Chapter 7: Production Functions

The primary function of a firm is to make things! In fact we can think of a firm as a process that transforms inputs into outputs. Specifically each firm implements one or more production functions to make its product or service. For the most part we will consider single production function examples, but other scenarios certainly exist. We can model the firm’s production process using any number of mathematical relationships but in Intermediate Microeconomics we generally focus on functional forms reminiscent of consumer theory: Cobb-Douglas, Perfect Substitutes, Perfect Complements (Leontief), and Quasilinear.

We select a particular production function to model our firm based on how its production process treats inputs in relation to its ability to generate its output. For instance, if a firm needs to use labor and capital in the same fixed proportion as would be the case where a worker needs to operate a machine, we use a perfect complements production function. If labor and capital are interchangable at some constant rate we use perfect substitutes. If labor and capital are interchangable at a rate dependent on how much of both the firm is currently using we use Cobb Douglas.

Is labor or capital on the vertical axis? I will always think of labor (L) as the first argument in a firm’s production function and capital (K) as the second. I will write F(L,K) to denote a production function. One reason why I prefer this convention is that when we’re thinking about a short run optimization problem and therefore set some constant for the level of capital this will sit at the end of the equation out of our way (and not immediately getting in the way of any scaling factor as would be the case with Cobb-Douglas). Each author needs to establish a convention for writing down these production functions and some have chosen differently. If you refer to a different textbook some authors write capital (K) as the first argument and labor (L) as the second argument. But following my convention I’ll always write these as:

Perfect substitutes:

\(F(L,K) = \alpha L + \beta K\)

Perfect complements (Leontief):

\(F(L,K) = min\{\alpha L, \beta K\}\)

Cobb-Douglas:

\(F(L,K) = AL^{\alpha}K^{\beta}\)

For concreteness the productive inputs we’re focusing on are generally labor and capital, but we are not restricted to just two inputs. In the cases where we don’t want to specifically refer to labor or capital we can use the more generic X1, X2, …Xn. How can we measure labor and capital, especially for mathematical comparison within our production function? While we won’t want to dwell on this issue, typically we’ll just think of L units of labor and K units of capital and understand that behind the scenes the interpretation is for inputs to be measured in flow units such as labor hours per week or machine hours per day.

We will define the production set as the collection of possible inputs and outputs that the firm could produce. Just as with the budget set where we most often focused on the budget line itself, here we will focus mainly on the collection of inputs that yield the maximal possible output. Conveniently this is specified precisely by the production function itself; the maximal boundary of the production set is the graph of the production function.

We define an isoquant to the set of all possible combinations of the inputs L and K that yield the same level of output. These are similar to indifference curves in the sense that we can get a whole collection of isoquants, one for each possible level of output. These differ from indifference curves in another sense: Typically in consumer theory we sought the highest level of utility the consumer could afford and therefore inspected a collection of indifference curves to find the best. But in producer theory often there’s a specific level of output that must be attained (in order to minimize costs) and therefore to the extent that its useful to graph isoquants we most often draw a single one corresponding to the desired level of output (much like we drew a single budget constraint corresponding to the actual level of income).

A further important observation is that whereas a consumer seeks to maximize utility our firm does not necessarily want to maximize output. Though this is sometimes a useful optimization exercise, most often the firm is seeking to maximize profits or minimize costs!

We have two key assumptions that help us focus our attention on valid production functions:

We assume production functions are monotonic which means that if we use more of one input we should be able to produce at least as much output as we could prior to that increase. This implies free disposal which basically refers to being able to costlessly discard unused inputs.

We assume technologies are convex which means if there are two alternative ways to produce a given level of output their weighted average will be able to produce at least that same amount.

In consumer theory an important concept was the consumer’s marginal utility which we generated as the partial derivative of the utility function with respect to that particular good. In producer theory the analogous entity is the firm’s marginal product of a particular input. This is just the partial derivative of the production function with respect to that input. Hopefully this is relatively natural as marginal product is a key term from microeconomic principles: It’s the additional output that is produced when we use one more unit of that input.

