1. BMayer@ChabotCollege.edu MTH15_Lec-10_sec_2-5_Incrementals_.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §2.5 Incrementals & Marginal Analysis

2. BMayer@ChabotCollege.edu MTH15_Lec-10_sec_2-5_Incrementals_.pptx 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §2.4 → Derivative Chain Rule  Any QUESTIONS About HomeWork §2.4 → HW-10 2.4

3. BMayer@ChabotCollege.edu MTH15_Lec-10_sec_2-5_Incrementals_.pptx 3 Bruce Mayer, PE Chabot College Mathematics §2.5 Learning Goals  Study marginal analysis in economics  Approximate derivatives using increments and the differential

4. BMayer@ChabotCollege.edu MTH15_Lec-10_sec_2-5_Incrementals_.pptx 4 Bruce Mayer, PE Chabot College Mathematics Example  RoC for Productivity  The productivity model (in Items per day) for a complex Engineered product: where w is the number of worker-days dedicated to making the products  For this Situation: a)Compute & interpret P(w+1) − P(w) b)Compute & Compare:  P(6) − P(5)  [dP/dw] w=5

5. BMayer@ChabotCollege.edu MTH15_Lec-10_sec_2-5_Incrementals_.pptx 5 Bruce Mayer, PE Chabot College Mathematics Example  RoC for Productivity  SOLUTION (a)  This expression is the difference between productivity at w+1 worker- days and at w worker-days.

6. BMayer@ChabotCollege.edu MTH15_Lec-10_sec_2-5_Incrementals_.pptx 6 Bruce Mayer, PE Chabot College Mathematics Example  RoC for Productivity  SOLUTION (b)  Recall from the §2.4 Lecture-slides that [dP/dw] w=5 which is approximately equal to the actual change in productivity when moving from 5 to 6 worker-days (calculated above). Items/day for 1 added WorkerDay

7. BMayer@ChabotCollege.edu MTH15_Lec-10_sec_2-5_Incrementals_.pptx 7 Bruce Mayer, PE Chabot College Mathematics Working on The Margin  Is it worth it?  A thing worth doing may NOT be worth doing well.  Know when it’s time to move on!  Look forward, not back!  When is enough enough?

8. BMayer@ChabotCollege.edu MTH15_Lec-10_sec_2-5_Incrementals_.pptx 8 Bruce Mayer, PE Chabot College Mathematics Marginal Analysis  Marginal analysis is used to assist people in allocating their scarce resources to maximize the benefit of the output produced That is, to Simply obtain the most value for the resources used.  What is “Marginal” Marginal means additional, extra, or incremental (usually ONE added “Unit”) Every choice has cost and benefit

9. BMayer@ChabotCollege.edu MTH15_Lec-10_sec_2-5_Incrementals_.pptx 9 Bruce Mayer, PE Chabot College Mathematics Marginal Analysis  A technique widely used in business decision-making and ties together much of economic thought  Specifically, in any situation, people want to maximize net benefits:

10. BMayer@ChabotCollege.edu MTH15_Lec-10_sec_2-5_Incrementals_.pptx 10 Bruce Mayer, PE Chabot College Mathematics The Control Variable  To do marginal analysis, we can change a variable, such as the: quantity of a good you buy, the quantity of output you produce, or the quantity of an input you use.  This variable is called the independent, or, CONTROL variable Marginal analysis focuses upon whether the control variable should be increased by one more unit or NOT

11. BMayer@ChabotCollege.edu MTH15_Lec-10_sec_2-5_Incrementals_.pptx 11 Bruce Mayer, PE Chabot College Mathematics Marginal Analysis GamePlan 1.Identify the control variable (cv). 2.Determine what the increase in total benefits would be if ONE more unit of the control variable were added. This is the Marginal BENEFIT of the SINGLE added unit

12. BMayer@ChabotCollege.edu MTH15_Lec-10_sec_2-5_Incrementals_.pptx 12 Bruce Mayer, PE Chabot College Mathematics Marginal Analysis GamePlan 3.Determine what the increase in total cost would be if one more unit of the control variable were added This is the Marginal COST of the SINGLE added unit 4.If the unit's marginal benefit exceeds (or equals) its marginal cost, it SHOULD BE ADDED.

13. BMayer@ChabotCollege.edu MTH15_Lec-10_sec_2-5_Incrementals_.pptx 13 Bruce Mayer, PE Chabot College Mathematics ∆C vs dC Tangent Line (slope)

