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Lecture 14: Firm Production Functions and Cost Minimization

Click here for the quiz on this reading.

Overview of the Model

Just as we derived demand from consumers’ utility functions, the purpose of this analysis will be to derive a firm’s supply decision from its production function. A production function describes the technology available to the firm as it transforms inputs (capital and labor) into some output.

In the consumer model, we analyzed how a consumer would spend a given budget on two goods to maximize a utility function. This was a constrained optimization problem: the agent was trying to maximize an objective function (in that case, a utility function) subject to an exogenously given constraint (a budget line). The consumer took their income $m$, as well as the prices $p_1$ and $p_2$, as given.

In the producer model, we will analyze an unconstrained optimization problem: the firm will take the prices of inputs ($w$ for labor, $r$ for capital) and the price of its output ($p$) as given, and simultaneously choose the amount of output to produce ($q$) and the way in which to produce it ($L$ and $K$), subject only to the constraint that $q = f(L,K)$. In other words, the firm is trying to maximize total revenue (price of output times quantity of output sold, or $py$) minus total cost of inputs (cost of labor $wL$, plus cost of capital $rK$): \(\pi(L,K,q) = pq - (wL + rK)\) subject to the constraint \(q = f(L,K)\) We could actually write down this in two different ways. One would be to simply make the choice one of capital and labor: \(\pi(L,K) = pf(L,K) - (wL + rK)\) The other would be to make the choice one of output \(\pi(q) = pq - c(q)\) and separately solve for $c(q)$ as a “cost function” describing the least-cost method of producing $q$ units of output. This second approach will be the one we develop: essentially, it means solving the cost-minimization problem \(\min wL + rK\) \(\text{s.t. }f(L,K) = q\) We can think of this as the cost-minimization problem of an operations manager within a firm, who receives an order from “management” to produce $q$ units of output and has the job of choosing the lowest-cost way of fulfilling that order.

In fact, this lecture is going to contain almost no new mathematical techniques: we’re going to use the same kinds of functional forms as we did with consumer theory, and the cost minimization problem is fundamentally identical to the one we did last lecture. So most of this lecture is going to be spent understanding the new context of firms.

The Production Function

The production function describes how inputs may be transformed into outputs. For simplicity, we will generally consider a production function with just two inputs: labor ($L$) and capital ($K$, after the German kapital).

In many ways the production function is the analog of the utility function, with one important exception: whereas we didn’t actually care about the “quantity” of utility being “produced” (because it was measured in “utils” or some other such nonsense), we now take seriously the notion of the quantity of output being produced by the production function $f(L,K)$. Let’s look at some of the attributes of a production function

Marginal Products

Another word for the output of a production function is the total product — i.e., $f(L,K)$ is the “total amount produced” using $L$ hours of labor and $K$ units of capital.

Marginal products are to production functions what marginal utilities are to utility functions: the partial derivatives of the function with respect to each of its inputs. In particular, for a production function, the partial derivatives of a represent how much additional output is generated by increasing the level of an input. For example, if the two inputs for a production function are labor and capital, the marginal product of labor ($MP_L$) is the amount by which the total product increases per additional hour of labor, holding $K$ constant: \(MP_L = {\partial f \over \partial L} = \lim_{\Delta L \rightarrow 0} {f(L + \Delta L, K) - f(L, K) \over \Delta L}\) Likewise, the marginal product of capital ($MP_K$) is the amount by which the total product increases per additional unit of capital, holding $L$ constant: \(MP_K = {\partial f \over \partial K} = \lim_{\Delta K \rightarrow 0} {f(L, K + \Delta K) - f(L, K) \over \Delta K}\) Visually, can be seen as the slopes of tangent lines in the $L$ and $K$ directions at a point along the production function:

See interactive graph online here.

It’s important to bear the units of marginal products in mind. In each case, they measure the additional output per additional input. So the $MP_L$ is measured in units of output per hour of labor, and the $MP_K$ is measured in units of output per unit of capital.

