Download TOPIC V: UTILITY AND DEMAND I. Preferences: A. Total utility (TU

TOPIC V: UTILITY AND DEMAND
I.
Preferences:
A.
B.
Total utility (TU)
Marginal utility (MU)
II.
Budget Constraint
III.
Maximizing Utility
A.
B.
Law of diminishing marginal utility
Equal marginal rule (understand the common sense meaning of this rule): MU x/P x =
MUy/Py
CONCEPTS AND PRINCIPLES
The theory we develop for investigating how consumers make “purchasing” decisions is called
optimization subject to a budget constraint. The basic idea is to choose the combination of goods that
would maximize utility (satisfaction, happiness, etc), constrained by what one can afford to purchase.
Thus, the theory has two basic components: (1) preferences and (2) a budget.
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CONSUMER PREFERENCES
Basically we assume that individuals know what is available and can rank “bundles” of goods in terms
of a ranking system we will call utility.
Don’t think of what the goods will cost to purchase, just what satisfaction (utility) they will bring. Two
fundamental measurements we consider are: (1) total utility and (2) marginal utility.
Total Utility (TU) refers to the total utility (satisfaction) one would receive from a bundle of goods.
For example: 2 bananas, 1 apple, 3 sodas, 4 bags of chips, and 2 hotdogs might yield me 1,200 “utils.”
Marginal Utility (MU) refers to the additional utility one would receive from an additional unit of
some good. It is measured as “the change in TU”/”the change in number of units of the good” MUX = )TU X /)QX.
For example what if my utility from adding 1 soda to the example above was an extra 50 utils - my MU
from that soda would be 50 and my total utility would now be 1,250.
A simple example:
Q sodas
0
1
2
3
4
5
TU sodas
0
100
300
380
430
440
MU sodas
100
200
80
50
10
A standard simplifying assumption that we often make is that as more and more units of a good are
consumed, the MU of additional units will eventually decrease. In the example above, this occurs at the
3rd soda and continues after that. We call this the assumption of “diminishing marginal utility.”
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BUDGET CONSTRAINT
The idea here is to determine what is feasible to purchase given existing market prices and existing
income. Consider the following example. Assume I have $50 to spend on sodas and pizza. Assume the
price of a soda is $1 and the price of a piece of pizza is $2. What combinations of soda and pizza
would be available to me? Consider the graph shown below.
Any combination of soda and pizza that would cost $50 would be on my budget constraint - for
example, 50 sodas and no pizza, 25 pizza slices and no soda, 30 soda and 10 pizza slices, etc.
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MAXIMIZING UTILITY
Maximizing Utility means finding that combination of goods that is affordable and yields the highest
total utility. How do we determine that combination? One way is to understand something the
“marginal utility per dollar spent on a good” and something called the “equal marginal principle.”
The marginal utility per dollar spent on a good X = MUX /PX
The equal marginal principle states that one should buy the combination of goods where the marginal
utility per dollar spent on each good is the same.
Consider the following example.
Assume there are two goods, X and Y. Also, assume that you are considering buying a combination of
those goods where MUX/PX > MU y/Py. If this is true, this means that you are getting more additional
utility per dollar spent on X than on Y. Thus, if you took a dollar from the consumption of Y and
moved it to the consumption of X, your total utility would increase.
Note the following. If you are buying X and Y at a point where you face diminishing MU, then as you
increase the amount of X you buy, the MUX decreases and as you buy less of Y, the MU of Y
increases. This tends to move you in the direction of MUX becoming equal to the MU Y.
Similarly, what if you are considering buying a combination of those goods where MU X/P X <
MUy/Py. If this is true, this means that you are getting less additional utility per dollar spent on X than
on Y. Thus, if you took a dollar from the consumption of X and moved it to the consumption of Y,
your total utility would increase.
Note the following. If you are buying X and Y at a point where you face diminishing MU, then as you
decrease the amount of X you buy, the MUX increases and as you buy more of Y, the MU of Y
decreases. This tends to move you in the direction of MU X becoming equal to the MU Y.
This leads us to the next “principle” - the equal marginal utility per dollar principle.
Principle 9 - Total Utility is maximized when a combination of goods (X,Y,Z) is chosen that is “on” the
budget constraint and one’s preferences are such that MUx/Px = MUy/Py = MUz/Pz.
