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CB046/Starr
LN0401˙2011
March 24, 2011
10:21
1
Lecture Notes for April 1, 2011
An elementary general equilibrium model – The Robinson Crusoe econ
omy
Robinson Crusoe is endowed with 168 manhours per week. On his island
there is only one production activity, harvesting oysters from an oyster bed,
and only one input to this production activity, Robinson’s labor.
q
=
F
(
L
)
,
(2.1)
where
F
is concave,
L
is the input of labor, and
q
is the output of oysters.
Denote Robinson’s consumption of oysters by
c
and his consumption of
leisure by
R
.
R
= 168

L,
(2.2)
u
(
c, R
)
F
p
(
·
)
>
0
, F
pp
(
·
)
<
0
,
∂u
∂R
>
0
,
∂c
>
0
,
∂
2
u
2
<
0
,
∂
2
u
2
<
0
,
∂
2
u
∂R∂c
>
0
,
and that
F
p
(0) = +
∞
.
Centralized allocation
u
(
c, R
) =
u
(
F
(
L
)
,
168

L
)
.
(2.3)
We now seek to choose
L
to maximize
u
:
max
L
u
(
F
(
L
)
,
168

L
)
.
(2.4)
The Frstorder condition for an extremum then is
d
dL
u
(
F
(
L
)
,
168

L
) = 0
.
(2.5)
That is,
u
c
F
p

u
R
= 0
,
(2.6)
where
u
c
and
u
R
denote partial derivatives. Hence, at an optimum — a
utility maximum subject to resource and technology constraint — we have
u
R
u
c
=

dq
dR
=
F
p
.
(2.7)
Restating (2.7),
MRS
R,c
=

∂
c
∂
R

u
=
constant
=
u
R
u
c
=

dq
dR
=
F
p
=
MRT
R,c
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LN0401˙2011
March 24, 2011
10:21
2
Equations (2.5), (2.6), and (2.7) represent conditions evaluated at the opti
mizing allocation, fulFlling (2.4).
Pareto e±cient :
•
the allocation makes technically e±cient use of produc
tive resources (labor) to produce output (that the inputoutput combination
is on the production frontier)
•
the mix of outputs (oysters and leisure) is the best possible among
the achievable allocations in terms of achieving household utility. Equation
(2.7), which shows the equality of slopes of the production function and the
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This note was uploaded on 11/30/2011 for the course ECON 311 taught by Professor Zambrano during the Fall '08 term at Cal Poly.
 Fall '08
 ZAMBRANO
 Microeconomics, Robinson Crusoe, Robinson Crusoe

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