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KWAME NKRUMAH UNIVERSITY OF SCIENCE AND
TECHNOLOGY, KUMASI
INSTITUTE OF DISTANCE LEARNING
MATH 473: MATHEMATICAL ECON0MICS 1
[Credits 3]
By
F.T. ODURO & C. SEBILDEPARTMENT OF MATHEMATICS, KNUST
OCTOBER, 2011
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ii
Contact Address
Dean
Institute of Distance Learning
New Library Building
Kwame Nkrumah University of Science and Technology
Kumasi, Ghana
Phone: +233-51-60013+233-51-60014
Fax: +233-51-60023
+233-51-60014
E-mail:
[email protected] [email protected] [email protected]
Web: www.fdlknust.edu.gh
mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]://www.fldknust.edu.gh/http://www.fldknust.edu.gh/http://www.fldknust.edu.gh/mailto:[email protected]:[email protected]:[email protected]
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About the Authors
F.T. ODURO is, currently, a senior lecturer at the Department of
Mathematics of the Kwame
Nkrumah University of Science and Technology, Kumasi. Dr.
F.T. Oduro has almost twodecades experience in the teaching of
application oriented mathematical courses such as
engineering mathematics, classical fields, mathematical
economics and graduate courses in
control theory and stochastic processes.
He has also supervised dozens of graduate research projects
involving the mathematical
modeling and control of environmental, health and economic
systems. He has held a number
of administrative positions at the university including Head of
the Department of
Mathematics, Coordinator of the Actuarial Science programme and
Head of the Kumasi
Virtual Center for Information Technology which is a department
of the Faculty of Distance
Learning.
Dr. Oduro is a member of the Ghana Science Society, a member of
the Mathematical
Association of Ghana and executive member of the Ghana Chapter
of the International
Biometric Society
Email: [email protected]
C. SEBIL is currently a lecturer at the Department of
mathematics of the Kwame Nkrumah
University of Science and Technology, Kumasi. Mr. C. SEBIL
teaches Optimisation,
mathematical economics, Engineering mathematics, Algebra and
statistical methods.
Mr. C. SEBIL is a member of the Ghana Science Society, a member
of the Mathematical
Association of Ghana.
Email: [email protected]
mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]
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Course I ntroduction
This course is designed to present fourth year students of
mathematics, actuarial science and
statistics students with the fundamental principles of
mathematical economics. It focuses on
basic concepts and uncovers the simplicity and directness
of a mathematical approach to
economics theory.
At the end of the course, students are expected to be able to
appreciate the constrained
optimizing behaviour of consumers and producers as well as the
key structures of the
marketplace.They should also be able to formulate and solve a
lot of economic problems in a
mathematical context.
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v
Table of Content
Contact Address
.........................................................................................................................iiAbout
the Author
.....................................................................................................................
iiiCourse Introduction
..................................................................................................................
ivTable of Content
........................................................................................................................
vList of Figures
...........................................................................................................................
vi
UNIT 1
.......................................................................................................................................
1GENERAL CONCEPTS
...........................................................................................................
1
Session 1-1: Introductory Remarks
........................................................................................
1Session 2-1: Special
Processes............................................. Error!
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UNIT 2
.....................................................................................Error!
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.........................................................Error!
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Session 1-2: N-dimensional and Complex Processes ..........
Error! Bookmark not defined. Session 2-2: Stationary
Processes ........................................Error! Bookmark
not defined.
UNIT 3
.....................................................................................Error!
Bookmark not defined. CALCULUS OF STOCHASTIC PROCESSES
...................... Error! Bookmark not defined.
Session 1-3: Stochastic Continuity and Differentiability.....
Error! Bookmark not defined. Session 2-3: Stochastic
Differential Equations ....................Error! Bookmark not
defined.
UNIT 4
.....................................................................................Error!
Bookmark not defined. CALCULUS OF STOCHASTIC PROCESSES
...................... Error! Bookmark not defined.
Session 1-4: Stochastic Integrals and Time Averages .........
Error! Bookmark not defined. Session 2-4: Ergodicity
........................................................Error!
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L ist of F igures
Figure 1-1
.................................................................................
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...............................................................................
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...............................................................................
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Figure 4-1
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Error! Bookmark not defined.
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1
UNIT 1
PRELIMININARY MATHEMATICAL CONCEPTS
Introduction to Unit 1
In this introductory unit, we briefly review classical
optimization which is the basis of
consumer optimizing behaviour. Next we deal with the basic
concept of axioms of choice and
indifference curves. After the preliminary notions of
probability theory, we present the main
ideas and fundamental properties. We continue with definitions
and discussions on the
concepts of transformations, continuity, and differentiation.
Finally we discuss simple
differential equations, as well as stochastic integrals, time
averages, and ergodicity.
Session 1- 0: Review of Classical Optimization
1.1 Local Extrema of Funct ions on R
Consider a differentiable function
: f . f is said to have a
local maximum at a point
* x if for some 0 , *)()*(
x f h x f
h . * x is then called a local maximum point
Similarly, f is said to have a local minimum
at a point * x if for some 0 ,
*)()*(
x f h x f
h . * x is then called a local minimum point.
A local extremum refers to either a local maximum or a local
minimum.
Theorem (Necessary condition for a local extremum)
If f has a local extremum point at
* x then 0*)(' x f
Note
The converse of the above theorem is not true
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Example
Consider the function 3)(
x x f , we note that although
0)0(' f , 0 x is
neither a maximum
nor a minimum point; in fact, it is a point of inflection.
Theorem (Sufficient condition for a local extremum)
If 0*)('
x f and 0*)(''
x f , then
f has a local maximum point at * x .
If 0*)('
x f and 0*)(''
x f , then
f has a local minimum point at * x .
1.2 Local Extrema of Funct ions on R n
Consider a differentiable function
n f : . f is said to have a
local maximum at a point
n
x * if for some 0 , *)()*(
x f h x f
h , * x is then called a local maximum
point
Similarly, f is said to have a local minimum
at a point n x * , if, for some 0 ,
*)()*(
x f h x f
h , * x is then called a local minimum
point.
A local extremum refers to either a local maximum or a local
minimum.
Theorem (Necessary condition for a local extremum)
If f has a local extremum point
at n
x * , then 0*)(
x f ; i.e.
0*)(
...*)(*)(
21
n x
x f
x
x f
x
x f
Note
The converse of the above theorem is not true
Theorem (Sufficient condition for a local extremum)
If n
x * is a critical point of f
and the Hessian *)( x H of
f is negative definite, then
f has a
local maximum point at * x .
If n x * is a critical
point of f and the Hessian
*)( x H of f is positive definite,
then f has a
local minimum point at * x .
Note that the Hessian of f is the matrix of its second
order derivatives and is given by
,ijh H where, ji
ij x x
f h
2
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Positive/Negative definiteness of the Hessian
A Symmetric Matrix is positive definite if its principal minors
are all positive. I.e.
011 h , 0
2221
1211
hh
hh, 0
333231
232221
131211
hhh
hhh
hhh
,…
A Symmetric Matrix is negative definite if its principal minors
alternate in sign as follows:
011
h , 02221
1211
hh
hh, 0
333231
232221
131211
hhh
hhh
hhh
,…
Summary
Thus, for a multivariable function such as 2: f
with 2),( y x ; for (x,y) to be a
local
minimum or maximum three conditions must be met
1) The first order partial derivatives must equal zero
simultaneously. This indicates that
at the given point (a, b) is a critical point at which the
function is neither increasing
nor decreasing with respect to the principal axes but is at a
relative plateau
2) The second-order direct partial derivatives when
evaluated at the critical point must
both be positive for a minimum and negative for a
maximum
3) The product of the second-order direct partials
evaluated at the critical point should
exceed the product of the cross partials evaluated at the
critical point.
Local Maximum Local Minimum
1. f x = 0 and f x = 0
2. f xx < 0 and f yy < 0
3. f xx f yy > (f xy)2
1. f x = 0 and f x = 0
2. f xx > 0 and f yy > 0
3. f xx f yy > (f xx)2
1.3 Constrained Optim izat ion on R n
Consider a differentiable function
n f : . Subject to a constraint 0),...,,( 21
n x x x g is
said to A local extremum point n x * can be
found if the function f to be optimized (the
objective function) is replaced by 1
: n
L called the Lagrangian and given by
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),...,,(),...,,(),,...,,( 212121 nnn
x x x g x x x f x x x L
is called a Lagrange multiplier. Similar necessary and
sufficient conditions for local
extrema then apply
For a linear constraint on R 2
the Hessian is said to be bordered and the sufficient
(secondorder) conditions are given by
011 h , 0
2221
1211
hh
hh, 0
021
22221
11211
p p
phh
phh
for a minimum
and
011
h , 02221
1211
hh
hh, 0
021
22221
11211
p p
phh
phh
for a maximum
Again note that here the Hessian of L is the matrix
of its second order derivatives and is given
by
,ijh H where ,
i
i
ji
ij x
L p ji
x x
Lh
22
and 2,1,for
And the linear constraints are given by
0),( 221121
C x p x p x x g
Example
1) Optimize the function z = 4x2 + 3yx + 6y 2 subject
to the constraint x + y = 56
Solution
Set the constraint equal to zero: 56 + x – y =
0
Multiply it by λ and add it to the objective to
form the Lagrangian function Z
Z = 4x2
+ 3xy + 16y2 + λ (56 – x-y).
Take the first-order partials, set them equal to zero and solve
simultaneously
)1....('.........38
y x
x
Z Z x
)2.........(.....'....0123
y x
y
Z Z x
056 y x Z
Z x
From 1) and 2) y = 20 and x = 36
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Now from 1) 8x + 3y = λ
8(36) + 3(20) = λ
Λ = 348
Substituting the critical value in Z:
Z = 4(36)2 + 3(36)(20) + 6(20) 2 + 348(56
– 36 – 20) = 9744
Since Zxx > 0, Z yy > 0 the optimal value is a
minimum
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7
Session 1-1: Consumer Optimizing Behaviour .
SOME AXIOMS OF CONSUMER BEHAVIOUR
The ranking of goods by the consumer is called his preference
function
1) The axiom of completeness or comparison. Given two
commodities
X and Y the consumer should be able to state one and only one of
the following
Y X : ( X is preferred to
Y)
X Y : ( Y is preferred to
X)
Y X : ( X, Y equally
satisfying, the consumer is indifferent between X and Y)
2) Axiom of Transitivity
Z X Z Y Y X
,
Z X Z Y Y X
,
3) Axiom of non-saturation or non-satiety
The consumer prefers more to less
Indifference Curves
An indifference curve is the locus of points or particular
combination of goods each of which
gives the same satisfaction.
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Utility, which is the measure of consumer satisfaction, is that
function the level curves of
which are the indifference curves. Along a particular
indifference curve, utility is constant.
Utility can thus be represented by a utility surface.
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9
Properties of Indifference Curves
Indifference curves
1)
are everywhere dense i.e. an indifference curve passes through
any point on in the
commodity space
2) are negatively sloped
3)
cannot intersect
4)
are convex to the origin
Marginal Rate of Substitution (MRSyx )
The marginal rate of substitution of Y for X measures the number
of units of Y that a
consumer is willing to sacrifice for a unit of X so as to
maintain a constant level of utility or
satisfaction and it is given as the negative of the slope of an
indifference curves
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NB It is always defined along a particular indifference
curves
The Constrained Maximization of Utility
The rational consumer desires to purchase a combination of X and
Y from which he derives
the highest level of satisfaction. His problem is one of
maximization. However, his income
is limited and he is not able to purchase unlimited amounts of
the commodities. The
consumer’s budget constraint can thus be written as
XPX + YP Y = I ……………………………………………………………….. (1)
Where, I is his income and PX and P Y are the
respective prices of X and Y. The amount he
spends on commodity X (X PX) plus that spent on Y (Y PY) equals
his income I.
The First Order and Second Order Conditions
The consumer desires to maximize his utility U = U(X,Y) subject
to XP X + YP Y = I
We form the Lagrangian
L (X,Y,λ ) = f(X,Y) - λ (XPX + YP Y - I
)………………………………(2)
Where λ is the Lagrange multiplier
The first-order conditions are obtained by setting the first
partial derivatives of L with respect
to X, Y and λ equal to zero. We obtain
0
()
()
I YP XP L
P Y
U
Y
L
P X
U
X
L
y x
y
x
…………………………………………….(3)
From the first two equations of (3) we have
x x
P
X L
P
X L //
Marginal utility divided by price must be the same for all
commodities. The ratios give the
rate at which satisfaction would increase if an additional cedi
were spent on a particular
commodity. If more satisfaction could be gained by spending an
additional cedi on X rather
than Y, the consumer would not be maximizing utility. He could
increase his satisfaction by
shifting some of his expenditure from Y to X.
The Lagrange multiplier λ is the marginal utility of income. The
marginal utility of income is
positive.
Again, from (3), the first order condition for the optimization
problem can also be written as
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y
x
P
P
Y L
X L
/
/
Thus, the ratio of the marginal utilities must equal the ratio
of prices for maximum.
Now U(X,Y) = constant along an indifference curve and
hence
0
dY
Y
U dX
X
U
0dY
dY MU
dX
dX MU y x
dX
dY MU MU y x
y
x yx
y
x
MU
MU
dX
dY MRS
MU
MU
dX
dY
Graphical solution of the Consumer’s Optimization Problem
As one moves from A to D the MRSyx decreases. MRS tends to
turn against the commodity
that is abundant and in favour of the commodity that is scarce.
The budget line equation with
y as subject is given by y
x
P
I X
Py
P Y
The consumer attains equilibrium on the budget line. At point F,
the slope of I2 and the slope
of the budget line are the same i.e.
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Y
X
YX
P
P MRS
Y
X
Y
X
Y
X
Y
X
P
MU
P
MU
P
P
MU
MU
Linear Indifference Curves
Suppose I is an Indifference curve MRS is constant the
commoditiesare perfect substitutes. If the slope of budget is
different from the indifferent curves then wehave
specialization.
Income Consumption Curve (ICC)
Consider a consumer who receives a permanent rise in his income.
If income rises and
prices remain constant, the budget line shifts line shifts
parallel to the first one..
All ICC start at the origin. This is because at the origin the
individual’s income is zero and
hence cannot purchase any one of Y and X.
In most cases ICC are upward sloping if the commodities are
normal or inferior. These are
defined in terms of income elasticities. If income elasticity is
negative, the commodity is
inferior:
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= income elasticity
0 inferior goods
102
normal goods
Second Order Conditions
The second-order condition as well as the first-order condition
must be satisfied to ensure that
a maximum is actually reached.
Denoting the second direct partial derivatives of the utility
function by Uxx and U yy and the
second cross partial derivatives by Uxy and U yx , the
second order condition for a constrained
maximum requires that the relevant bordered Hessian determinant
be positive.
0
0
y x
y yy yx
x xy xx
P P
P U U
P U U
Expanding, we get
0 y x
xy xx
y
y x
yy yx
x P P
U U P
P P
U U P
0
xy x xx y y yy x yx y x
U P U P P U P U P P
022
xy x y xx y yy x yx y x
U P P U P U P U P P
02 22
xx y yy x xy y x
U P U P U P P
Substituting
X U P x
/ and
Y U P y
/
we have
0////
2
22
Y U U
X U U
Y U X U U
xx yy xy
02
22
Y
U U
X
U U
Y
U
X
U U xx yy xy
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Demand Function
Ordinary Demand Function
A Consumer’s ordinary function sometimes called a Marshallan
demand function give the
quantity of a commodity that he will buy as a function of
commodity prices and his income.
The demand function can be derived from the analysis of utility
maximization. Using the
first-order conditions of maximization the demand functions can
be obtained.
Example
Let us assume that the utility function is U = XY and the budget
constraint.
I – XP y – YP x = 0.
From the expression
L = XY + λ(I - XPy – YP x) and its partial
derivatives equal to zero
0
y P Y X
L
0
x P X
Y
L
Solving for X and Y gives the demand functions
x P
M X
2 and
y P
M Y
2
Properties of Demand Functions
1)
The demand for a commodity is a single-valued function of prices
and
income
2) Demand function is homogenous of degree zero in income.
That is if all
prices and income change in the same proportion, the
quantities demanded
remained unchanged. We now look at the proofs of these
properties
1a) The first property follows from the strict quasi-concavity
of the utility function, a
single maximum, and therefore a single commodity combination
corresponds to a given set of prices and income
NB
If the utility function were quasi-concave, the indifference
curves would possess straight-line
portions and maxima would not need to be unique. In this
case more than one value of the
quantity demanded may correspond to a given price, and the
demand relationship is called a
correspondence rather than a demand function.
2b) To prove the second property, assume that all prices and
income change in the
proportion K. The budget constraint becomes
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KI – KXP x - KYP y = 0 where K is
the factor of proportionality and
L = U(X,Y) + λ (KI – KXPx – KYP y)
and the first-order conditions are
0
y x KP U X
L
0
x y KP U
Y
L
0
Y Y KYP KXP KI
L
KI - KXPX - KYP Y can be written as K(I
– XP X - YP y) = 0 . Since K 0, I - XP
x – YP y = 0
Eliminating K from the first two equations of the first-order
conditions for a maximum
we have
y
x
y
x
y
x
x
y
P
P
U
U
KP
KP
U
U
Hence we have I - XPx – YP y = 0
and y
x
y
x
P
P
U
U which are like the original equations
Therefore the demand function for the price-income set (KPx,KPy,
KI)) is derived from the
same equations as for the set (Px,Py, I)
Compensated Demand Function
Imagine a situation in which some public authority taxes or
subsidies to a consumer in such a
way as to leave his utility unchanged after a price change.
Assume that this is done by
providing a lump-sum payment that will give the consumer’s
compensated demand function
the quantities of the commodities. They are obtained by
minimizing the consumer
expenditure subject to the constraint that his utility is at the
fixed level U (This is the dual
optimization problem)Assume again that the utility function is U
= XY. From the expression
Z = XPy + YP y + λ (U - XY) and setting its partial
derivatives equal to zero, we get
0
UY P
X
Z x
02
UX P
Y
Z
0
XY U
Z
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Now Px = UY , P y = UX
X
Y
P
P
x
x and Y =
y
x
P
XP
y x YP XP
x
x
P
YP X
Compensated demand for X
0
y
x
P
XP X U
U P
P X
y
y
2
y
y
y
y
P
UP Y
P
UP X ..
Demand Curves
In general the consumer’s ordinary demand function Qy is
written as q 1 = φ(P1, I) or
assuming that P2
Given parameters, p2 and I as fixed, q 1 = D(P 1) is
the demand curve for commodity 1. It is
often assumed the function possesses an inverse such that price
may be expressed as unique
function of quantity.
Generally demand curves are negatively sloped which implies that
the lower the prices, the
greater the quantity demanded.
In exceptional cases the opposite may hold. An example is
provided by ostentatious
consumption. If the consumer derives utility from a high price,
the demand function may
have a positive slope.
Price Elasticity of Demand
The quantity demand of a commodity depends upon its price. It is
of interest to measure the
relative change in quantity demanded as a result of given
proportional change in price. This
measure is called the price elasticity of demand.
Definition
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The price elasticity of demand is the relative responsiveness of
quantity demanded to change
in commodity price in other words price elasticity is the
proportional change to quantity
demanded divided by the proportional change in price.
Let e be own price elasticity of demand for the commodity X,
then e
Let the demand curve for commodity be q1 = f (P 1, P2
,..., Pn, I) where q is the quantity
demanded, p j the price of the jth commodity, I is
the income and we assume there are n
commodities. By definition, the own price elasticity of demand
is
)(ln
)(ln
i
i
i
i
i
i
ii p
q
q
p
p
qe
A numerically large value for an elasticity implies that the
quantity is proportionately very
responsive to price changes. If e0 < -1, then the good
is a luxury good. (A numerically high
value). If e0 > -1, then the good is a necessity. (A
numerically small value)
Price Elasticity of Demand and Expenditure
The rate of change of consumer expenditure on q1 wrt p 1
is given by
1
1
1
11
1
111
1
11 1)(
p
q
q
pq
p
q pq
p
q p
= q1 (1 + e 11)
)1()(
111
1
11 eq p
q p
Thus the consumer expenditure on q1 will
i)
Increase with p1 if e 11 > -1 (necessity)
ii) Remain unchanged if e11 = -1
iii) Decrease if e11 < -1 (luxury)
Cross-Price Elasticity of Demand
DEFINITION The price cross-elasticity of demand measures
the relative
responsiveness of quantity demanded of a given commodity to
changes in the price of a
related commodity. In other words, it is the proportional change
in the price of a good)
1
1
1
1
1
1
1
111 ,
q
P
P
q
P
p
q
qe
OR
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x
y
y
x
y
x
xyq
p
p
q
p
qe
)(ln
)(ln
I)
If exy > 0 then X and Y are substitutes
II)
If exy < 0 then X and Y are complements
The Cournot Aggregation Condition
Taking the total differential of the budget constraint
XPy + YP x= I and letting
dI = dpy = 0 and multiplying through by MXYdp
NY P x
pxdX + Xdpx + p ydY = 0. Multiplying through by
IXYdp
XY P x we get
0
x
y
y x
x
x
x
x x
IXYdp
XY P dY p Xdp
IXYdp
XY P
IXYdp
XY P dX p
0
2
IX
Y P
Y
P
dp
dY
IXY
Y P X
P
dp
IY
XYp
X
p
dp
dX y y
x
x
x
x x x
x
0 I
Y P e
I
Xp
I
Xpe x x yx
x x xx
x yx y xx x ee
where
I
YP
I
XP y y
x
x , are the
proportions of total expenditures
for the two goods. Given e11 (own price elasticity of
demand) for q 1 the formula
1212111 ee can be
used to calculate the cross-elasticity of demand.
i) If e11 = -1, e 21 = 0 i.e.
12121 )1( e
11212 e
0021212
ee
ii) If e11 < -1, e 21 > 0
0,121111 ee
Then -1
+1
e 11 > 0 hence 00 21212
ee
Similarly,
iii) If e11 > -1, e 21 < 0
1 e 11 + 1212
e
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1111212 ee
)1()1(
11
1
1
2
11121 e
ee
021 e
Income Elasticity of Demand
Income elasticity of demand for an ordinary demand function is
defined as the proportionate
change in income with prices constant
)(
)(
InI
InX x
Engel’s Aggregation Condition Taking the total
differential of the budget constraint, XPx + YP
y – I, we have
Px dX + Py dY = dI multiplying through by I
I the first term on the left by
,
X
X the second by
Y
Y and dividing by dI, we get
I
I dI dY P
Y
Y
I
I dX P
X
X
I
I y x
I
I
dI
dI
dI
dY P
Y
Y
I
I
dI
dX P
X
X
I
I y x
1 I
YP
Y
I
dI
dY
I
XP
X
I
dI
dX y x
1Y
I
dI
dY
I
YP
X
I
dI
dX
I
XP y x
12211
where M
YP
M
XP y x 21 ,
Is Engel’s aggregation condition
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Price Elasticity and Marginal Revenue
By definition, Total Revenue (TR) is TR = pq where p is the
price of the good q is the
commodity bought.
TR = pq and MR dq
dTR
dq
dp
p
q P
dq
dpq p
dq
dTR MR 1
Butedq
dp
p
qe
dp
dq
q
p 1
where e is the price elasticity of demand of the commodity.
e P MR
1
1
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SOLVED PROBLEMS
Example 1
A consumer spends $360 per week on two goods and X
and Y and P x = $3 and P y =$2.
His utility function is U = 2X2 Y. What quantities of X
and Y does he buy each week in
equilibrium. Check whether the second-order condition of maximum
is satisfied
Solution
We have: I = $360, U = 2X2Y, PX = $3, P Y = $2
Thus XY x
U MU
x 4
and 22 X
y
U MU y
yx xy U x
U
y X y
U
xU
4
0,4
y
U
yU Y
x
U
xU yy xx
At equilibrium
x y
v
x
y
x XP YP P
P
X
XY
MU
MU 2
2
42
And since I XP YP x y
We have I YP y 3
And therefore 603
x
P
I Y
Also
I XP XP XP YP
x x x y 2
1
2
1
803
2
x P
I X
The second-order condition is given by
02
22
Y
U U
X
U U
Y
U
X
U U xx yy xy
Substituting values of the first and second order derivatives of
U on the LHS, we get
2222 4)4(4)0(24)4(2
X Y XY X XY X
Which, simplifies to
01664 22
Y X Y X 00
Y
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Example 2
A consumer spends $450 per week on two goods X and Y, with
PX = $5 and P Y = $3.
His utility function is U = 0.5XY2.
i)
Find his demand functions for X and Y
ii) Find the optimal values of X and Y
Solution
I = $450, PX = $3, P Y = $2, U = 0.5XY2
Thus, XY y
U MU Y
x
U MU y x
,5.0
2
At equilibrium
x
x
x
x
P
P
MU
MU x y
y
x XP YP P
P
XY
Y 5.0
5.0 2
Thus, I XP YP x y
becomes I YP YP y y
5.0
i)
Hence y P
I Y 3
2 is his demand function for Y
Also, I XP YP x y
becomes I XP XP
x x 2
Hence, x P
I X
3 is his demand function for X
ii)
Numerically, 3015
450
3
x P
I X and 100
9
900
3
2
y P
I Y
Example 3
A rational utility-maximizing individual lives in a world with
only two goods: X and Y.
His utility function is given by U(X,Y) = XY .
His money income is $256 per week and
PY = $8
a)
Derive the equation for his demand curve for Y
b)
Find the equilibrium quantity of Y
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Solution
XY U P I y x
,8$256$
X
Y
X
U MU x
2
1
Y
X
Y
U MU y
2
1
At equilibrium
y
x
y
x
P
P
MU
MU x y
y
x XP YP P
P
X
Y
Thus, I XP YP x y
becomes I YP y 2
And his demand curve for Y is given by y P
I Y
2
The equilibrium quantity of Y: 1616
256
2
y P
I Y
Example 3
Total Revenue from the sale of a commodity is given by the
equation
22100 QQTR
Calculate the point elasticity of demand when marginal revenue
is 20.
Solution
Marginal revenue MR is given by
Q
P
Q
P
dP
dQe
2
Where 20Q and 60)20(21002100 Q P
5.140
60
2
Q
P e (demand is elastic)
Example 4
A rational utility-maximizing individual lives in a world with
only two goods X and Y.
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His utility function is
XY Y X U ),( . His
money income is $256 per week and Px = $8
a) Derive the equation for his demand curve for Y
b) If the price of Y is $2
i)
Calculate his cross-elasticity of demand for Y w.r.t the price
of X
ii)
Calculate his income elasticity of demand for X
c) He is given the option of joining a club, for dues of
$176 per week which would
give him one but only one, of the following rights concerning
purchases for his
own consumption:
i)
he would buy X at 50% of the normal price
ii)
He could buy Y at 50% of the normal price
iii) He could buy both X and Y at 75% of the normal
prices. The normal prices
are still Px = $8 and P y = $2 and his income before
payment of dues is still at
$256 per week.
Will he join the club, and if so, will he choose option (i),
(ii) or (iii)?
Solution
X
Y
X
U MU x
2
1
Y
X
Y
U MU y
2
1
At equilibrium
y
x
y
x
P
P
MU
MU
x y y
x XP YP P
P
X
Y
Thus, I XP YP x y
becomes I YP y 2
And his demand curve for Y is given by y P
I Y
2
b) i) From the demand equation y P
I Y
2 the demand for Y is independent of the price
of
X and hence the cross-elasticity is zero
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Note also that x y XP YP
implies I XP x 2
Now x x
P dI
dX
P
I X
2
1
2
Income elasticity for
Therefore, Income elasticity of demand X
I
dI
dX e I = 1
2/2
1
2
1
x x x P I
I
P X
I
P
Therefore, income elasticity of demand is unity
c) Using the demand equations derived above with given income
and prices we know
that
if he does not join the club his position is
642
256
2
1
2
y P I Y
168
256
2
1
2
x P
I X
32)64)(16( U
i) If joins and buys X at 50% of the normal price his position
is:
4$8100
50 x P x
I = 256 – 176 = 180
5.224
180
2
1
2
x P
I X
452
180
2
1
2
x P
I Y
)45)(5.22( U = 25.22
iii)
If he joins and buys Y at 50% of the normal price, his
position
1$2100
50 x P y
901
180
2
1
2
y P
I Y
25.11
8
180
2
1
2
x P
I X
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)90)(25.11( U = 25.22
iii) If he joins and buys both X and Y at 75% of normal price
his position is
6$8
100
75 x P x and 3$2
100
75 x P y
156
180
2
1
2
x P
I X
303
180
2
1
2
y P
I Y
)30)(15( U = 215
Thus he will join and be indifferent to options (i) and (ii)
because his utility in each case is
25.22 which is greater than the utility before joining the
club which is 32 .
It is thus not worth joining in respect of option (iii).
Example 5
The relationship between a consumer’s income and the quantity of
X he consumes is given
by the equation
21000Q I
Calculate his point income elasticity of demand for X when his
income is 64,000.
Solution
81000
64000
1000
2 Q
I Q
AndQdI
dQ
2000
1
2
1
8
64000
)8)(2000(
1
Q
I
dI
dQ
His point income elasticity of demand for X is 0.5.
Example 6
A town of 2,000 households constitutes a market for eggs.
Current sales are 2,400 dozen
eggs per week of $1.25 per dozen 1,200 households living on the
west side of the river buy
1,600 dozen and their elasticity of demand is 1.5. The remaining
households live on the east
of the river, buy the rest of the eggs, and have an elasticity
of demand of -3 calculate the
elasticity of the market demand curve for the town as a
whole.
Solution
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If two-thirds of eggs are subject to an elasticity of 1.5
and-third to -3 the combined elasticity
is the weighted average.
3
2
3
13
3
25.1
x x
Therefore the elasticity of the market demand curve is -2.
Example 7
An individual spends his income on three goods. He buys 550
units of X at $1 per units,425
units of Y at $2 per month units, and 200 units of Z at $3 per
unit. He now buys 440 units of
Y and 190 units of Z. Calculate his price elasticity of demand
for X.
Solution
Old expenditure on Z = 200 x 3 = $600
Old expenditure on Y = 425 x 2 = $850
Total expenditure on Y and Z = $1450
New expenditure on Y = 440 x 2 = $880
New expenditure on Z = 190 x 3 = $570
Total expenditure of Y and Z = $1450
Expenditure on Y and Z remains unchanged hence expenditure on X
remains unchanged.
Therefore the price elasticity of demand for X is -1.
Example 8
An individual lives in a world where there are only two goods X
and Y. His utility function
per period is:
U = 50X – 0.5X 2 + 100Y
– Y
The price of X is 4 and his income per period is 672.
b) derive his demand function for Y
c) If the price of Y is 14, how much X does he buy?
d)
At equilibrium. Calculate his income point elasticity of demand
for X
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e) The individual is given the opportunity to join a
society whose members can buy
Y at z price of 5, this would be individuals only benefit from
membership. What
is the maximum amount that he would just be prepared to pay in
membership dues
each period to join the society?
f)
Suppose the membership dues are 222 per period. Will he join?
What then would
be the marginal utility of money to him?
Solution
a)
the equilibrium condition are
I YP XP and P
P
MU
MU
x x
x
x
x
x
..,..
I YP XP and P
P
MU
MU y x
y
x
y
x ..,..
U = 50X – 0.5 X 2 + 100Y
– Y 2
Y X
U MU X
X
U MU y x 2100,50
Py (50 – X) = P x (100
– 2Y)
(50Py – X) =P y (100P
x – 2YP x)
50Py – XP y = 100P
x – 2YP x
-XPx = 100P x – 2YP
x – 50P x
X = y
y x x
P
P YP P
502100
X = y
y x y
P
P YP P
50250
But XPx + YP y = I so P x
I YP P
P YP P y
y
x x y
100250
50PxPy + 2YP x – 100P x = P
y I
22
2
2
50100
y x
y x x y
P P
P P P I P Y
b) Given that Py = 14 P x = 4
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22
2
2
50100
y x
y x x y
P P
P P P I P Y
22
2
1442
4450410067214
x x
x x x x
3619632
280016009408
Y
Now from MPx + YP y = I, we have
X(4) + 36 x 14 = 672
4X + 504 – 672
4X = 672 – 594 = 168
X = 42
There equilibrium quantity of X bought is 42
e) We find the demand function for x
y
x
y
x
y
x
P
P
Y
X so
P
P
MU
MU
2100
50.....
Hence
50Py - XP y = 100P x – 2YP
x
2YPx = 100P x – 50P y + XP
y
x
y y x
P
XP P P Y
2
50100
Substituting into the budget constraint we get
I P
P P P XP
x
y x
y
2
50100
2XPx + 100P x – 50P
x – 2P x I
222
501002
x x
y x y x
P P
P P P I P X
Hence the demand function for X is
222
501002
x x
y x y x
P P
P P P I P X
Income elasticity X
I
dI
dX
But 222
2
y x
x
P P
P
dI
dX
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Therefore income elasticity = X
I
P P
P
y x
x
222
2
substituting the values of
Px, Py and X we have income elasticity
).4(561456103508.042
672
19632
8
42
672,
1442
42
2
2 pd x x x
Income elasticity of demand for X=0.5614
d) If joins he would not like his utility to fall at least
he would want t0 maintain his
original utility. When X = 42 and Y = 36 his
U = 50(42) – 0.5(42) 2 + 100(36)
– 36 2 = 3522
The new equilibrium will be
5
4
2100
50
Y
X
P
P
MU
MU
y
x
y
x
i.e. Px = 4, P y = 5
250 – 5X = 400 – 8Y
8Y = 400 + 5X – 250
150 + 5X
But U = 3522
85150
851501005.0503522 2
X X X X U
352264
25)5)()150(2150
8
500
8
150005.050
222
X X X X X
64(50X) – (0.5)(64)X -2 + (15000)(8) +
500X(8) + 22500 – 1500X 25 2 3522(64)
57.X2 - 5700X + 127908 = 0
X-2 – 100X + 2244 = 0
(X-34)(X-66)=0
X = 34 or 66
From (b) X = 42, hence X = 34
But = 408
)5(34150
8
5150
X
Therefore his new expenditure is
XPx + YP y = 34(4) + 40(5) = 336
To remain at the same level of utility he spends $336. Therefore
he will be prepared to use
the balance to pay his dues. Hence maximum membership dues =
$672 - $336 = $3
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e) Since 222 is less than 336 he will join. His new
remaining income is $672-$222 =
$450
The marginal utility of money is y
y
x
y
P
MU
P
MU
THEORY OF PRODUCTION
The two fundamental concepts behind supplies decision are:
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i) Production ii) Cost
All economic goods come to existence through the process of
production. This includes
production. This includes production of goods and services
e.g. Legal service, medical etc.
In the theory of the firm, the theory of consumer behaviour
consumer maximize satisfaction
and firms maximizing profits. In production we have Marginal
rate of Technical Production
between inputs. In the theory of consumer behaviour
consumers maximize ordinal utility. In
the theory of the firm, the firms maximize cardinal variables.
The production process utilizes
the production of inputs e.g. Capital goods i.e. intermediate
products, all capital goods come
into existence through an act of production.
Assumption
1) We have one variable input, one fixed input and they
may be combined in
various proportions. Fixed-an input whose quantity can’t be
readily
changed when market conditions indicates immediate change in
output. In
actual fact, no input is fixed. The cost of varying might be too
high.
Short Run is one in which one or more input is fixed. In the
Short Run, change in product
can’t be accomplished by varying the variable inputs.
In the Long Run all inputs are variable.
Fixed proportion production: There is only one ratio of input
that can be used to produce aninput.
Production Function If a schedule showing the maximum output
that can be produced from
any specified set of inputs.
Example
Q= f(K,L),
K is fixed capital; L, labour is assumed variable
The figure below depicts total output at alternative units of
variable inputs. At L=0, Q = 0
since capital alone can’t produce an output: we need L to
combine.
The Average product of L:
APP = Q/L
The Marginal product of L:
MPP = dQ/d/L
As the variable input remaining is increased a point is reached
where maximum product is
achieved and after that point it reduces or diminishes. As we
increase the input from the
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origin, MP rises to a maximum at L0 and then it begins to
fall. L 0 is called a point of
inflexion: a point where the curve changes its concavity. At L2,
MP is zero. The Average
product is increasing from the origin and MP remains
greater than AP before the point of
intersection L1 where AP achieves a maximum.
Stage I : APPL is rising (L is between 0 and L 0 )
Stage II: MPPL is falling but it is positive (L is between
L 0 and L 2 )
Stage III: MPPL is falling and it is negative (L is greater
than L 2 )
No rational producer produces in stage III because MPPL is
negative. Also no rational
producer produces in stage I because in stage I we have
too few labourers on a large plant
(and in stage III we have too many labourers the plant).
ISOQUANTS
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Isoquants are curves input space showing all positive
combinations (bundles) of inputs that
are capable of producing a given output. Isoquants are downward
sloping. They don’t
intersect. Isoquants are convex to the origin. A higher isoquant
is preferred to a lower one.
