Consumer Preferences, Utility Functions and Budget Lines Overheads.

102
Consumer Preferences, Utility Functions and Budget Lines Overheads

Transcript of Consumer Preferences, Utility Functions and Budget Lines Overheads.

Page 1: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Consumer Preferences, Utility Functions

and Budget Lines

Overheads

Page 2: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Utility is a measure of satisfaction or pleasure

UtilityUtility is defined as the pleasure or satisfactionpleasure or satisfaction obtainedfrom consuming goods and services

Utility is defined on theentire consumption bundle of the consumer

Page 3: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Mathematically we define the utility function as

u u(q1 , q2 , q3 , , qn)

u represents utility

qj is the quantity consumed of the jth good

(q1, q2, q3, . . . qn) is the consumption bundle

n is the number of goods and services available to the consumer

Page 4: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Marginal utility

Marginal utility is defined as theincrement in utility an individual enjoys from consuming an additional unit of a good or service.

Page 5: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Mathematically we define marginal utility as

MUqj MUj Δu(q1 , q2 , q3 , , qn )

Δqj

If you are familiar with calculus, marginal utility is

MUqj u(q1 , q2 , q3 , , qn)

qj

Page 6: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Data on utility and marginal utility

q1 q2 utility marginal utility1 4 8.00

2.082 4 10.08

1.463 4 11.54

1.164 4 12.70

0.985 4 13.68

0.866 4 14.54

0.777 4 15.31

0.698 4 16.00

0.659 4 16.65

0.5910 4 17.24

0.5611 4 17.80

0.5212 4 18.32

Change q1 from 8 to 9 units

MU1 ΔuΔq1

16.65 169 8

0.65

Page 7: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Marginal utility

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 2 4 6 8 10 12 14

q1

Mar

gin

al u

tili

ty mu1(q1,q2=3)

mu1 (q1, q2=4)

Page 8: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Law of diminishing marginal utility

The law of diminishing marginal utility says that as the consumption of a good of service increases, marginal utility decreases.

The idea is that the marginal utility of a good diminishes,with every increase in the amount of it that a consumer has.

Page 9: Consumer Preferences, Utility Functions and Budget Lines Overheads.

The Consumer Problem

maxq1 , q2 , q3 , , qn

[ u(q1 , q2 , q3 , , qn ) ]

As the consumer chooses more of a given good,utility will rise,

but because goods cost money, the consumerwill have to consume less of another good

because expenditures are limited by income.

subject to

p1q1 p2q2 p3q3 pnqn I

Page 10: Consumer Preferences, Utility Functions and Budget Lines Overheads.

The Consumer Problem (2 goods)

maxq1 , q2

[ u(q1 , q2) ]

subject to

p1q1 p2q2 I

Page 11: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Notation

Income - I

Number of goods - n

u - utility

Quantities of goods - q1, q2, . . . qn

Prices of goods - p1, p2,. . . pn

Page 12: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Optimal consumption is along the budget line

Given that income is allocated amonga fixed number of categories

Why?

and all goods have a positive marginal utility,

the consumer will always choose a pointa pointon the budget lineon the budget line.

Page 13: Consumer Preferences, Utility Functions and Budget Lines Overheads.

1

5 6 74321

2

3

4

5

Budget Constraint - 0.3q1 + 0.2q2 = $1.20

Affordable

Not Affordable

q1

q2

Page 14: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Marginal decision making

To make the best of a situation, decision makers

should consider the incremental or marginalincremental or marginal effects

of taking any action.

In analyzing consumption decisions,

the consumer considers small changesconsiders small changes in the quantities consumed,

as she searches for the “optimal”searches for the “optimal” consumption bundle.

