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Lecture 2
MGMT 7730 - © 2011 Houman Younessi
Derivatives
Derivative of a constant
Y
X
Y=3Y1
012
0lim
0
XXdX
dYX
X1 X2
12
11limlim
00 XX
YY
X
Y
dX
dYXX
X
Y
dX
dYX
0lim
Lecture 2
MGMT 7730 - © 2011 Houman Younessi
Derivative of a line
Y
X
Y=5X
Y1=10
51
5
23
1015lim
0
XdX
dY
X1=2 X2=3
12
12limlim
00 XX
YY
X
Y
dX
dYXX
Y2=15
Lecture 2
MGMT 7730 - © 2011 Houman Younessi
Derivative of a polynomial function
1)3(
2)2(
1)1( ......)2()1( aXanXanXna
dX
dY nb
nb
nb
0)2(
2)1(
1 ...... aXaXaXaY nb
nb
nb
23XY
Examples:
XXdX
dY623 12
1323 XCXKXY 323 2 CXKXdX
dY
Lecture 2
MGMT 7730 - © 2011 Houman Younessi
Derivatives of sums and differencesIn general:
For )()()( xZxWxY )()()( xZxWxY
dX
dZ
dX
dW
dX
dY
or
dX
dZ
dX
dW
dX
dY
Lecture 2
MGMT 7730 - © 2011 Houman Younessi
Derivatives of products and quotientsIn general:
For )()()( xZxWxY )(/)()( xZxWxY
dX
dWZ
dX
dZW
dX
dY
or
2
)()(
ZdX
dZWdXdWZ
dX
dY
Examples:
)1(181818
61812
6)3()2(6
)3(6
3)(
6)(
)3(6)(
22
22
2
2
2
2
XX
XX
XXXdX
dYdX
dWX
dX
dZX
dX
dY
XxZ
XxW
XXxY
2
2
2
32
2
32
2
3
3
)43(
)4045(
)43(
4045
)43(
)4(5)15)(43(
4
15
43)(
5)(
43
5)(
X
XX
X
XX
X
XXX
dX
dY
dX
dZ
XdX
dW
XxZ
XxW
X
XxY
Lecture 2
MGMT 7730 - © 2011 Houman Younessi
Derivative of a derivative
0 Q1
Profit
Number of units of output
A
Q1
Profit
Number of units of output
A
0
)(2
2
dX
dY
dx
d
dX
Yd
6
46
143
2
2
2
dX
Yd
XdX
dY
XXY
8
78
374
2
2
2
dX
Yd
XdX
dY
XXY
Lecture 2
MGMT 7730 - © 2011 Houman Younessi
Partial DerivativeA derivate with respect to only one variable when the function is the
function of more than just that variable
A single variable function:
A multi-variable function:
181080
449
1243),(
22
223
XWXWW
Y
XWXX
Y
WXXWXwxY
49
143)(
2
3
XdX
dY
XXxY
Lecture 2
MGMT 7730 - © 2011 Houman Younessi
Optimization TheoryUnconstrained Optimization
Unconstrained optimization applies when we wish to find the maximum or minimum point of a curve. In other words we wish to find the value of the independent variable at which the dependent variable is maximized or minimized without any other external conditions restricting it.
Let us assume that there is an activity x which generates both value V(x) and cost C(x).
Net value would therefore be:
The necessary condition to find the optimal level is:
Or:
)()()( xCxVxNV
dx
xdC
dx
xdV
dx
xdNV )()(0
)(
dx
xdC
dx
xdV )()(
Lecture 2
MGMT 7730 - © 2011 Houman Younessi
Unconstrained Optimization: Multiple variables
In the case where there are more than one activity, say when the value function is a function of x and y, we take the derivative of the function with respect to each variable and set them all to zero.
0),(),(),(
0),(),(),(
),(),(),(
y
yxC
y
yxV
y
yxNVx
yxC
x
yxV
x
yxNV
yxCyxVyxNV
As such we have:
y
yxC
y
yxV
andx
yxC
x
yxV
),(),(
),(),(
Lecture 2
MGMT 7730 - © 2011 Houman Younessi
Example:
ABCO LLC has two product lines: gadgets and widgets.
ABCO produces G of gadgets and W of widgets annually
The profit made by ABCO is of course related to their quantity of widgets and gadgets sold. The following equation shows this relationship:
2075.1138010105),( 22 GWWGGWWGP
Find the derivative (partial derivative) of profit with respect to G.
