Consumer Theory - Microeconomics

8/10/2019 Consumer Theory - Microeconomics

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ESS, BRAC University; Fall 2012, THN

ECO 206: Intermediate Microeconomics IConsumer Theory

1 Preference:

People dont just choose to buy goods arbitrarily (or blindly), they are usually guided by some form

of underlying preferences. Consumers do appear to make systematic choices, e.g., we probably buy

more or less of the same specific items every time we go to the grocery store. To start with, assume

that there are two goods, x and y , and that there are three bundles, A, B, and C. Each bundle has

a different combination of the two goods. Economists generally assume that consumers preferences

follow these properties:

(a) Completeness: Between two bundles, A and B, the consumer either prefers A to B or B to A

or is indifferent between A and B. It is never possible that the consumer cannot decide between

two bundles, such as A and B.

(b) Transitivity: Consider a situation where it is true that the consumer prefers A to B and B

to C. Then the assumption of transitivity implies that the consumer prefers A to C. In other

words, the consumers preferences are consistent over a range of choices, also known as the

rationality assumption.

(c) Nonsatiation or more is better: Consumers always prefer more of a good than less of it.

In economics, a good is defined as a product for which more is preferred to less and a bad is a

product for which less is preferred than more (such as noise pollution).

(d) Continuity: Consider a situation where it is true that the consumer prefers A to B. Then

under this assumption, the consumer will prefer C to B if C is sufficiently close to A. In other

words, this assumption rules out the phenomenon of preference reversals.

(e) Strict convexity: Consider a situation where it is true that the consumer prefers both goods

A and B to C. Then, under this assumption, the consumer will prefer a weighted average of

the two goods A and B, such as A + (1 )B, to good C; where 0< 1.

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(f) Reflexivity: Each of the bundles, A or B or C, is as good as itself.

With these assumptions, a lot can be explained about a persons preferences. A preference map is

a graphical representation of peoples preferences. For simplicity, in most cases, we will consider a

world with only two goods, x and y ... will make the math easier ...

2 Indifference Curve:

It is the set of all the possible combinations of the two goods, xandy, that are deemed to be equally

desirable by the consumer. In other words, the consumer is indifferent between any combinations

of goods x and y that lie along the same indifference curve. We can draw an entire indifference

map for a particular consumer purchasing combinations of the two goods,x and y . An indifferencemap is then the set of all indifference curves that summarizes a persons tastes.

Indifference curves also have to display a set of properties:

(a) Bundles of goods on indifference curves that are further away from origin are preferred more

than those on indifference curves closer to the origin.

(b) There is an indifference curve through all possible combinations of goods and indifference curves

cannot cross.

(c) Indifference curves slope downward.

(d) Indifference curves cannot be thick.

2.1 Marginal Rate of Substitution (MRS):

The slope of an indifference curve has a special meaning it is the consumers marginal willingness

to substitute one good for an incremental change in the consumption of the other good. In a world

with two goods, xandy:

MRSU=U=dy

dx=

UxUy

(1)

In equation (1),dy is the change in the amount consumed of good y due to a change in the amount

consumed of good x equal to dx. In most cases, indifference curves are convex to the origin we

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have what we call diminishing marginal rate of substitution. In simple words, convex indifference

curves mean that a consumer is willing to give up less of y (for x) as we move down along an

indifference curve. MRS tends to zero as we move towards the bottom right-hand corner of the

curve. In general, indifference curves can have various shapes:

(a) Ifx and y are perfect substitutes, indifference curves will be downward sloping with a slope

of exactly 1. In other words, MRS is equal to 1.

(b) If x and y are perfect complements, indifference curves will be L-shaped. In other words,

there is no substitution between the two goods.

(c) Ifx and y are imperfect substitutes (intermediate or between perfect substitutes and perfect

complements), indifference curves will have the standard downward sloping convex to the origin

shape. In other words, MRS is falling as we move down along the indifference curve.

3 Utility Function:

Underlying all the analysis so far we have assumed that individuals can actually assign numerical

values to the satisfaction derived from different combinations of the two goods and then compare

those numerical values. The term utility refers to such a set of numbers (or values) that represent

the relative rankings of the different combinations. Simply, it can be thought of as a measure of

satisfaction. Saying that John prefers x to y is the same as saying consuming x gives John more

utility or satisfaction than consumingy .