Page 15: Consumer Preferences, Utility Functions and Budget Lines Overheads.

q1 q2 Utility MarginalUtility

4 3 11.000.85

5 3 11.850.74

6 3 12.59

3 4 11.54 1.16

4 4 12.70 0.98

5 4 13.68 0.86

6 4 14.54

4 5 14.201.10

5 5 15.300.96

6 5 16.26

Implementing the small changes approach - p1 = p2

Consider the point (5, 4) with utility 13.68

Now raise q1 to 6 and reduce q2 to 3. Utility is 12.59

q = (4, 5) is preferred toq = (5, 4) and q = (6, 3)

Now lower q1 to 4 and raise q2 to 5. Utility is 14.20

Page 16: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Budget lines and movements toward higher utility

Given that the consumer will consume along the budget line,the question is

Example

p1 = 5 p2 = 10 I = 50

q1 = 2 q2 = 4 (5)(2) + (10)(4) = 50

q1 = 6 q2 = 2 (5)(6) + (10)(2) = 50

q1 = 4 q2 = 3 (5)(4) + (10)(3) = 50

which point will lead to a higher level of utility.

Page 17: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Budget Constraint

(6,2)

(4,3)

(2,4)

0123456789

1011

0 1 2 3 4 5 6q2

q1

q1 q2 utility6 2 10.280

2 4 10.080

4 3 10.998

Exp = I = 50

Exp = I = 50

Exp = I = 50

p1 = 5 p2 = 10 I = 50

Page 18: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Indifference Curves

An indifference curveindifference curve representsall combinations of two categories of goods

that make the consumer equally well off.

Page 19: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Example data and utility level

q1 q2 utility8 1 82.83 2 8

1 4 8

0.72 5 8

1.54 3 8

Page 20: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Indifference Curve

0

2

4

6

8

10

12

14

0 1 2 3 4 5 6 7

q2

q1

u = 8

Graphical analysis

Page 21: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Example data with utility level equal to 10

q1 q2 utility15.625 1 10.008 1 8.00

Page 22: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Example data with utility level equal to 10

q1 q2 utility15.625 1 10.005.524 2 10.00

3.007 3 10.00

1.953 4 10.001.398 5 10.001.063 6 10.000.844 7 10.00

Page 23: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Graphical analysis with u = 10

Indifference Curves

02468

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q2

q1

u = 10

Page 24: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Graphical analysis with several levels of u

Indifference Curves

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q2

q1u = 8u = 10u = 12u = 15

Page 25: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Slope of indifference curves

The slope of an indifference curve is called the marginal rate of substitution (MRS) between good 1 and good 2

Indifference curves normally have a negative slope

If we give up some of one good, we have to getmore of the other good to remain as well off

Page 26: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Indifference Curves

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0 1 2 3 4 5 6

q2

q1

u = 12

Page 27: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Slope of indifference curves (MRS)

The MRS tells us the decrease in the quantityof good 1 (q1) that is needed to accompanya one unit increasein the quantity of good two (q2),in order to keep the consumer indifferent to the change

Page 28: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Indifference Curves

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0 1 2 3 4 5 6

q2

q1

u = 12

Page 29: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Shape of Indifference CurvesShape of Indifference Curves

Indifference curves are convex to the originIndifference curves are convex to the origin

This means that as we consume more and more of a good,

its marginal value in terms of the other good becomes less.

Page 30: Consumer Preferences, Utility Functions and Budget Lines Overheads.

05

10152025303540

0 1 2 3 4 5 6q2

q1

u = 12

The Marginal Rate of Substitution (MRS)

The MRS tells us the decrease in the quantity of good 1 (q1)that is needed to accompany a one unit increasein the quantity of good two (q2),in order to keep the consumer indifferent to the change

Page 31: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Algebraic formula for the MRSAlgebraic formula for the MRS

MRSq1,q2

Δq1

Δq2

u constant

The marginal rate of substitution of good 1 for good 2 isThe marginal rate of substitution of good 1 for good 2 is

We use the symbol - |We use the symbol - | u = constant u = constant - - to remind us that the to remind us that the

measurement is along a constant utility indifference curvemeasurement is along a constant utility indifference curve

Page 32: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Example calculations

q1 q2 utility5.524 2 10.003.007 3 10.001.953 4 10.001.398 5 10.001.063 6 10.00

MRSq1,q2

Δq1

Δq2

u constant

Change q2 from 4 to 5

1.953 1.3984 5

0.555 1

0.555

Page 33: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Example calculations

q1 q2 utility5.524 2 10.003.007 3 10.001.953 4 10.001.398 5 10.001.063 6 10.00

Change q2 from 2 to 3

2.517 1

2.517

MRSq1,q2

Δq1

Δq2

u constant

5.524 3.0072 3

Page 34: Consumer Preferences, Utility Functions and Budget Lines Overheads.

