Econ 3102  Competitive Equilibrium Notes
John Seliski
6/19/11
1
The Economic Agents in our Economy
1.1
Consumers
Suppose that there exists a
representative consumer
in this economy whose preferences over consumption
(
C
) and leisure (
‘
) are represented by the utility function
U
:
R
+
°
°
0
;
°
h
±
!
R
Assume that:
±
C
and
‘
are both
normal
goods
±
U
is continuous, twicedi/erentiable,
strictly
concave (consumers prefer variety), and
U
C
>
0
;
U
‘
>
0
U
CC
<
0
;
U
‘‘
<
0
In words, the marginal utility of consumption and leisure are assumed to be positive and strictly
decreasing.
That is, there is diminishing returns to consuming more of the consumption good and
leisure. From the above, we can also conclude that
U
is
strictly increasing
(consumers prefer more to
less) in both its arguments.
±
U
satis°es the Inada conditions. Namely,
lim
C
!
0
U
C
= +
1
;
lim
‘
!
0
U
‘
= +
1
Both the factor and output markets are assumed to be
competitive
, so the representative consumer will be a
pricetaker. That is, the representative consumer will take the price of the consumption good and wages as
given. Since the consumer is representative and our model is a closed economy, all pro°ts generated by °rms
will be distributed back to the consumer (think of our circular ±ow diagram). In other words, the consumer
owns
the °rms but does not have a say in the °rms²decision making process. There also exists a government
in this economy that uses lumpsum taxation on the consumer to °nance government spending. Therefore,
the consumer²s budget constraint in this economy (letting the consumption good be the numeraire) is:
C
²
wn
+ (
°
³
T
)
where
C
:
Consumption good
n
:
Labor units supplied to the factor market
w
:
Real wage rate (consumption units per labor unit)
°
:
Firms²pro°ts
T
:
Lumpsum taxes
1
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Thus, given wages
w
, °rms²pro°ts
°
, and taxes
T
, the problem of the representative consumer is:
max
C;‘;n
U
(
C; ‘
)
s.t.
C
²
wn
+ (
°
³
T
)
‘
+
n
²
°
h
C
´
0
; ‘
´
0
; n
´
0
(Nonnegativity constraints)
where
°
h >
0
is a parameter that determines the upper bound on how much labor/leisure the consumer
can feasibly choose.
For example, I cannot feasibly work more than 24 hours in one day.
Similarly, I
cannot enjoy more than 24 hours of leisure time in one day. Note that this problem would make no sense if
w
°
h
+ (
°
³
T
)
<
0
, since that would imply that consumption would always be negative. Also, I assume that
wages are strictly positive,
w >
0
, which will always be true in an equilibrium. Given our assumptions on
U
, we can simplify this problem.
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 Summer '10
 ECON
 Economics, representative, representative consumer, market clearing condition

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