The downward slope implies that if one increases one input one
has to reduce the other and if
one decreases the one, one should increase the other. The
further away from the origin an
isoquant is, the higher the associated level of output. MR of
Technical
Substitution
, KL MRTS KL =dL
dK measures the number of units of K that replaces
a unit of L at a point
so as to produce a given level of output. Along an isoquant the
level of output is constant and
therefore
Q = F(K,L)
dQ = Fk dk + FdL
Along the isoquant Q = constant
dQ = 0
0 = Fk dK F1dL
1k
k
L MRTS
F
F
dL
dK
As L is substituted for K along an isoqunt the
MRTSKL declines law of diminishing
MRTSKL
L
ISOCOST CURVE:
R – rental rate of capital
W – wage rate of labourOptimal way in which the
firm combines various input is given below
We assume that the firm purchases from a P.C input market
(prices are given) The T.C of
purchasing K and L is C = Pk K + PLL i.e C = rK +
wL
C = rK + wL
r
C L
r
w K isocost curve (or equation
It is a line showing all
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r
wSlope combination of the various
inputs that the firm can
isocost curve purchase at the given costs
The totality of all the isocost curves is the isocost map. We
first consider profit maximization
- Profit = Revenue-Cost
- = PQ – (rK + wL)
If firm is a Perfect Competitor in the input market and in the
output markets and in this
case P is given, then when C is minimum would be
maximum output for given cost or
minimum cost for a given output. Implies we are moving along a
given isoquant. An
isocost curve further away form the origin corresponds to a
higher TC. It is not possible
to produce at C. The cost minimization point is E where the
isocost curve is tangential to
the isoquant. At E therefore
r
w
MPP
MPP MRTS
K
L
If the above question is not satisfied, the producer will
substitute one of the inputs for the
other. From the equation
L
L
L
L
K MP
MP
MP
MP
MP
r , marginal cost of labour per unit of output
W is the unit cost of labour. MPL is how much output increases
if we increase the number of
labour by 1.
L
MP
r marginal cost of capital per unit of output
MC of output is the increase in output due to an
increase MP w
MP r K x
We substitute L for K because MC of output due to L is cheaper
and this is same as
r
w
MP
MP
L
L
w
MP
r
MP
w
MP L K L
output per units of labour
r
MP K output per unit of K
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Instead of minimizing cost we can maximize output in the dual
optimization problem. Now
output varies and cost is
Qx is not attainable Q 1 is attainable but not the
right equilibrium position
At E, MRTS = ratio of prices. Equilibrium is the same in both
cases.
DEFINITION OF SOME CONCEPTS
Isocline is the locus of points along which MRTS is constant. It
connects points along which
MRTS is the same.
K
Expansion path
L
Suppose inputs prices are given and the firm wants to expand
output. When all the
equilibrium points are joined together we have Expansion Path.
An Expansion
Path is an isocline along which output will expand if factor
prices remain constant. An
Expansion Path has a positive slope so that if we want to
increase output then employment of
both inputs should be increased.
‘RETURNS TO SCALE’ refers to a relationship between the
proportionate change in all
inputs and the resultant proportionate change in output. If
output changes by the same
proportion we have constant return to scale (CRS). If
output changes by more than the
proportionate change in input, we have increasing Returns
to scale (IRS). If output changes
by less than proportionate change in inputs we have
decreasing Returns to Scale (DRS)
There is a presumption that production functions exhibit CRS
Reasons
1)
The production process can be duplicated
2)
IRS implies indivisibility (efficiency)
3) Division of labour (efficiency)
Reasons for DRS
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1 There are certain cases where inputs can be doubled
example extractive
industries-mining
2 Difficult in supervision e.g as scale of operations
increase management becomes
less efficient
THEORY OF COST
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Social cost of production, when resources are used to produce a
commodity it implies the
society in curs a loss of production of another commodity
Opportunity cost the opportunity cost of producing one
units of X is the amount Y that must
be sacrificed for producing X
Explicit Cost. Are payments made by firms for purchase or hired
factors of production
Implicit Cost: Are the imputed cost of self-owned factors of
production
Total Cost Explicit: Implicit Costs
Implicit and Explicit costs are private costs of production.
In economics, cost means opportunity cost unless otherwise
stated.
LONG-RUN COST CURVES
The least cost of producing a given a quantity of output is
summarized in the long-run total
cost curve. And it shows how total cost varies as the level of
output varies. It is cost-output
equivalent of the expansion path.
L
TC is concave to the origin at low levels of output and becomes
convex. It starts from the
origin. The shape of the curve reflects the characteristics of
the production function. It
reflects return to scale.
1st – TRS, 2nd – CRS
– 3 rd – DRS
SHORT-RUN TOTAL COST CURVE
In the long-run the firm operates along the expansion path but
in the short-run some of the
factors are fixed. Once the tangency condition is violated in
the short-run, it implies that it
costs more to produce a given quantity of output in the
short-run than in the long-run
In the long-run all costs are variable
SRTC = SRFC + SRVC
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Corresponding to TC’s we have Average costs and marginal costs
SAC = AFC + AVC
Q
TFC AFC
dQ
dQd
dQ
STC d SMC ,
()('
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The AFC is U-shaped Assume labour is the only variable
factor
TFC = w,L where w is the wage rate
Q
Lw
Q
Lw
Q
TVC AVC .
.
L MPP
w
dL
dQ
dL
TVC d
dQ
dL
dL
TVC d SMC
(,
)(
dL
dQ Marginal product of labour
SRMC and thedL
dQ are inversely related
Both the SRVC , STC start from the origin is asymptotic to the
cost axis. The gradient of the
ray from the origin to the STC gives the ATC at that point
LONG-RUN COST CURVES
In the long run, all inputs are variable. It is the planning of
the firm. The long-run cost
curves are always below the short-run cost curves. If a firm
wants to take a decision it
happens in the long-run. LCRC ≤ SRC>
All production take place in the short-run. As we move towards
the LR, the fixed factor
becomes variable. We have a series of SRC curves. The LTC
is the envelope of the possible
short-run TC curves. The long-run LTC is also the envelope of
all the possible short-run, ACcurves.
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FUNCTION COEFFICENT
It shows the proportionate change in output when all inputs are
changed by the same
proportion. If λ is the change in input then function
coefficient
Q
Q
If є > 1 implies increasing return to scale. IRS or economics
of scale
Є < 1, constant returns to scale (CRS)
Є = 1 , decrease returns to scale or diseconomies of scale
(DR S)
C
C
Q
Q
C
C
LMC
LAC
,
LMC
LAC
C
Q LAC
C
C
Q
Q
,
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Q
C Average cost
LMC
LAC
LMC LAC 1
LMC LAC 1
LMC LAC 1
If the production function exhibits CRS then the expression path
is linear i.e. through origin.
If the production function is linear, it implies CRS
If expansion path is linear then it belongs to homothetic
production functions.
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THEORY OF THE FIRM MARKET STRUCTURES
How do we classify market structure? We consider two
criteria:
1)
Number of firms in the industry
2)
The nature of the product produced by the firms in the
industry:
1. The basis of the first criteria
We classify industry whether there are many firms, few or one
firm in the industry
2. We categorize the industries by whether the products of the
firms are homogeneous or
different
Number 0f Firms
Number of product Many Few One
Homogeneous PC Pure Oligopoly
Differentiated MC Differentiated Monopoly
Oligopoly
PC Perfect competition
MC Monopolistic competition
We use the world competition to imply many firms in the
industry
Oligopoly implies few firms in the industry.
