11.3 Long-Run Costs

In the long run, when a firm can vary all inputs, the long-run total cost of producing $q$ units of output is the cost of the cost-minimizing combination of inputs.

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). 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. It is constructed in the same way as an income offer curve in consumer theory: for each potential level of output, you plot the amount of labor and capital used, and connect the dots:

In fact, the expansion path for a firm’s production function and the income offer curve for a consumer’s utility function are exactly the same mathematically and conceptually.

Total cost of producing a given level of output

If there are just two inputs (labor and capital), then given conditional demand functions $L^c(w,r,q)$ and $K^c(w,r,q)$, the cost of producing $q$ units of output when facing input prices $w$ and $r$ is just the cost of the labor plus the cost of capital: \(c(w,r,q) = wL^c(w,r,q) + rK^c(w,r,q)\) Note that this is exactly the same as an expenditure function from consumer theory.

Example: Cobb-Douglas Production Function

For the production function $f(L,K) = \sqrt{LK}$, we found that the conditional demand functions were \(\begin{aligned} L^c(q) &= \sqrt \frac{r}{w} \times q\\ K^c(q) &= \sqrt \frac{w}{r} \times q \end{aligned}\)

Given these, we can find the long-run total cost of producing $q$ units, for general $w$ and $r$, when we can vary both labor and capital: \(\begin{aligned} c^{LR}(q) &= wL^c(q) + rK^c(q)\\ &= w \times \left(\sqrt \frac{r}{w} \times q\right) + r \times \left(\sqrt{\frac{w}{r}}\times q\right)\\ &= 2\sqrt{rw}q \end{aligned}\) For example, if $w = 8$ and $r = 2$, the total cost of producing $q$ units of output would be $c^{LR}(q) = (2\sqrt{8 \times 2})q = 8q$. Note that because this particular production function exhibits constant returns to scale, the long-run cost function is linear: doubling output doubles inputs, and therefore doubles total cost.

You can use the graph below to see how the cost-minimizing combination of inputs and the total cost of producing a certain amount $q$ varies with $w$, $r$, and $q$:

Notice that this tangency condition is exactly the same as the expansion path!