In applying this principle, the idea is that individuals will make decisions in line with moving toward
combinations of goods where marginal utility per dollar is equal across all goods. In practice problems
(or even in the real world) the key idea is that individuals consider the marginal value per dollar spent
and make purchasing decisions that reflect the best value for the dollar.
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Applying the Marginal Utility per dollar principle
Assume you have $100/month to spend.
Assume the table below displays your preferences for three goods in terms of the marginal utility you
receive from each good.
MU is the marginal utility from each extra unit of coke, burgers, or movies.
MU/$ is the marginal utility per dollar received from expenditures on each extra unit of these goods.
Pcoke = $.50
Pburger = $2.00
Pmovie = $5.00
Q
MUC
MU/$
Q
MUB
MU/$
Q
MUM
MU/$
1
30
60
1
400
200
1
2000
400
2
30
60
2
200
100
2
1800
360
3
30
60
3
140
70
3
1300
260
4
30
60
4
50
25
4
1000
200
5
30
60
5
20
10
5
800
160
6
30
60
6
10
5
6
200
40
7
....
....
....
30
....
....
....
60
....
....
....
7
0
0
7
100
20
Using the marginal utility per dollar principle, we evaluate the extra benefit from extra
expenditures on each good. Start with the good that gives the greatest MU/$. “Purchase” that
good, subtract the price from income, go to the next good (or next unit of the same good) with the
greatest MU/$, purchase that good, subtract the price from income, continue the process until all
income is spent.
This is the “essence” of the marginal utility per dollar principle.
When we do this, we can show that the optimal purchase of these good would be:
Quantities:
QC = 138
Expenditures: $ 69
QB = 3
$6
(138x$.50)
QM = 5
$25
(3x$2.00)
Total= $100
(5x$5)
We can now show how the choices above relate to the demand for each of these goods.
The information we gain from solving the optimization problem is depicted in the graphs below.
V5
Solving the problem shows us one point on each demand curve.
This figure shows us what we have learned about the demand for
coke when the Pburger= $2, Pmovie= $5, and the person has $100
to spend.
We really only know about the single point shown.
This figure shows us what we have learned about the
demand for burgers when the Pcoke= $.50, Pmovie= $5,
and the person has $100 to spend.
We really only know about the single point shown.
This figure shows us what we have learned about the demand for
movies when the Pcoke= $.50, Pburger= $5, and the person has
$100 to spend.
We really only know about the single point shown.
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Now, consider what happens when the price of one of the goods changes.
Let:
Pmovie increase to $20.
Pcoke = $.50
Pburger = $2.00
Pmovie = $20.00
Q
MUC
MU/$
Q
MUB
MU/$
Q
MUM
MU/$
1
30
60
1
400
200
1
2000
100
2
30
60
2
200
100
2
1800
90
3
30
60
3
140
70
3
1300
65
4
30
60
4
50
25
4
1000
50
5
30
60
5
20
10
5
800
40
6
30
60
6
10
5
6
200
10
7
30
60
7
0
0
7
100
5
Using the equal marginal principle, we evaluate the extra benefit from extra expenditures on each
good.
When we do this, we can show that the opimal combination of these goods would be:
Quantities:
Expenditures: $ 34
QC = 68
QB = 3
$6
QM = 3
$60
Note that we now have:
1) a new demand curve for cokes
2) a new demand curve for burgers
3) and we have a new point on the demand curve for movies.
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The new demand curve shows a smaller quantity demanded
(than before) at the price of $.50. In other words, when the
price of movies increased - the demand for cokes decreased making cokes and movies complements.
The new demand curve shows the same quantity
demanded (than before) at the price of $2.00. In other
words, when the price of movies increased - the demand
for burgers did not change - making movies and burgers
neither substitutes nor complements for this particular
example.
Notice that in the case of the demand for movies - we are
moving along the original demand curve. That is, as the
price of movies increased, the consumer moved up along
the original demand (reducing the quantity demanded).
V8
QUESTIONS
Short Answer
1.
Complete the following table:
Qx
0
1
2
3
4
2.
TUx
_____
_____
_____
_____
_____
MUx
_____
1000
800
300
80
Complete the following table:
Qx
0
1
2
3
4
TUx
_____
_____
_____
_____
_____
MUx
_____
0
100
0
80
Can you think of any goods that might lead to such a strange set of preferences?
3.