14. BMayer@ChabotCollege.edu MTH15_Lec-10_sec_2-5_Incrementals_.pptx 14 Bruce Mayer, PE Chabot College Mathematics MATLAB Code MATLAB Code % Bruce Mayer, PE % MTH-15 07Jul13 % XYfcnGraph6x6BlueGreenBkGndTemplate1306.m % % The Limits xmin = 0; xmax = 0.3; ymin =0; ymax = 1.4; % The FUNCTION x = linspace(xmin,xmax,1000); y1 = x.*(12-10*x-100*x.^2); y2 = - 4*(x-.2) +1.2 % % The ZERO Lines zxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax]; % % the 6x6 Plot axes; set(gca,'FontSize',12); whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green plot(x,y1, 'LineWidth', 5),axis([.15.3.6 1.4]),... grid, xlabel('\fontsize{14}p ($k/Ph)'), ylabel('\fontsize{14}R ($M)'),... title(['\fontsize{16}MTH15 \DeltaC vs dc',]),... annotation('textbox',[.15.05.0.1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'XYfcnGraph6x6BlueGreenBkGndTemplate1306.m','FontSize',7) hold on plot(x,y2, '-- m', 0.2,1.2, 'd r', 'MarkerSize', 6,'MarkerFaceColor', 'r', 'LineWidth', 2) plot([0.2,.2], [0.8,1.2], 'k', [0.2,.25], [.8,.8], 'k', [0.2,.25], [1.2,1.2], '-.k', [0.25,.25], [1,1.2], '-.k', 'LineWidth', 3) set(gca,'XTick',[xmin:.05:xmax]); set(gca,'YTick',[ymin:.2:ymax]) hold off

15. BMayer@ChabotCollege.edu MTH15_Lec-10_sec_2-5_Incrementals_.pptx 15 Bruce Mayer, PE Chabot College Mathematics ∆C vs dC  If x 0 is large, say 10 3 = 1 thousand, then adding 1 to the 1-thousand base makes ∆x ≈ dx  From Graph Observe

16. BMayer@ChabotCollege.edu MTH15_Lec-10_sec_2-5_Incrementals_.pptx 16 Bruce Mayer, PE Chabot College Mathematics ∆C vs dC  If x 0 becomes VERY LARGE, say 10 9 = 1 billion, then adding 1 to the 1-billion base makes ∆x = dx for all Practical Purposes  From Graph Observe

17. BMayer@ChabotCollege.edu MTH15_Lec-10_sec_2-5_Incrementals_.pptx 17 Bruce Mayer, PE Chabot College Mathematics Marginal Cost  If x is the Production-Rate (Units/Time) and C(x) is the Unit-Cost ($/Unit) then for very large x 0, Then the Cost to Produce ONE MORE UNIT of OutPut  Where dC/dx taken at x 0 is the Cost to produce the NEXT UNIT of output; i.e., the Marginal Cost →

18. BMayer@ChabotCollege.edu MTH15_Lec-10_sec_2-5_Incrementals_.pptx 18 Bruce Mayer, PE Chabot College Mathematics Marginal: Revenue & Profit  By Similar Reasoning The Marginal REVENUE from SELLING one additional unit: The Marginal PROFIT from SELLING one additional unit:

19. BMayer@ChabotCollege.edu MTH15_Lec-10_sec_2-5_Incrementals_.pptx 19 Bruce Mayer, PE Chabot College Mathematics Example  Marginal Cost  A Model for the total cost to farm “a” acres of soybeans is approximately  Paridhi would like to expand her 400- acre SoyBean farm  For this Situation Use marginal cost to estimate the increase in cost incurred from increasing the farm’s acreage by one. What is the marginal average cost to farm the 401 st acre?

20. BMayer@ChabotCollege.edu MTH15_Lec-10_sec_2-5_Incrementals_.pptx 20 Bruce Mayer, PE Chabot College Mathematics Example  Marginal Cost  SOLUTION  (a) The marginal cost  Approximate Paridhi’s increase in cost by computing the marginal cost at 400 acres:

21. BMayer@ChabotCollege.edu MTH15_Lec-10_sec_2-5_Incrementals_.pptx 21 Bruce Mayer, PE Chabot College Mathematics Example  Marginal Cost  SOLUTION  (b) The AVERAGE cost in $/acre:

22. BMayer@ChabotCollege.edu MTH15_Lec-10_sec_2-5_Incrementals_.pptx 22 Bruce Mayer, PE Chabot College Mathematics Example  Marginal Cost  So the marginal average cost is  At 400 Acres  So the average cost per acre is estimated to increase by 25¢ per acre when increasing total acreage by one

23. BMayer@ChabotCollege.edu MTH15_Lec-10_sec_2-5_Incrementals_.pptx 23 Bruce Mayer, PE Chabot College Mathematics Approximation by Increments  As long as a function f(x) is differentiable at x = x 0, then values of f near x 0 can be approximated by  where ∆x is a small value called the (finite) difference of x

24. BMayer@ChabotCollege.edu MTH15_Lec-10_sec_2-5_Incrementals_.pptx 24 Bruce Mayer, PE Chabot College Mathematics Increment GeoMetry Tangent Line (slope)