A declining marginal rate of substitution

The marginal rate of substitution becomeslarger in absolute value,as we have more of a product.

The amount of a good we are willing to give up to keep utility the same,is greater when we already have a lot of it.

Page 35: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Indifference Curves

05

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0 1 2 3 4 5 6

q2

q1

u = 10

-2.517

-0.555

Give up lots of q1 to get 1 q2

Give up a little q1 to get 1 q2

Page 36: Consumer Preferences, Utility Functions and Budget Lines Overheads.

05

10152025303540

0 1 2 3 4 5 6q2

q1

u = 10

A declining marginal rate of substitution

q1 q2 utility3.007 3 10.001.953 4 10.001.398 5 10.001.063 6 10.00

When I have 1.953 units of q1, I can give up 0.55 units for a one unit increase in good 2 and keep utility the same.

-0.555

-0.555

Page 37: Consumer Preferences, Utility Functions and Budget Lines Overheads.

05

10152025303540

0 1 2 3 4 5 6q2

q1

u = 10

-2.517

A declining marginal rate of substitutionWhen I have 5.52 units of q1, I can give up 2.517 units for an increase of 1 unit of good 2 and keep utility the same.

q1 q2 utility5.524 2 10.003.007 3 10.001.953 4 10.00

-2.517

Page 38: Consumer Preferences, Utility Functions and Budget Lines Overheads.

05

10152025303540

0 1 2 3 4 5 6q2

q1

u = 10

A declining marginal rate of substitutionWhen I have 15.625 units of q1, I can give up 10.101 unitsfor an increase of 1 unit of good 2 and keep utility the same.

q1 q2 utility15.625 1 10.00 5.524 2 10.00 3.007 3 10.00 1.953 4 10.00

-10.101

-10.101

Page 39: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Break

Page 40: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Indifference curves and budget lines

We can combine indifference curves and budget linesto help us determine the optimal consumption bundle

The idea is to get on the highest indifference curve allowed by our income

Page 41: Consumer Preferences, Utility Functions and Budget Lines Overheads.

u = 8

u = 10

Indifference Curves

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q2

q1

u = 12

Budget Line

q1 q2 cost utility8 1 50.00 8.0002.828 2 34.14 8.0003.007 3 45.04 10.000

Budget Lines

4 3 50.00 10.998

3.375 4 56.88 12.000

Page 42: Consumer Preferences, Utility Functions and Budget Lines Overheads.

02468

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0 1 2 3 4 5 6 7q2

q1

u = 8

Budget Line

q1 q2 cost utility8 1 50.00 8.0002.828 2 34.14 8.000

At the point (1,8) all income is being spent and utility is 8

The point (2, 2.828) will give the utility of 8, but at a lessor cost of $34.14.

Page 43: Consumer Preferences, Utility Functions and Budget Lines Overheads.

u = 8

02468

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0 1 2 3 4 5 6 7

q2

q1

u = 10

Budget Line

q1 q2 cost utility8 1 50.00 8.0002.828 2 34.14 8.0003.007 3 45.04 10.000

The point (3, 3.007) will give a higher utility level of 10,but there is still some income left over

Page 44: Consumer Preferences, Utility Functions and Budget Lines Overheads.

u = 8

u = 10

02468

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0 1 2 3 4 5 6 7

q2

q1

Budget Line

q1 q2 cost utility8 1 50.00 8.0002.828 2 34.14 8.0003.007 3 45.04 10.0004 3 50.00 10.998

The point (3,4) will exhaust the income of $50 and give a utility level of 10.998

Page 45: Consumer Preferences, Utility Functions and Budget Lines Overheads.

u = 10

u = 8

The point (4, 3.375) will give an even higher utility level of 12, but costs more than the $50 of income.

