# How to determine the optimal price of a product as a monopolist

In business, finding the right price for your product is crucial. Price it too high, and few people will buy; price too low and the business leaves money on the table. But how do you determine the right price? We will explore this question with a simplified example, looking at Umbrella Corp (UCorp). Note that in this simplified case, UCorp is a monopolist and cannot charge different prices to different customers (i.e. is unable price discriminate).

## Step 1: Estimate the demand curve

The first step is to estimate the demand curve of UCorp’s customers. In real-world settings, companies can use several tools to estimate the demand curve for a product such as customer surveys, market experiments, regression analysis and others. In our example, to estimate the demand for umbrellas, Ucorp asked a class of 80 students to raise hands if they are willing to buy an umbrella at various price points on a rainy day.

## Step 3: Find profit maximizing quantity/price

If the goal was to maximize revenue we would be done, but the goal is to maximize profit, so we have to introduce costs. Let’s assume that UCorp has some fixed costs (F) to manufacture umbrellas and that variable costs (labor, capital and material) per umbrella are constant (c).

Cost equation:  C=F+c×q Note: In this case F = 100, c = 5;

Profit equation: P=p*q-F+c×q=p-c*q-F

Intuitively, we know that UCorp should produce umbrellas as long as the revenue from each umbrella is greater or equal to its variable cost (p=c), or in Econ language, Ucorp’s optimal output is determined at a quantity/price combination where marginal revenue is equal to marginal cost (MR =MC).

In this case marginal revenue is equal to the price (the revenue for each additional unit sold is the price of one umbrella) and marginal cost is the variable cost (the cost of each additional umbrella produced is marginal cost), so we can substitute MR with p and MC with c.

### Graph 3: UCorp’s profit at various quantities

In this case with F=100 and c=5, UCorp would choose to only sell 18 umbrellas at a price of $30 per umbrella to generate revenue of$540. Even though this leaves revenue lower than the maximum of $600, UCorp generates higher profit ($350 vs. $300) as it saves 22 x 5$ = \$110 in costs.

In summary, to determine the optimal price to charge, a business needs to understand its customers, in particular their demand behavior, and its own cost structure.

## Appendix: Data table excerpt

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