[email protected] mth15_lec-10_sec_2-5_incrementals_.pptx 1 bruce mayer, pe chabot college...
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[email protected] • MTH15_Lec-10_sec_2-5_Incrementals_.pptx1
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics§2.5
Incrementals&
Marginal Analysis
[email protected] • MTH15_Lec-10_sec_2-5_Incrementals_.pptx2
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
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Bruce Mayer, PE Chabot College Mathematics
§2.5 Learning Goals
Study marginal analysis in economics Approximate derivatives using
increments and the differential
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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
wwwP 303 2
[email protected] • MTH15_Lec-10_sec_2-5_Incrementals_.pptx5
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.
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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).
97.1 Items/day for 1 added WorkerDay
1597.16
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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?
[email protected] • MTH15_Lec-10_sec_2-5_Incrementals_.pptx8
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
[email protected] • MTH15_Lec-10_sec_2-5_Incrementals_.pptx9
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:
NETbenefits=TOTALbenefits−TOTALcosts
[email protected] • MTH15_Lec-10_sec_2-5_Incrementals_.pptx10
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
[email protected] • MTH15_Lec-10_sec_2-5_Incrementals_.pptx11
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 theMarginal BENEFIT
of theSINGLE added unit
[email protected] • MTH15_Lec-10_sec_2-5_Incrementals_.pptx12
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 COSTof the SINGLE added unit
4. If the unit's marginal benefit exceeds (or equals) its marginal cost, it SHOULD BE ADDED.
[email protected] • MTH15_Lec-10_sec_2-5_Incrementals_.pptx13
Bruce Mayer, PE Chabot College Mathematics
∆C
vs dC
0.15 0.2 0.25 0.30.6
0.8
1
1.2
1.4
p ($k/Ph)
R (
$M
)MTH15 • C vs dc
0.15 0.2 0.25 0.30.6
0.8
1
1.2
1.4
p ($k/Ph)
R (
$M
)MTH15 • C vs dc
0.15 0.2 0.25 0.30.6
0.8
1
1.2
1.4
p ($k/Ph)
R (
$M
)MTH15 • C vs dc
0.15 0.2 0.25 0.30.6
0.8
1
1.2
1.4
p ($k/Ph)
R (
$M
)MTH15 • C vs dc
0.15 0.2 0.25 0.30.6
0.8
1
1.2
1.4
p ($k/Ph)
R (
$M
)MTH15 • C vs dc
0.15 0.2 0.25 0.30.6
0.8
1
1.2
1.4
p ($k/Ph)
R (
$M
)MTH15 • C vs dc
0.15 0.2 0.25 0.30.6
0.8
1
1.2
1.4
p ($k/Ph)
R (
$M
)MTH15 • C vs dc
XYf cnGraph6x6BlueGreenBkGndTemplate1306.mXYf cnGraph6x6BlueGreenBkGndTemplate1306.mXYf cnGraph6x6BlueGreenBkGndTemplate1306.mXYf cnGraph6x6BlueGreenBkGndTemplate1306.mXYf cnGraph6x6BlueGreenBkGndTemplate1306.mXYf cnGraph6x6BlueGreenBkGndTemplate1306.mXYf cnGraph6x6BlueGreenBkGndTemplate1306.m
C
dC
0x 10 x x0
y
xCy
Tangent Line (slope)
11 00 xxx
11 00 xxdx
[email protected] • MTH15_Lec-10_sec_2-5_Incrementals_.pptx14
Bruce Mayer, PE Chabot College Mathematics
MA