02468

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0 1 2 3 4 5 6 7q2

q1

u = 12

Budget Line

q1 q2 cost utility8 1 50.00 8.0002.828 2 34.14 8.0003.007 3 45.04 10.0004 3 50.00 10.9983.375 4 56.88 12.000

Page 46: Consumer Preferences, Utility Functions and Budget Lines Overheads.

The utility function depends on quantitiesof all the goods and services

u u(q1 , q2 , q3 , , qn)

u u(q1 , q2 )

For two goods we obtain

We can graph this function in 3 dimensions

Page 47: Consumer Preferences, Utility Functions and Budget Lines Overheads.

3-dimensional representation of the utility function

0 2 4 6 8 10 12 14

q1

0

5

10

15

20q2

0

10

20

30

40

u

0 2 4 6 8 10 12 14

q1

0

5

10

15

20q2

Page 48: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Another view of the same function

0

2

4

6

8

10

12

14

q1

0

5

10

15

20

q2

0

10

20

30

40

u0

2

4

6

8

10

12

14

q1

0

10

20

30

40

u

Page 49: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Contour lines are lines of equal height or altitude

If we plot in q1 - q2 space all combinations of q1 and q2 that lead to the same (value) height for the utility function,we get contour lines

For the utility function at hand, they look as follows:

similar to those you see on a contour map.

Page 50: Consumer Preferences, Utility Functions and Budget Lines Overheads.

0 5 10 15 20q1

0

5

10

15

202q

Contour lines

Page 51: Consumer Preferences, Utility Functions and Budget Lines Overheads.

0

2

4

6

8

10

12

14

q1

0

5

10

15

20

q2

0

10

20

30

40

u0

2

4

6

8

10

12

14

q1

0

10

20

30

40

u

Function

Page 52: Consumer Preferences, Utility Functions and Budget Lines Overheads.

0 5 10 15 20q1

0

5

10

15

202q

Contour lines

Page 53: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Representing the budget line in 3-space

p1q1 + p2 q2 = I

5q1 + 10 q2 = 50

q1 = 10 - 2q2

Page 54: Consumer Preferences, Utility Functions and Budget Lines Overheads.

The budget line in q1 - q2 - u (3) space

All the points directly above the budget line create a plane

02.5

57.5

10

0

2

4

0

2

4

02.5

57.5

10

0

2

4

Page 55: Consumer Preferences, Utility Functions and Budget Lines Overheads.

02.5

57.5

10

0

2

4

0

2

4

02.5

57.5

10

0

2

4

Another view of the budget line (q1 - q2 - u (3) space)

Page 56: Consumer Preferences, Utility Functions and Budget Lines Overheads.

We can combine the budget linewith the utility functionto find the optimal consumption point

Page 57: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Combining the budget line and the utility function

02.5

57.5

10

q1

0

2

4

6 q2

0

5

10

15

20

u

02.5

57.5

10

q1

0

2

4

6 q2

Page 58: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Along the budget “wall” we can find the highest utility point

0

5

10

q1

02

46

q2

0

5

10

15

20

u

0

5

10

15

20

u

Page 59: Consumer Preferences, Utility Functions and Budget Lines Overheads.

The plane at the level of maximum utility

02.5

57.5

10

q1

0

2

4

6q2

0

5

10

15

20

u

02.5

57.5

10

q1

0

2

4

6q2

All points at the height of the plane have the same utility

Page 60: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Another view of the plane at the level of maximum utility

0

2.5

5

7.5

10

q1

02

46

q2

0

5

10

15

20

u

0

5

10

15

20

u

Page 61: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Combining the three pictures

02.5

57.5

10

q1

0

2

4

6 q2

0

5

10

15

20

uHq1, q2L

02.5

57.5

10

q1

0

2

4

6 q2

Page 62: Consumer Preferences, Utility Functions and Budget Lines Overheads.

