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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).

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).

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