5.2 Short-Run and Long-Run Costs

Recall how we have defined the difference between the short run and the long run. In the short run, some input (usually capital) is fixed, while in the long run all inputs are variable. Intuitively, this means that in the long run you can produce any quantity at its cost-minimizing combination of outputs (i.e. its lowest possible cost), while in the short run you are “stuck” with some fixed amount of one good.

It follows that for any quantity $q$, the long-run cost of producing $q$ units of output must be no greater than the cost in the short run. As a consequence, we say that the long-run total cost is the lower envelope of the short-run total cost. This is actually where the name of the Envelope Theorem comes from.

Think about a situation in which a firm can choose three discrete levels of capital: a small factory, a medium-sized factory, and a large factory. If we plot out its short-run cost curves for each of these, we can see that as factory size increases, fixed costs increase while variable costs decrease. This means that for each potential quantity, of the three sizes is optimal; so the long-run cost curve traces out the “lower envelope” of the short-run cost curves:

Now assume that the size of the factory ($K$) is continuous. Now each $q$ has an optimal $K^\star(q)$. For example, in the diagram below, the optimal factory size for $q = 270$ is $K^\star = 70$. If you drag the quantity left and right, you can see that the optimal amount of capital changes: the more you want to produce, the more capital you would optimally use.

Let’s think about the relationship between the long-run cost curve and the short-run cost curve for a particular level of capital: say, $\overline K = 70$. We saw above that these two curves meet at $q = 270$; this means that if you were “stuck” with $\overline K = 70$ and were lucky enough to want to produce $q = 270$ units of output, you wouldn’t want to change a thing: you’re already using your cost-minimizing level of captial. But if you wanted to produce any other quantity, you could do so at a lower cost if you used a different amount of capital.

Let’s reinterpret the objects here to see the Envelope Theorem at work. Take the short-run cost $c^{SR}(q \mid \overline K)$ as our objective function. Usually both $\overline K$ and $q$ are parameters of short run cost minimization, but let’s think of $\overline K$ as a choice variable and $q$ as a parameter just for this note. Picking $\overline K$ optimally for each $q$ then gives the value function, which is exactly the long-run cost: \(c^{LR}(q) = \min_{\overline K}\ c^{SR}(q \mid \overline K).\) The Envelope Theorem then tells us that \(\frac{d c^{LR}(q)}{d q} = \left. \frac{\partial c^{SR}(q \mid \overline K)}{\partial q} \right|_{\overline K = \overline K^\star(q)}.\) Cool, right? At the quantity where the two costs are equal, i.e. where we have optimized the short run cost, long-run cost equals short-run cost in value and slope. So in the lower graph the two curves do not just touch at that quantity: by the envelope theorem, they are also tangent.