For a given production function F(L,K), formally the marginal product of labor and capital are just:

\(\huge \substack{MP_L = \frac{\partial F}{\partial L} \\\\\\\\\\\\\\\\\\\\\ MP_K = \frac{\partial F}{\partial K} \\\\\\\\\ }\)

Just like with the Marginal Rate of Substitution (MRS) that describes the tradeoff between the goods governed by the slope of the indifference curve, here we have the Technical Rate of Substitution that describes the tradeoff between the goods governed by the production function.

\(TRS = \frac{dK}{dL} = - \frac{MP_L}{MP_K}\)

Note: Other authors use a variety of terms to all mean this same thing. You may also see Rate of Technical Substitution (RTS), Marginal Rate of Technical Substitution (MRTS), and possibly Marginal Technical Rate of Substitution (MTRS). I will only use TRS.

You might wonder why it’s dK/dL but then MPL/MPK and not dL/dK? Remember from indifference curves, the slope of the indifference curve is dY/dX (the derivative of the IC) and the MRS is MPx/MPy.

Just like with consumer theory where we assumed diminishing marginal rate of substitution (implied by convex preferences), here we assume diminishing marginal product. Diminishing marginal product occurs when the additional output generated from using one more unit of the input is decreasing as additional inputs are employed. It’s basically due to a bottleneck issue, typically due to capital being fixed.

Here’s an example: Suppose I have a lawn-mowing business and I have one riding lawnmowers and no workers. Without any labor we’re not able to mow any lawns (the unit of output). If I hire the first worker perhaps I can mow 2 lawns per hour. The marginal product of the first worker is therefore 2 lawns. If I hire a second worker suppose I can mow 3 lawns per hour. The marginal product of the second worker is now just 1 lawn. That’s perhaps because while I only have a single mower, the second person is able to assist the first worker and allow them to take a break to attend to other tasks while we keep the machine continuously operating by moving on to the next lawn. If I hire a third worker maybe I can still only mow 3 lawns per hour. The marginal product of the third worker was 0. They didn’t add anything because now at any given time we have 1 person just standing around idle. Maybe I hire a fourth worker and now I can only do 2 lawns per hour. The marginal product of the fourth worker is negative (-1). Maybe now worker 3 and worker 4 are getting in the way of those actually working and/or distracting them! We can solve this whole problem by hiring fewer workers. But we can also boost the marginal product of labor by alleviating the bottleneck of capital. If we add 3 more mowers now we can mow 8 lawns per hour (each of 4 workers are individually able to mow 2 lawns!).

Diminishing technical rate of substitution is the condition where as we increase the amount of labor and, in order to stay on the same isoquant, we reduce the amount of capital used, the TRS falls (because MPk/MPl). This implies convex isoquants just like the condition of diminishing marginal rate of substitution implied convex preferences.

Lastly we consider returns to scale: which refers to what happens to the proportion of output we’re able to produce as we increase the level of all inputs.

Constant returns to scale (CRS) implies that if we double all inputs we’ll get exactly double the output in return. Formally this is:

\(f(tx_1, tx_2) = tf(x_1, x_2)\)

Decreasing returns to scale (DRS) implies if we double all inputs we’ll get less than double the amount of output in return. Formally this is:

\(f(tx_1, tx_2) < tf(x_1, x_2)\)

Increasing returns to scale (IRS) implies if we double all inputs we’ll get more than double the amount of output in return. Formally this is:

\(f(tx_1, tx_2) > tf(x_1, x_2)\)

At bit of algebra and careful mathematical reasoning is required in order verify returns to scale for a specific production function. Take for example a Cobb-Douglas production function like: F(L,K) = LK

We scale each input by some constant factor, t, like this:

\(F(tL, tK)\)

But if this is true, we have to propagate this scaling through the function itself:

\(\huge \substack{ =(tL) \cdot (tK) \\\\\\\\\\\\\\\ =t^2 \cdot LK \\\\\\\\\\\\ }\)

But that’s clearly larger than the original function, =LK so we write lastly:

\(F(tL, tK) =t^{2}LK > LK\)

And we conclude the production technology, F(L,K) = LK has increasing returns to scale. It turns out for Cobb-Douglas there’s a bit of a shortcut. Note in this example we set the exponents equal to 1. In general if ab > 1 we have IRS, if ab =1 we have CRS, and if ab < 1 we have DRS!

Here’s the YouTube video version: