Description
It explains various models using graphical representation.
Economic Models:
Basic Mathematical Tools applied in
economics
? Simplified representations of reality play a crucial role
in economics.
Why models?
Models in Economics
?A model is a simplified representation of a real
situation that is used to better understand real-life
situations.
? Create a real but simplified economy
? Simulate an economy on a computer
? Ex.: Tax models, money models
?The “other things equal” assumption means that
all other relevant factors remain unchanged.
Functional Relationships
? Relationship between two variables, for e.g. price
and output sold, expressed in various ways
? Table or graph
? Use of equations – Quantity sold depends on the
price, in other words quantity sold is a function of
price.
? P is the independent value and Q is the
dependent value
p p f Q 5 200 ) ( ÷ = =
Marginal Concepts & Slope of a
Curve
? Marginal Value is defined as change in a
dependent value associated with a 1-unit change
in an independent value.
? For e.g. change in total revenue earned by a firm
associated with an increase in output sold by one
unit, is the marginal revenue
? MR=Change in TR associated with change in Q
PQ TR =
Tabular form Representation
Q P=100-10Q TR=100Q-10Q
2
AR MR
0 100 0 - 0
1 90 90 90 90
2 80 160 80 70
3 70 210 70 50
4 60 240 60 30
5 50 250 50 10
6 40 240 40 -10
Graphical Representation &
Concept of Slope & Curvature
TR
TR
Q
O
A
B
C
? Slope of TR Curve at a particular point
represents MR at a particular output, i.e.,
change in TR for an infinitesimal change in
output level
? Implication of slope for any variable implies
marginal value of the same variable
? Curvature depends on changes in slope or
changes in marginal value
Changes in Slope
Changes in Curvature
? Linear Curve – Marginal value constant, no
change in curvature
? Curve Convex to the origin – Marginal value
(Slope) changing at an increasing rate
? Curve Concave to the origin – Marginal value
( Slope) changing at a decreasing rate
Average and Marginal
? Graphically Average value can be derived from the
total value curve.
? Average at a point on the Total value curve is equal
to the slope of the ray from the origin to that particular
point
? To increase (decrease) the average value, Average
value should be less (more) than the Marginal value
? Average Value constant implies its equality with
Marginal Revenue
Find out from Total Cost,
Average, & Marginal Cost
Q TC AC MC
0 20 - -
1 140 140 120
2 160 80 20
3 180 60 20
4 240 60 60
5 480 96 240
AC = TC/Q
MC = ATC/AQ
Average Cost (AC)
Q TC AC MC
0 20 - -
1 140 140 120
2 160 80 20
3 180 60 20
4 240 60 60
5 480 96 240
AC = TC/Q
Total, Average, and
Marginal Cost
Q TC AC MC
0 20 - -
1 140 140 120
2 160 80 20
3 180 60 20
4 240 60 60
5 480 96 240
AC = TC/Q
MC = ATC/AQ
Total, Average, and Marginal Cost
0
60
120
180
240
0 1 2 3 4
Q
TC ($)
0
60
120
0 1 2 3 4
Q
AC, MC ($)
AC
MC
Optimization Techniques
? In Economics different optimization techniques as a
solution to decision making problems
? Optimization implies either a variable is maximized or
minimized whichever is required for efficiency purposes,
subject to different constraints imposed on other
variables
? E.g. Profit Maximization, Cost Minimization, Revenue
Maximization, Output Maximization
? A problem of maxima & minima requires the help of
differential calculus
Profit Maximization
Q TR TC Profit
0 0 20 -20
1 90 140 -50
2 160 160 0
3 210 180 30
4 240 240 0
5 250 480 -230
Profit Maximization
0
60
120
180
240
300
0 1 2 3 4 5
Q
($)
MC
MR
TC
TR
-60
-30
0
30
60
Profit
Profit Maximization
? Total Profit Approach for Maximization
? ?=TR-TC=> The difference to be maximized
in order to Max. Profit
TR
TC
Q
O
TR, TC
A
B
Marginal Analysis to profit
maximization
? Marginal Analysis requirement for profit
Maximization,
Marginal Revenue = Marginal Cost
(MR) (MC)
? Marginal Value represents slope of Total
value curves,
? Thus slopes of TR &TC should be equal
Two output level showing same
slope, i.e. MR=MC
TR
TC
Q
O
TR, TC
A
B
Q
2
Q
1
Interpretation of the previous
diagram
? MR=MC is a necessary condition for Maximization,
not a sufficient one as this condition also hold for loss
maximization
? Sufficient condition requires that reaching a point of
maximization, profit should start declining with any
further rise in output, i.e. Slope of TC should rise &
Slope of TR must fall after reaching the point of
Maximization,
? Change in MC>Change in MR
*Case Study to be discussed: An alleged blunder in the
stealth bomber’s design
Concept of the Derivative
The derivative of Y with respect to X
is equal to the limit of the ratio
AY/AX as AX approaches zero.