A unit of utility is referred to as a util. So a utility function is basically a function showing the

satisfaction values that are generated from different combinations of the two goods consumed. An

example of a utility function is given by:

U(x, y) =x y= x1

2 y1

2 (2)

We can create a schedule of numerical utility values (or utils) for the various possible mixtures of

the two goods x and y. For example, calculate the utility or util value when (x, y) = (9, 16) and

(x, y) = (13, 13). Also remember that in this analysis, we are in the world of cardinal utility as

opposed to ordinal utility. Cardinal is when we can compute and compare numeric util values for

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different (x, y) combinations. In an ordinal world, we can only say which is better betweenx and

y, but not how much better.

3.1 Marginal Utility and MRS:

Along one specific indifference curve, Joe receives a constant level of satisfaction (i.e. utility or

util) at all the points (or combinations ofx and y). Let us hold Joes consumption ofy constant

at 9 units and try to see what happens to Joes utility as he consumes more and more of the other

good, x. Substitute y= 9 in the utility function:

U(x|y= 9) =x 12 (9) 12 = 3x 12 (3)

The slope of the utility function is now given by:

dU

dx = 3

1

2

(x)

1

21 =

3

2x

1

2 = 3

2

x (4)

This slope of the utility function, dUdx

, is known as the marginal utility ofx. It is the gain in total

utility or satisfaction due to a marginal increase in the consumption of good x while we hold the

consumption ofy constant at 10 units.

Note that the marginal utility of x, dUdx

, is a decreasing function of the consumption ofx. This

negative slope of marginal utility refers to the phenomenon called diminishing marginal utility.Each extra unit ofx gives Joe more satisfaction, but the additional satisfaction becomes smaller

and smaller as xincreases.

This slope ofM Ux can be computed as:

d

dx

dU

dx

=

3

2 d

dx

x

1

2

=3

4x

3

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4 Budget Constraint:

Utility functions and indifference curves constitute just one part of the consumers constrained op-

timization problem, the other part is the constraints faced by the consumer and the most important

constraint is the budget constraint. For simplicity, we exclude any savings or borrowing behavior

for the consumer. We say that each consumer has a fixed (or constant) sum of money to spend

right now the consumers income. And again for graphical simplicity, we continue to use a two

good world.

Let Ibe the consumers money income, px is the per unit price of good x, and py is the per unit

price of good y. So the budget constraint can be written as:

I=xpx+ ypy (7)

where xpx is the total share of income spent on consuming x and ypy is the total share of income

spent on consuming y. If we fix the level of income atI=I, we can solve equation (7) to draw the

budget constraint in the (x, y)-plane. The equation of the budget line is:

y= I

py

pxpy

x (8)

where pxpy is the slope of the budget line; this slope is also known as the relative price of

x in terms of the price of y. This ratio of the two prices is also known as the marginal rate

of transformation and the negative sign in the front illustrates the fact that budget line will be

negatively sloped. Buying more of good xwill mean that less of good y can be afforded.

5 The Consumers Problem Utility Maximization:

So the consumer has a budget set (or opportunity set) and an indifference map that helps us

determine which combination of purchases gives him the highest amount of pleasure (utils of utility).

Consumer will try to maximize utility subject to his budget constraint (without a budget constraint

he/she will definitely want to consume infinite amounts so the problem becomes unrealistic and

unbounded).

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Mathematically, the consumers problem can be stated as:

Maximize:{x,y}

U=U(x, y) (9)

s.t.: I=xpx+ ypy (10)

In the above maximization problem, consumer is maximizing his/her utility function (given by

U(x, y)) by choosing the quantities ofx and y to consume, keeping in mind that total expenditure

cannot go beyond I. This is the classic constrained utility maximization problem in consumer

theory.

At the optimum, the following condition has to be satisfied:

MRS =MUx

MUy=

px

py= MRT (11)

Condition (11) above implies that the slope of the budget line must equal the slope of the indifference

curve at the optimum. In other words, the indifference curve has to be tangent to the budget line

at the optimum. Condition (11) can also be rearranged to:

M Uxpx

=M Uy

py(12)

which is thelast dollar rulethat was covered in the principles of microeconomics class. According

to condition (12), marginal utility per dollar spent must be equal for both goods, x and y. For

example, if MUxpx

> MUypy

, consumer is getting more utility by spending the last dollar in buying x

thany and he/she will continue to buy more x. And, buying morexleads to lowerM Uxand higher

M Uy (because of lower y being purchased); eventually, MUxpx

will fall and will be equal to MUypy

.