Isoquants

One of the core questions we might consider regarding producing something is how to produce it. For example, suppose a firm could produce 20 units of some good using many hours of labor and just a little bit of capital, or just a few hours of labor and a large automated factory, or any number of other combinations of labor and capital. The set of all such combinations that could produce a given amount of output is of interest to the firm, because it provides a set of options to choose from.

For a production function with two inputs — such as labor ($L$) and capital ($K$) — the visual representation of this menu of options called an isoquant, from the Greek iso meaning “same,” and quant meaning “amount.” An isoquant shows the set of all combinations of labor and capital that could be used to produce $q$ units of output: \(\text{Isoquant for }q = \{(L,K)\ |\ f(L,K) = q\}\) Mathematically, this is just the level set for quantity $q$, just as an indifference curve is a level set for some utility $U$. And just as we could draw an indifference map for a utility function, we can draw an “isoquant map” for a production function. For example, the following diagram shows the surface plot, isoquant, and isoquant map for $f(L,K) = 4L^{1 \over 2}K$:

See interactive graph online here.

The Marginal Rate of Technical Substitution

Fundamentally, the isoquant illustrates a tradeoff. Suppose a firm is currently using some amount of labor and capital $(L,K)$. It might consider adding some capital and firing some workers, while keeping its output constant. To do so, it would need to figure out how much more capital it would need ($\Delta K$) to reduce its workforce by some amount ($\Delta L$). The rate of additional capital needed per labor reduced, $\Delta K / \Delta L$, is called the marginal rate of technical substitution between labor and capital. (Note: Some textbooks refer to this as the “Technical Rate of Substitution.”)

Visually, the MRTS is represented by the magnitude of the slope of an isoquant:

See interactive graph online here.

How do we calculate the MRTS? As with the MRS, we can use implicit function theorem to simply assert that the slope along an isoquant is given by \(\left.{dK \over dL}\right|_{f(L,K) = q} = - {\partial f/\partial L \over \partial f/\partial K}\) Since the MRTS is the magnitude of the slope, it’s therefore given by the formula \(MRTS = \left|- {\partial f/\partial L \over \partial f/\partial K}\right| = {MP_L \over MP_K}\) This is, of course, the equivalent of the consumer theory definition of the MRS as the ratio of the marginal utilities.

There’s a good economic interpretation of this formula. Let’s think about this in terms of the marginal products of labor and capital. The marginal product of labor says that the change in output due to a small change in labor is given by \(MP_L = {\Delta q \over \Delta L}\) Suppose the firm were to reduce its labor by just enough to drop its production by $\Delta q$. Solving the above equation for $\Delta L$, this would mean it would need to reduce its labor by \(\Delta L = -{\Delta q \over MP_L}\) To stay along the isoquant, the firm then needs to add capital to bring its output back up by $\Delta q$. By the same argument, the additional amount of capital it would need to use to raise its output by $\Delta q$ would be \(\Delta K = {\Delta q \over MP_K}\) Therefore \(MRTS = \left|{\Delta K \over \Delta L}\right| = \left|\frac{\Delta q / MP_K}{-\Delta q / MP_L}\right| = {MP_L \over MP_K}\)

Different functional forms for different production technologies

So far we’ve analyzed the features of a production function: given $f(L,K)$, we can find the marginal products, plot the isoquants, and calculate the MRTS.

Just as with utility functions, we can use different functional forms to describe different production processes. In fact, the utility functions we looked at in the first part of this course were “born” as production functions: as you might imagine, economists were first interested in firms, and developed some ways to model production; and when it came time to look at preferences they basically said to themselves, “hmmm, what if we just think of this like consumers are producing some good called utility…”

For example, a linear production function (of the sort that we would call “perfect substitutes” for a consumer) has the form \(f(L,K) = aL + bK\) For example, a human cashier and a self-checkout kiosk perform approximately the same job; you might be able to swap one out easily for the other.

On the other hand, the analogue of a “perfect complements” utility function is called a “Leontief” or “fixed-proportions” production function, and takes on the form \(f(L,K) = \min\{aL, bK\}\) This might be thought of as a human cashier and the cash register they use; the cashier would be useless without a cash register, while a cash register would also be useless without a human to operate it.

