Transcript Demand, Utility and Expenditure

Demand, Utility and
Expenditure
Chapter 5, Frank and Bernanke
Key Concepts
Law of Demand – other things equal, when price
goes up, the quantity demanded goes down.
Utility maximization – consumers determine the
quantity demanded of each of two goods by equating
their marginal utility per dollar.
Demand and expenditure – the Law of Demand
makes no prediction on the relation of price and
expenditure. When price goes up, expenditure may
go up, go down or stay the same.
Expenditure = price times quantity purchased.
Elasticity = responsiveness of quantity demanded to
price changes.
Utility Maximization
• Consumers apply the equimarginal
principle and find the point at which the
marginal benefit of spending another dollar
on Good X equals the marginal cost of
NOT spending another dollar on Good Y.
MUx
MUy
Px
Py
• This equation is the “rational spending
rule”
Application of the Rational Spending Rule
• Suppose good X (“pizza”) is at a price of $10 a
pie, and good Y (“concert tickets”) is at a price of
$30 a concert.
• You have a budget of $ 130 for entertainment,
and want to rationally allocate it among the two
goods.
• You know that the marginal utility of either good
declines with the amount consumed (though
TOTAL utility continues to increase)
• You know your utility tables – see the next slide
Units
of X
1
T.Ux M.Ux M.Ux Units T.Uy M.Uy M.Uy
per $ of Y
per $
70
1
140
2
110
2
220
3
140
3
280
4
161
4
322
5
179
5
358
6
195
6
390
7
208
7
416
8
220
8
440
• Finding marginal utility:
– MU of X = change in total utility from X with 1 more X
– MU of Y = change in total utility from Y with 1 more Y
• Finding marginal utility per dollar:
– MU per dollar of X is the MU of X divided by the price
of X
– And likewise for the MU per dollar of Y.
• The last is the crucial step – we are changing
our choices dollar by dollar, not unit by unit.
Units
of X
1
T.Ux M.Ux M.Ux Units T.Uy M.Uy M.Uy
per $ of Y
per $
70
70
1
140 140
2
110
40
2
220
80
3
140
30
3
280
60
4
161
21
4
322
42
5
179
18
5
358
36
6
195
16
6
390
32
7
208
13
7
416
26
8
220
12
8
440
24
Units
of X
1
T.Ux M.Ux M.Ux Units T.Uy M.Uy M.Uy
per $ of Y
per $
70
70
7.0
1
140 140 4.67
2
110
40
4.0
2
220
80
2.67
3
140
30
3.0
3
280
60
2.00
4
161
21
2.1
4
322
42
1.4
5
179
18
1.8
5
358
36
1.3
6
195
16
1.6
6
390
32
1.07
7
208
13
1.3
7
416
26
0.87
8
220
12
1.2
8
440
24
0.80
•
•
•
•
Using the table
The problem can’t be easily solved by
considering all possible choices.
Reduce it to the simpler problem: what do
I buy next?
Since for the first unit MU of x per $ is 7.0,
and MU of y per $ is 4.67, buy X first.
Now compare the MU per dollar of the
second unit of X ( 4.0) with the MU per
dollar of the first unit of Y (4.67). Buy Y
next.
Units
of X
1
T.Ux M.Ux M.Ux Units T.Uy M.Uy M.Uy
per $ of Y
per $
70
70
7.0
1
140 140 4.67
2
110
40
4.0
2
220
80
2.67
3
140
30
3.0
3
280
60
2.00
4
161
21
2.1
4
322
42
1.4
5
179
18
1.8
5
358
36
1.3
What do you buy next??
Remember, go for the highest MU per dollar.
Units
of X
1
T.Ux M.Ux M.Ux Units T.Uy M.Uy M.Uy
per $ of Y
per $
70
70
7.0
1
140 140 4.67
2
110
40
4.0
2
220
80
2.67
3
140
30
3.0
3
280
60
2.00
4
161
21
2.1
4
322
42
1.4
5
179
18
1.8
5
358
36
1.3
Remember to ask yourself how much you have left –
So far, you’ve spent $ 30 on X and $ 30 on Y,
Leaving you with $ 70 from the budget of $ 130.
Units
of X
1
T.Ux M.Ux M.Ux Units T.Uy M.Uy M.Uy
per $ of Y
per $
70
70
7.0
1
140 140 4.67
2
110
40
4.0
2
220
80
2.67
3
140
30
3.0
3
280
60
2.00
4
161
21
2.1
4
322
42
1.4
5
179
18
1.8
5
358
36
1.3
6
195
16
1.6
6
390
32
1.07
7
208
13
1.3
7
416
26
0.87
8
220
12
1.2
8
440
24
0.80
Checking the rational spending rule
At X = 4 and Y = 3, we’ve exhausted our
budget ($ 40 on X, $ 90 on Y)
Go back to the table to calculate our utility
score:
Utility of 4 X = 161 utils
Utility of 3 Y = 280 utils
Total utility = 441 utils
• Could we do better? If we bought one less Y, we
would have $ 30 more and could buy 3 more X :
• The new consumption bundle is 2 Y and 7 X
– Utility of 7 X = 208 utils
– Utility of 2 Y = 220 utils
– Total utility = 428 utils (less than 441 utils)
• Try buying one more Y and 3 less X:
– Utility of 1 X = 70 utils
– Utility of 4 Y = 322 utils
– Total utility = 392 utils
Utility maximization, functionally
speaking
• A common economic model for a utility
function is the logarithmic function:
• TUx = 100 ln X
• TUy = 200 ln Y
(it wasn’t an accident that the tables just used are almost
the same as you would get from computing 100 ln 2,
100 ln 3, etc. The only slight difference is that ln 1 = 0, so
the table was shifted back 1 level to avoid TU of 1 = 0).
Marginal Utility, functionally speaking
• It can be shown that if TUx = A ln X,
MUx = A divided by X
Full proof requires calculus, but you should be able to see that the
formula works by a few examples:
If TUx = 100 ln X, what is the marginal utility between 50 and 51
units of X?
MUx = 100 ln 51 minus 100 ln 50
MUx = 393.1826 minus 391.2023 = 1.9803
Using the formula for marginal utility,
MUx = 100 / X = 100 / 50.5 = 1.9802
[should you divide by 50 or 51?
Dividing by 50 gets MUx = 2, by 51 gets MUx = 1.96
The difference is never too important in practice]
Rational spending, functionally speaking
• Let TUx = 150 ln X and TUy = 300 ln Y
• Then MUx = 100 / X and MUy = 300 / Y
• Hence the rational spending rule is:
MUx / Px = MUy / Py
or 150 / Px X = 300 Py Y
or Py Y = 2 Px X