0

5

10

q1

02

46

q2

0

5

10

15

20

uHq1, q2L0

5

10

15

20

uHq1, q2L

Another view

Page 63: Consumer Preferences, Utility Functions and Budget Lines Overheads.

We can also depict the optimum in q1 - q2 space

Different levels of utility are representedby indifference curves

The budget wall is represented by the budget line

Page 64: Consumer Preferences, Utility Functions and Budget Lines Overheads.

The optimum in q1 - q2 space

0 2 4 6 8 10q1

0

2

4

6

82q

Page 65: Consumer Preferences, Utility Functions and Budget Lines Overheads.

0 2 4 6 8 10q1

0

2

4

6

82q

Raise p1 to 10

Page 66: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Characteristics of an optimum

From observing the geometric properties of the optimum levelsof q1 and q2, the following seem to hold:

a. The optimum point is on the budget line

b. The optimum point is on the highest indifference curveattainable, given the budget line

c. The indifference curve and the budget line are tangent at the optimum combination of q1 and q2

d. The slope of the budget line and the slopeof the indifference curve are equal at the optimum

p2

p1

MRSq1q2

Page 67: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Intuition for the conditions

The budget line tells us the rate at whichthe consumer is able to trade one good for the other,given their relative prices and income

Page 68: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Budget Line

Slope of Indifference Curvesand the Budget Line

02468

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0 1 2 3 4 5 6 7q2

q1 For example in this case, the consumer must give up 2 units of good 1 in order to buy a unit of good 2

Page 69: Consumer Preferences, Utility Functions and Budget Lines Overheads.

The indifference curve tells us the rateat which the consumer could trade one goodfor the other and remain indifferent.

Page 70: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Budget Line

Slope of Indifference Curvesand the Budget Line

02468

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0 1 2 3 4 5 6 7q2

q1

u = 10

For example on the indifference curve whereu = 10, the slope between the points (2, 5.524)and (3, 3.007) is approximately -2.517.

Page 71: Consumer Preferences, Utility Functions and Budget Lines Overheads.

The consumer is willing give up 2.517 units of good 1 for a unit of good 2,

but only has to give up 2 units of good 1for 1 unit of good 2 in terms of cost

So give up some q1

Page 72: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Budget Line

u = 8

Slope of Indifference Curvesand the Budget Line

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0 1 2 3 4 5 6 7q2

q1

On the indifference curve where u = 8,the slope between the points (1, 8)and (2, 2.828) is approximately -5.172

Page 73: Consumer Preferences, Utility Functions and Budget Lines Overheads.

q1 q2 cost utility8 1 50.00 8.0002.828 2 34.14 8.0003.007 3 45.04 10.0004 3 50.00 10.998

q1 = 2.828 - 8 = -5.172

Where did -5.172 come from?

Page 74: Consumer Preferences, Utility Functions and Budget Lines Overheads.

The consumer is willing give up 5.172 units of good 1 for a unit of good 2,

but only has to give up 2 units of good 1for 1 unit of good 2 in terms of cost

So give up some q1

Page 75: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Budget Line

u = 8

Slope of Indifference Curvesand the Budget Line

02468

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0 1 2 3 4 5 6 7q2

q1

u = 10

Move down the line

Page 76: Consumer Preferences, Utility Functions and Budget Lines Overheads.

If the consumer is willing give up 5.172 units of good 1 for a unit of good 2,

but only has to give up 2 units (in terms of cost),

the consumer will make the movedown the budget line,and consume more of q2

Page 77: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Budget Line

u = 8

02468

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q2

q1u = 10.998u = 10.28

Slope of Indifference Curvesand the Budget Line

Move down

Page 78: Consumer Preferences, Utility Functions and Budget Lines Overheads.

