# Lesson 7: Cost, Revenue and Profit Functions

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1 Lesson 7: Cost, Revenue and Profit Functions OBJECTIVE In today s lesson, we will study study three functions fundamental in economy and business: the cost, the revenue and the profit functions. Pay close attention to these three functions. We will use them throughout the year in our study of differential calculus. The Total Cost Function, C(q) or TC, gives the total cost of producing a quantity q of some product. There are two components of cost: Fixed cost: This is cost incurred even if nothing is produced. Ex. rent, utilities, or cost of the factory and machinery needed for production. Also called overheads. Variable cost: This is cost that depends on how many units are produced. Ex. Costs of labor and raw materials. Total cost for a company, C(q) =Fixed costs + Variable costs, i.e.,. TC = FC + VC What does a cost function graph look like? Observe: C(q) is an increasing function. That is, as the amount of goods produced increases, so does the cost to produce them. The y-intercept of the cost function represents the fixed costs. So, Fixed Costs = C(0) 1

2 Example/Practices 1. Suppose that the weekly cost to produce q widgets is given by C(q) = 16, q 1.6q q 3 where 0 apple q apple 250. What are the weekly overheads (FC)? What are the variable costs (VC)? Average cost, AC(q), is the total cost divided by the number of goods produced. We say: AC(q) = C(q) q. Marginal cost, MC(q), at the level of production q, describes the variable costs for one additional product, i.e., the added cost of producing one more unit of product. We say, MC(q) =C(q + 1) C(q). Remark If C is a linear function, i.e. C(q) =mq + b, then the marginal cost represents the slope of the line. MC(q) =C(q + 1) C(q) =[m(q + 1)+b] [mq + b] =mq + m + b mq b = m 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? (c) What is the marginal cost of the producing the 51st radio when 50 have been produced? The 101st radio when 100 have been produced? (b) What is the graph of the cost function? 2

3 (d) What is the average cost when 50 items are produced? When 100 items are produced? (e) What happens to the average cost function when the number of radios produced increases indefinitely? 3. Suppose that the weekly cost to produce q widgets is given by C (q) = 16, q 1.6q q3 where 0 q 250. Find the marginal and the average cost at a level of production of: q = 20, q = 100, q = 150. The Total Revenue Function, R(q), describes the revenue received by a firm from selling a quantity q of some product. If p is the price per unit then: R(q) = p q The Marginal Revenue, MR(q), is the revenue generated by selling one more unit. So, MR(q) = R(q + 1) R(q) If R(q) is a linear function, then MR(q) is the slope of the graph of R(q) and MR(q) = p. 3

4 4. Consider again the company making radios in the Example 2 above. Assume that the radios are selling for \$15 each. (a) What is the revenue function? (c) What is the marginal revenue of the producing the 51st radio when 50 have been produced? The 101st radio when 100 have been produced? (b) Sketch a graph of the revenue function? (d) In the same system of axis, sketch the graphs of the cost function and that of the revenue function. (Use 0 q 7000 and 0 C (q) 70000). For what value of q does the company start making money? The Profit Function, p (q) (we use the letter p to differentiate it from the price p), is defined as: p (q) = R(q) C (q) or simply: p = R C. The break-even point is the point where the profit is 0, i.e. where R(q) = C (q). 4

5 5. Consider the radio manufacturer once more. 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. q C (q), in \$ R(q), in \$ (a) Estimate the company s break-even point? (b) Find the company s profit if 1000 units are produced. (c) What price do you think the company is charging for its product? (d) What level of production would you advise to the company? 5

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