A couple of notes on the Midterm exam.
I just want to point out that if the agent is risk averse
than RRA > 0, if the agent is risk neutral (linar utility function) than RRA = 0
(because u00 (c) = 0), if the agent is risk lover than RRA < 0. Therefore, since
you know that logarithmic utility implies risk aversion, you should know that
RRA > 0. This way of reasoning maybe useful in order to avoid silly mistakes
like messing up with the signs and getting RRA = −1.
Exercise 1 point a
Exercise 1 point c - Lagrangian method
L = ln (c1 ) +
FOCs
1
5
3
1
ln (c2 ) + ln (c2 ) + λ1
− c1 − c2 + λ2
− c1 − c2
2
2
2
2

1


c1 = λ1 + λ2


1


 2c2 = λ1
1
2c2 = λ2


5



2 − c1 − c2 = 0

3
2 − c1 − c2 = 0
By substituting the second and the second line inside the rst we get rid of the
Lagrange multipliers and we restrict our problem to the system
1
1
1

 c1 = 2c2 + 2c2
5
2 − c1 = c2

3
2 − c1 = c2
Now we can substitute the second and the third line into the rst to solve for
c1 :

1
1

 c1 = 5−2c1 +
5
2 − c1 = c2

3
2 − c1 = c2
1
3−2c1
Actually, since we know that s = 1 − c1 we are actually only interested in c1 .
Applying some simple algebra to the rst line we have
c1 (3 − 2c1 )
c1 (5 − 2c1 )
(3 − 2c1 ) (5 − 2c1 )
+
=
c1 (5 − 2c1 ) (3 − 2c1 ) c1 (5 − 2c1 ) (3 − 2c1 )
c1 (5 − 2c1 ) (3 − 2c1 )
3c1 − 2c21 + 5c1 − 2c21 = 15 − 6c1 − 10c1 + 4c21
8c21 − 24c1 + 15 = 0
√
√
24 ± 96
24 ± 576 − 480
=
c1 =
16
16
√
√
6 · 4 ± 6 · 16
6
6
=
= ±
16
4
4
1
Therefore we have
√
1
6
s1,2 = − ±
2
4
which is exactly the solution that we get by substituting directly the budget
constraints in the expected utility function and solving the unconstrained maximization problem.
Now remember, why we may rule out the negative solution
in this case? The
√
negative solution implies that the agent today borrows 12 + 26 from the bank. If
the realization of√income today is 23 than the agent will be able to payback the
debt in full 12 + 26 (the interest rate is zero); if the realization of his income is
1
2 than he will not have enough money to payback the debt and he will default
at least partially on his debt. Assume that the bank is risk neutral, than it will
not be willing to lend money to the agent in the rst place because it expects
to receive less than the amount it lends out.
Furthermore, notice that the function ln (c)is not dened for c = 0 and tends
to −∞ as c → 0.
2