Perfect and Pure implies homogeneity of product
Competition does not mean competition in the real sense of the
word. Here rivalry is absent.
All other market structures other than perfect are classified as
imperfect competition. We are
talking about the sellers side of the product. We have the
counterpart of this that is those who
do the buying.
If the buyers are many implies perfective competitive buyer
If there is one buyer, then the market is monopsonistic
If a monopolist faces a single buyer, we have a bilateral
monopoly
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Π = 0
At A and B, the tangents to the STC is parallel to TR,BT
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At the min point of the SAC the firm is indifferent about
production i.e. it can continue to
produce or shut down. That is at this point the firms
covers its variables costs.
SUPPLY CURVE OF THE FIRM
The industry supply curve can be the summation of the individual
supply curves of the firms
of the expansion does not affect the cost curves of the
firms.
If resource prices do not change the supply curve of the
industry will be the horizontal
summation of the supply curves of the individual firm. But if
the prices changes then this
cannot be done.
LONG-RUN EQUILIBRIUM OF PC
The long-run adjustment of a single firm
In the long-run the firm will operate at point E. At q, the firm
will be making profits hence
new firms will enter into the industry.
LONG-RUN EQUILIBRIUM THE INDUSTRY
At qi, the firm will be making profits hence new firms will
enter hence the curve will shifts to
right that is 21 S S .
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At q, firm will be making losses firms will be leaving the
industry. At q, supply to S1. As
firms leave the industry the long-run supply will be S2
.
MONOPOLY
If there is only one producer in the market then we have
monopoly. Monopoly and PC are
extreme cases.
1} PC has many rivals. We have competition in technical
sense
2} In the case of monopoly rivalry does not exist. There is only
firm in the in the market.
However there are indirect forms of competition that the
monopolist faces.
i)
That is competition for the consumer income. That is he has to
secure a
market for his products.
ii)
The existence of substitutes for his products. His power depends
on the
exact goods. Reasons why monopolists exist
1) Control of the basic input. That is the Aluminum Company of
USA owns the Bauxite
in the production of A1 in USA.
2) A firm can obtain the property right to produce a commodity.
They cannot prevent
other firms from producing close substitute example Pata and
Club, IBM and the rest
3. The Average Cost of producing a product reaches its minimum
at an output rate that
is ideal for one firm
If more than one firm does the production they produce a higher
AC. There will be a price
warfare and some of them will be driven out of production, hence
leaving only one firm in
the market, this termed as natural monopoly which means the
emergence of one firm as the
sole producer of a commodity due to the price warfare
4) Demand curve facing the monopoly. The dd curve facing the
monopolist is the same
as the market dd curve.
5) Under monopoly the firm chooses the price. The MR is not the
supply curve as is the
case.
SHORT-RUN EQUILIBRIUM OF THE MONOPOLIST
A firm may be a monopolist in the product market and may be a PC
in the input market. If itis a PC in the input market then the cost
curves will be the same as the PC.
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Let us consider a monopolist who owns two plants. Each of the
plants has its own cost
curves. He decides on output and price. The allocation of
optimal output in each plant are as
follows:
Total Market
MC MC MC MC 21
P
P MC = MR
D
Q
At the optimal level MC1 = MC 2,
When MC1 < MC2 then there will be a shift of production from
MC1 to MC 2,
21
21
)()(
qqQ
Q FR pqq F P
)( 21 qqqF pq R
)()( 1q f Q R
C=fq1),C2=g(q2)
)()( 1121 qC qq R
= R(q1 + q 1) – g(q 2)
1..)(')(2
1
1
21
1
q
Qeiq f
q
Qqq R
q
Mc MR f R f Rq P
q
Q
Q
R ''0'')(, 1
1
0')('1
21
2
g q
Qqq R
q
MC MR R R ,0''
321 MC MC MC
These are first order conditions. The second order
conditions
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2
2
12
2
21
2
2
1
2
qqq
qqq
Second order conditions
0.0,2221
1211
11
2221
1211
The second order condition implies MC must be rising.
LONG-RUN EQUILIBRIUM
Since entry is blocked in the LR profit is not reduced to zero.
If the monopolist incurs a SR
loss and there is a revenue for expanding output then in the LR
it will close down. If the
monopolist makes profits in the SR, he can expand output. The
maximum-maximum plant is
the plant which gives the maximum of the maxima profit.
That is LMC = MR
SAC is the optimal plant of the LR. The maximum-mximum plant is
the plant which gives
the maximum of all profits. In the LR, monopolist need not
operate at the minimum point of
the LAC, so reso@ e perfectly utilized in PC. To the monopolist.
P < MC price
or dd represents the marginal social evaluation of a product LM
MC shows the marginal
social cost i.e. society wants more of product but the
monopolist will not increase output. PC
promotes social welfare more than monopoly. That is P = MC
in PC and P > MC in
monopoly.
LONG-RUN EQUILIBRIUM OF A TWO PLANT MONOPOLIST In the
long-run, the monopolist can alter the plant size.
Cost curve for one plant
q
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LONG-RUN EQUILIBRIUM OF A MULTIPLANT MONOPOLIST
P C
P
P LMC = LAC
A B
D
Number of firms in a PC market will beq
Q p
Number of firms in a monopoly market will beq
Q
q
Q
q
Q p . And the monopolist will change a higher price.
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QUESTIONS
Construct a short-run supply function for an entrepreneur
whose short-run cost function
is TC = 0.04q3 – 0.8q 2 + 5 +
10q
1) Solution
TC = 0.04q3 – 0.8q 2 + 5 + 10q
TVC = 0.04q3 – 0.8q 2 + 10q
108.006.0 2
qqq
TVC AVC
For Min AVC set 0
dq
dAVC
08.008.0 qdq
dAVC
108
80q
When q = 10, AVC = 0.04 x 100 – 0.8(10) + 10 =
6
MC = 0.12q2 – 1.6q + 10
pqq 106.112.0 3
Multiply through by 12.5
5q2 – 20 + 125 – 12.5p =
0
3
5.12125(640020 pq
3
143520
pq
The positive branch gives output at which MC is increasing,
hence dS/dp >0 and
S =3
143520 p.if..p ≥ 6
I,e. minimum AVC = 6, below 6, S = 0) min P = 6
2) The long-run cost function for each firm that supplies Q is C
= q3 – 4q + 5q.
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Firms will enter the industry if profits are positive and leave
the industry if profits are
negative. Describe the industry’s long-run supply function.
Assume that the corresponding
demand function is D = 2000 – 100p.
Determine equilibrium price of industry and number of firms
OTHER QUESTIONS
1) A firm has the following long-run production function
X = 25.05.05.0 C B A where X is
weakly output,
is positive constant A, B and C are the weekly
outputs of the three factors used.
The price of A is $1, the price of B is $ and the price of C is
$8
a) Derive the following:
i) the firms long-run total cost function
ii) the firms short-run average cost function
iii) the firm’s short-run average variable cost function
b) if the short-run factor C is fixed, while factors A and
B are variable, derive the
following
i) the firm’s short-run average function
ii) the firm’s short-run average cost function
iii) the firm’s short-run average cost function
iv) the short-run average marginal cost function
c) Derive an equilibrium in the form C = f(x) showing the
optimum quantity of the fixed
factor
C for the firm to acquire as a function of the intended output
of X
Solution
The equilibrium or cost-minimum conditions are
b
a
b
a
b
a
b
a
P
P
MPP
MPP
P
P
MPP
MPP ,
From th given data
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MPPa = o.5 αA-0.5 B0.3 C 0.25
MPPb = 0.5 α A-0.5 B0.3 C 0.25
MPP = o.25 αA0.5 B 0.5 C 0.25
25.05.05.0
25.05.05.0
25.0
5.0
C B A
C B A
P
P
MPP
MPP
b
a
b
a
=
9
1
So )...(..........168
12ii
AC
A
e
Substituting (i) and (ii) into production function we get
25.05.0
5.0
169
A A A X
2
)(
3
)( 25.05.05.0 A A A
).......(..........6
6
.0
8.0.
