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Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

Chabot Mathematics. §2.5 Incrementals & Marginal Analsysis. Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu. 2.4. Review §. Any QUESTIONS About §2.4 → Derivative Chain Rule Any QUESTIONS About HomeWork §2.4 → HW-10. §2.5 Learning Goals.

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Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

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  1. Chabot Mathematics §2.5 Incrementals&Marginal Analsysis Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

  2. 2.4 Review § • Any QUESTIONS About • §2.4 → Derivative Chain Rule • Any QUESTIONS About HomeWork • §2.4 → HW-10

  3. §2.5 Learning Goals • Study marginal analysis in economics • Approximate derivatives using increments and the differential

  4. 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: • Compute & interpret P(w+1) − P(w) • Compute & Compare: • P(6) − P(5) • [dP/dw]w=5

  5. Example  RoC for Productivity • SOLUTION (a) • This expression is the difference between productivity at w+1 worker-days and at w worker-days.

  6. 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. 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. 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. Marginal Analysis • A technique widely used in businessdecision-making and ties together much of economic thought • Specifically, in any situation, people want to maximize net benefits: NETbenefits=TOTALbenefits−TOTALcosts

  10. 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 increasedby one more unit or NOT

  11. Marginal Analysis GamePlan • Identify the control variable (cv). • Determine what the increase in total benefits would be if ONE more unit of the control variable were added. This is theMarginal BENEFITof theSINGLEadded unit

  12. Marginal Analysis GamePlan • Determine what the increase in total cost would be if one more unit of the control variable were added This is the Marginal COSTof the SINGLEadded unit • If the unit's marginal benefit exceeds (or equals) its marginal cost, it SHOULD BE ADDED.

  13. Tangent Line (slope) ∆CvsdC

  14. % 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 = [xminxmax]; zyh = [0 0]; zxv = [0 0]; zyv = [yminymax]; % % 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 • \DeltaCvs 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 MATLAB Code

  15. ∆CvsdC • If x0 is large, say 103 = 1 thousand, then adding 1 to the 1-thousand base makes ∆x ≈ dx • From Graph Observe

  16. ∆CvsdC • If x0 becomes VERY LARGE, say 109 = 1 billion, then adding 1 to the 1-billion base makes ∆x = dxfor all Practical Purposes • From Graph Observe

  17. Marginal Cost • If x is the Production-Rate (Units/Time) and C(x) is the Unit-Cost ($/Unit) then for very large x0, Then the Cost to Produce ONE MORE UNIT of OutPut • Where dC/dx taken at x0 is the Cost to produce the NEXT UNIT of output; i.e., the Marginal Cost →

  18. Marginal: Revenue & Profit • By Similar Reasoning • The Marginal REVENUE from SELLING one additional unit: • The Marginal PROFIT from SELLING one additional unit:

  19. Example  Marginal Cost • A Model for the total cost to farm “a” acres of soybeans is approximately • Paridhiwould 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 401st acre?

  20. Example  Marginal Cost • SOLUTION • (a) The marginal cost • Approximate Paridhi’s increase in cost by computing the marginal cost at 400 acres:

  21. Example  Marginal Cost • SOLUTION • (b) The AVERAGE cost:

  22. 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. Approximation by Increments • As long as a function f(x) is differentiable at x = x0, then values of fnearx0 can be approximated by • where ∆x is a small value called the (finite) difference of x

  24. Increment GeoMetry Tangent Line (slope)

  25. 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 = [xminxmax]; zyh = [0 0]; zxv = [0 0]; zyv = [yminymax]; % % 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. Example  Increment Calc • Let f(x) = x3. 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. 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) = 180K1/3 (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. 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. Example  Incremental Analysis • For Q(a+∆K) • Then

  30. 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. WhiteBoard Work • Problems From §2.5 • P20 → Production Decision • P28 → Balloon Catheter Volume

  32. All Done for Today BurgerBenefitAnalysis

  33. Chabot Mathematics Appendix Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu –

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