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Economia V; Instructor: Marco Del Negro
Problem set 6
Solutions
a) In order to know the amount invested (K2) we just have to remember the no-arbitrage
condition:
which in this case becomes:
The amount invested is therefore K2=1.
b) Missing.
c)
K2+C1=Y1+D
d) There are two ways to solve this problem. The ``nose-to-the-grindstone'' solution is the standard one:
substitute the constraints into the objective function and maximize:
s.t.
K2+C1=Y1+D
or
FOC w.r.t K2
FOC w.r.t. D
From the two FOCs we get the no-arbitrage condition:
from which we get that K2=1. Substitute this solution
into the second FOC to get D (remembering that
and Y1=2)
ad obtain
.
From the constraints you get:
.
The ``quicker'' way is to notice that with K2=1 the constraints become:
C1=Y1-K2+D
or (substituting for D)
But this is the typical permanent income model! And we know the solution!
Since C1<Y1, the solution satisfies the constraint.
e)If we have Y1=0.5, then consumption in period 1 would be:
so this violates the constraint. We know that the agent can borrow to finance
investment. So the no-arbitrage condition still holds:
which implies K2=1. We also know that the agent would like to borrow but it cannot. Therefore she will choose
.
So D=1 (the necessary amount to finance investment only), and C2=2-1=1.
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Marco Del Negro
2000-02-26