TL
AB
Co
de
% Bruce Mayer, PE% MTH-15 • 07Jul13% XYfcnGraph6x6BlueGreenBkGndTemplate1306.m%% The Limitsxmin = 0; xmax = 0.3; ymin =0; ymax = 1.4;% The FUNCTIONx = linspace(xmin,xmax,1000); y1 = x.*(12-10*x-100*x.^2); y2 = -4*(x-.2) +1.2% % The ZERO Lineszxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax];%% the 6x6 Plotaxes; set(gca,'FontSize',12);whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Greenplot(x,y1, 'LineWidth', 5),axis([.15 .3 .6 1.4]),... grid, xlabel('\fontsize{14}p ($k/Ph)'), ylabel('\fontsize{14}R ($M)'),... title(['\fontsize{16}MTH15 • \DeltaC vs dc',]),... annotation('textbox',[.15 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'XYfcnGraph6x6BlueGreenBkGndTemplate1306.m','FontSize',7)hold onplot(x,y2, '-- m', 0.2,1.2, 'd r', 'MarkerSize', 6,'MarkerFaceColor', 'r', 'LineWidth', 2)plot([0.2,.2], [0.8,1.2], 'k', [0.2,.25], [.8,.8], 'k', [0.2,.25], [1.2,1.2], '-.k', [0.25,.25], [1,1.2], '-.k', 'LineWidth', 3)set(gca,'XTick',[xmin:.05:xmax]); set(gca,'YTick',[ymin:.2:ymax])hold off
[email protected] • MTH15_Lec-10_sec_2-5_Incrementals_.pptx15
Bruce Mayer, PE Chabot College Mathematics
∆C vs dC If x0 is large, say 103
= 1 thousand, then adding 1 to the 1-thousand base makes ∆x ≈ dx
From Graph Observe
0.15 0.2 0.25 0.30.6
0.8
1
1.2
1.4
p ($k/Ph)
R ($
M)
MTH15 • C vs dc
0.15 0.2 0.25 0.30.6
0.8
1
1.2
1.4
p ($k/Ph)
R ($
M)
MTH15 • C vs dc
0.15 0.2 0.25 0.30.6
0.8
1
1.2
1.4
p ($k/Ph)
R ($
M)
MTH15 • C vs dc
0.15 0.2 0.25 0.30.6
0.8
1
1.2
1.4
p ($k/Ph)
R ($
M)
MTH15 • C vs dc
0.15 0.2 0.25 0.30.6
0.8
1
1.2
1.4
p ($k/Ph)
R ($
M)
MTH15 • C vs dc
0.15 0.2 0.25 0.30.6
0.8
1
1.2
1.4
p ($k/Ph)
R ($
M)
MTH15 • C vs dc
0.15 0.2 0.25 0.30.6
0.8
1
1.2
1.4
p ($k/Ph)
R ($
M)
MTH15 • C vs dc
XYf cnGraph6x6BlueGreenBkGndTemplate1306.mXYf cnGraph6x6BlueGreenBkGndTemplate1306.mXYf cnGraph6x6BlueGreenBkGndTemplate1306.mXYf cnGraph6x6BlueGreenBkGndTemplate1306.mXYf cnGraph6x6BlueGreenBkGndTemplate1306.mXYf cnGraph6x6BlueGreenBkGndTemplate1306.mXYf cnGraph6x6BlueGreenBkGndTemplate1306.m
C
dC
0x 10 x
11 00 xxx
11 00 xxdx
00 1 xCxCC
10
0
xx dx
dCdxmdC
[email protected] • MTH15_Lec-10_sec_2-5_Incrementals_.pptx16
Bruce Mayer, PE Chabot College Mathematics
∆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
xx
111 00
0
0
0.2
1.2
p ($k/Ph)
R (
$M
)
MTH15 • C vs dc
C dC
xCy 11 00 xxdx
11 00 xxx
[email protected] • MTH15_Lec-10_sec_2-5_Incrementals_.pptx17
Bruce Mayer, PE Chabot College Mathematics
Marginal Cost
If x is the Production-Rate (Units/Time) and C(x) is the Unit-Cost ($/Unit) then for very large 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 →
000
11 00xxx dx
dC
dx
dCdx
dx
dCdCCxCxC
00 '0
xCdx
dCxC
xM
[email protected] • MTH15_Lec-10_sec_2-5_Incrementals_.pptx18
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:
00
11 00xx dx
dR
dx
dRdRRxRxR
00
11 00xx dx
dP
dx
dPdPPxPxP
[email protected] • MTH15_Lec-10_sec_2-5_Incrementals_.pptx19
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 401st acre?