In terms of second order conditions, utility will be maximized if the indifference curves are

strictly convex to the origin, that is, MRS is falling along the indifference curve. More discussion

on the second order conditions will be done in class. More formally, the Lagrange Multiplier

method is most often used in economics to solve constrained optimization problems. Recall that,

mathematically, the consumers problem can be stated as:

Maximize:{x,y}

U=U(x, y) (13)

s.t.: I=xpx+ ypy (14)

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The Lagrangemultiplier technique proceeds by, first, writing down the Lagrange function:

L(x,y,) =U(x, y) + [I xpx+ ypy] (15)

where, is the Lagrange multiplier and is being added as an extra variable to be solved from

the optimization problem. According to Lagranges theorem, first order necessary conditions for

optimization requires that the following three equations are satisfied:

Lx

=Lx

: Ux(x, y) px= 0 (16)

Ly

=Ly

: Uy(x, y) py = 0 (17)

L

=L

: I xPx+ yPy = 0 (18)

Now, the above first order conditions can be solved simultaneously to obtain the solutions (x, y, ),

where,{x, y} are also knows as the Marshallian (or, uncompensated) demand functionsfor the two goods. Also, divide equation (16) by (17) to obtain the tangency condition.

5.1 Marshallian Demand Curves:

Theoretically, the first order conditions from the constrained utility maximization problem can

be solved to obtain closed-form solutions or the demand functions for each good. These optimal

quantities are the consumers quantity demanded of the various goods, in this case,{

x, y

}. In

general, the demand curves are given by:

xm =x(px, py, I) (19)

ym =y(px, py, I) (20)

where, themdenotes the fact that these are Marshallian demand functions. The above two demand

functions are the equations of demand curves that we have studied in principles classes. If we assume

specific forms for the utility function we will have a closed form solution for the demand curves

in equation (19) and (20). For example, if we assume a Cobb-Douglas utility function of the form

U=xy, the resultant demand equations will be as follows:

xm =I

px(21)

ym =I

py(22)

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Some properties of these demand functions should be noted:

(a) The demand for each good depends only on its own price and income (only valid in case of this

Cobb-Douglas utility function). In this case, the two goods are independent because:

xm

py= y

m

px= 0

For certain utility functions, this property may not hold.

(b) In general, if:

xm

py>0 goods xandy are gross substitutes

xm

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Equation (24) is the indirect utility function (IUF); it shows the maximum amount of utility

that can be achieved for any combination of the two prices (pxandpy) and income (m). Important

properties of the IUF has been discussed in class. In addition, using Roys Identity, one can

obtain the Marshallian demand functions directly from the IUF as follows:

x(px, py, I) =vpxvI

and y(px, py, I) =vpxvI

Obviously, the above trick is applicable only if you already know the IUF. You should check,

mathematically, that using Roys Identity, you can go from equation (2) to the two demand functions

in equation (1).

6 Expenditure Minimization and Hicksian Demand:

An alternative way to look at the consumers problem is to ask the question: given that the consumer

is facing a given set of prices, what is theminimum expenditure neededto achieve a target level

of utility? Look at the following two figures to understand the fundamental difference between the

utility maximization and the expenditure minimization problems:

y

xx

y

e0

A

(a)Utility Maximization (b) Expenditure Minimization

B

u0

u4

u3

u2

u1e1 e2 e3

e4

The figure on the left explains the utility maximization problem, where, the objective is to maximize

utility (or, reach the indifference curve furthest from the origin,u4> u3> u2> u1) given that the

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consumer faces a fixed level of income (or, one particular budget line given by e0). The one on the

right depicts the expenditure minimization problem, where, the objective is to minimize spending

(or, reach the budget line closest to the origin, e4> e3> e2> e1) given that the consumer achieves

a given level of utility (or, one particular indifference curve given by u0). Keep in mind that, in

each figure, the consumer is facing a given set of prices of the two goods (px andpy). And in both

problems, the tangency condition remains the same, that is, optimality is achieved when the slope

of an indifference curve equals the slope of a budget line:

MRS =MUxMUy

=pxpy

= MRT

Formally, the expenditure minimization problem is:

Minimize:{x,y} e= I=xpx+ ypy

s.t.: u= u(x, y) =xy

where,xpx+ypy =Iis the budget constraint or the total spending for the consumer. Again, xandy

are the two choice variables (or, instruments) andpx, py, uare the parameters (or, constants). This

problem is also solved using similar techniques as before, that is, forming the Lagrange function

and setting up the first order conditions. But, the solutions forx and y will now be of the form

