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Individual Choice
Principles of MicroeconomicsProfessor Dalton
ECON 202 – Fall 2013
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Marginal Utility approaches• Ordinal analysis• Cardinal analysis• problem of handling complements and
substitutes
Indifference Curve approach• Ordinal utility• Handles complements and substitutes
well
Models of Consumer Behavior
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Economists use the terms value, utility and benefit interchangeably when speaking of individual choice.
Marginal utility =Marginal value =Marginal benefit
Terminology Warning!
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Ordinal Analysis: Marginal Utility
Alternative Uses for horses (in order of declining value)
1st Pull plow2nd Pull wagon3rd Ride for farmer4th Ride for farmer’s wife
5th Ride for farmer’s children
Most valuable use of a horse
Least valuable use of a horse
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Law of DiminishingMarginal Utility
For all human actions, as the quantity of a good
increases, the utility from each additional unit
diminishes.
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Ordinal Analysis: Marginal Utility
Suppose the farmer owns three horses
1st Pull plow2nd Pull wagon3rd Ride for farmer4th Ride for farmer’s wife
5th Ride for farmer’s children
Farmer will use one horse to pull plow, one horse to pull wagon, and one to ride himself
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Ordinal Analysis: Marginal Utility
The farmer rides the “third” horse because the marginal benefit from riding the horse himself is greater than the marginal benefit from having his wife ride a horse.
The marginal cost of his riding the horse is the foregone marginal benefit from his wife riding the horse.
The marginal benefit from riding the horse himself is greater than the marginal cost of his riding the horse.
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Uses cardinal measure of utility Makes distinction between Total
and Marginal utility “law of diminishing Marginal
Utility” still holds Produces the Equimarginal rule
and allows for utility maximization
Cardinal Analysis: Marginal Utility
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Total utility [TU] is defined as the amount of utility an individual derives from consuming a given quantity of a goodduring a specific period of time. TU = f (Q, preferences, . . .)
1 2 3 4 5 6 7 Q/t
20
40
60
100
80
120
Utility
Q1
2
4
8
5
6
7
3
TU
30
55
75
90
100105
105
100
.. . . . . ..TU
TU
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Marginal utility [MU] is the change in total utility associated with a 1 unit change in consumption.
As total utility increases at a decreasing rate, MU declines.
As total utility declines, MU is negative.
When TU is a maximum, MU is 0.• “Satiation point”
Marginal Utility
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Marginal Utility [MU] is the change in total utility [ΔTU] caused by a one unit change in quantity [ΔQ] ;
MU = ΔTUΔQ
Utility
Q
12
4
8
567
3
TU
30
55
75
90
100105
105
100
MUQ=1 TU=30
The first unit consumed increases TU by 30.
.
The marginal utility is associated with the midpoint between the units as each additional unit is added.
30Q=1 TU=25
.25
Q
The 2nd unit increases TU by 25.
25Q=1 TU=20
.20
.15
10
.
5
0
-5
1 2 3 4 5 6 7 Q/t
10
20
30
MU
. . .MU
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If there are no costs associated with choice, the individual consumes until MU = 0, thereby maximizing TU.
Typically, individuals are constrained by a budget [or income] and the prices they pay for the goods they consume.
Net benefits are maximized where MU = MC; as long as the MU of the next unit of good purchased exceeds the MC, it will increase net benefits.
Individual Choice
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The individual purchases more of a good so long as their expected MU exceeds the price they must pay for the good:
Buy so long as MU (MB) > MC; Don’t buy if MU (MB) < MC. The maximum net utility
(consumer surplus) occurs where MU (MB) = MC.
Individual Choice
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Individual choices become a function of the price of the good, income, prices of related goods and preferences.
QX = f (PX , I, PY, Preferences, . . . )• Where:
• PX = price of good X
• I = income
• PY = prices of related goods
• “preferences” is the individual’s utility function
Constrained Optimization
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Utility X
Qx
1
2
4
8
5
6
7
3
TUx
30
55
75
90
100105
105
100
20
15
10
5
0
-5
30
25
MUx
Utility YQy
1
2
4
8
5
6
7
3
TUy
60
90
110
120
128
128
120
100
60
30
20
10
8 0
- 8
- 20
MUy
Consider an individual’s utility preference for 2 goods, X & Y;
If the two goods were “free,”[ or no budget constraint],the individual would consume each good until the MU ofthat good was 0, 7 units
of good X and 6 of Y.
Once the goods have a priceand there is a budget constraint, the individualwill try to maximize the utility from each additionaldollar spent.
