This is not the document you are looking for? Use the search form below to find more!

Report home > Art & Culture

a

0.00 (0 votes)
Document Description
a
File Details
  • Added: January, 22nd 2012
  • Reads: 178
  • Downloads: 1
  • File size: 87.52kb
  • Pages: 7
  • Tags: a, b, c
  • content preview
Submitter
  • Name: a
Embed Code:

Add New Comment




Related Documents

A BRIEF HISTORY OF VOLLEYBALL

by: rika, 47 pages

1895 William G. Morgan invented the game of volleyball in Holyoke, Massachusetts; there were no limits to the number of players on a team or the number of times the ball could be ...

Creating a Blog with mBlog

by: rika, 15 pages

A weblog, or blog, is a frequently updated website consisting of dated entries arranged in reverse chronological order so the most recent post appears first (see temporal ordering). ...

HOME SCHOOLING: A GUIDE FOR PARENTS

by: rika, 4 pages

There is no developed consensus on the definition of home schooling although it generally refers to an alternative or supplemental form of education in which a parent teaches children at home. The U ...

PLASMA LEVELS OF VITAMINS A, E AND CAROTENE IN COWS IN LATE PREGNANCY AND IN THEIR FOETUSES

by: shinta, 5 pages

Experiments involving 20 cows in late pregnancy and their foteuses aged 8-9 months were conducted to study the blood plasma content of the vitamins A. E and carotene and the liver vitamin ...

Caroten rich orange-fleshed sweet potato improves the vitamin A status of primary school children assessed with the modified-relative-dose-response test

by: shinta, 8 pages

Carotene rich orange-fleshed sweet potato (OFSP) is an excellent source of provitamin A. In many developing countries,sweetpotatoisasecondarystaplefoodandmayplayarole in controlling ...

Serum Progesterone, Vitamin A, E, C and Beta Carotene Levels in Pregnant and Nonpregnant Cows Post-Mating

by: shinta, 4 pages

The aim of the study was to detect the progesterone, vitamin A, E, C and ß-carotene levels in blood serum and the relationship between them in pregnant and nonpregnant cows after ...

Radiation effects on vitamin A and beta carotene contents in liver products

by: shinta, 3 pages

The movement toward commercialization of the process and the recent actions of various governmental agencies to accept irradiated foods as wholesome, even when high doses are applied has ...

Genetic polymorphism of the adenosine A 2A receptor is associated with habitual caffeine consumption

by: shinta, 5 pages

Caffeine is the most widely consumed stimulant in the world with an estimated 80–90% of adults reporting regular consump- tion of caffeine-containing beverages and foods. ...

Vitamin A Deficiency, Iron Deficiency, and Anemia Among Preschool Children in the Republic of the Marshall Islands

by: shinta, 7 pages

In many developing countries worldwide, young children are at a high risk of vitamin A deficiency and iron deficiency. Vitamin A deficiency affects an estimated 253 million preschool ...

Chemopreventive efficacy of curcumin and piperine during 7,12-dimethylbenz [a]anthracene-induced hamster buccal pouch carcinogenesis

by: shinta, 8 pages

Oral carcinoma accounts for 40–50 percent of all cancers in India. Tobacco chewing, smoking and alcohol consumption are the major risk factors associated with the high ...

Content Preview
Econ 105: Lecture 3
1
Consumer Optimality
Budget Constraints
The main factor constraining choice for most individuals in a market economy is that they have
limited wealth to spend on goods.
Suppose that a consumer's income is I and that the prices of goods X and Y (the only two
things he consumes) are pX and pY . Then, the consumer can afford to purchase any bundle (x, y)
such that
pX x + pY y I.
(1)
This is called the individual's budget constraint. The frontier of this constraint (sometimes called
the budget line) is the set of bundles that completely exhausts the individual's income. This line
can be written in slope-intercept form as
pX
I
y = -
x +
.
(2)
pY
pY
The vertical intercept, I/pY , tells how many units of good Y the consumer could get if he spent
all his income on it, and likewise the horizontal intercept, I/pX , is the maximum amount of X the
individual could purchase. The price ratio pX /pY is called the relative price of good X. It specifies
how many units of good Y the consumer has to give up in order to get enough money to buy one
additional unit of good X; e.g., if pX = $10 and pY = $5, then a consumer must give up 10/5 = 2
units of good Y to get one more unit of good X. One can think of pX /pY as the economic or market
rate of exchange of good Y for good X whereas the MRS, U/x , is an individual consumer's psychic
U/y
rate of exchange of Y for X.
A rise (fall) in I causes the budget line to shift out (in) in a parallel manner. A rise (fall) in
pX causes the budget line to pivot in (out) holding the vertical intercept fixed. A rise (fall) in pY
causes the budget line to pivot in (out) holding the horizontal intercept fixed.
The Consumer's Optimization Problem
Economists typically assume that the objective of a consumer is to maximize his utility. He cannot,
however, consume as much of every good as he wants because scarce goods have positive prices
and consumers have limited wealth. Hence, a consumer must pick the best bundle of commodities
he can afford. That is, he maximizes his utility subject to his budget constraint. Formally this
maximization program is written:
max U (x, y) subject to pX x + pY y I.
(3)
(x,y)
To solve a constrained optimization problem like (3), we use the method of Lagrange and write
max L = U (x, y) + (I - pX x - pY y).
(4)
(x,y,)
The variable, , is called the Lagrange multiplier. It can be shown that = 0 if the budget
constraint does not bind and > 0 only if it does.1 In fact, if preferences are continuous and
1 Formally the KarushKuhnTucker conditions necessary for a maximum are: (Primal feasibility) I -pXx-pY y 0,
(Dual feasibility) 0, and (Complementary slackness) (I - pX x - pY y) = 0.

Econ 105: Lecture 3
2
non-satiated, then the consumer will always spend all his income (i.e., the budget constraint will
always bind and > 0). There are two possible types of solution to (4), an interior solution in
which the consumer buys positive amounts of both goods or a corner solution in which he buys
zero units of one of the goods.
Interior solutions
If the utility function is differentiable, then the first-order conditions for an interior solution are :
L
U
=
- pX = 0,
(5)
x
x
L
U
=
- pY = 0,
(6)
y
y
and
L = I - pXx - pY y = 0.
(7)

These three equations must be solved for the three optimal values of the (endogenous) choice
variables (x, y, and as functions of the (exogenous) parameters pX , pY , and I. Notice that
condition (7) is just the budget constraint. Now, in order to eliminate from conditions (5) and
(6), write:
U = pX
(5 )
x
and
U = pY .
(6 )
y
Dividing equation (5') by (6') yields:
U/x
pX
=
.
(8)
U/y
pY
This is called the tangency condition. The left side is the MRS and the right side is the absolute
value of the slope of the budget line. The tangency condition (8) along with the budget con-
straint (7) can be solved for x and y, which is the optimal commodity bundle the consumer can
afford to purchase. (Technically we need to check the second-order conditions for a constrained
max, but these are messy! If preferences are strictly convex, then the second-order conditions are
automatically satisfied.)
In essence, the math above says that a consumer is at a constrained utility maximum when he
purchases a bundle on his budget frontier where his MRS equals the price ratio pX /pY . In other
words, the highest indifference curve he can reach is the one that is just tangent to his budget line.
Corner Solutions
A corner solution obtains at x = 0 and y = I/pY if
U/x
pX
<
U/y
pY

Econ 105: Lecture 3
3
at every point (x, y) on the budget frontier. Likewise, a corner solution with x = I/pX and y = 0
obtains if
U/x
pX
>
U/y
pY
for every bundle (x, y) on the budget frontier. At a corner solution, the consumer in question is
always willing to trade less or more of Y for X than the market requires.
Ordinary Demand
The solution (x, y) gives the consumers ordinary (also called Marshallian or uncompensated)
demand for goods X and Y given prices pX and pY , and income I. In fact, if we let pX vary holding
fixed pY , I, and preferences, then x will vary yielding the consumer's demand function for X.
Example 1 (Deriving Ordinary Demand). As an example, suppose an individual has sym-
metric Cobb-Douglas preferences
U (x, y) = x x y.
His budget constraint is
pX x + pY y = I.
The Lagrangean is
L = x x y + (I - pX x - pY y).
The first-order conditions are
L = y - pX = 0,
x
L = x - pY = 0,
y
and
L = I - pXx - pY y = 0.

