Cost, Revenue, and Profit Functions In this lecture, we will study three functions fundamental in economy and business: the cost, the revenue and the profit function. Throughout the year, we will use these quantities to model some of the tools we will study, in particular when we start differential calculus. Definition The total cost function, denoted (or ( ), describes the total cost of producing a quantity of some goods. The two components of the cost function are the fixed costs ( ) and the variable costs ( ). Fixed costs describe the cost generated before producing any goods (rent, utilities ). They are also called overheads. Variable costs are the costs that change in proportion to the amount of goods produced (raw material, labor ). or ) What does a cost function graph look like? As the amount of goods produced increases, so does the cost to produce them, i.e. is an increasing function. Below are three possible graphs of cost functions. The y-intercept of the cost function represents the fixed costs. I.e. lecture 7 unit 1 cost revenue and profit 1
Example/Practice: 1. Suppose that the weekly cost to produce widgets is given by,.. where. What are the weekly overheads ( )? What are the variable costs ( )? Definitions The marginal cost function, denoted, at the level of production, describes the variable costs for one additional product, i.e. the added cost of producing one more unit of good. So The average cost function, denoted, is the total cost divided by the number of goods produced. Thus Example/Practice: 2. Consider a company that makes radios. Assume that the fixed costs (raw material, machinery ) are $24,000 and that the variable cost (raw material, labor ) are $7 per radio. a. What is the cost function? b. What is the graph of the cost function? c. What is the marginal cost of the producing the 51 st radio when 50 have been produced? The 101 st radio when 100 have been produced? d. What is the average cost when 50 items are produced? When 100 items are produced? What happens to the average cost function when the number of radios produced increases indefinitely? lecture 7 unit 1 cost revenue and profit 2
3. Suppose that the weekly cost to produce widgets is given by,.. where. Find the marginal and the average cost at a level of production of:,,. Remark: If is a linear function, i.e., then the marginal cost represents the slope of the line. Definitions The total revenue function, denoted, describes the revenue received by a firm from selling a quantity of some goods. If is the price per unit then: When (the price of an item) is not a function of, then describes a linear relation. When (the price is a function of the number of items produced), then is no longer a linear relation. The marginal revenue, denoted, is the revenue generated by selling one more unit. So lecture 7 unit 1 cost revenue and profit 3
Example/Practice: 4. For the company making radios in the example 2 above, assume that the radios are selling for $15 each. What is the revenue function? Sketch the graph of the revenue function. What is the marginal revenue from selling the 51 st radio when 50 have been sold? The 101 st radio when 100 have been sold? In the same system of axis, sketch the graphs of the cost function and that of the revenue function. For what value of does the complany starts making money? lecture 7 unit 1 cost revenue and profit 4
Definition The profit function, denoted (we use the letter to differentiate it from the price ), is defined as: or simply:. The break-even point is the point where the profit is 0, i.e. where. Examples/Practices: 5. In the example of the radios manufacturer, find the profit function. Graph it, and find the break-even point. 6. The following table gives a company's estimates of cost and revenue for a product. 500 600 700 800 900 1000 1100, in $ 5000 5500 6000 6500 7000 7500 8000, in $ 4000 4800 5600 6400 7200 8000 8800 Estimate the break-even point. What is the company's profit if 900 units are sold? What level of production would you advise to the company? lecture 7 unit 1 cost revenue and profit 5