If total utility from good X were to reach a maximum and then start to decline for additional
units of good X, what would this imply about the marginal utility from additional units of good
X? Make up a numerical example to show this.
4.
Assume Z is a complement to good X, and Y is a substitute for good X. Also assume Joe
Brown is currently maximizing utility with respect to consumption of each of these goods.
What might we expect to happen to Joe Brown' s purchases of X, Y, and Z if the price of good
X falls? Why?
5.
Assume Z is a complement to good X, and Y is a substitute for good X. Also assume Joe
Brown is currently maximizing utility with respect to consumption of each of these goods.
What might we expect to happen to Joe Brown' s purchases of X, Y, and Z if Joe' s income
increases from $1,000 per month to $1,500 per month? Why? State precisely what assumptions
your answer depends upon.
V9
Multiple Choice
1.
The equal marginal principle of consumer behavior suggests that consumers will make purchase
decisions based on comparing the:
A.
B.
C.
D.
2.
total benefits of all goods.
additional utility per dollar of each good.
rate of diminishing total utility from successive units of a given good.
all of the above
Use the information below to answer the next question.
Qx
MUx
1
2
3
50
20
0
The total utility from 3 units of good X would be:
A.
B.
C.
D.
3.
Assume: (1) that the price of cola is $2 and the price of tacos is $1; (2) that at your current
consumption levels the MU of cola is 50 and the MU of tacos is 30; and (3) that you are
spending all of your income. The equal marginal principle would imply that you should:
A.
B.
4.
0
20
70
none of the above
buy additional units of tacos and less cola.
buy additional units of cola and less tacos.
Think through this problem carefully - use equal marginal value per dollar. Assume you
have $5,000 to invest. Assume stock in IBM costs $50 per share and pays a yearly dividend of
$10 per unit of stock. Assume stock in TI costs $20 per share and pays a yearly dividend of $5
per unit of stock. (Ignoring all other factors) to maximize your investment return you should:
A.
B.
C.
D.
invest $5,000 in IBM.
invest $5,000 in TI.
split your investment so that the ratio of your shares of TI relative to IBM is 2.5 to 1.
split your investment so that the ratio of your shares of IBM relative to TI is 2.5 to 1.
V 10
5.
Assume you are throwing a party and have $150 to spend. You are considering buying 10 cases
of coke and 10 boxes of chips. The cost of coke is $5 per case and the cost of chips is $10 per
box. The MU from the 10th case of coke is 132 and the MU from the 10th box of chips is 200.
To maximize the value (utility) from your purchases, you should - NOTE: assume you can buy
partial cases:
A.
B.
C.
D.
6.
The equal marginal principle implies that:
A.
B.
C.
D.
7.
demand curves will be downward sloping (the law of demand).
consumers will consume all goods at a point where the extra satisfaction from
additional units of each good is the same.
the marginal utility per dollar spent on each good is equal.
all of the above
If the consumer is maximizing utility and the marginal utility per dollar spent on good X is 4
and the marginal utility per dollar spent on good Y is 2, we know that
A.
B.
C.
D.
8.
buy more coke and less chips.
buy more chips and less coke.
buy more coke and more chips.
the answer is indeterminate without knowing the total utility received from coke and
chips.
the price of Y must be twice the price of X.
the MU of X must equal that of Y.
total utility can be increased by increasing the consumption of X and decreasing the
consumption of Y.
all of the above must be true
To maximize utility:
A.
B.
C.
D.
the marginal utility is the same for all goods.
the marginal utility per dollar spent is the same for all goods.
the marginal cost per dollar spent is the same for all goods.
all of the above
V 11
Use the following table for the next two questions:
9.
Number of
Colas
Total
Utility
1
Total
Utility
6
1
10
2
10
2
14
3
13
3
15
4
15
4
15
5
16
5
7
MU/$
MUB
MU/$
Which of the two goods demonstrates the law of diminishing marginal utility?
A.
B.
C.
D.
10.
Number
of
Burgers
MUC
cola only
burgers only
both cola and burgers
neither cola nor burgers
If the consumer has only $8 to spend, the price of cola is $1 each, and burgers cost $2 each,
what would be the optimal consumption of both goods for this consumer?
A.
B.
C.
D.
4 colas and
5 colas and
2 colas and
4 colas and
2 burgers
1 burger
3 burgers
4 burgers
V 12