If the consumer is willing give up 2.517 units of good 1 for a unit of good 2,

but only has to give up 2 units (in terms of cost),

the consumer will make the movedown the budget line,and consume more of q2

Page 79: Consumer Preferences, Utility Functions and Budget Lines Overheads.

When the slope of the indifference curve is steeperthan the budget line,the consumer will move down the line

When the slope of the indifference curve is less steepthan the budget line,the consumer will move up the line

Page 80: Consumer Preferences, Utility Functions and Budget Lines Overheads.

02468

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0 1 2 3 4 5 6 7q2

q1

Slope of Indifference Curvesand the Budget Line

Budget Line

u = 8

u = 10

Page 81: Consumer Preferences, Utility Functions and Budget Lines Overheads.

02468

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0 1 2 3 4 5 6 7q2

q1

Slope of Indifference Curvesand the Budget Line

Budget Line

u = 8

u = 10

Page 82: Consumer Preferences, Utility Functions and Budget Lines Overheads.

02468

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0 1 2 3 4 5 6 7q2

q1 u = 8

u = 10

u = 10.998Budget Line

Slope of Indifference Curvesand the Budget Line

Page 83: Consumer Preferences, Utility Functions and Budget Lines Overheads.

02468

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0 1 2 3 4 5 6 7q2

q1

Slope of Indifference Curvesand the Budget Line

u = 10

Budget Line

When an indifference curve intersects a budget line,the optimal point will lie between thetwo intersection points

Move down the line

Move up the line

Page 84: Consumer Preferences, Utility Functions and Budget Lines Overheads.

02468

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0 1 2 3 4 5 6 7q2

q1

u = 10

u = 10.998

Slope of Indifference Curvesand the Budget Line

Page 85: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Alternative interpretation of optimality conditions

Marginal utility is defined as the incrementincrement in utility an individual enjoys from consuming an additional unitadditional unit of a good or service.

MUqj MUj Δu(q1 , q2 , q3 , , qn )

Δqj

Page 86: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Marginal utility and indifference curves

All points on an indifference curve are associated with the same amount of utility.

Hence the loss in utility associated with q1

MUq1Δq1 MUq2

Δq2 0

must equal the gain in utility from q2 ,

as we increase the level of q2 and decrease the level of q1.

Page 87: Consumer Preferences, Utility Functions and Budget Lines Overheads.

MUq1Δq1 MUq2

Δq2

Rearrange this expression by subtracting MUq2 q2

from both sides, MUq1

Δq1 MUq2Δq2 0

Then divide both sides by MUq1

Δq1 MUq2

Δq2

MUq1

Then divide both sides by q2

Δq1

Δq2

MUq2

MUq1

Page 88: Consumer Preferences, Utility Functions and Budget Lines Overheads.

The left hand side of this expression is the marginal rate of substitution of q1 for q2, so we can write

MRSq1q2

Δq1

Δq2

MUq2

MUq1

So the slope of an indifference curve is equal tothe negative of the ratio of the marginal utilitiesof the two goods at a given point

Page 89: Consumer Preferences, Utility Functions and Budget Lines Overheads.

MRSq1q2 MRS12

Δq1

Δq2

MUq2

MUq1

So the slope of an indifference curve ( MRSq1q2 )

is equal to the negative of the ratioof the marginal utilities of the two goods

Page 90: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Optimality conditions p2

p1

MRSq1q2

Substituting we obtain

p2

p1

MRSq1q2

Δq1

Δq2

MUq2

MUq1

The price ratio equals the ratio of marginal utilities

p2

p1

MUq2

MUq1

Page 91: Consumer Preferences, Utility Functions and Budget Lines Overheads.

We can write this in a more interesting form

Multiply both sides by MUq1

p2

p1

MUq2

MUq1

and then divide by p2

MUq1

p1

MUq2

p2

MUq1p2

p1

MUq2

Page 92: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Interpretation ?

The marginal utility per dollar for each goodThe marginal utility per dollar for each good must be equal at the optimum point of consumption.must be equal at the optimum point of consumption.

