Economics 5
D. Richards
Principles of Microeconomics
Fall, 2007
Third Problem Set
**Note: For some reason, my computer changed some of the ‘>’s to ‘
’s and it wouldn’t let me
change them back and I don’t know how to fix it**
1.
The market for widgets has fifty (50) identical consumers.
Each chooses that combination of
widgets,
X,
and a composite commodity,
Y
, (representing an agglomeration of all other goods) that
maximizes his or her budget subject to a budget constraint. The marginal utility of
all
other goods
taken as a collection is constant.
That is, within the range of changes considered here, the amount a
consumer spends on these other commodities as a group does not affect the marginal utility of
Y
because so little is spent on any one commodity that the overall effect is insignificant.
To make this
concept operational, assume that the marginal utility of
Y
is fixed at 20. The same does
not
hold for
the marginal utility for widgets or
X.
This is just one good—not a collection—and it is assumed to
obey the natural assumption of diminishing marginal utility.
Specifically, the marginal utility of
X
is
assumed to obey the following equation:
MU
X
= 400 – 20X.
Each consumer has a budget of $10,000
(think of this as an amount per month) and the price index for
Y
is given and is equal to $10.
a.
Use the information you have and the equimarginal condition for utility maximization to determine
how many widgets an individual widget consumer will buy at a price of:
P
X
= $200; P
X
= $180; P
X
=
$160; P
X
= $140; P
X
= $120; P
X
= $100; P
X
= $80; P
X
= $60; P
X
= $40; P
X
= $20; and P
X
= 0.
b.
Multiply the amount purchased by one widget consumer at each price by 50 (the total number
of such consumers) to obtain the market demand at each of the prices noted in part (a).
MUx/P=2>MU=2P>40020X=2P
Px
X (a)
50X (b)
200
0
0
180
20
1000
160
40
2000
140
60
3000
120
80
4000
100
100
5000
80
120
6000
60
140
7000
40
160
8000
20
180
9000
0
200
10000
c.
Show that the market (inverse) demand curve just obtained may be represented by the equation:
P = 200 – Q/5. MU=2p, (X=Q/50), 2P=40020Q/50>2P=4002Q/5>P=200Q/5
2.
An industry is comprised of 60 firms each of which has the following Total Cost function:
TC = 15
q
+ 7.5
q
2
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a.
One way to define Marginal Cost is as the
increase
in Total Cost when output is increased by
one unit.
That is, one definition of Marginal Cost for the typical firm is as follows:
MC
A
= 15(
q
+ 1) + 7.5(
q
+ 1)
2
– [15
q
+ 7.5
q
2
]
Show that this measure of Marginal Cost yields a value of MC
A
= 15
q
+ 22.5.
MC
A
=15q+15+7.5q
2
+15q+7.515q7.5q
2
15q+15q15q+7.5+15
15q+22.5
b.
A second way to define Marginal Cost is as the
decrease
in Total Cost when output is reduced
by one unit.
This alternative definition of MC thus is:
MC
B
= 15
q
+ 7.5
q
2
– [15(
q
1) + 7.5(
q
 1)
2
]
Show that this measure of Marginal Cost yields a value of MC
B
= 15
q
+ 7.5
MC
B
=15q+7.5q215q+157.5
2
15q7.5
15q+7.5
c.
Since the two measures of marginal cost are not identical, the best way to proceed is to take an
average of MC
A
and MC
B
.
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 Spring '08
 RICHARDS
 Economics, Microeconomics, Supply And Demand, Ben Williams

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