In this case, the tangency condition is
y
pX
=
,
x
pY
or
pY y = pX x.
In other words, this consumer always spends equal amounts of his income on X and Y. Using this
to substitute for pY y in the budget constraint gives
pX x + pX x = I
or
I
x(pX , I) =
.
2pX
Similarly,
I
y(pY , I) =
.
2pY
These expressions are the demands for X and Y respectively by the consumer in question. For
instance, if pX = 1 and I = 50, then x = 25, and if pX = 2.5, and I = 50 then x = 10. Notice
that this is a unit-elastic demand function; i.e., the consumer always spends half his income on
good X and half on Y.

Econ 105: Lecture 3
4
Indirect Utility
When the solution to an optimization problem is substituted into the objective function, the result
is called a value function. It specifies how the optimal value of the objective function depends
on the underlying parameters. In the current context, the solution to a consumer's optimization
problem yields his ordinary demand functions x(pX , pY , I) and y(PX , pY , I). Specifically, these
functions specify the optimal levels for the endogenous choice variables x and y as a function of the
exogenous parameters pX , pY , and I. If we substitute the demand functions into the consumer's
utility function we obtain the value function known as the consumer's indirect utility function:
V (pX , pY , I) U (x(pX , pY , I), y(pX , pY , I)).
(9)
The indirect utility function tells us how an individual's welfare is related to the environmental
variables he faces, namely prices and income. For instance if pX rises, the consumer will optimally
adjust his consumption of X and Y and will obtain a new level of utility which is given by his
indirect utility function evaluated at the new price of X.
Example 2 (Deriving Indirect Utility). The indirect utility function from Example 1 is found
by substituting the consumer's demand functions into his utility function
I2
V (pX , pY , I) = x(pX , pY , I) x y(pX , pY , I) =
.
4pX pY
For example, if pX = pY = 1 and I = 50, then this consumer would buy 25 units of X and 25
units of Y and obtain utility of
502
V (1, 1, 50) =
= 625.
4 x 1 x 1
If pX then rises to $2.5, her demand for X will fall to 10 units, her consumption of Y will stay
constant at 25 units, and her new level of utility will fall to
502
V (2.5, 1, 50) =
= 250.
4 x 2.5 x 1
Of course, the values for indirect utility are only ordinal numbers. Hence, the only thing we can
say is that a rise in pX leaves the consumer worse off since her utility falls from 625 to 250.
The Lagrange Multiplier
At an interior optimum, it is possible to solve equations (5), (6), and (7) above not only for the
optimal consumption levels x and y but also for the optimal value for the Lagrange Multiplier
. In general, the optimal value for the multiplier in a constrained optimization program gives the
marginal change in the value function from relaxing the constraint. This is sometimes called the
shadow price of the constraint. In a utility maximization problem, is, thus, the increase in the
optimal level of utility from a marginal increase in income; i.e., it represents the marginal (indirect)
utility of income:
V
=
.
(10)
I
In the context of utility maximization, this result is admittedly of limited use since marginal
utility has no meaning beyond its sign (positive or negative) because utility is only ordinal. The
interpretation of the Lagrange multiplier in other problems is, however, very useful!

Econ 105: Lecture 3
5
Example 3 (The Shadow Price of the Constraint). For the example U (x, y) = x x y, equa-
tion (5') is
u = pX y = pX.
x
Substituting the demand for Y
I
y(pY , I) =
.
2pY
Gives
I
= pX
2pY
Dividing both sides by pX gives the optimal value of the multiplier as a function of the parameters
I
(pX , pY , I) =
.
2pX pY
Recall from Example 2 that indirect utility in this case is given by
I2
V (pX , pY , I) =
.
4pX pY
Hence, the marginal (indirect) utility of income is
V
I
=
= (pX , pY , I).
I
2pX pY
This illustrates the claim that the optimal value of the multiplier gives the marginal change in the
value function (indirect utility) from relaxing the constraint (income). For instance, if pX = pY = 1
and I = 50, the marginal utility from giving the consumer another dollar of income would be
50/2 = 25.
Duality
Associated with every constrained maximization problem is a conjugate constrained minimization
problem and vice versa. We call the original program the primal and the conjugate program the
dual. For instance, if the primal program is to maximize utility subject to a budget (or expenditure)
constraint, then the dual program is to minimize expenditure subject to a utility constraint. That
is, rather than thinking about finding the highest indifference curve on a given budget line, the
dual is to find the lowest budget (isoexpenditure) line on a given indifference curve; i.e., to solve:
min pX x + pY y subject to U (x, y) u,
(11)
(x,y)
where u is some specified level of utility. The Lagrangean for this dual program is written
min M = pX x + pY y + (u - U (x, y)),
(12)
(x,y,)
where is the Lagrange multiplier. The first-order conditions are:
M
U
= pX -
= 0,
(13)
x
x

