Download Demand and Utility

Demand and Utility
Diminishing Marginal Utility:
The more you have of a good, the less an
additional unit of the good is worth to you.
Utility Maximization Condition:
The additional satisfaction obtained from
purchasing another dollar’s worth of a good
should be equal for all goods.
Example
apples
plums
price = $0.50
price = $0.25
MU = 4
MU = 3
How can you tell your utility isn’t maximized?
$1 worth of apples
$1 worth of plums
4u
3u
3u
4u
3u
3u
8 units of satisfaction
12 units of satisfaction
If you spend $1 less on apples, you would lose
2 apples.
4u
4u
That would reduce your satisfaction by 8 units.
8
If you spend $1 more on plums, you would gain
4 plums.
3u
3u
3u
3u
That would increase your satisfaction by 12 units.
12
So,
8
+
12
=
increase of 4
units of
satisfaction.
If before making the adjustment, your total
utility was 20 units, your total utility after
the adjustment would be 20 + 4 = 24.
Once again: In order to maximize utility, the
additional satisfaction from another dollar’s
worth of a good should be equal for all goods.
In other words: The marginal utility per
dollar should be equal for all goods.
MUa / Pa = MUb / Pb
Let’s apply the formula
MUa / Pa = MUb / Pb
directly to our example.
For apples,
For plums,
MU / P = 4/.50 = 8.
MU / P = 3/.25 = 12.
Since 8 does not equal 12, utility was not
maximized.
As you eat fewer apples, the value to you of
an apple increases, perhaps from 4 to 5.
As you eat more plums, the value to you of a
plum decreases, perhaps from 3 to 2.5.
Then for apples,
MU / P = 5/.50 = 10.
and for plums,
MU / P = 2.5/.25 = 10.
Now the marginal utilities per dollar are
equal for the two goods and no adjustments
will make you any happier.
The Connection between
Demand and Utility
Instead of thinking in terms of utils, let’s
think in terms of dollars.
Suppose the purchase of one unit of a good
gives you $10 worth of satisfaction.
In other words, the marginal utility of that
first unit of the good is $10.
Then you would be willing to pay up to $10
for it.
If a second unit of the good contributes $8
more of satisfaction, the marginal utility of
your second unit is $8 and you would be
willing to pay up to $8 for it.
If a third unit of the good contributes $6 more
of satisfaction, the marginal utility of your
third unit is $6 and you would be willing to
pay up to $6 for it.
Remember that the demand curve tells you
what people are willing to pay for various
amounts of a good or, equivalently, how
many units of a good they are willing to
purchase at various prices.
So, since we just found that the marginal
utility tells us what we are willing to pay for
a good, the marginal utility provides us with
information that we can use to determine
our demand curve.
In our example, we had the following:
Number of units
purchased (Q)
Price you are
willing to pay (P)
Marginal
Utility
0
-
-
1
10
10
2
8
8
3
6
6
If we graph that information,
we get our demand curve.
Number of units Price you are
purchased (Q)
willing to pay (P)
Marginal
Utility
price
0
-
-
10
1
10
10
2
8
8
3
6
6
8
6
Demand
1
2
3
quantity
Notice that by adding our MU values we can determine
our total utility at different consumption levels.
Number of units Price you are
purchased (Q)
willing to pay (P)
Marginal
Utility
Total
Utility
0
-
-
0
1
10
10
10
2
8
8
18
3
6
6
24
Let’s combine our new information
with the ideas we had earlier.
Recall that we maximize our utility where
the marginal utility per dollar is equal for all
goods.
MU a
Pa

MU b
Pb
.
In addition, we concluded that we increase our
purchases of a good as long as our marginal
utility is greater than the good’s price.
Recall marginal utility drops as we consume
more and more of a good.
So we would consume a good until we get to
the point where our marginal utility is equal to
its price.
That is, MU = P
or MU/P = 1.
So, we purchase up to the point where
MU/P is equal for all goods and MU/P=1
for each good.
So to maximize our utility, we consume
such that our marginal utility per dollar is
equal for all goods, and we do that up to the
point where we either run out of money or
MU a
Pa

MU b
Pb
 1.
Consumer Surplus
the difference between what you have to pay
for a good and what you would have been
willing to pay.
Example
Suppose that you would have been willing to
pay $25 for a pair of jeans. When you get to
the store, you find them selling for $20. What
is your consumer surplus?
consumer surplus = 25 - 20 = 5
In terms of a supply and demand graph, the
consumer surplus is the area under the demand
curve and above the price.
Graphing Consumer Surplus
price
supply
demand
quantity
Graphing Consumer Surplus
price
supply
P*
demand
Q*
quantity
Graphing Consumer Surplus
price
supply
P*
demand
Q*
quantity
Example: Using the graph below,
determine the consumer surplus.
Price
10
supply
consumer surplus
= (1/2)(base)(height)
= (1/2) (8)(4)
= 16
P* = 6
demand
0
Q* = 8
quantity
We can also calculate consumer surplus using the
information in our earlier utility table.
Number of units Price you are
Marginal Total
purchased (Q)
willing to pay (P) Utility
Utility
0
-
-
0
1
10
10
10
2
8
8
18
3
6
6
24
Remember that consumer surplus is the difference between
what you are willing to pay for a good (or what it’s worth to
you) and what you have to pay for it.
Number of units Price you are
Marginal Total
purchased (Q)
willing to pay (P) Utility
Utility
0
-
-
0
1
10
10
10
2
8
8
18
3
6
6
24
What the good is worth to you is its TU, which is the sum of the
MUs, or equivalently the sum of the prices that you would be willing
to pay for each unit of a good.
Suppose the price is $6. You purchase three units of the good, and
their total worth to you is TU=$24. But since you only have to pay
6x3=$18, your consumer surplus would be 24-18=$6.