience increasing returns to scale when it’s small, constant returns to scale as it grows to a medium size, and then decreasing returns to scale if it becomes very large.
Constant returns to scale (CRS) is often considered a reasonable long-run expectation for many industries. This is because of the “replication argument.” The idea is that if a firm is operating at its most efficient scale, it should be able to simply replicate its entire operation to double its output. For instance, if a factory is operating efficiently, building an identical factory right next to it should double the output. However, it’s important to note that while CRS might be a good theoretical baseline, real-world factors like limited resources, market size, or management complexities can lead to increasing or decreasing returns to scale in practice.
When a production process uses two inputs, we can represent their relationship graphically using an isoquant. An isoquant shows all the different combinations of the two inputs that can produce a specific level of output.
For example, consider a farm that produces wheat using two inputs: labor (measured in hours of work) and fertilizer (measured in kilograms). The farm might be able to produce 1000 bushels of wheat in several ways:
100 hours of labor and 500 kg of fertilizer
150 hours of labor and 300 kg of fertilizer
200 hours of labor and 200 kg of fertilizer
300 hours of labor and 100 kg of fertilizer
These combinations all lie on the same isoquant. They show how the farmer can substitute between labor and fertilizer while maintaining the same level of output (1000 bushels). Using more labor allows the use of less fertilizer, and vice versa.
Isoquants are similar to indifference curves in consumer theory, but there’s an important difference:
Indifference curves show combinations of goods that give a consumer the same level of satisfaction.
Isoquants show combinations of inputs that produce the same level of output.
The key distinction is that while the “utility” on indifference curves is an arbitrary measure, the output represented by isoquants is typically measured in concrete, meaningful units (like bushels of wheat in this example).
Mathematically, isoquants are level curves of the production function. Each isoquant represents a different level of output, and as you move to isoquants further from the origin, they represent higher levels of output.
For a very familiar example: if you’re a cookie producer and suppose in a very simplistic case, you just have those two inputs. If you need a cup of butter or a cup of margarine (or half a cup of both) to make a batch of cookies, here’s your production function: \(f(x_1, x_2) = x_1 + x_2\). If I want to draw isoquants for 1, 2, and 3 batches, I’d set the output level to 1, 2, or 3, and then solve for \(x_2\) on the y-axis to draw it using ggplot:
library(tidyverse)
ggplot() +
stat_function(fun = function(x1) 1 - x1, color = "green") +
stat_function(fun = function(x1) 2 - x1, color = "purple") +
stat_function(fun = function(x1) 3 - x1, color = "blue") +
xlim(0, 3.5) +
ylim(0, 3.5)
Of course, this example is a lot like the perfect substitutes example from consumer theory.
Consider a production process like this: you want to dig holes and to dig a hole, you need one person and one shovel. Extra shovels wouldn’t help, and extra people wouldn’t help. Then if you had just one shovel, it doesn’t matter how many people you have, if you have at least one, you can produce one hole. You’d write down the production process as \(f(x_1, x_2) = \min (x_1, x_2)\) and of course isoquants would look like right angles.
ggplot() +
geom_segment(aes(x = 1, xend = 1, y = 1, yend = 4), color = "pink") +
geom_segment(aes(x = 1, xend = 4, y = 1, yend = 1), color = "pink") +
geom_segment(aes(x = 2, xend = 2, y = 2, yend = 4), color = "orange") +
geom_segment(aes(x = 2, xend = 4, y = 2, yend = 2), color = "orange") +
geom_segment(aes(x = 3, xend = 3, y = 3, yend = 4), color = "limegreen") +
geom_segment(aes(x = 3, xend = 4, y = 3, yend = 3), color = "limegreen") +
xlim(1, 4) +
ylim(1, 4)
Consider the production function \(f(x_1, x_2) = \min(x_1, x_2)\).
Use ggplot to sketch an isoquant.
Find the marginal product of input 1. That is, if you hold \(x_2\) constant and increase \(x_1\), how much does output increase? The marginal product is also the partial derivative.
Does this production function have increasing, decreasing, or constant returns to scale? That is, if you double all inputs, is the resulting output greater than double, less than double, or exactly double?
Consider the Cobb-Douglas production function \(f(x_1, x_2) = x_1^2 x_2^2\).
Use ggplot to sketch an isoquant.
Does this production function have increasing, decreasing, or constant returns to scale?
The Cobb-Douglas production function is given by \(f(x_1, x_2) = A x_1^a x_2^b\). Show that the returns to scale depend on the magnitude of \(a + b\). Which values of \(a + b\) can be associated with which types of returns to scale?
The marginal rate of technical substitution (MRTS) is the slope of the isoquant (rise \(\Delta x_2\) over run \(\Delta x_1\)). If the MRTS = 2, then you know output can be held constant either by increasing \(x_2\) by 2 units (that’s the rise \(\Delta x_2\) part), or by increasing \(x_1\) by 1 unit (the run \(\Delta x_1\) part). Let’s derive the formula for the MRTS: start with the total differential of the production function \(df = \frac{\partial f(x_1, x_2)}{\partial x_1} dx_1 + \frac{\partial f(x_1, x_2)}{\partial x_2} dx_2\). Letting output \(f\) stay constant (\(df = 0\)), solve for \(\frac{dx_2}{dx_1}\) to get a formula for the MRTS.
Use your answer to question 4 to find the MRTS for the Cobb-Douglas production function \(f(x_1, x_2) = 3 x_1^{.5} x_2^{.5}\), evaluated at \((x_1, x_2) = (3, 1)\). Interpret the MRTS by filling in the blank: output can be held constant either by increasing \(x_2\) by _____ units or by increasing \(x_1\) by _____ units.