25. BMayer@ChabotCollege.edu MTH15_Lec-10_sec_2-5_Incrementals_.pptx 25 Bruce Mayer, PE Chabot College Mathematics MATLAB Code MATLAB Code % Bruce Mayer, PE % MTH-15 07Jul13 % XYfcnGraph6x6BlueGreenBkGndTemplate1306.m % % The Limits xmin = 0; xmax = 0.3; ymin =0; ymax = 1.4; % The FUNCTION x = linspace(xmin,xmax,1000); y1 = x.*(12-10*x-100*x.^2); y2 = -4*(x-.2) +1.2 % % The ZERO Lines zxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax]; % % the 6x6 Plot axes; set(gca,'FontSize',12); whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green plot(x,y1, 'LineWidth', 5),axis([.15.3.6 1.4]),... grid, xlabel('\fontsize{14}p ($k/Ph)'), ylabel('\fontsize{14}R ($M)'),... title(['\fontsize{16}MTH15 Incrementals',]),... annotation('textbox',[.15.05.0.1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'XYfcnGraph6x6BlueGreenBkGndTemplate1306.m','FontSize',7) hold on plot(x,y2, '-- m', 0.2,1.2, 'd r', 'MarkerSize', 6,'MarkerFaceColor', 'r', 'LineWidth', 2) plot([0.2,.2], [0.6,1.2], 'k', [.15,.2], [1.2,1.2], 'k',[0.25,.25], [0.6,0.8], 'k',... [.15,.25], [.8,.8], 'k', [0.2,.25], [1.2,1.2], '-.k', [0.25,.25], [1,1.2], '-.k', [.15,.25], [1,1], '-.k', 'LineWidth', 2) set(gca,'XTick',[xmin:.05:xmax]); set(gca,'YTick',[ymin:.2:ymax]) hold off

26. BMayer@ChabotCollege.edu MTH15_Lec-10_sec_2-5_Incrementals_.pptx 26 Bruce Mayer, PE Chabot College Mathematics Example  Increment Calc  Let f(x) = x 3. Then we can get a good idea of the value of f(4.02) by using the value of f(4) and then approximating using increments:  Note that f(4.02) = 64.96481 so we have a fair approximation

27. BMayer@ChabotCollege.edu MTH15_Lec-10_sec_2-5_Incrementals_.pptx 27 Bruce Mayer, PE Chabot College Mathematics Example  Incremental Analysis  Jeong-Bin (JB to his Friends), owner of a small frozen yogurt stand, is considering upgrading his infrastructure. A model for similar businesses is that each month he can expect to produce about Q(K) = 180K 1/3 (K in hundreds of $) gallons/month of frozen yogurt when investing a hundred dollars in capital. JB currently spends $500 dollars/month on capital (K = 5).  Approximate the increase in JB’s production if he invests an additional $50 in capital.

28. BMayer@ChabotCollege.edu MTH15_Lec-10_sec_2-5_Incrementals_.pptx 28 Bruce Mayer, PE Chabot College Mathematics Example  Incremental Analysis  SOLUTION  An estimate of the increase in production uses the derivative of the production function:  Note that the input on production is in hundreds of dollars of capital, so we have a = 5 and ∆K = 0.5 so we get:

29. BMayer@ChabotCollege.edu MTH15_Lec-10_sec_2-5_Incrementals_.pptx 29 Bruce Mayer, PE Chabot College Mathematics Example  Incremental Analysis  For Q(a+∆K)  Then

30. BMayer@ChabotCollege.edu MTH15_Lec-10_sec_2-5_Incrementals_.pptx 30 Bruce Mayer, PE Chabot College Mathematics Example  Incremental Analysis  The 318.055 value is the new predicted level of production, as compared to Q(5) = 180(5) 1/3 ≈ 307.976 an estimated increase of 318.055−307.976 = 10.259 gallons.  Thus the investment metrics

31. BMayer@ChabotCollege.edu MTH15_Lec-10_sec_2-5_Incrementals_.pptx 31 Bruce Mayer, PE Chabot College Mathematics Marginal vs Incremental  Marginal & Incremental Analysis BOTH Use  For the MARGINAL case:  For the INCREMENTAL CASE

32. BMayer@ChabotCollege.edu MTH15_Lec-10_sec_2-5_Incrementals_.pptx 32 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §2.5 P20 → Production Decision P28 → Balloon Catheter Volume

33. BMayer@ChabotCollege.edu MTH15_Lec-10_sec_2-5_Incrementals_.pptx 33 Bruce Mayer, PE Chabot College Mathematics All Done for Today Burger Benefit Analysis

34. BMayer@ChabotCollege.edu MTH15_Lec-10_sec_2-5_Incrementals_.pptx 34 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

35. BMayer@ChabotCollege.edu MTH15_Lec-10_sec_2-5_Incrementals_.pptx 35 Bruce Mayer, PE Chabot College Mathematics

36. BMayer@ChabotCollege.edu MTH15_Lec-10_sec_2-5_Incrementals_.pptx 36 Bruce Mayer, PE Chabot College Mathematics

37. BMayer@ChabotCollege.edu MTH15_Lec-10_sec_2-5_Incrementals_.pptx 37 Bruce Mayer, PE Chabot College Mathematics

38. BMayer@ChabotCollege.edu MTH15_Lec-10_sec_2-5_Incrementals_.pptx 38 Bruce Mayer, PE Chabot College Mathematics

39. BMayer@ChabotCollege.edu MTH15_Lec-10_sec_2-5_Incrementals_.pptx 39 Bruce Mayer, PE Chabot College Mathematics