Econ 105: Lecture 3
6
M
U
= pY -
= 0,
(14)
y
y
M = u - U(x, y) = 0.
(15)

Again, at an interior solution these three conditions can be solved for the expenditure minimizing
consumption levels xc and Y c and the optimal value for the multiplier c as a function of the
parameters: pX , pY , and u. We identify the solution to this problem with a superscript c rather
than a star in order to distinguish this solution to the dual (expenditure-min problem) from the
solution to the primal (utility-max) problem. The c stands for compensated for reasons explained
below. To find the expenditure-minimizing bundle we rewrite (13) and (14) in the form
U
pX =
(13 )
x
and
U
pY =
.
(14 )
y
Dividing (13') by (14') results in the tangency condition
pX
U/x
=
.
(16)
pY
U/y
Observe that (16) is the same condition as (8). That is , both solutions are marked by a tangency
between an indifference curve and a budget (or expenditure) line. The difference is in the con-
straints! In the primal (utility-max) problem the tangency condition (8) is solved along with the
budget constraint (7) for the ordinary demand functions x(pX , pY , I) and y(pX , pY , I). In the dual
(expenditure-min) problem the tangency condition (16) is solved along with the utility constraint
(15) for the compensated (also called Hicksian) demand functions xc(pX , pY , u) and yc(pX , pY , u).
Compensated demand functions are a useful but purely fictional construction because consumer's
solve utility-max problems not expenditure-min problems. Compensated demand functions rep-
resent the bundles of goods a consumer would buy if his income was adjusted to keep his utility
fixed at some initial level. He would receive some extra income after a price rise and give up
some income after a price fall, so as to remain on the same indifference curve! Compensated de-
mand functions have some interesting properties and are related in important ways to the ordinary
(uncompensated) demand functions in ways we will explore in the next lecture.
Example 4 (Deriving Compensated demand). If U (x, y) = x x y, then the tangency condi-
tion (16) (or (8)) is
pX
y
=
.
pY
x
or
pX
y =
x.
pY
Substituting for y in the constraint x x y = u gives
pX
u =
x2,
pY
or
pyu
xc(pX , pY , u) =
.
pX

Econ 105: Lecture 3
7
Similarly
pX u
yc(pX , pY , u) =
.
pY
These are the compensated demand functions. They specify the least expensive way for the given
individual to achieve utility level u when prices are pX and pY . For instance, suppose pX = pY = 1

and we want the individual to achieve a utility level of u = 625. Then xc = yc =
625 = 25.
Observe that this is the same solution to the primal problem x = y = 25 given in Example 1
when pX = pY = 1 and income is constrained to be I = 50. This is because we selected the
constrained level of utility in the dual problem to be equal to the value of indirect utility in the
primal problem, u = 625. Notice, in fact, that the reverse is also true, namely that the constrained
level of income in the primal problem I = 50 equals the optimal level of expenditure in the dual
problem
pX xc + pY yc = 1 x 25 + 1 x 25 = 50.
To restate this property, the solutions to the primal and dual problems coincide when the constraint
of one problem equals the value of the other problem (and vice versa per force).

Download
a

 

 

Your download will begin in a moment.
If it doesn't, click here to try again.

Share a to:

Insert your wordpress URL:

example:

http://myblog.wordpress.com/
or
http://myblog.com/

Share a as:

From:

To:

Share a.

Enter two words as shown below. If you cannot read the words, click the refresh icon.

loading

Share a as:

Copy html code above and paste to your web page.

loading