MUq1

p1

MUq2

p2

Page 93: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Example

p1 = 5 p2 = 10 I = 50

q2 q1 u MU1 MU2 MU1/p1 MU2/p2

0 10 0.000 0.000 0.000 1 8 8.000 0.334 4.000 0.067 0.42 6 10.280 0.572 2.570 0.115 0.2573 4 10.998 0.917 1.833 0.184 0.1844 2 10.080 1.680 1.260 0.336 0.1265 0 0.000 0.000 0.000

Page 94: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Budget Constraint

0123456789

1011

0 1 2 3 4 5 6q2

q1

Budget Line

p1 = 5 p2 = 10 I = 50

0

1

2

3

4

5

Page 95: Consumer Preferences, Utility Functions and Budget Lines Overheads.

q2 q1 u MU1 MU2 MU1/p1 MU2/p2

0 10 0.000 0.000 0.000 1 8 8.000 0.334 4.000 0.067 0.4

Consider the consumption point where q2 = 0 and q1 = 10.

Thus we should clearly move to the point q2 = 1, q1 = 8.

The marginal utility (per dollar) of an additional unitof q1 is 0.00,

while the utility of an additional unit (per dollar)of q2 is is infinite

Page 96: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Consider q2 = 1 and q1 = 8.

The marginal utility (per dollar) of an additional unitof q1 is 0.067,

Thus we should clearly move to the point q2 = 2, q1 = 6

q2 q1 u MU1 MU2 MU1/p1 MU2/p2

0 10 0.000 0.000 0.000 1 8 8.000 0.334 4.000 0.067 0.42 6 10.280 0.572 2.570 0.115 0.257

while the utility of an additional unit (per dollar)of q2 is 0.4

Page 97: Consumer Preferences, Utility Functions and Budget Lines Overheads.

q2 q1 u MU1 MU2 MU1/p1 MU2/p2

0 10 0.000 0.000 0.000 1 8 8.000 0.334 4.000 0.067 0.42 6 10.280 0.572 2.570 0.115 0.2573 4 10.998 0.917 1.833 0.184 0.184

At the consumption point where q2 = 3 and q1 = 4,the marginal utility (per dollar) of an additional unit of q1 is 0.184,and the utility of an additional unit (per dollar) of q2 is 0.184.

We should stay hereWe should stay here

Page 98: Consumer Preferences, Utility Functions and Budget Lines Overheads.

q2 q1 u MU1 MU2 MU1/p1 MU2/p2

0 10 0.000 0.000 0.000 1 8 8.000 0.334 4.000 0.067 0.42 6 10.280 0.572 2.570 0.115 0.2573 4 10.998 0.917 1.833 0.184 0.1844 2 10.080 1.680 1.260 0.336 0.1265 0 0.000 0.000 0.000

And we stop!And we stop!

The other way

Because > 0, we move from q2 = 5, q1 = 0 to q2 = 4, q1 = 2

Because 0.336 > 0.126, we move from q2 = 4, q1 = 2 to q2 = 3, q1 = 4

Page 99: Consumer Preferences, Utility Functions and Budget Lines Overheads.

The End

Page 100: Consumer Preferences, Utility Functions and Budget Lines Overheads.

02468

1012141618

0 1 2 3 4 5 6 7q2

q1

Slope of Indifference Curvesand the Budget Line

Budget Line

u = 8

u = 10

Page 101: Consumer Preferences, Utility Functions and Budget Lines Overheads.

u = 8

Budget Line

Slope of Indifference Curvesand the Budget Line

02468

1012141618

0 1 2 3 4 5 6 7q2

q1

u = 10

Move down the line

Move down the line

Page 102: Consumer Preferences, Utility Functions and Budget Lines Overheads.

Budget Line

u = 8

Slope of Indifference Curvesand the Budget Line

02468

1012141618

0 1 2 3 4 5 6 7q2

q1

u = 10

Move down the line