75.113205
GWG
P
Lecture 2
MGMT 7730 - © 2011 Houman Younessi
Find the derivative (partial derivative) of profit with respect to W.
80205
WGW
P
Now, using this information find the quantities of G and W that ABCO must manufacture to maximize profit.
To answer this question, we remember that a point is either a maximum or minimum when the derivative for that point is zero. For P to be maximized both derivatives with respect to G and W must be zero.
75.2
0.5
080205
075.113205
W
G
WG
GW
Lecture 2
MGMT 7730 - © 2011 Houman Younessi
Constrained Optimization
Constrained optimization applies when we wish to find the maximum or minimum point of a curve but there are also other limiting factors. In other words we wish to find the value of the independent variable at which the dependent variable is maximized or minimized with other external conditions restricting it.
y
yxVand
x
yxV
),(),(Let us start – without loss of generality -with
the marginal value for a two variable case:
The constraint is that the total cost must equal a specified level of cost relating to the price and quantities of the two components x, and y:
CyPxPyxC yx ),(
Lecture 2
MGMT 7730 - © 2011 Houman Younessi
There are two equivalent ways of solving such problems:
1. Simple simultaneous equations:
In this approach we solve the set of equations:
0
0),(
0),(
CyPxP
y
yxVx
yxV
yx
2. Lagrangian method:
The Lagrangian method works on the basis of adding “meaningful zeros” to the original equation and then assess their impact.
Lecture 2
MGMT 7730 - © 2011 Houman Younessi
The first thing we do is to form the Lagrangian function.
To do so, we first rearrange our constraint formula or formulas so that they all evaluate to zero:
0
CyPxP
CyPxP
yx
yx
Then, we add “zero” to the original value function:
)(),( yPxPCyxVL yx
Now we take partial derivatives of the value function wrt x, y and λ, set these to zero and solve.
Lecture 2
MGMT 7730 - © 2011 Houman Younessi
Example:Cando Co wishes to minimize the cost of their production governed by:
2122
21 54 QQQQTC
The constraint is that the company can only make 30 units of product in total
03030 2121 QQorQQ
The Lagrangian becomes: )30(54 212122
21 QQQQQQL
As such we have:
030),,(
010),,(
08),,(
2121
212
21
211
21
QQQQL
QQQ
QQL
QQQ
QQL
Lecture 2
MGMT 7730 - © 2011 Houman Younessi
Solving
030
010
08
21
21
21
For Q1, Q2 and λ
We get: Q1=16.5
Q2=13.5
λ= 118.5
What does λ mean?
It means that if the constraint were to be relaxed so that more than 30 units could be produced, the cost of producing the 31st is $118.5
Lecture 2
MGMT 7730 - © 2011 Houman Younessi
Imagine that you are running a manufacturing plant. This plant has the capacity of making 30 units of either widgets or gadgets. Furthermore, the total cost of the manufacturing operation is:
GWWGC 22 54How many widgets and how many gadgets should you manufacture to minimize cost?
To minimize cost, we must find the minimum of the cost function above. We also must make sure that the total units manufactures equals 30.
As such:
WG
therefore
WG
30
30
Example 2:
Lecture 2
MGMT 7730 - © 2011 Houman Younessi
Substituting: WG 30 GWWGC 22 54
360027010
)30(5)30(42
22
WW
WWWWC
Into:
Taking the derivative and setting it to zero, we get:
5.165.1330
5.13
027020
G
W
or
WdW
dC
To make sure this is a minimum point:
020)( dW
dC
dW
d
Lecture 2
MGMT 7730 - © 2011 Houman Younessi
Market Demand and the Demand Function
Market Demand Schedule for laptops
Price per unit ($) Quantity demanded per year (‘000)
3000
2750
2500
2250
2000
800
975
1150
1325
1500
Lecture 2
MGMT 7730 - © 2011 Houman Younessi
2000
2500
3000
800 1000 1200 1400 1600
Demand CurvePrice
Quantity
Lecture 2
MGMT 7730 - © 2011 Houman Younessi
Influences on Demand
2000
2500
3000
800
1000
1200
1400
1600
Price
Quantity
Increase in customer preference for laptops
2000
2500
3000
800
1000
1200
1400
1600
Price
Quantity
Increase in customer per capita income
2000
2500
3000
800
1000
1200
1400
1600
Price
Quantity
Increase in advertising for laptops 2000
2500
3000
800
1000
1200
1400
1600
Price
Quantity
reduction in cost of software
Lecture 2
MGMT 7730 - © 2011 Houman Younessi
Demand Function
Q=f( price of X, Income of consumer, taste of consumer, advertising expenditure, price of associated goods,….)