(again, assuming = = 0.5):

xc(px, py, u) =u

pypx

0.5and yc(px, py, u) =u

pxpy

0.5(25)

where,xc and yc are the Hicksian or compensateddemand curves. Along a compensated demand

curve, total spending is changing, but, utility is held constant. In other words, the consumer is

constantly being compensated such that utility remains the same and hence, the name compen-

sated. Substitute and complement goods can also be defined in terms of the Hicksian demand

functions. In general, if:

xc

py>0 goods x and y are net substitutes

xc

py


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minimization problem) to obtain the minimized value of spending:

e(px, py, u) =px u

pypx

0.5+pyu

pxpy

0.5

=up10.5x p0.5y + up

0.5x p

10.5y = 2p

0.5x p

0.5y (26)

which is the expenditure function; it shows the minimum spending necessary to attain a par-

ticular level of utility given that the consumer faces the prices (px and py) and income (I). Using

Sheppards Lemma, one can obtain the Hicksian demand functions directly from the expenditure

function as well:

xc(px, py, u) = e

pxand yc(px, py, u) =

e

py

Obviously, the above trick is applicable only if you already know the expenditure function. You

should check, mathematically, that using Sheppards Lemma, you can go from equation (4) to the

two demand functions in equation (3). Lastly, you should also note that the IUF (equation (2))

and the expenditure function (equation (4)) are inverses of each other.

7 Change in Income and Type of Goods:

We know that changes in the consumers income (while prices of both goods remain the same) shift

the demand curve for a good. But how does it happen? Consider there is an increase in income and

we know as a result the budget line will shift outward or to the right. When the budget line shifts to

the right we will have a new optimal bundle of goods at the same level of prices as before. This shift

in demand will continue if there is another subsequent increase in income and the demand curve

for both goods will continue to shift to the right. An income consumption curve traces all such

optimal consumption bundles as the consumers income changes. And an Engel curve shows therelationship between quantity demanded of an individual good and income, holding prices constant.

For normal goods, Engel curve is positively sloped.

Superior vs. Inferior Good: Relationship between changes in income and changes in consump-

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tion defines two types of goods:

(i) If xm

I >0, good x is a normal or superiorgood; and

(ii) if xm

I

1, e.g., the 2009 BMW 335xi Coupe.

(d) Keep in mind that the above definitions of all the goods are relevant for a representative rational

consumer whose underlying preferences always follow economic theory. Also, a good that is

luxury for on individual could be a necessity for another individual depending on their relative

perception of things and a host of other subjective factors.

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8 Change in Price and the SlutskyEquation:

Remember we are still in the world with two goods (x and y). Price of good x is given by px and

price of goody is given bypy. Now, consider a decrease in the price of good x (that ispxhas fallen)

and the total effect consists of two smaller effects:

(a) Substitution effect: If utility is held constant, the consumer will now substitute more of good

x with the cheaper good y. Substitution effect is always negative, that is, the consumer will

always buy more of the cheaper product, good xin this case.

(b) Income effect: A decrease in px means an increase in the consumers real income (that is

purchasing power), so he or she will buy more of at least one good (either x or y or both).

Income effect can be positive and negative depending on the type of the good.

The Slutskyequation is a mathematical equation that shows the decomposition of the total effect

of a price change into the above two effects. For a change in the price ofx, the Slutsky equation is

given by:

xm

px=

xc

px

xm xm

m

(28)

wherepx is the price ofx, xm is the Marshallian demand,xc is the compensated demand, and I is

income. In the above Slutsky equation, the partial derivative on the left hand side is the total effect

of the change in px. This is the slope of the Marshallian demand curve. The first partial derivative

term on the right hand side is the substitution effect and this term is always negative; this term

is, basically, the slope of the compensated demand curve. The second term inside brackets on the

right hand side is the income effect and this can be either positive and negative:

(a) If x is a normal good, then, xm/I > 0, which means the overall income effect term is

negative. Therefore, together with the negative substitution effect, the total effect will be

negative. Hence, Marshallian demand curve will be negatively sloped.

(b) If x is an inferior good, then, xm/I < 0, which means the overall income effect term is

positive. But, the positive income effect is smaller than the negative substitution effect, so, the

total effect will still be negative. Hence, Marshallian demand curve will again be negatively

sloped.

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8.2 Decrease in Price of x(Inferior Good):

The following figure shows the income and substitution effects for a decrease in px when x is an

inferior good:

"##$ %

"##$&

'()&

*

'()%

+'()%

*

%*

%,

%+

-+

-*

.

