Supply and demand

Understanding supply and demand through the lens of optimization

Ashish Gaurav · 2 August 2026 · 2 min read · Last updated 4 August 2026

Setup

Suppose there is a single company selling a single product and a single consumer buying that specific product. Can we understand how the market interaction works in this scenario? 11 Need a visual example in this post.

Supply as profit maximization

Let’s have

$p =$ price per unit item
$q =$ quantity
$C(q)$ is the cost of producing $q$ items

Then, the company’s profit is total price it sells at, minus total cost of production:

\text{Profit}(q) = pq - C(q)

The company’s objective is to produce exactly that many units which maximizes its profit:

q^* = \argmax_q \text{Profit}(q)

Note that $q^*$ is a function of price per unit item $p$ .

This is the supply curve: for every price $p$ how much should the company produce to maximize its profits?

Demand as utility maximization

In the same way, a consumer is trying to maximize the utility of the goods they buy. But what exactly is utility? Utility is a scoring function that tells the benefit of the bought goods for the consumer. 22 How should we quantify utility? An unanswered question for later.

So if we have a utility function $U(q)$ , then the consumer’s benefit is the utility minus the cost of buying the goods 33 The buying and selling of goods $pq$ is the common part between the company and the consumer. :

\text{ConsumerBenefit}(q) = U(q)-pq

The consumer’s objective (also a function of $p$ ) is the maximize their own benefit:

q^{**} = \argmax_q \text{ConsumerBenefit}(q)

The is the demand curve: for every price $p$ how much should the consumer buy the maximize their benefit.

Market equilibrium

We know that:

The company is trying to maximize their profit, and in that process, it will produce a (supply of) $q^*(p)$ quantity of product if the product price in the market is $p$ .
The consumer is trying the maximize their benefit, and in that process, they will try to buy a (demand of) $q^{**}(p)$ quantity of product if the product price in the market is $p$ .

If the system reaches an equilibrium, then the company and consumer are both maximizing their respective profit and benefit. That means:

\frac{\partial}{\partial q} \bigg(pq-C(q)\bigg)\bigg|_{q=q^*}=0,\,\,\,\,\,\,\,\,\,\, \frac{\partial}{\partial q} \bigg(U(q)-pq\bigg)\bigg|_{q=q^{**}}=0

This gives us:

p = \frac{\partial C(q)}{\partial q} \bigg|_{q=q^*},\,\,\,\,\,\,\,\,\, p = \frac{\partial U(q)}{\partial q} \bigg|_{q=q^{**}}