There are many other possible functional forms, and you probably have some sense from our analysis of utility functions how they might affect the decisions a firm makes. However, we have enough to worry about in this unit without thinking about a bunch of different forms of production functions. So for this course, we’re going to focus on just one that’s very familiar to us: Cobb-Douglas.

Cobb-Douglas

In 1928, mathematician Charles Cobb and economist (and future politician!) Paul Douglas published an article in the American Economic Review titled “A Theory of Production.” In it, they attempted to write down a single production function which might capture the relationship between capital and labor in an economy. They suggested the functional form \(q = f(L,K) = AL^aK^b\) This wasn’t based on their observation of any particular production process, but rather because the mathematical properties of the function implied that (a) the shares of GDP accruing to labor and capital would remain relatively constant over time, even if the prices of those goods varied, and (b) it could be estimated using linear regression by taking the log of both sides: \(\ln q = \ln A + a \ln L + b \ln K\) Their work was so influential that it bears their name to this day — even though other economists such as Leon Walras had previously used the functional form.

The marginal products of labor and capital are given by \(\begin{aligned} MP_L &= aAL^{a - 1}K^b\\ MP_K &= bAL^aK^{b-1}\\ \end{aligned}\) Here, both the $MP_L$ and the $MP_K$ depend on the amount of labor and capital used.

The following diagram allows you to play around with the parameters of $A$, $a$, and $b$, to see how they affect the isoquant map:

See interactive graph online here.

One counterintuitive aspect about isoquants is the fact that an increase in productivity leads to an inward shift of the isoquants. (You can see this if you raise $A$ in the diagram above.) This is because better technology means you can produce a given quantity with fewer inputs (less capital and labor); so the isoquant for any given quantity shifts in toward the origin when technology improves!

As usual with Cobb-Douglas, each of the isoquants is bowed in toward the origin: that is, the MRTS is decreasing as you move a long an isoquant by using more labor and less capital. This is true regardless of whether there are increasing, decreasing, or constant marginal returns to labor and capital. What’s the intuition behind this for a production function?

Intuitively, what’s happening here is that the levels of labor and capital affect both the $MP_L$ and the $MP_K$: as you increase labor, you make capital more productive, and vice versa.

As you move down and to the right along an isoquant, you’re adding labor, which makes capital more productive. At the same time, you’re reducing capital, which makes labor less productive. So as you shed capital, you have to hire more and more labor to make up for the lower productivity.

We can also think of it in the opposite direction: suppose you have a lot of labor, and you’re thinking of firing some people and replacing their jobs with automation. As your labor force decreases, the people who remain are doing more and more work, so they become increasingly valuable; so as you continue reducing your labor force, you need to use more and more machines to do the work of the people you’ve just fired.

Properties of Production Technologies

Fundamentally, there are two aspects of any production technology which are interesting to us: what happens as you move along an isoquant (i.e. substitute labor for capital), and what happens as you move between isoquants (i.e. scale production). The first concerns the elasticity of substitution; the second, returns to scale.

Elasticity of Substitution

Thinking about the different kinds of production functions, one important feature concerns how substitutable capital and labor are. For example, with a linear production function, capital and labor are what we might call perfect substitutes: if $f(L,K) = L + K$, for example, you can always replace one hour of labor with one unit of capital. On the other hand, with a Cobb-Douglas production function, as we just saw, the MRTS changes as you move along an isoquant.

One measure of the substitutability of capital and labor is called the elasticity of substitution. A CES (or “constant elasticity of substitution”) function, similar to consumer theory, takes the form \(f(L,K) = (aL^\rho + bK^\rho)^{1 \over \rho},\)where the Greek letter $\rho$ is a parameter related to the elasticity of substitution: in particular, the term “elasticity of substitution” is defined as \(\sigma = {1 \over 1 - \rho}\) If you drag the blue dot along the isoquant in the graph below, you can see that the MRTS changes; and by using the slider, you can see how the elasticity of substitution affects the behavior of the MRTS:

See interactive graph online here.

