Section 4.4 Applications to Marginality

Many decisions in business are based on maximizing profits We have used the derivative to find maximums Profits are determined by revenue and production cost The cost function, C(q), gives the total cost of producing a quantity q of some good The revenue function, R(q), gives the total revenue received by a firm from selling a quantity q of some good The profit function, π(q), gives the total profit from producing and selling q items and is given by π(q) = R(q) - C(q)

Let’s think about the shape of these functions What would a cost function look like? What would a revenue function look like? What would the resulting profit function look like?

Marginal Analysis Often a companies decision to continue to produce goods is based on how much additional revenue they gain versus the additional cost The Marginal Cost is the average rate of change of adding one more unit Therefore it can be approximating by the instantaneous rate of change Marginal Cost = MC = C’(q) Marginal Revenue = MR = R’(q) Where is profit maximized (or minimized)? Where C’(q) = R’(q)

Example What is the marginal cost of q if fixed costs are $3000 and the variable cost is $225 per item? What is the marginal revenue if you charge $375 per item?

Example Find the quantity q which maximizes profit if the total revenue, R(q), and total cost, C(q) are given in dollars by Does the value of q give you a local max or a local min? How can you tell?

Example A hotel if they charge $300 per night for a hotel room, they can rent out a total of 20 rooms. They find that for each $25 decrease in price, they can rent an additional room. How many rooms should they rent out to maximize their revenue? What is their maximum revenue?