/

0

1

1

'23 )%

*%*4 )

&&

*

'53 )%

+%+4 )

&&

+

The consumer is initially at point A, where, the bundle (x0, y0) is bought at prices (p0x, p

0y) and

income m0

. When the price ofx decreases to p1x, the budget line rotates to the line m

1

. To find

the substitution effect, draw a budget line (given by line II) parallel to the new budget line, but,

tangent to the old indifference curve (u0). So, substitution effect is the movement (or distance)

fromx0 toxs and this effect is always negative. And, since x is an inferior good, the income effect

will decrease the consumption from xs to x1. Remember, the decrease in px means larger real

income (or purchasing power), but, the consumer lowers consumption from xs, because, x is an

inferior good. So, income effect is the movement (or distance) from xs tox1. Lastly, note that the

decrease inx (income effect) is smaller than the initial increase inx (substitution effect) and hence,

the net (total) effect is still negative.

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9 Revealed Preference:

So far, in consumer theory, we have derived demand curves from the consumers preferences (under

a certain set of assumptions guaranteeing the existence of a real valued utility function). In this

chapter, our objective is to understand whether we can identify the underlying preferences from

the consumers observed (or, revealed) behavior (or, demand). Keep in mind that, in most real life

situations, peoples behavior is directly observable, while the preferences are not. In the discussion

that follows, to keep things simple, assume that our consumer is living in a two good world (x1 and

x2), where, prices arep1 andp2. And, assume that the consumers current income ism. Then, the

Principle of Revealed Preference states that:

Let (x1, x2) be the chosen bundle when prices are (p1, p2), and let (x01, x02) be some other

bundle such that p1x1+ p2x

2 p1x01+ p2x02 (which means that bundle (x01, x02) was also

affordable at p1, p2). Then, if the consumer is choosing the most preferred bundle she

can afford, it must be that (x1, x2)(x01, x02).

What does it all mean? Imagine that the consumer can afford to buy both the bundles, (x1, x2)

and (x01, x02), at the given prices and income; and, it has been observed that the consumer bought

the bundle (x1, x2). So, it is being observed through the consumers purchase decision that the

consumer revealed prefers bundle (x1, x2) over bundle (x

01, x

02). In other words, consumers

preference ordering is revealed through their buying decisions in the market. In earlier chapters,

we started with the fact that we know the consumers preferences (and, hence, a utility function)

and, then, derived the demand functions. Let us the understand the above in a simple figure:

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"##$ %&

"##$%'

(%&)* %

')+

(%&

,* %'

,+

(%&

,* %'

,+

A

B

C Budget line

The above figure shows a usual downward sloping budget line, where, the quantity of good x1 is

on the horizontal axis and quantity of good x2 is on the vertical axis. This budget line is drawn

for prices (p1, p2) and income m. It is observed in the market that, at these prices and income, the

consumer purchases the bundle (x1, x2) when bundle (x

01, x

02) was clearly affordable. Two possible

locations for this affordable bundle is shown all income is spent at B, while, there is excess income

at C. Mathematically:

p1x1+p2x

2= p1x

01+p2x

02 between A and B

p1x1+p2x

2> p1x

01+p2x

02 between A and C

The theory of revealed preference postulates that since the consumer is buying (x1, x2) when other

bundles were also affordable, it must be the case that bundle (x1, x

2) is revealed preferred than

all the other affordable bundles. It is not important whether the other bundles exhaust all the

income (such as at point B) or leaves the consumer with some excess income (such as at point C).

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9.1 Direct and Indirect Revealed Preference:

From the above analysis, bundle (x1, x2) is directly revealed preferred to bundle (x

01, x

02). To

understand indirect revealed preference, we need to introduce another set of prices and use the

idea of transitivity of preferences. Imagine that at a different set of prices, (p1, p2), the consumer

purchases the bundle (x01, x02) when bundle (x

1, x

2) was also affordable. This means that the con-

sumer directly revealed prefers bundle (x01, x02) over bundle (x

1, x

2). Based on the assumption

of transitivity, it must be that the consumer indirectly revealed prefersthe bundle (x1, x2) over

the bundle (x1, x2).