In this case, it diminishes, or gets smaller in absolute value, as you move to the right. This means that as you use more and more labor, the amount of capital it would take to replace one unit of labor gets smaller. The elasticity of substitution is defined as the percentage change in the MRTS due to a $1\%$ change in the ratio of capital to labor, $K/L$, as one moves along an isoquant.

Use the slider to change the elasticity of substitution. As you can see, if you drag the elasticity all the way to the left, the isoquants are extremely “bendy,” even to the point of being almost L-shaped; in the middle, the isoquants have the “bowed” shape of the Cobb-Douglas production function; and as it becomes perfectly elastic, the isoquants become linear, indicating that capital and labor are perfectly substitutable.

Just as we saw that goods could be complements or substitutes in consumer theory, the elasticity of substitution reflects whether labor and capital are complements in production or substitutes in production. The substitutability of labor and capital has profound political and moral implications. A centerpiece of Andrew Yang’s 2020 Presidential campaign centered around solving the problem of structural job loss caused by automation — an example of companies shifting their production processes away from labor and toward capital. Semi-autonomous truck convoys may be able to deliver goods more safely, more quickly, and with lower emissions than human-driven trucks; but such automation threatens thousands of long-haul truck driving jobs, which are some of the best-paying jobs for workers without a college degree. So the elasticity of substitution isn’t just a dry mathematical formula: at its core, it’s a measurement of how much people’s jobs are at risk of being lost to automation. That doesn’t make the concept “bad,” any more than an earthquake is “bad;” but it does mean that if you’re interested in solving problems of income inequality, understanding the substitutability of labor and capital has got to figure prominently in your analysis.

Returns to scale

The other way to analyze a production technology is to think about what happens to the total output of a production function as we “scale up” production. While there are many useful ways to consider this, one particularly important one is to analyze what happens if we scale all of those inputs proportionally: for example, increasing all inputs by 10%, or doubling all inputs: that is, if we double all inputs, what happens to the output? Does it also double, or less than double, or more than double? In other words, does output scale proportionally to inputs?

Mathematically, the way we can analyze this is by comparing the amount that’s produced with the input combination $(L,K)$ and the combination $(2L,2K)$. For example, for a Cobb-Douglas production function of the form \(f(L,K) = AL^aK^b\) If we double all inputs, we have \(f(2L, 2K) = A(2L)^a(2K)^b = 2^{a+b}AL^aK^b = 2^{a+b}f(L,K)\) Since $2^1 = 2$, this means the function is decreasing returns to scale if $a + b < 1$; constant returns to scale if $a + b = 1$; and increasing returns to scale if $a + b > 1$.

You can use the following graph to examine how doubling just labor or doubling both labor and capital affect output for a Cobb-Douglas production function. The solid blue isoquant shows $q = f(L,K)$; the dashed blue isoquant shows double that quantity, $\hat q = 2f(L,K)$:

See interactive graph online here.

How can we observe returns to scale in this diagram?

One thing you can also do is to rotate the 3D graph so that the orange line appears on the top of the hill. You can see that when $a + b = 1$, it has a constant slope; when $a + b > 1$, it has an increasing slope; and when $a + b < 1$, it has a decreasing slope.

In the real world, production functions may scale differently at different levels of input use: many startups may have increasing returns to scale when they are small and expanding can allow them to achieve greater efficiencies, but start experiencing a loss of efficiency as they grow and become more corporate in nature. This is just one more example of a case in which we’ll start out by looking at mathematically simple but unrealistic functions to get our start in economic modeling; if you continue in your studies of economics, you’ll see more interesting and realistic mathematical structures.

Cost Minimization for Firms

If an isoquant shows a “menu of options” for producing a certain quantity $q$, the natural question to ask is: what is the optimal point along an isoquant? Our assumption is going to be that whatever quantity a firm wants to produce, it wants to do so at the lowest possible cost; hence, if isoquant as a “menu” of options for producing a certain level of output, we’re looking for the cheapest “item” (i.e. combination of inputs) on that menu.