120048001.0 2 aaaC
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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:
48002.0 a
480)400(02.0)400('400
Cdz
dC
a
488
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Bruce Mayer, PE Chabot College Mathematics
Example Marginal Cost
SOLUTION (b) The AVERAGE cost in $/acre:
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Bruce Mayer, PE Chabot College Mathematics
Example Marginal Cost
So the marginal average cost is
At 400 Acres
So the average cost per acre is estimated to increase by 25¢ per acre when increasing total acreage by one
2120001.0 a
2
400
)400(120001.0400'
ACda
dAC
a0025.0
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Bruce Mayer, PE Chabot College Mathematics
Approximation by Increments
As long as a function f(x) is differentiable at x = x0, then values of f near x0 can be approximated by
where ∆x is a small value called the (finite) difference of x
xdx
dfxfxxf
xx
0
00
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Bruce Mayer, PE Chabot College Mathematics
Increm
ent G
eoM
etry
0.15 0.2 0.25 0.30.6
0.8
1
1.2
1.4
p ($k/Ph)
R (
$M
)MTH15 • Incrementals
XYf cnGraph6x6BlueGreenBkGndTemplate1306.m
xdx
dff
x
0
y
xfy
Tangent Line (slope)
x
0x xx 0 x0
0xf
xxf 0
fxf 0
[email protected] • MTH15_Lec-10_sec_2-5_Incrementals_.pptx25
Bruce Mayer, PE Chabot College Mathematics
MA
TL
AB
Co
de
% Bruce Mayer, PE% MTH-15 • 07Jul13% XYfcnGraph6x6BlueGreenBkGndTemplate1306.m%% The Limitsxmin = 0; xmax = 0.3; ymin =0; ymax = 1.4;% The FUNCTIONx = linspace(xmin,xmax,1000); y1 = x.*(12-10*x-100*x.^2); y2 = -4*(x-.2) +1.2% % The ZERO Lineszxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax];%% the 6x6 Plotaxes; set(gca,'FontSize',12);whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Greenplot(x,y1, 'LineWidth', 5),axis([.15 .3 .6 1.4]),... grid, xlabel('\fontsize{14}p ($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
[email protected] • MTH15_Lec-10_sec_2-5_Incrementals_.pptx26
Bruce Mayer, PE Chabot College Mathematics
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
96.64
[email protected] • MTH15_Lec-10_sec_2-5_Incrementals_.pptx27
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) = 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 $50 in capital.
[email protected] • MTH15_Lec-10_sec_2-5_Incrementals_.pptx28
Bruce Mayer, PE Chabot College Mathematics
Example Incremental Analysis
SOLUTION An estimate of the increase in
production uses the derivative of the production function:
Note that the input on production is in hundreds of dollars of capital, so we have a = 5 and ∆K = 0.5 so we get:
KdK
dQaQKaQ
a
[email protected] • MTH15_Lec-10_sec_2-5_Incrementals_.pptx29
Bruce Mayer, PE Chabot College Mathematics
Example Incremental Analysis
For Q(a+∆K)
Then
5.055.055
KdK
dQQQ
,603
1180 3/23/2 KKKQ
dK
d
055.318
[email protected] • MTH15_Lec-10_sec_2-5_Incrementals_.pptx30
Bruce Mayer, PE Chabot College Mathematics
Example Incremental Analysis
The 318.055 value is the new predicted level of production, as compared to Q(5) = 180(5)1/3 ≈ 307.976 an estimated increase of 318.055−307.976 = 10.259
gallons. Thus the investment metrics
mon$
Gal 2058.0
$50
monGal 259.10
K
Q
3.33% monGal 307.976
monGal 259.10for %
BaseLIneQ
[email protected] • MTH15_Lec-10_sec_2-5_Incrementals_.pptx31
Bruce Mayer, PE Chabot College Mathematics
Marginal vs Incremental
Marginal & Incremental Analysis BOTH Use
For the MARGINAL case:
For the INCREMENTAL CASE
xdx
dfxfxxf
xx
0
00
(Exactly) 1x
10%0
x
x
[email protected] • MTH15_Lec-10_sec_2-5_Incrementals_.pptx32
Bruce Mayer, PE Chabot College Mathematics
WhiteBoard Work
Problems From §2.5• P20 → Production Decision• P28 → Balloon
Catheter Volume
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Bruce Mayer, PE Chabot College Mathematics
All Done for Today
BurgerBenefitAnalysis
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Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
Appendix
–
srsrsr 22
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Bruce Mayer, PE Chabot College Mathematics
[email protected] • MTH15_Lec-10_sec_2-5_Incrementals_.pptx36
Bruce Mayer, PE Chabot College Mathematics
[email protected] • MTH15_Lec-10_sec_2-5_Incrementals_.pptx37
Bruce Mayer, PE Chabot College Mathematics
[email protected] • MTH15_Lec-10_sec_2-5_Incrementals_.pptx38
Bruce Mayer, PE Chabot College Mathematics
[email protected] • MTH15_Lec-10_sec_2-5_Incrementals_.pptx39
Bruce Mayer, PE Chabot College Mathematics