# How To Make a Profit Maximization Model

One of the first things that comes up in a Managerial Economics class is the decision on how to maximize profits and establish a profit maximization model. That's a different issue than maximizing revenue, since it's profits that we're usually looking to maximize.

The sweet spot in maximizing profits is to find the point where your marginal revenue equals your marginal costs. Your marginal costs are the cost of making an additional item; generally, that will also be the variable cost of making an item. For instance, if it costs you \$0.50 in supplies and labor to make a hamburger, the marginal cost of a hamburger is \$0.50.

Your marginal revenue is a trickier beast. If you are in a very competitive industry and the product is generic or "fungible" in econ-speak, you often will have to take whatever the going rate is for your product. In that case, the going price is your marginal revenue.

If you sell something that isn't generic such that customers have a demand for your product in particular, you'll have a demand function (or demand curve) for your product, where we measure the demand for a product as a function of the price. If we have P as price and Q as quantity, we'll have a demand function of P=a-bQ; as quantity goes up, the price goes down, as it will take a lower price to get people to buy more of a product.

Your revenue is price times quantity, or P*Q. Since we now P, we can state revenue as aQ-bQ^2 (using ^ as the exponential term as in Excel). Marginal revenue, using basic a calculus derivative, is a-2bQ. Your marginal revenue function will always have twice the slope of your demand curve.

Let's say we have a demand function for burgers of P=4-0.1Q. Our marginal revenue function would 4-0.2Q, giving things twice the slope of the demand function.

We then set up shop where marginal revenue equals marginal cost. If we have a marginal cost of 0.5, then 4-0.2Q=0.5, 0.2Q=3.5, or Q=17.5. We'd then shoot to sell 17.5 burgers a period and then plug 17.5 into our demand function, we'd have a target price of 4-0.1*17.5 or \$2.25 a burger.

The spot where marginal revenue equals marginal cost gives us the point where marginal profit equal zero, and your marginal profit equals zero where you're maxing out your profit. Finding that spot is how you do profit maximization.