Isocost Lines

Let’s start by thinking about how much any combination of labor and capital costs. If the firm has to pay $w$ for each unit of labor and $r$ for each unit of capital, then the total cost of some combination of inputs $(L,K)$ is \(c(L,K) = wL + rK\) Conversely, the set of all combinations of labor and capital that cost some amount $c$ may be given by the equation \(wL + rK = c\) If we plot this line in a graph with labor on the horizontal axis and capital on the vertical axis, we get what we might call an isocost line: that is, all combinations of labor and capital that cost the same amount:

See interactive graph online here.

The cost minimization problem

The goal of the firm’s cost minimization problem is to produce a given quantity at the lowest possible cost: that is, find the point along an isoquant which is along the lowest possible isocost line.

The key thing here is that we’re treating the amount of output as fixed; that is, some target amount $q$. In other words, the firm faces a constrained optimization problem to produce this amount $q$ at the lowest possible cost. We can write this problem as \(\begin{aligned} \min_{L,K} \ \ & wL + rK\\ \text{s.t. } \ & f(L,K) = q \end{aligned}\) If the firm is solving this problem correctly, then it is choosing a point along its isoquant that costs as much or less than every other point along an isoquant.

Visually, this occurs at a point along the isoquant where the relevant isocost line never crosses the isoquant: that is, every other point along the isoquant costs more than that point:

See interactive graph online here.

The “gravitational pull” along an isoquant

Suppose a firm is not currently solving the problem correctly: it’s either using too much labor (and too little capital) or too much capital (and too little labor). If that’s the case, then it could save money by shifting its production mix around.

In particular, suppose the firm is currently using some production combination $(L,K)$ to produce $q$ units of output, and is considering using a bit more labor and a bit less capital. In particular, let’s assume that if it used some amount $\Delta L$ more labor and $\Delta K$ less capital, it would stay along the isoquant. If it did so, its labor expenditures would increase by $w \times \Delta L$, and its capital expenditures would decrease by $r \times \Delta K$. It should do this if the amount it saved on capital would be greater than the amount it cost in additional labor: \(r \times \Delta K > w \times \Delta L\) or \({\Delta K \over \Delta L} > {w \over r}\) By construction, since they’re staying along the isoquant, the left-hand side of this is just the marginal rate of technical substitution (MRTS), or the magnitude of the slope of the isoquant; so this is just another way of saying that at such a point, the isoquant would be steeper than the isocost line: \(MRTS > {w \over r}\) Intuitively, if the slope of the isoquant is not equal to the slope of the isocost line passing through that point, then there is an area of overlap between the set of input combinations that cost less, and the set of input combinations that produce more output.

Solving for the cost-minimizing bundle

The “objective” of the firm is to minimize the expenditure $wL + rK$; the “constraint” is that it wishes to produce $q$ units of output. Therefore, if calculus works to find the solution, relevant Lagrangian for this problem is \(\mathcal{L}(L,K,\lambda) = wL + rK + \lambda [q - f(L,K)]\) which has the three first-order conditions \(\begin{aligned} \frac{\partial \mathcal{L}}{\partial L} &= w - \lambda \times MP_L = 0 \Rightarrow \lambda = \frac{w}{MP_L}\\ \frac{\partial \mathcal{L}}{\partial K} &= r - \lambda \times MP_K = 0 \Rightarrow \lambda = \frac{r}{MP_K}\\ \frac{\partial \mathcal{L}}{\partial \lambda} &= q - f(L,K) = 0 \Rightarrow q = f(L,K) \end{aligned}\) Setting the $\lambda$ for the first two conditions equal to each other gives us the condition \(\frac{MP_L}{MP_K} = \frac{w}{r}\) The left-hand side of this is just the expression for the MRTS. Therefore, this represents a tangency condition between the isoquant constraint and the isocost lines defined by the price ratio $w/r$.

What is our interpretation of the Lagrange multiplier $\lambda$? As always, the Lagrange multiplier represents the effect on the objective function of relaxing the constraint by one unit. In this case, the constraint is defined by the quantity $q$, and the objective function is the cost of producing $q$ units; so $\lambda$ represents the marginal cost of producing an additional unit.