0
lim
X
dY Y
dX X
A ÷
A
=
A
Rules of Differentiation
Constant Function Rule: The derivative of a
constant, Y = f(X) = a, is zero for all values
of a (the constant).
( ) Y f X a = =
0
dY
dX
=
Rules of Differentiation
Power Function Rule: The derivative of
a power function, where a and b are
constants, is defined as follows.
( )
b
Y f X a
X
= =
1 b
dY
b a
X
dX
÷
= ·
Rules of Differentiation
Sum-and-Differences Rule: The derivative
of the sum or difference of two functions U
and V, is defined as follows.
( ) U g X =
( ) V h X =
dY dU dV
dX dX dX
= ±
Y U V = ±
Rules of Differentiation
Product Rule: The derivative of the product
of two functions U and V, is defined as
follows.
( ) U g X =
( ) V h X =
dY dV dU
U V
dX dX dX
= +
Y U V = ·
Rules of Differentiation
Quotient Rule: The derivative of the
ratio of two functions U and V, is
defined as follows.
( ) U g X =
( ) V h X =
U
Y
V
=
( ) ( )
2
dU dV
V U
dY
dX dX
dX
V
÷
=
Rules of Differentiation
Chain Rule: The derivative of a function
that is a function of X is defined as
follows.
( ) U g X =
( ) Y f U =
dY dY dU
dX dU dX
= ·
Using derivatives to solve max and min problems
Optimization With Calculus
To optimize Y = f (X):
First Order Condition:
Find X such that dY/dX = 0
Second Order Condition:
A. If d
2
Y/dX
2
> 0, then Y is a minimum.
OR
B. If d
2
Y/dX
2
< 0, then Y is a maximum.
CENTRAL POINT
The dependent variable is maximized when its
marginal value shifts from positive to
negative, and vice versa
The Profit-maximizing rule
Profit( ) = TR – TC
At maximum profit
ot/dQ = oTR/dQ - oTC/dQ = 0
So,
oTR/dQ = oTC/dQ (1
st
.O.C.)
==> MR = MC
o
2
TR/ oQ
2
= o
2
TC/oQ
2
(2
nd
O.C.)
==> oMR/oQ < oMC/dQ
This means
slope of MC is greater than slope of MR function
Constrained Optimization
To optimize a function given a
single constraint, imbed the
constraint in the function and
optimize as previously defined
doc_126221151.ppt
It explains various models using graphical representation.
Economic Models:
Basic Mathematical Tools applied in
economics
? Simplified representations of reality play a crucial role
in economics.
Why models?
Models in Economics
?A model is a simplified representation of a real
situation that is used to better understand real-life
situations.
? Create a real but simplified economy
? Simulate an economy on a computer
? Ex.: Tax models, money models
?The “other things equal” assumption means that
all other relevant factors remain unchanged.
Functional Relationships
? Relationship between two variables, for e.g. price
and output sold, expressed in various ways
? Table or graph
? Use of equations – Quantity sold depends on the
price, in other words quantity sold is a function of
price.
? P is the independent value and Q is the
dependent value
p p f Q 5 200 ) ( ÷ = =
Marginal Concepts & Slope of a
Curve
? Marginal Value is defined as change in a
dependent value associated with a 1-unit change
in an independent value.
? For e.g. change in total revenue earned by a firm
associated with an increase in output sold by one
unit, is the marginal revenue
? MR=Change in TR associated with change in Q
PQ TR =
Tabular form Representation
Q P=100-10Q TR=100Q-10Q
2
AR MR
0 100 0 - 0
1 90 90 90 90
2 80 160 80 70
3 70 210 70 50
4 60 240 60 30
5 50 250 50 10
6 40 240 40 -10
Graphical Representation &
Concept of Slope & Curvature
TR
TR
Q
O
A
B
C
? Slope of TR Curve at a particular point
represents MR at a particular output, i.e.,
change in TR for an infinitesimal change in
output level
? Implication of slope for any variable implies
marginal value of the same variable
? Curvature depends on changes in slope or
changes in marginal value
Changes in Slope
Changes in Curvature
? Linear Curve – Marginal value constant, no
change in curvature
? Curve Convex to the origin – Marginal value
(Slope) changing at an increasing rate
? Curve Concave to the origin – Marginal value
( Slope) changing at a decreasing rate
Average and Marginal
? Graphically Average value can be derived from the
total value curve.
