Copyright © Cengage Learning. All rights reserved. 14 Further Integration Techniques and Applications of the Integral

Copyright © Cengage Learning. All rights reserved Applications to Business and Economics: Consumers’ and Producers’ Surplus and Continuous Income Streams

33 Consumers’ Surplus

44 Consider a general demand curve presented, as is traditional in economics, as p = D(q), where p is unit price and q is demand measured, say, in annual sales (Figure 15). Thus, D(q) is the price at which the demand will be q units per year. The price p 0 shown on the graph is the highest price that customers are willing to pay. Figure 15

55 Consumers’ Surplus Suppose, for example, that the graph in Figure 15 is the demand curve for a particular new model of computer. When the computer first comes out and supplies are low (q is small), “early adopters” will be willing to pay a high price. This is the part of the graph on the left, near the p-axis. As supplies increase and the price drops, more consumers will be willing to pay and more computers will be sold. We can ask the following question: How much are consumers willing to spend for the first units?

66 Consumers’ Surplus Consumers’ Willingness to Spend We can approximate consumers’ willingness to spend on the first units as follows. We partition the interval [0, ] into n subintervals of equal length, as we did when discussing Riemann sums. Figure 16 shows a typical subinterval, [q k − 1, q k ]. Figure 16

77 Consumers’ Surplus The price consumers are willing to pay for each of units q k – 1 through q k is approximately D(q k – 1 ), so the total that consumers are willing to spend for these units is approximately D(q k – 1 )(q k – q k – 1 ) = D(q k – 1 )  q, the area of the shaded region in Figure 16. Thus, the total amount that consumers are willing to spend for items 0 through is which is a Riemann sum.

88 Consumers’ Surplus The approximation becomes better the larger n becomes, and in the limit the Riemann sums converge to the integral This quantity, the area shaded in Figure 17, is the total consumers’ willingness to spend to buy the first units. Figure 17

99 Consumers’ Surplus Consumers’ Expenditure Now suppose that the manufacturer simply sets the price at,with a corresponding demand of, so D( ) =. Then the amount that consumers will actually spend to buy these is, the product of the unit price and the quantity sold. This is the area of the rectangle shown in Figure 18. Notice that we can write as suggested by the figure. Figure 18

10 Consumers’ Surplus The difference between what consumers are willing to pay and what they actually pay is money in their pockets and is called the consumers’ surplus. Consumers’ Surplus If demand for an item is given by p = D(q), the selling price is, and is the corresponding demand [so that D( ) = ], then the consumers’ surplus is the difference between willingness to spend and actual expenditure:

11 Consumers’ Surplus Graphically, it is the area between the graphs of p = D(q) and p =, as shown in the figure.

12 Example 1 – Consumers’ Surplus Your used-CD store has an exponential demand equation of the form p = 15e −0.01q where q represents daily sales of used CDs and p is the price you charge per CD. Calculate the daily consumers’ surplus if you sell your used CDs at $5 each.

13 Example 1 – Solution We are given D(q) = 15e −0.01q and = 5. We also need. By definition, D( ) = or 15 e−0.01 = 5 which we must solve for :

14 Example 1 – Solution We now have cont’d

15 Example 1 – Solution = $ per day cont’d

16 Producers’ Surplus

17 Producers’ Surplus We can also calculate extra income earned by producers. Consider a supply equation of the form p = S(q), where S(q) is the price at which a supplier is willing to supply q items (per time period). Because a producer is generally willing to supply more units at a higher price per unit, a supply curve usually has a positive slope, as shown in Figure 19. Figure 19

18 Producers’ Surplus The price p 0 is the lowest price that a producer is willing to charge. Arguing as before, we see that the minimum amount of money producers are willing to receive in exchange for items is. On the other hand, if the producers charge per item for items, their actual revenue is The difference between the producers’ actual revenue and the minimum they would have been willing to receive is the producers’ surplus.

19 Producers’ Surplus The producers’ surplus is the extra amount earned by producers who were willing to charge less than the selling price of per unit and is given by where Graphically, it is the area of the region between the graphs of p = and p = S(q) for 0 ≤ q ≤, as in the figure.

20 Example 2 – Producers’ Surplus My tie-dye T-shirt enterprise has grown to the extent that I am now able to produce T-shirts in bulk, and several campus groups have begun placing orders. I have informed one group that I am prepared to supply T-shirts at a price of p dollars per shirt. What is my total surplus if I sell T-shirts to the group at $8 each?

21 Example 2 – Solution We need to calculate the producers’ surplus when = 8. The supply equation is but in order to use the formula for producers’ surplus, we need to express p as a function of q. First, we square both sides to remove the radical sign: q 2 = 400(p − 4) so giving

22 Example 2 – Solution We now need the value of corresponding to = 8. Substituting p = 8 in the original equation, gives Thus, cont’d

23 Example 2 – Solution Thus, I earn a surplus of $ if I sell T-shirts to the group at $8 each. cont’d

24 Continuous Income Streams

25 Example 4 – Continuous Income An ice cream store’s business peaks in late summer; the store’s summer revenue is approximated by R(t) = t – 0.05t 2 dollars per day (0 ≤ t ≤ 92) where t is measured in days after June 1. What is its total revenue for the months of June, July, and August? Solution: Let’s approximate the total revenue by breaking up the interval [0, 92] representing the three months into n subintervals [t k − 1, t k ], each with length  t.

26 Example 4 – Solution In the interval [t k − 1, t k ] the store receives money at a rate of approximately R(t k − 1 ) dollars per day for  t days, so it will receive a total of R(t k − 1 )  t dollars. Over the whole summer, then, the store will receive approximately R(t 0 )  t + R(t 1 )  t + · · · + R(t n − 1 )  t dollars. As we let n become large to better approximate the total revenue, this Riemann sum approaches the integral cont’d

27 Example 4 – Solution Substituting the function we were given, we get cont’d

28 Continuous Income Streams Total Value of a Continuous Income Stream If the rate of receipt of income is R(t) dollars per unit of time, then the total income received from time t = a to t = b is

29 Continuous Income Streams Future Value of a Continuous Income Stream If the rate of receipt of income from time t = a to t = b is R(t) dollars per unit of time and the income is deposited as it is received in an account paying interest at rate r per unit of time, compounded continuously, then the amount of money in the account at time t = b is

30 Continuous Income Streams Present Value of a Continuous Income Stream If the rate of receipt of income from time t = a to t = b is R(t) dollars per unit of time and the income is deposited as it is received in an account paying interest at rate r per unit of time, compounded continuously, then the value of the income stream at time t = a is

31 Continuous Income Streams We can derive this formula from the relation FV = PVe r(b − a) because the present value is the amount that would have to be deposited at time t = a to give a future value of FV at time t = b.