# 11.2 Cost Minimization

If a firm has multiple variable inputs, it faces a *cost minimization* problem: what is the least-costly way of producing a given level of output? That is, if we view a firm’s isoquant as a “menu” of options for producing a certain level of output, what is 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:

## 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, (Note: In fact, this is identical to the cost minimization problem we used when finding the Hicks decomposition bundle for a consumer. Just as the utility function maps goods into utility, the production function maps inputs into the quantity of goods produced. In each case, the agent takes the prices of the goods being bought as given, and the agent is trying to minimize the amount spent.) 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:

## 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.

## When calculus works

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 amonut 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.

## Kinks and corners

For production functions that don’t have a smoothly decreasing MRTS, the Lagrange method will not work. For example, if we have the Leontief production function $f(L,K) = \min{2L, 3K}$, the cost-minimizing way to produce any quantity of output will be to produce at the base of the L-shaped isoquant: that is, $q = 2L = 3K$. In this case the conditional demands for labor and capital will be $L(q) = {1 \over 2}q$ and $K(q) = {1 \over 3}q$. Likewise, if we have the linear production function $f(L,K) = 2L + 3K$, firm will cost minimize by producing either entirely with labor or entirely with capital (or be indifferent between the two). All the techniques we derived in Part I apply here.

## Worked example

For this chapter, we’re going to use the 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 $L = 8, K = 32$:

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 \(\begin{aligned} q &= \sqrt{L\times\left[\frac{w}{r}L\right]}\\ q &= \sqrt{\frac{w}{r}} \times L\\ L^c &= \sqrt \frac{r}{w} \times q \end{aligned}\) and therefore \(K^c = \frac{w}{r}L^c = \sqrt \frac{w}{r} \times q\) For example, if $w = 8$, $r = 2$, and $q = 16$, the cost-minimizing combination of labor and capital is \(\begin{aligned} L^c &= \sqrt \frac{r}{w} \times q^2 = \frac{1}{2} \times 16 = 8\\ K^c &= \sqrt \frac{w}{r} \times q^2 = 2 \times 16 = 32 \end{aligned}\) 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 $q = 16$:

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.