Intuitively, both $w/MP_L$ and $r/MP_K$ are such marginal costs: the first is the marginal cost of producing another unit using labor, and the second is the marginal cost of producing another unit using capital. More specifically, $1 / MP_L$ is the amount of labor required to produce an additional unit of output by increasing labor; multiplying that by $w$ gives the cost of producing that unit with labor. Likewise, $1 / MP_K$ is the amount of labor required to produce an additional unit of output by increasing capital; multiplying that by $r$ gives the cost of producing that unit with capital.

Worked example

Throughout this unit, we’re going to use the Cobb-Douglas production function \(f(L,K) = \sqrt{LK}\) Let’s look at this algebraically and visually first, and then get to the analytical solution. The graph below shows five different ways of producing $q = 16$ units of output given our production function $f(L,K) = \sqrt{LK}$. The lines show green isocost lines given the wage rate $w$ and the rental rate of capital $r$; the table evaluates the cost of each of the possible input combinations. Note that the lowest-cost combination of output corresponds to the farthest-in isocost line. For $w = 8$ and $r = 2$, this combination is $L = 8, K = 32$:

See interactive graph online here.

We can see that, in this case, the lowest-cost combination of labor and capital is the point along the isoquant where the isoquant is tangent to the isocost line. For the production function $q = \sqrt{LK}$, the $MRTS = K/L$, so the tangency condition is \(\frac{K}{L} = \frac{w}{r} \Rightarrow K = \frac{w}{r}L\) Plugging this into the constraint (i.e., the isoquant) gives us \(\begin{aligned} q &= \sqrt{L\times\left[\frac{w}{r}L\right]}\\ q &= \sqrt{\frac{w}{r}} \times L\\ L^c(w,r,q) &= \sqrt \frac{r}{w} \times q \end{aligned}\) and therefore \(K^c(w,r,q) = \frac{w}{r}L^c = \sqrt \frac{w}{r} \times q\) For example, if $w = 8$, $r = 2$, and $q = 16$, the cost-minimizing combination of labor and capital is \(\begin{aligned} L^c(w,r,q) &= \sqrt \frac{r}{w} \times q^2 = \frac{1}{2} \times 16 = 8\\ K^c(w,r,q) &= \sqrt \frac{w}{r} \times q^2 = 2 \times 16 = 32 \end{aligned}\) as we found before. Visually, we can see that this occurs at the intersection of the line representing the tangency condition $K = \frac{w}{r}L$ and the isoquant for $q = 16$:

See interactive graph online here.

Next, let’s see how this optimal bundle changes as prices and target output change, and how we can use the solution to the cost minimization problem to derive the firm’s long-run total cost of production.

Conditional Demands for Labor and Capital in the Long Run

Obviously the cost-minimizing combination of inputs depends on the prices of the inputs ($w$ and $r$) and the amount the firm wants to produce ($q$). Therefore, we can write the solution to the firm’s cost minimization problem as a function of these exogenous variables: \(L^c(w,r,q)\) \(K^c(w,r,q)\) We call these the conditional demands for labor and capital: they are the amount of labor and capital the firm will buy, given prices $w$ and $r$, holding $q$ fixed. The reason this is called the conditional is because it’s “conditional” on the target output level $q$ (i.e., the isoquant). Students in 50Q might realize that this is the same concept as the Hicksian demand for a consumer, which is conditional on some target utility.

Expansion path

One way of visualizing a firm’s conditional demands for labor and capital is plot out how much of each unit it would use to produce various levels of output. Because this illustrates the firm’s choice as it “expands,” this is called an expansion path. To construct it, for each potential level of output, you plot the amount of labor and capital used, and connect the dots:

See interactive graph online here.

Summary

This was a lot to read, but my anticipation is it will go fairly quickly in lecture, for the simple reason that there isn’t actually much new here – it’s just a new way to look at the same kind of mathematical objects we have already seen, just in a new context.

To sum up:

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Copyright (c) Christopher Makler / econgraphs.org