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Utility X
Qx
1
2
4
8
5
6
7
3
TUx
30
55
75
90
100105
105
100
20
15
10
5
0
-5
30
25
MUx For PX = $3, the MUX per dollar spent on good X is…
Given the budget constraint, individuals will attempt to gain the maximum utility for each additional dollar spent,“the marginal dollar.”
MUX
PX
10.8.33
6.67
5.00
3.331.67
0
For PY = $5, the MUY per dollar spent on good Y is…
Utility YQy
1
2
4
8
5
6
7
3
TUy
60
90
110
120
128128
120
100
60
30
20
10
8 0
- 8
- 20
MUy
MUY
PY
12
6
4
21.6
0
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Utility X
Qx
1
2
4
8
5
6
7
3
TUx
30
55
75
90
100105
105
100
20
15
10
5
0
-5
30
25
MUx For PX = $3, the MUX per dollar spent on good X is…
Given the budget constraint, individuals will attempt to gain the maximum utility for each additional dollar spent,“the marginal dollar.”
MUX
PX
10.8.33
6.67
5.00
3.331.67
0
For PY = $5, the MUY per dollar spent on good Y is…
Utility YQy
1
2
4
8
5
6
7
3
TUy
60
90
110
120
128128
120
100
60
30
20
10
8 0
- 8
- 20
MUy
MUY
PY
12
6
4
21.6
0
If the objective isto maximize utilitygiven prices, preferences, andbudget, spend eachadditional $ on thegood that yieldsthe greater utility for that expenditure.
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MUX
PX
10.8.33
6.67
5.00
3.331.67
0
MUY
PY
12
6
4
21.6
0
$5$3
$3
$3 $5
Continue to maximize the MU per $ spent until the budget of $30 has been spent.
$3 $5
$3
MUX
PX
<MUY
PY
, BUY Y !
Constrained Optimization
, BUY X !MUX
PX
>MUY
PY
if
if
Budget = $30
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Constrained Optimization
If MUx/Px > MUy/Py then an additional dollar spent on good X increases TU by more than an additional dollar spent on good Y.
If MUx/Px < MUy/Py then an additional dollar spent on good X increases TU by less than an additional dollar spent on good Y.
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Constrained Optimization
When the entire budget is spent, if MUx/Px > MUy/Py, then one should buy more X and less Y.
When the entire budget is spent, if MUx/Px < MUy/Py, then one should buy less X and more Y.
When the entire budget is spent, if MUx/Px = MUy/Py, then one has “maximized utility subject to the budget constraint”.
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Constrained Optimization
MUx/Px = MUy/Py
is an equilibrium condition
for individual choice.
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PX X + PY Y = I = MUX
PX
MUY
PY
subject to the constraint:
insures the individual has maximized their total utility andhas not spent more on the two goods than their budget.
This model can be expanded to include as many goods asnecessary:
= MUX
PX
MUY
PY
= = . . . . . . . = MUZ
PZ
MUN
PN
subject to
PX X + PY Y + Pz Z + . . . + PN N = I
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Constructing a Demand Curve
From the information of utility maximization, given prices and income, one can construct a demand curve for a good by varying the price of that good, with other information held constant (ceteris paribus).
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MUX
PX
10.8.33
6.67
5.00
3.331.67
0
MUYPY
12
6
4
21.6
0
Given preferences, prices [PX = $3, PY = $5] and budget [$30], the individual’s choices were:
$5$3
$3
$3 $5
$3 $5
$3
Five units of X and 3 units of Y were purchased
Graphically…
1 2 3 4 5 6 7 QX/t
PX
1
2
3
4
5
PX =
5
This point lies on thedemand curve for good X.
.
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MUY
PY
12
6
4
21.6
0
1 2 3 4 5 6 7 QX/ut
PX
1
2
3
4
5
.
MUX
PX
10.8.33
6.67
5.00
3.331.67
0
[$3]
Now, suppose the price of X [PX ] increases to $5. The MUx/Px falls, and now at the combination of 5 Xand 3 Y, the MUx/Px < MUy/Py. There is now an incentiveto buy less X and more Y.MUX
PX
6 5
4
3
2 1
0
[$5]Choices about spending the $30 are now:
$5
$5$5
$5
$5$5
= MUX
PX
MUY
PY
At PX = $5,ceteris paribus,3 units of X arepurchased.
.DemandThat
portionof demandbetween $3 and $5is mapped!
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By continuing to change the price of good X (and holding all other variables constant) the rest of the demand for good X can be mapped.
All price and quantity combinations on the demand curve for X are equilibrium points, or points of maximized utility for the consumer.
Demand
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1 2 3 4 5 6 7 QX/t
PX
1
2
3
4
5
By changing the price of the good and holding allOther variables constant, the demand for the goodcan be mapped.
