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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics §5.5 Int Apps Biz & Econ

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 2 Bruce Mayer, PE Chabot College Mathematics Review § Any QUESTIONS About §5.4 → Applying the Definite Integral Any QUESTIONS About HomeWork §5.4 → HW

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 3 Bruce Mayer, PE Chabot College Mathematics §5.5 Learning Goals Use integration to compute the future and present value of an income ﬂow Deﬁne consumer willingness to spend as a deﬁnite integral, and use it to explore consumers’ surplus and producers’ surplus

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 4 Bruce Mayer, PE Chabot College Mathematics Time Value of Money We say that money has a time value because that money can be invested with the expectation of earning a positive rate of return In other words, “a dollar received today is worth more than a dollar to be received tomorrow” –That is because today’s dollar can be invested so that we have more than one dollar tomorrow

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 5 Bruce Mayer, PE Chabot College Mathematics Terminology of Time Value Present Value → An amount of money today, or the current value of a future cash flow Future Value → An amount of money at some future time period Period → A length of time (often a year, but can be a month, week, day, hour, etc.) Interest Rate → The compensation paid to a lender (or saver) for the use of funds expressed as a percentage for a period (normally expressed as an annual rate)

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 6 Bruce Mayer, PE Chabot College Mathematics Time Value Abreviations PV → Present value FV → Future value Pmt → Per period payment $-amount N → Either the total number of cash flows or the number of a Payment periods i → The interest rate per period Usually %/Yr expressed as a fraction

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 7 Bruce Mayer, PE Chabot College Mathematics TimeLines A timeline is a graphical device used to clarify the timing of the cash flows for an investment Each tick represents one time period PVFV Today5 TimePeriods Later

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 8 Bruce Mayer, PE Chabot College Mathematics Finding Future Value by Arith Consider $100 ($1 cNote) invested today at an interest rate of 10% per year or 0.10/yr as decimal Then the $-Value expected in 1 year $100?

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 9 Bruce Mayer, PE Chabot College Mathematics Finding Future Value by Arith Now Extend the Investment for another Year (“Let it Ride”) Then the $-Value expected after the 2 nd year with no additional investment $110?

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 10 Bruce Mayer, PE Chabot College Mathematics Recognize Future Value Pattern Engaging in the EXTREMELY VALUABLE Practice of PATTERN RECOGNITION surmise The Pattern that is developing for FV in year-N If the $1c investment were extended for a 3 rd Year then the FV

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 11 Bruce Mayer, PE Chabot College Mathematics Present Value by Arithmetic What is the $Amount TODAY (the Present Value, PV) needed to realize a FV $Goal after N-years invested at interest rate i per year? Solve the FV equation for PV

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 12 Bruce Mayer, PE Chabot College Mathematics Example Present Value Tadesuz has a 5-year old daughter for which he now plans for college $-expenses. Tadesuz lives in San Leandro, and he develops this college plan for his daughter She can live at home until she is 22 Attends Chabot and takes the Lower-Division Courses needed for University Transfer Transfers to UCBerkeley (Go Bears!) where she earns her Bachelors Degree

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 13 Bruce Mayer, PE Chabot College Mathematics Example Present Value Tadesuz estimates that he will need about $30k on her 18th birthday to pay for her bacaluarate Education. If he can earn 8% per year on his ONE- Time Initial investment, then how much must he invest today to achieve the $30k Future-Value Goal? SOLUTION: After 18-5 periods the PV:

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 14 Bruce Mayer, PE Chabot College Mathematics Annuities An annuity is a series of nominally equal $-payments equally spaced in time Annuities are very common: Bldg Leases, Mortgage payments, Car payments, Pension income The timeline shows an example of a 5- year, $100 ($1 cNote) annuity

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 15 Bruce Mayer, PE Chabot College Mathematics Principle of Value Additivity To find the value (PV or FV) of an annuity first consider principle of value additivity: The value of any stream of cash flows is equal to the sum of the values of the components Thus can move the cash flows to the same time period, and then simply add them all together to get the total value

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 16 Bruce Mayer, PE Chabot College Mathematics PRESENT Value of an Annuity Use the principle of value additivity to find the present value of an annuity, by simply summing the present values of each of the components:

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 17 Bruce Mayer, PE Chabot College Mathematics Present Value of an Annuity Using the example, and assuming a discount rate (a.k.a., interest rate) of 10% per year, find that the PV A as:

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 18 Bruce Mayer, PE Chabot College Mathematics Present Value of an Annuity Actually, there is no need to take the present value of each cash flow separately Using Convergent Series Analysis find a closed-form of the PV A equation instead:

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 19 Bruce Mayer, PE Chabot College Mathematics Present Value of an Annuity Using the PV A equation in the $1c example Thus a 5yr constant yearly income Annuity of $100/yr, discounted by 10% has PV of $379 The PV A equation works for all regular annuities, regardless of the number of payments

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 20 Bruce Mayer, PE Chabot College Mathematics FUTURE Value of an Annuity Use the principle of value additivity to find the Future Value of an annuity, by simply summing the future values of each of the components

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 21 Bruce Mayer, PE Chabot College Mathematics Future Value of an Annuity Again consider a $1c annuity, and assume a discount rate of 10% per year, find that the future value: } = at year 5

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 22 Bruce Mayer, PE Chabot College Mathematics Future Value of an Annuity As was done for the PV A equation, use series convergence to find a closed- form of the FV A equation: As with The PV A, the FV A eqn works for all regular annuities, regardless of the number of payments

