# The math behind supply curves

When I took introductory microeconomics towards the end of my undergraduate I expected almost everything in the course to be nonsense. Most of the claims of the economics profession that I had encountered seemed to me to be so far from the world around me that I expected the course to be full of errors. However my initial expectations were somewhat subverted during the beginning portion of the class. Having taken many math courses I could see that many of the concepts that I was learning were simply basic calculus proofs. Having taken physics I was used to simplifying assumptions, and many of the assumptions that were made seemed like good candidates for a first approximation of reality.

Midway through the course however an assumption was made that confirmed my views that most of the conclusions of the course did not apply to reality. The assumption made was not explicitly stated as an assumption, but rather stated as if it were an empirical fact. The assumption was that firms face decreasing returns to scale. This assumption is crucial to deriving supply curves as the analysis below will show.

We start with the equation for the profit of a firm, where $\pi$ is profit, p is price, q is the quantity produced, and $c(q)$ is the cost to produce quantity q $\pi=pq-c(q)$

Firms are assumed to maximize profits. We can find the maximum of the above profit function using basic calculus. First take the derivative and set it equal to zero. Since we are trying to find the profit maximizing quantity we take the derivative with respect to quantity. $\frac{d\pi}{dq}=p+q\frac{dp}{dq}-\frac{dc(q)}{dq}=0$

Since we assume price taking the rate of change of a firm’s price is independent of quantity, and so $\frac{dp}{dq}=0$

This gives us the critical points of the profit function, which are when $p=\frac{dc(q)}{dq}$, which is the familiar price=marginal cost from econ 101. However the critical points could be either maxima or minima of the profit function. We can test which by using the second derivative test. If $\frac{d^2\pi}{d^2q}<0$ then we have a maximum, if it is greater than zero we have a minimum, and if it is equal to zero the test is inconclusive.

Since $\frac{d^2\pi}{d^2q}=-\frac{d^2c(q)}{d^2q}$ the profit function has a maximum where $p=\frac{dc(q)}{dq}$ when $\frac{d^2c(q)}{d^2q}>0$, or when marginal cost is increasing. Otherwise $p=p=\frac{dc(q)}{dq}$ is a minimum of the profit function and the function has its maximum at a quantity of either zero or infinity. If marginal costs are constant firm behaviour depends on the price. If the price is equal to marginal cost the firm does not care how much is produced. If the price is greater than marginal cost the firm wants to sell as much as possible, and if the price is less than marginal cost the firm wants to produce a quantity of zero.

This means that as long as marginal costs are increasing, or as long as it gets more expensive to produce goods the more is being produced, the usual supply curves from econ 101 apply. Firms produce an amount where price is equal to marginal cost. However, if it gets cheaper to produce a good as more is produced, then firms want to sell as much as possible or nothing at all, depending on whether the market price is less than or more than the minimum it costs them to produce a good. Drawing a supply curve implies that costs are increasing as more of a good is produced.

While it might seem odd to say that a firm wants to produce an infinite amount at a given price, in practice all we are saying is that a firm wants to produce as much as possible. Some graduate textbooks, for example Macroeconomic Theory, by Mass-Cowell and Whinston, use the fact that firms want to produce as much as possible to say that price taking makes no sense for firms without increasing costs. However there is no problem for the theory, all that happens is that firms in that situation are limited by demand. Readers who are familiar with Keynesian macroeconomic ideas will have an inkling of how this could be important to macroeconomics. I will explore this more in later posts.

Despite the simplicity of the above derivation many economists seem to forget the limited situations in which supply curves exist when teaching and thinking about economics. Most students of economics also do not come away with an understanding of the extremely limited situations in which the economics 101 material they are learning applies. In fact, as I hope to show in future posts, assuming that cost functions are increasing is a problem even for papers that are currently being published, for example many macroeconomics papers assume increasing marginal costs.

## 5 thoughts on “The math behind supply curves”

1. Arilando says:

Where does the +q dp/dq come from in the first derivative? Shouldn’t it simply be p – dc(q)/dq ?

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2. The Mountain Goat says:

Product rule. We aren’t treating p as a constant here. If it is constant, as in perfect competition, dp/dq=0 and our answer reduces to what you have above.

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