Demand
The demand function is a schedule of the quantities that individuals are willing and able to buy at a
schedule of prices during a specific period of time, ceteris paribus.
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1 2 3 4 5 6 7 QX/t
PX
1
2
3
4
5
Demand
The demand function has a negative slope because of theincome and substitution effects.
Income effect: As the price of a good that you buy increasesand money income is held constant, your real income decreasesand you can not affordto buy as much as youcould before.Substitution effect: As
the price of one good risesrelative to the prices ofother goods, you will substitute the goodthat is relatively cheaperfor the good that is relatively more expensive.
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Elasticity
Elasticity - measure of responsiveness
Measures how much a dependent variable changes due to a change in an independent variable
Elasticity = %Δ X / %Δ Y • Elasticity can be computed for any two
related variables
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Price Elasticity of Demand
Can be computed at a point on a demand function or as an average [arc] between two points on a demand function
ep, are common symbols used to represent price elasticity of demand
Price elasticity of demand, ε, is related to revenue• “How will a change in price effect the total
revenue?” is an important question.
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Price Elasticity of Demand
The “law of demand” tells us that as the price of a good increases the quantity that will be bought decreases but does not tell us by how much.
The price elasticity of demand, ε, is a measure of that information
“If you change price by 5%, by what percent will the quantity purchased change?
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ε % Q
% P
At a point on a demand function this
can be calculated by:
ε =
Q2 - Q1
Q1
P2 - P1
P1
Q2 - Q1 = Q
P2 - P1 = P=
QQ1
PP1
Price Elasticity of Demand
=(ΔQ/ΔP) x (P1/Q1)
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For a simple demand function: Q = 10 - 1P
price quantity ep TotalRevenue
$0 10
$1 9
$2 8
$3 7
$4 6
$5 5
$6 4
$7 3
$8 2
$9 1
$10 0
using our formula,
ε =Q P1
Q1*P
ε =Q P1
Q1P *
the slope is -1,
(-1)
price is 7
7
at a price of $7, Q = 3
3= -2.3
-2.3
Calculate ε at P = $9Q = 1
ε = (-1) 91
= -9
Calculate ε for all other price and quantity
combinations. -9
0-.11
-.25-.43-.67
-1.
-1.5
-4.
undefined
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For a simple demand function: Q = 10 - 1P
price quantity ep TotalRevenue
$0 10
$1 9
$2 8
$3 7
$4 6
$5 5
$6 4
$7 3
$8 2
$9 1
$10 0
-2.3
-9
0-.11
-.25-.43-.67
-1.-1.5
-4.
undefined
Notice that at higher prices the absolute value of the price
elasticity of demand, ε is greater.
Total revenue is price times quantity; TR = PQ.0
9162124
2524
211690
Where the total revenue [TR]is a maximum, εis equal
to 1
In the range where ε< 1, [less than 1 or “inelastic”], TR increases as
price increases, TR decreases as Pdecreases.
In the range where ε> 1, [greater than 1 or “elastic”], TR
decreases as price increases, TR increases as P decreases.
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Q/t
Price
10
10
ε = -15
5
|ε | > 1 [elastic]
The top “half” of the demand function is elastic.
|ε | < 1inelastic
The bottom “half” of the demand function is inelastic.
Graphing Q = 10 - P,
TR
TR is a maximumwhere ep is -1 or TR’s
slope = 0When ε is -1 TR is a maximum.
When |ε | > 1 [elastic], TR and P move in opposite directions. (P has
a negative slope, TR a positive slope.)
When |ε | < 1 [inelastic], TR and P move in the same direction. (P and TR
both have a negative slope.)
Arc or average ε is the average elasticity between two point [or prices]
pointε is the elasticity at a point or price.
Price elasticity of demand describeshow responsive buyers are to change
in the price of the good. The more “elastic,” the more responsive to P.
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Use of Price Elasticity
Ruffin and Gregory [Principles of Economics, Addison-Wesley, 1997, p 101] report that:• short run εof gasoline is = .15 (inelastic)• long run εof gasoline is = .78 (inelastic)• short run εof electricity is = . 13
(inelastic)• long run εof electricity is = 1.89 (elastic)
Why is the long run elasticity greater than short run?
What are the determinants of elasticity?
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Determinants of Price Elasticity
Availability of substitutes• greater availability of substitutes makes a good
more elastic Proportion of budget expended on good
• higher proportion – more elastic Time to adjust to the price changes
• longer time period means more adjustments possible and increases elasticity
Price elasticity for “brands” tends to be more elastic than for the category
