How do calculate whether selling a product leads to profit?

Question

To simplify things, imagine a company which sells two products, A, and B, and it has two types of revenues, two types of variable costs, and 1 type of fixed costs.

I can calculate margins by taking revenue A - variable cost A, and revenue B - variable cost B.

This tells me how much I make by selling each product.

However, then I have fixed costs, and I can't really "attribute" them to A and B.

My question is, how do I then decide which product is profitable to sell or not?

How do I "apply" fixed costs to my products?

Do I give use weights? For example, if 50 % of my sales are of product A, then it gets 50% of fixed costs attributed to it, and thus my profits from selling A is

revenue from A - variable costs from A - 50 % of fixed costs = profit from A

My interest lies in deciding whether I should scrap a product, or invest more in a product.

Answer 1

The process for solving this type of problem is very general--it's not a set of rules. You don't need to attribute fixed costs to one good or the other (if there is only one fixed cost).

Here's the procedure:

1. Write out the whole profit function
2. Find the maximum of the profit function

If the maximum is less than the scrap value, then scrap it.

If your profit function is continuous, then you can maximize it by differentiating with respect to the quantity produced of the two inputs, setting the first order conditions to 0, and solving. Just be sure to check that you have found the maximums and not the minimums. If it is not continuous you will need to use a different method to find the maximums.

How to write the profit function?

$$ \pi= TR(q_1,q_2) -TC(q_1,q_2) $$

$TR$ is usually easy. It's

$$ TR(q_1,q_2) = P_1(q_1,q_2)\cdot q_1 + P_2(q_1,q_2)\cdot q_2 $$

Insert your price functions (which will be constants if you are a price-taker) and your cost function so that $\pi$ is a function only of $q_1$ and $q_2$. Then find its max. The $q_1$ and $q_2$ that maximize $\pi$ are what you want.

This type of problem seems hard if you get caught up letting your right brain trying to figure out how to solve it heuristically. Don't do that. Write out the profit function and maximize it.

Answer 2

You can take 2 options: fixed costs are attributed (evenly or with a weight) to each unit produced, or use the fixed costs only to calculate the break-even point.

Example:

I'm a taxi driver, I pay 300$ just to have a car every month (say insurance+license+depreciation of the car).

1. Fixed costs can be attributed to each ride (a "weight", as you mentioned). I would distribute by each of the ~1000 rides I make per month, so I know that for every trip I need to discount (300$/1000 customers=)30 cent, in average for fixed costs. Of course, this only works when you can estimate +/- accurately your sales for a given month.

2. Knowing that I will have 300\$ of fixed costs, I will keep an eye at my contribution margins and know that I need to make at least 300\$ every month to cover it. Simplifying, I can make short rides for 5\$, with a margin of 3\$, or long rides for 10\$, with a margin of 5$. This means I need to make 100 short rides or 60 long rides just to cover my fixed costs.

The approach you take depends from what makes more sense for your business. In this case, I would take the approach 2, which also makes it more intuitive to understand that I making long rides will cover the fixed costs with less rides.

If you have a positive contribution margin in a product, you should only stop producing it if you can produce and sell more of another product with the same resources and a higher margin.

You can also use approach 2 and continuously verify the results of approach 1 for the previous month.