"##$ %&

"##$%'

(%&)* %

')+

(%&

,* %'

,+

(%&-* %

'-+

A

C

B

BL1

BL2

In the above figure, two budget lines are shown. Budget line BL1is drawn for the prices (p1, p2) and

budget line BL2 is drawn for the prices (p1, p

2). For the situation depicted above, bundle (x

1, x

2)

is directly revealed preferred to bundle (x01, x02), and bundle (x01, x02) is directly revealed preferred

to bundle (x1, x2). Therefore, based on the assumption of transitivity, bundle ( x

1, x

2) is indirectly

revealed preferred over the bundle (x1, x2). In addition, the chain of transitive preferences can be

much longer than just three bundles (as explained above). If, for example, bundle A is directly

revealed preferred to bundle B, B to C, C to D, , N to O, then, bundle A is still indirectly

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revealed preferred to bundle O. Then, the theory of revealed preference states that if a bundle is

either directly or indirectly revealed preferred to another bundle, we will say that the first bundle is

revealed preferred to the second bundle. And, if the consumeralways chooses the best bundles

that he or she can afford, then revealed preference implies preference. In the presence of

certain additional assumptions, we may derive a consumers indifference curves from the theory

of revealed preference. We will cover this in appropriate detail in class, both graphically and

intuitively.

10 Weak Axiom of Revealed Preference:

The Weak Axiom of Revealed Preference (WARP) states that: if bundle(x1, x2) is directly

revealed preferred to bundle(x01, x02), and the two bundles are not the same, then, it cannot be that

(x01, x02) is directly revealed preferred to bundle(x

1, x

2).

"##$ %&

"##$%'

(%&)* %

')+

(%&

,* %'

,+

A

B

BL1

BL2

It is easiest to understand the implications of WARP using a figure like the above. Two budget

lines are shown in the figure, BL1 and BL2, for two different sets of prices. What can we conclude

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from this figure? First, in case of BL1, the consumer is buying bundle (x1, x

2) when bundle (x

01, x

02)

was affordable, which means (x1, x2) is preferred to (x

01, x

02). Second, in case of BL2, the consumer

is buying bundle (x01, x02) when bundle (x

1, x

2) was affordable, which means (x

01, x

02) is preferred to

(x1, x2). WARP simply implies than this type of cyclical preferences are inconsistent (not valid)

and this situation cannot occur for a utility maximizing consumer. In contrast, the following figure

shows a situation, where, WARP is satisfied:

"##$ %&

"##$%'

(%&)* %

')+

(%&

,* %'

,+

A

B

BL1

BL2

Why is WARP satisfied in the above figure? In case of BL1, the consumer is buying bundle

(x1, x2) when bundle (x

01, x

02) was affordable, which means (x

1, x

2) is preferred to (x

01, x

02) (same

as before). But, now, in case of BL2, the consumer is buying (x01, x

02) when the bundle (x

1, x

2) is

not affordable. So, we can still conclude that, in case of BL2, the consumer is simply buying the

cheapest bundle and, so, it is not the case that (x0

1, x0

2) is preferred to (x1, x

2).

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11 Exercises:

1. CalculateM Ux, MUy, and MRS for each of the following utility functions; also, check whether

there is diminishing marginal utility for each good:

(a) U(x, y) =x1

3 y2

3

(b) U(x, y) =x + Axy + y

(c) U(x, y) =

x2 y2

(d) U(x, y) = x

+ y

, where 1 > , , > 0

2. Vasco consumes two goods: (i) pizza,q1 ; and (ii) burrito, q2 . His utility function is given by

U(q1, q2) = 10q1q22 . The price of pizza is p1 = $5, the price of burritos is p2 = $10, and his

income isY= $150. Mathematically solve for Vascos optimal consumption bundle and show it

in a graph.

3. Anns utility function is given by U(x, y) = xyx+y . Solve for her optimal values ofx and y as a

function ofPx,Py, and M.

4. Diego consumes two goods: (i) chewing gums, G ; and (ii) butter, B. His utility function is

given by U(G, B) =G3

4 B1

4 . The price of gums is PG= $1, the price of butter is PB = $2 , and

his income is Y = $100. Mathematically solve for Diegos optimal demand for chewing gums

and butter.

5. Suppose in a three good world that quantities and prices of the goods are labeledxi = (xi1, xi2, x

i3)

and pi = (pi1, pi2, p

i3) respectively, where, i = 1, 2, 3. Using the following information, check

whether the consumers preferences satisfy assumptions of WARP and transitivity:

p1 = (4, 3, 2) x1 = (2, 2, 2)

p2 = (5, 3, 3) x2 = (1, 3, 3)

p3 = (5, 2, 3) x3 = (1, 3, 2)

22

Consumer Theory - Microeconomics

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