25.0iii X A A X
x
CP BP AP LTC ba
c
= )8($16
)9($9
)1 A A
AS
2
5
2
A A A A
But 8.08.0
6 X A
Hence LTC =
8.0
8.06
2
5
X
(i) LTC = 8.08.0
6
2
5 X
(ii)
LAC = X
LTC 18.0
8.0
6
2
5
X
= 8.08.0
6
2
5 X
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(iii)
LMC = 2.08.0
2.0
8.0
2.0
8.06
26
2
46)8.0(
2
5 X X X
dX
dTC
b) From a) we have X =3
.....5.05.025.0 A
Band A BC
39
25.05.0
5.0
25.0 AC A
AC X
25.0
25.0 3
3 C
X A
AC X
SFC = AP + BP + CP
But (i) STC = d($1) + C A A A
8)8$)9($9
STC = 2A + 8C
But25.0
3
C
X A
Hence STC = C C
X C
C
X 8
68
32
25.025.0
(ii) X C
C X
X STC SAC 86
25.0
= X
C
C
8625.0
(iii)
SAVC =25.0
6...,
C
X TVC but
X
TVC
Hence SAVC 25.025.06
/6
C X
C X
v) SMC =25.025.0
68
6
C C
C
X
dX
d
dX
dSTC
SMC =25.0
6
C
b) From part C BC A9
16,16
X = 25.05.05.0 C B A substituting A
and B we have
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25.0
5.0
9
16)16( C C C X
25.125.05.05.0
316
34 C C C AC X
8.0
8.0
16
3 X C
2) A firm uses a number of factors to produce a single product.
X. In the short-run plant
is fixed, while all other factors are variable. We are concerned
with two of the possible
plants. In the long long-run of course all factors are
variable. The cost functions are
LTC = 0.005 X3 – 1.4X 3 + 280X
Plant 1 : STC1 = 0.006X2 -1.33X2 + 6860
Plant II STC2 = 0.0057 X 2 -1.424 X-2 + 205.6X +
10240
a) Derive equation for the following LAC, LMC, SAC,
SATC,..SMC,,SMC
b) At what output does the firm achieve minimum
LAC?
c)
Does either plant permit achievement of minimum LAC?
d)
At what output is SAC, minimized?
e) What is the level of SAR at 160?
f) What is the level of SAC at X = 160?
g)
At what is the SAC minimized?
h)
Which of these two plants will the firm use if it intends to
produce the
output in (g)
i)
For what output would plant 2 be the best of all possible
plants
j)
Would plant 1 operate in the short- run if the product price
were 120?
k) Would plant 2 operate in the short-run if the product
price were 120?
l) Which plant would be more profitable if product price
were 120?
m)
At what product price would the firm produce the same positive
output inthe short-run which ever of these two plants it had?
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Solution
LTC = 0.005X3 -1.4X 2 + 280X
a)
LAC = 2804.1005.0 2
X X X
LTC
LAC = 0.005X2 – 1.4X + 280
LAC = 2802815.0 2
X X X
LTC
STC1 = 0.006X3 -1.33X 2 + 201.6X + 6860
X
STC SAC
1
SAC1 = 0.006X2 – 1.33X + 201.6 +
6860X -1
SAVC = X
STC 1
SAC = 0.006X3 – 1.33X 2 +
201.6X
SAC = 0.006X2 – 1.33X + 201.6
SAFC1 = X
STC 1 but SFC = 6860
SAFC1 = 6860X-1
SMFC1 = 6.20166.2018.0 21
x xdX
dSTC
SMC = 0.018.X2 – 2.66x + 201.6
SAC = X
STC 1 = 0.0057X -1.424X + 205.6 + 10240X -1
SAC = 0.0057X2 -1.424X + 205.6 + 10240X -1
SALC2 = 6.205848.20171.0 2
x xdX
dSTC
Either set the derivative of LAC - or set LAC = LAFC
LAC = 0.005X2 - 1.4X + 280
4.101.0 X dX
dLAC Set 140,4.101.00
X X
dX
dLAC
For min 02
2
dX
LAC d
01.02
2
dX LAC d minimum, hence X = 140
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or
LAC = LMC
LAC = 0.005X2 – 1.4 + 280
LMC = 0.015X2 – 2.8X + 280
0.005X2 – 1.4X + 280 = 0.015X 2
– 2.8X + 280
(0.005 – 0.015) X 2
– 1.4X + 2.8X = 0
-0.01X2 + 1.4X = 0
X(-0.01X + 1.4) = 0
X = 0 or X = 140
X = 140
C) SAC1 at X = 140 is
SAC1 = 0.003(140)2 - 1.33(140) + 201 + 182140
6860
SAC at X = 140 is
SAC = 0.0057(140)2 – 1.424(140) + 205.6 +
1.191140
10240
LAC at X = 140 is
LAC = 0.005 (1402) – 1.4 (140) + 28 = 182
Hence plant 1 achieves minimum LAC
d) Find the minimum of SAC. Thus we find the derivative of SAC ,
and set it equal to zero
and check whether it is a minimum
SAC1 = 0.006X2 -1.33X + 201.6 + 6860X -1
06860
33.1012.00...686033.1,912.02
121
X x
dX
dSAC Set x x
dX
dSAC
0.012 X3 – 1.33X2
– 6860 = 0
Solving we get
X1 = 140,X2, = -14.58 + 62.21, X, =
-14.58 – 62.21
For minimum, 02
1
1
2
dX
SAC d
332
1
1
213750
012.0)6860(2
012.0
X X dX
SAC d
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0005.0012.0
140
1
2
x
dX
SAC d
X = 140 gives a minimum
0)21.6258.14(.0 3
,
2
1
1
2
12
E I dX
SAC d
X X X
X = 140 minimizes SAC1
SMC2 = 0.0171X2 – 2.848X + 205.6
= 0.0171 (1602) – 2.848 (160)
= 437.76 – 455.68 + 205.6
= 187.68
SMC2 = 187.68 at = 160
g) From theory SMC cuts SAC at its minimum point. But at
X = 160, SAC2 = SAC 2 = 187.68 hence SAC2 is
minimized at the
output level of X = 160
SMC2 = 0.0171X 2 -2.848.X + 205.6
02
dX
dSMC
848.20342.02
X dX
dSMC
624.2848.2472.5848.2)160(0342.0160
2
xdX
dSMC
Hence SMC2 is rising
X = 160 is the output level at which SAC is minimised
h) That SAC2 is minimized at X = 160 does not mean that SAC <
SAC1 at X = 160At X = 160
SAC = 0.006X2 = 1.33X + 201.6 + 6860X -1
i.e. SAC1 = 0.006 (1602) – 1.33 (160) +
201.6 +
160
6860
SAC1 = 185.275 and SAC 2 = 187.68
Thus plant 1 has a lower average cost than 2 therefore plant 1
will be used to produce the
output of X = 160
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i) It is necessary that SAC1 = LAC at tangency and
therefore SMC 2 = LMC setting
the average cost equal yields a cubic equation i.e.
SAC2 = 0.0057X2 -1.424X + 205.6 + X
10240
LAC = 0.005X2 1.4X + 280
