Consumer Preferences, Utility Functions and Budget Lines Overheads
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Transcript of 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
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
Marginal utility
Marginal utility is defined as theincrement in utility an individual enjoys from consuming an additional unit of a good or service.
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
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
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)
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.
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
Notation
Income - I
Number of goods - n
u - utility
Quantities of goods - q1, q2, . . . qn
Prices of goods - p1, p2,. . . pn
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.
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.
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
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.
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
Indifference Curves
An indifference curveindifference curve representsall combinations of two categories of goods
that make the consumer equally well off.
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
Graphical analysis with several levels of u
Indifference Curves
02468
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0 1 2 3 4 5 6
q2
q1u = 8u = 10u = 12u = 15
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
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
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.
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
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
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
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
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.
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
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
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
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
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
u = 8
u = 10
Indifference Curves
02468
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0 1 2 3 4 5 6 7
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
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.
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
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
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
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
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
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
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.
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
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
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
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
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
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
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
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
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
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
The indifference curve tells us the rateat which the consumer could trade one goodfor the other and remain indifferent.
Budget Line
Slope of Indifference Curvesand the Budget Line
02468
1012141618
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.
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
Budget Line
u = 8
Slope of Indifference Curvesand the Budget Line
02468
1012141618
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
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?
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
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
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
Budget Line
u = 8
02468
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0 1 2 3 4 5 6 7
q2
q1u = 10.998u = 10.28
Slope of Indifference Curvesand the Budget Line
Move down
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
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
02468
1012141618
0 1 2 3 4 5 6 7q2
q1
Slope of Indifference Curvesand the Budget Line
Budget Line
u = 8
u = 10
02468
1012141618
0 1 2 3 4 5 6 7q2
q1
Slope of Indifference Curvesand the Budget Line
Budget Line
u = 8
u = 10
02468
1012141618
0 1 2 3 4 5 6 7q2
q1 u = 8
u = 10
u = 10.998Budget Line
Slope of Indifference Curvesand the Budget Line
02468
1012141618
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
02468
1012141618
0 1 2 3 4 5 6 7q2
q1
u = 10
u = 10.998
Slope of Indifference Curvesand the Budget Line
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
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.
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
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
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
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
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
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
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
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
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
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
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
02468
1012141618
0 1 2 3 4 5 6 7q2
q1
Slope of Indifference Curvesand the Budget Line
Budget Line
u = 8
u = 10
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