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 23 Bruce Mayer, PE Chabot College Mathematics PV of an Income Stream Now assume the Pmt is divided into k payments per year (say 12) and then the discount is also applied k times a yr Call this an Income Stream as the payments are NO Longer made annually Then the FV A → FV IS eqn Note that Pmt kt may VARY in time; e.g., Pmt 73 ≠ Pmt 74

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 24 Bruce Mayer, PE Chabot College Mathematics PV of an Income Stream The discounts Occur infinitely often, and the Pmt kt becomes continuously variable in time, then the PV IS equation becomes Or textbook notation

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 25 Bruce Mayer, PE Chabot College Mathematics Example PV of Income Stream Yasiel’s grandfather promises to contribute continuously at a rate of $10,000 per year to a trust fund earning 3% interest as long as the boy maintains a 3.0 GPA in school. If Yasiel maintains the required grades for 8 years during high school and college, what is the value of the trust at the end of that period?

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 26 Bruce Mayer, PE Chabot College Mathematics Example PV of Income Stream The trust’s value is the future value of the annuity into which Yasiel’s grandfather is paying. Since the money accrues at a rate of $10,000 per year, and is simultaneously invested, its future value is

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 27 Bruce Mayer, PE Chabot College Mathematics Example PV of Income Stream Running the Numbers Thus the fund is worth about $90,416 at the end of eight years.

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 28 Bruce Mayer, PE Chabot College Mathematics Present Value of an Income Stream From the PV discussion, taking payment infinitely often, and the payments to becomes continuously variable in time, then the FV IS equation becomes

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 29 Bruce Mayer, PE Chabot College Mathematics Example Present Value Instead of investing at a continuous rate of $10,000 per year over the eight years, Yasiel’s grandfather in could have invested for eight years a lump sum of

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 30 Bruce Mayer, PE Chabot College Mathematics CAL Football Tickets Value of Ticket to Potential Demanders Peter$200 Paul$150 Mary$100 Jack$50 Jill$50 Value of Ticket to Potential Suppliers: Professor V$50 Professor W$50 Professor X$100 Professor Y$150 Professor Z$ Price Tickets Mary Peter Paul Jack and JillV and W X Y Z

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 31 Bruce Mayer, PE Chabot College Mathematics CAL Football Tickets Equilibrium Price = $100 Peter, Paul and Mary buy tickets from Professors V, W and X. If they all Buy & Sell at the equilibrium price, does it matter who buys from whom? → No Gains: Peter= $200 - $100 = $100 Paul= $150 - $100 = $50 Mary= $100 - $100 = $0 V= $100 - $50 = $50 W= $100 - $50 = $50 X= $100 - $100 = $0 Total Gain: $ Price Tickets Mary Peter Paul Jack and JillV and W X Y Z Consumer Surplus Producer Surplus

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 32 Bruce Mayer, PE Chabot College Mathematics Consumers’ Surplus By Supply & Demand the Price settles at P 0, but SOME Consumers are willing to pay MUCH MORE, thus these consumers save $-Money Price ($/unit) Quantity (Units) D(Q) PoPo QoQo Maximum Willingness to Pay (or Spend) for Q o What is Actually Paid Consumer Surplus

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 33 Bruce Mayer, PE Chabot College Mathematics Consumers’ Surplus Thus the Surplus Total $-Funds kept by the consumers: With Reference to the Areas shown on the Supply-n-Demand Graph CS = [Amount Willing to Spend] − [Amount Actually Paid]

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 34 Bruce Mayer, PE Chabot College Mathematics Producers’ Surplus By Supply & Demand the Price settles at P 0, but SOME Producrs are willing to accept a LOWER Price, thus these Producers make extra $-Money Minimum $-Amount Needed to Supply Q o Price ($/Unit) Quantity (units) PoPo QoQo $-Amount Actually paid Producer Surplus S(Q)

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 35 Bruce Mayer, PE Chabot College Mathematics Consumer & Producer Surplus The Two Surpluses usually exist simultaneously Price ($/Unit) Quantity (Units) PoPo QoQo S(Q) Producer Surplus Consumer Surplus D(Q) Equilibrium Price-Point

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 36 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work Problems From §5.5 P18 → Supply & Demand P26 → Construction Decision P42 → Invention Profit

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 37 Bruce Mayer, PE Chabot College Mathematics All Done for Today Tree NPV

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 38 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics Appendix –

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 39 Bruce Mayer, PE Chabot College Mathematics

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 40 Bruce Mayer, PE Chabot College Mathematics

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 41 Bruce Mayer, PE Chabot College Mathematics

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 42 Bruce Mayer, PE Chabot College Mathematics

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 43 Bruce Mayer, PE Chabot College Mathematics

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 44 Bruce Mayer, PE Chabot College Mathematics

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 45 Bruce Mayer, PE Chabot College Mathematics

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 46 Bruce Mayer, PE Chabot College Mathematics

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 47 Bruce Mayer, PE Chabot College Mathematics

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 48 Bruce Mayer, PE Chabot College Mathematics

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 49 Bruce Mayer, PE Chabot College Mathematics

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 50 Bruce Mayer, PE Chabot College Mathematics

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 51 Bruce Mayer, PE Chabot College Mathematics

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 52 Bruce Mayer, PE Chabot College Mathematics P55_42_AirPurifiers_1307.mn

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 53 Bruce Mayer, PE Chabot College Mathematics

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MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx 54 Bruce Mayer, PE Chabot College Mathematics

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