? Average at a point on the Total value curve is equal
to the slope of the ray from the origin to that particular
point
? To increase (decrease) the average value, Average
value should be less (more) than the Marginal value
? Average Value constant implies its equality with
Marginal Revenue
Find out from Total Cost,
Average, & Marginal Cost
Q TC AC MC
0 20 - -
1 140 140 120
2 160 80 20
3 180 60 20
4 240 60 60
5 480 96 240
AC = TC/Q
MC = ATC/AQ
Average Cost (AC)
Q TC AC MC
0 20 - -
1 140 140 120
2 160 80 20
3 180 60 20
4 240 60 60
5 480 96 240
AC = TC/Q
Total, Average, and
Marginal Cost
Q TC AC MC
0 20 - -
1 140 140 120
2 160 80 20
3 180 60 20
4 240 60 60
5 480 96 240
AC = TC/Q
MC = ATC/AQ
Total, Average, and Marginal Cost
0
60
120
180
240
0 1 2 3 4
Q
TC ($)
0
60
120
0 1 2 3 4
Q
AC, MC ($)
AC
MC
Optimization Techniques
? In Economics different optimization techniques as a
solution to decision making problems
? Optimization implies either a variable is maximized or
minimized whichever is required for efficiency purposes,
subject to different constraints imposed on other
variables
? E.g. Profit Maximization, Cost Minimization, Revenue
Maximization, Output Maximization
? A problem of maxima & minima requires the help of
differential calculus
Profit Maximization
Q TR TC Profit
0 0 20 -20
1 90 140 -50
2 160 160 0
3 210 180 30
4 240 240 0
5 250 480 -230
Profit Maximization
0
60
120
180
240
300
0 1 2 3 4 5
Q
($)
MC
MR
TC
TR
-60
-30
0
30
60
Profit
Profit Maximization
? Total Profit Approach for Maximization
? ?=TR-TC=> The difference to be maximized
in order to Max. Profit
TR
TC
Q
O
TR, TC
A
B
Marginal Analysis to profit
maximization
? Marginal Analysis requirement for profit
Maximization,
Marginal Revenue = Marginal Cost
(MR) (MC)
? Marginal Value represents slope of Total
value curves,
? Thus slopes of TR &TC should be equal
Two output level showing same
slope, i.e. MR=MC
TR
TC
Q
O
TR, TC
A
B
Q
2
Q
1
Interpretation of the previous
diagram
? MR=MC is a necessary condition for Maximization,
not a sufficient one as this condition also hold for loss
maximization
? Sufficient condition requires that reaching a point of
maximization, profit should start declining with any
further rise in output, i.e. Slope of TC should rise &
Slope of TR must fall after reaching the point of
Maximization,
? Change in MC>Change in MR
*Case Study to be discussed: An alleged blunder in the
stealth bomber’s design
Concept of the Derivative
The derivative of Y with respect to X
is equal to the limit of the ratio
AY/AX as AX approaches zero.
0
lim
X
dY Y
dX X
A ÷
A
=
A
Rules of Differentiation
Constant Function Rule: The derivative of a
constant, Y = f(X) = a, is zero for all values
of a (the constant).
( ) Y f X a = =
0
dY
dX
=
Rules of Differentiation
Power Function Rule: The derivative of
a power function, where a and b are
constants, is defined as follows.
( )
b
Y f X a
X
= =
1 b
dY
b a
X
dX
÷
= ·
Rules of Differentiation
Sum-and-Differences Rule: The derivative
of the sum or difference of two functions U
and V, is defined as follows.
( ) U g X =
( ) V h X =
dY dU dV
dX dX dX
= ±
Y U V = ±
Rules of Differentiation
Product Rule: The derivative of the product
of two functions U and V, is defined as
follows.
( ) U g X =
( ) V h X =
dY dV dU
U V
dX dX dX
= +
Y U V = ·
Rules of Differentiation
Quotient Rule: The derivative of the
ratio of two functions U and V, is
defined as follows.
( ) U g X =
( ) V h X =
U
Y
V
=
( ) ( )
2
dU dV
V U
dY
dX dX
dX
V
÷
=
Rules of Differentiation
Chain Rule: The derivative of a function
that is a function of X is defined as
follows.
( ) U g X =
( ) Y f U =
dY dY dU
dX dU dX
= ·
Using derivatives to solve max and min problems
Optimization With Calculus
To optimize Y = f (X):
First Order Condition:
Find X such that dY/dX = 0
Second Order Condition:
A. If d
2
Y/dX
2
> 0, then Y is a minimum.
OR
B. If d
2
Y/dX
2
< 0, then Y is a maximum.
CENTRAL POINT
The dependent variable is maximized when its
marginal value shifts from positive to
negative, and vice versa
The Profit-maximizing rule
Profit( ) = TR – TC
At maximum profit
ot/dQ = oTR/dQ - oTC/dQ = 0
So,
oTR/dQ = oTC/dQ (1
st
.O.C.)
==> MR = MC
o
2
TR/ oQ
2
= o
2
TC/oQ
2
(2
nd
O.C.)
==> oMR/oQ < oMC/dQ
This means
slope of MC is greater than slope of MR function
Constrained Optimization
To optimize a function given a
single constraint, imbed the
constraint in the function and
optimize as previously defined
doc_126221151.ppt