bmayer@chabotcollege.edu mth15_lec-10_sec_2-5_incrementals_.pptx 1 bruce mayer, pe chabot college...

BMayer@ChabotCollege.edu • MTH15_Lec-10_sec_2-5_Incrementals_.pptx1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

BMayer@ChabotCollege.edu

Chabot Mathematics§2.5

Incrementals&

Marginal Analysis

Review §

Any QUESTIONS About• §2.4 → Derivative Chain Rule

Any QUESTIONS About HomeWork• §2.4 → HW-10

§2.5 Learning Goals

Study marginal analysis in economics Approximate derivatives using

increments and the differential

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

wwwP 303 2

SOLUTION (a)

This expression is the difference between productivity at w+1 worker-days and at w worker-days.

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).

97.1 Items/day for 1 added WorkerDay

1597.16

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?

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

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:

NETbenefits=TOTALbenefits−TOTALcosts

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

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 theMarginal BENEFIT

of theSINGLE added unit

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 COSTof the SINGLE added unit

4. If the unit's marginal benefit exceeds (or equals) its marginal cost, it SHOULD BE ADDED.

0.15 0.2 0.25 0.30.6

p ($k/Ph)

)MTH15 • C vs dc

0.15 0.2 0.25 0.30.6

p ($k/Ph)

)MTH15 • C vs dc

0.15 0.2 0.25 0.30.6

p ($k/Ph)

)MTH15 • C vs dc

0.15 0.2 0.25 0.30.6

p ($k/Ph)

)MTH15 • C vs dc

0.15 0.2 0.25 0.30.6

p ($k/Ph)

)MTH15 • C vs dc

0.15 0.2 0.25 0.30.6

p ($k/Ph)

)MTH15 • C vs dc

0.15 0.2 0.25 0.30.6

p ($k/Ph)

)MTH15 • C vs dc

XYf cnGraph6x6BlueGreenBkGndTemplate1306.mXYf cnGraph6x6BlueGreenBkGndTemplate1306.mXYf cnGraph6x6BlueGreenBkGndTemplate1306.mXYf cnGraph6x6BlueGreenBkGndTemplate1306.mXYf cnGraph6x6BlueGreenBkGndTemplate1306.mXYf cnGraph6x6BlueGreenBkGndTemplate1306.mXYf cnGraph6x6BlueGreenBkGndTemplate1306.m

0x 10 x x0

Tangent Line (slope)

11 00 xxx

11 00 xxdx

% Bruce Mayer, PE% MTH-15 • 07Jul13% XYfcnGraph6x6BlueGreenBkGndTemplate1306.m%% The Limitsxmin = 0; xmax = 0.3; ymin =0; ymax = 1.4;% The FUNCTIONx = linspace(xmin,xmax,1000); y1 = x.*(12-10*x-100*x.^2); y2 = -4*(x-.2) +1.2% % The ZERO Lineszxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax];%% the 6x6 Plotaxes; set(gca,'FontSize',12);whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Greenplot(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 onplot(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

∆C vs dC If x0 is large, say 103

= 1 thousand, then adding 1 to the 1-thousand base makes ∆x ≈ dx

From Graph Observe

0.15 0.2 0.25 0.30.6

p ($k/Ph)

MTH15 • C vs dc

0.15 0.2 0.25 0.30.6

p ($k/Ph)

MTH15 • C vs dc

0.15 0.2 0.25 0.30.6

p ($k/Ph)

MTH15 • C vs dc

0.15 0.2 0.25 0.30.6

p ($k/Ph)

MTH15 • C vs dc

0.15 0.2 0.25 0.30.6

p ($k/Ph)

MTH15 • C vs dc

0.15 0.2 0.25 0.30.6

p ($k/Ph)

MTH15 • C vs dc

0.15 0.2 0.25 0.30.6

p ($k/Ph)

MTH15 • C vs dc

XYf cnGraph6x6BlueGreenBkGndTemplate1306.mXYf cnGraph6x6BlueGreenBkGndTemplate1306.mXYf cnGraph6x6BlueGreenBkGndTemplate1306.mXYf cnGraph6x6BlueGreenBkGndTemplate1306.mXYf cnGraph6x6BlueGreenBkGndTemplate1306.mXYf cnGraph6x6BlueGreenBkGndTemplate1306.mXYf cnGraph6x6BlueGreenBkGndTemplate1306.m

0x 10 x

11 00 xxx

11 00 xxdx

00 1 xCxCC

dCdxmdC

∆C vs dC If x0 becomes VERY

LARGE, say 109 = 1 billion, then adding 1 to the 1-billion base makes ∆x = dx for all Practical Purposes

From Graph Observe

CxCxCdx

dCdxmdC

111 00

p ($k/Ph)

MTH15 • C vs dc

xCy 11 00 xxdx

11 00 xxx

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 →

11 00xxx dx

dCdCCxCxC

Marginal: Revenue & Profit

By Similar Reasoning• The Marginal REVENUE from SELLING

one additional unit:

• The Marginal PROFIT from SELLING one additional unit:

11 00xx dx

dRdRRxRxR

11 00xx dx

dPdPPxPxP

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 401st acre?

120048001.0 2 aaaC

Example Marginal Cost

SOLUTION (a) The marginal cost

Approximate Paridhi’s increase in cost by computing the marginal cost at 400 acres:

48002.0 a

480)400(02.0)400('400

SOLUTION (b) The AVERAGE cost in $/acre:

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

2120001.0 a

)400(120001.0400'

a0025.0

Approximation by Increments

As long as a function f(x) is differentiable at x = x0, then values of f near x0 can be approximated by

where ∆x is a small value called the (finite) difference of x

dfxfxxf

Increm

0.15 0.2 0.25 0.30.6

p ($k/Ph)

)MTH15 • Incrementals

XYf cnGraph6x6BlueGreenBkGndTemplate1306.m

Tangent Line (slope)

0x xx 0 x0

% Bruce Mayer, PE% MTH-15 • 07Jul13% XYfcnGraph6x6BlueGreenBkGndTemplate1306.m%% The Limitsxmin = 0; xmax = 0.3; ymin =0; ymax = 1.4;% The FUNCTIONx = linspace(xmin,xmax,1000); y1 = x.*(12-10*x-100*x.^2); y2 = -4*(x-.2) +1.2% % The ZERO Lineszxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax];%% the 6x6 Plotaxes; set(gca,'FontSize',12);whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Greenplot(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 onplot(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

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

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.

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:

dQaQKaQ

For Q(a+∆K)

5.055.055

1180 3/23/2 KKKQ

055.318

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

Gal 2058.0

monGal 259.10

3.33% monGal 307.976

monGal 259.10for %

BaseLIneQ

Marginal vs Incremental

Marginal & Incremental Analysis BOTH Use

For the MARGINAL case:

For the INCREMENTAL CASE

dfxfxxf

(Exactly) 1x

WhiteBoard Work

Problems From §2.5• P20 → Production Decision• P28 → Balloon

Catheter Volume

All Done for Today

BurgerBenefitAnalysis

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

BMayer@ChabotCollege.edu

Chabot Mathematics

Appendix

srsrsr 22

bmayer@chabotcollege.edu mth15_lec-10_sec_2-5_incrementals_.pptx 1 bruce mayer, pe chabot college...

pe chabot college mathematics2

w workerdays

incrementalsmarginal

n s j o h n s o n c

number of worker

quantity of output

orthe quantity

questions about2

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