Example:
Q= -700P+200I-500S+0.01Awhere
Demand for laptops in 2007 is estimated to be:
P is the average price of laptops in 2007I is the per capita disposable income in 2007
S is the average price of typical software packages in 2007
A is the average expenditure on advertising in 2007
Lecture 2
MGMT 7730 - © 2011 Houman Younessi
Now let us assume that in 2007:
I=$33,000 S=$400 and A=$50,000,000
What will be the relationship between price and quantity demanded?
Given that: Q= -700P+200I-500S+0.01A
We have:
Q= -700P+200(33,000)-500(400)+0.01(50,000,000)
Q= -700P+6,900,000
Lecture 2
MGMT 7730 - © 2011 Houman Younessi
Price Elasticity of Demand
By what percentage would the quantity demanded change as a result of one unit of change in price?
The percentage change of quantity would be:
The percentage change of price would be:
Dividing one by the other:
Q
Q
P
P
P
P
Q
Q
Rearranging: P
Q
Q
PE
)(
Lecture 2
MGMT 7730 - © 2011 Houman Younessi
At the limit:dP
dQ
P
Q
P
Q
Q
PE
)(
dP
dQ
Q
P)(Therefore: becomes
Example:
Determine the price elasticity of demand for laptops in 2007 when price is $3000.
We know that: Q= -700P+6,900,000
700dP
dQ
Q=-700(3000)+6900,000=4,800,000
000625.04800000
3000
Q
P 4375.0000625.0700
Lecture 2
MGMT 7730 - © 2011 Houman Younessi
b
P
b/a
P = -aQ+b
Q
1
1
1
00 Pas
0 Qas
Demand is price elastic
Demand is price inelastic
Lecture 2
MGMT 7730 - © 2011 Houman Younessi
P
Demand Curve
Q
Exercise: Show that the price elasticity of demand on a demand curve given by the equation is always P
kQ 1
1
1
1
1
)(
2
22
2
2
2
kP
kP
k
P
Pk
k
P
k
PP
Q
P
Pk
dP
dQ
pkQ
Q
P
dP
dQ
Lecture 2
MGMT 7730 - © 2011 Houman Younessi
Exercise: Given the price elasticity of demand and the price, find marginal revenue
1
1
))((1
PMR
dQ
dP
P
QP
dQ
dPQP
dQ
dPQ
dQ
dQP
dQ
dPQ
dQ
dTRMR
PQTR
Lecture 2
MGMT 7730 - © 2011 Houman Younessi
Exercise: Given price elasticity of demand and marginal cost, what is the maximum price we should charge?
1
1PMRWe said that:
We also know that in order for price to be maximum, MR=MC, so
MCPMRPfor
1
1max
111
11
MCMCP
or
PMC
is the maximum price you should charge
Lecture 2
MGMT 7730 - © 2011 Houman Younessi
Income Elasticity of DemandBy what percentage would the quantity demanded change as a result of one unit of change in consumer income?
The percentage change of quantity would be:
The percentage change of income would be:
Dividing one by the other:
Q
Q
I
I
I
I
Q
Q
Rearranging:I
Q
Q
IE
)(
dI
dQ
I
Q
I
Q
Q
IE
)(
dI
dQ
Q
II )(Therefore: becomes
At the limit:
Lecture 2
MGMT 7730 - © 2011 Houman Younessi
Given that: Q= -700P+200I-500S+0.01A
Example:
Determine the income elasticity of demand for laptops in 2007 when Income is $33000
200dI
dQ
S=$400 P=$3000 and A=$50,000,000
dI
dQ
Q
II )(
375.12004800000
33000)( dI
dQ
Q
II
Therefore one percent increase in income leads to 1.375 percent increase in demand for laptops.
Lecture 2
MGMT 7730 - © 2011 Houman Younessi
Cross Elasticity of DemandBy what percentage would the quantity demanded change as a result of one unit of change in the price of an associated product?
Y
X
X
YXY dP
dQ
Q
P)(
Example:
Determine the cross elasticity of demand for laptops in 2007 when price of software is $400
042.05004800000
400)(
Y
X
X
YXY dP
dQ
Q
P
Therefore one percent increase in price of software leads to 0.042 percent decrease in demand for laptops.
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