General Equilibrium
1.
A small island country produces and consumes goods X and Y.
The country has available
15,000 labor hours per day.
The production functions for producing X and Y are
X=(0.5L
X
)
0.5
and Y=(L
Y
)
0.5
, where L
X
and L
Y
are the number of hours of labor devoted to X
and Y, respectively.
All consumers have the same preferences of U=X
0.25
Y
0.75
.
a)
Find the formulas for the island’s production possibility frontier (PPF), rate of product
transformation (RPT) and its marginal rate of substitution (MRS).
b)
Suppose that the country trades with the rest of the world at a price ratio P
x
/P
y
=1.
Find the country’s optimal consumption and production choices.
What is the utility
level associated with this optimum?
c)
Now suppose that the country discovers a technological innovation changes the
production function for X to X=(L
X
)
0.5
while the production function for Y remains
unchanged.
World prices are also unchanged.
Find the country’s new optimal
consumption and production choices.
What is the utility level associated with this new
optimum?
d)
Discuss why the country adjusted its production and consumption decisions in part c
relative to part b.
Discuss how the innovation affected the country’s opportunity set
and wellbeing (utility).
Answer:
a)
2X
2
+Y
2
=15000, 4X dX + 2Y dY = 0, so RPT=dY/dX=(2X)/Y.
MRS=Y/(3X)
b)
Consumption:
X=37.5, Y=112.5.
Production:
X=50, Y=100.
U=85.48
c)
Production:
X=86.6, Y=86.6.
Consumption:
X=43.3, Y=129.9.
U=98.71
d)
The innovation makes it relatively cheaper for the country to produce X than before,
so the PPF fans out to the right.
The price vector is not tangent to the new PPF at the
old production combination, and the country adjust by producing more X and less Y
than before to reach a tangency where the new RPT equals world prices.
The budget
set shifts to the left – the country has more opportunities than before because the
innovation allows it to produce goods more cheaply (i.e., more goods with the same
resources as before).
With greater opportunities, the country can move to a higher
indifference curve and improve wellbeing.
2.
Otto lives alone in the mountains.
He consumes a diet of berries (B) and fish (F).
His
production functions for B and F are B=(0.5L
B
)
0.5
and F=(L
F
)
0.5
, where L
B
and L
F
are the
number of hours of labor devoted to picking berries and fishing, respectively.
Otto works
exactly 162 hours per month and his preferences are given by U=B
0.5
F
0.5
.
a)
Find the formulas for Otto's production possibility frontier (PPF), rate of product
transformation (RPT), and marginal rate of substitution (MRS).
b)
Suppose that Otto trades with a nearby city and the price ratio is P
B
/P
F
=0.5.
Find the
Otto's optimal consumption and production choices. What is the utility level
associated with this optimum?
c)
Now suppose that a flood closes the road between Otto's house and the city, so he can
no longer trade with the local community.
Find Otto's new optimal consumption and
production choices.
What is the utility level associated with this new optimum?
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 Spring '10
 Buddin
 Utility, optimal consumption, Pratt measure

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