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:

$$F_K(K,N)=A\alpha K2^{\alpha-1}=(1+r^w)$$

which in this case becomes:

$$2\frac12 K2^{-\frac12}=1$$

The amount invested is therefore K2=1.
b) Missing.
c) K2+C1=Y1+D

$$C2=F(K2,1)+(1-\delta)-(1+r^w)D$$

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:

$$Max \ln(C1)+\beta \ln(C2)$$

s.t.

K2+C1=Y1+D

$$C2=F(K2,1)+(1-\delta)-(1+r^w)D$$

or

$$Max_{K2,D} \ln(Y1+D-K2)+\beta \ln(F(K2,1)-(1+r^w)D)$$

FOC w.r.t K2

$$\frac{-1}{(Y1+D-K2)}+\frac{\beta\alpha AK2^{\alpha-1}}{AK2^{\alpha}-(1+r^w)D}=0$$

FOC w.r.t. D

$$\frac{1}{(Y1+D-K2)}-\frac{\beta(1+r^w)}{AK2^{\alpha}-(1+r^w)D}=0$$

From the two FOCs we get the no-arbitrage condition:

$$F_K(K,N)=A\alpha K2^{\alpha-1}=(1+r^w)$$

from which we get that K2=1. Substitute this solution into the second FOC to get D (remembering that $\beta=\frac1{1+r^w}$ and Y1=2)

$$\frac{1}{1+D)}-\frac{1}{2-D}=0$$

ad obtain $D=\frac12$. From the constraints you get: $C1=C2=\frac32$.
The ``quicker'' way is to notice that with K2=1 the constraints become: C1=Y1-K2+D

$$C2=F(K2,1)+(1-\delta)-(1+r^w)D$$

or (substituting for D)

$$C1+\frac{C2}{1+r^w}=Y1-K2+\frac{F(K2,1)}{1+r^w}$$

But this is the typical permanent income model! And we know the solution!

$$C1=\frac1{(1+\beta)}(Y1-K2+\frac{F(K2,1)}{1+r^w})=\frac12(1+2)=\frac32$$

$$C2=\frac{(1+r)\beta}{(1+\beta)}(Y1-K2+\frac{F(K2,1)}{1+r^w})=\frac12(1+2)=\frac32$$

Since C1<Y1, the solution satisfies the constraint.
e)If we have Y1=0.5, then consumption in period 1 would be: $C1=1/2(-0.5+2)=\frac34>Y1$ so this violates the constraint. We know that the agent can borrow to finance investment. So the no-arbitrage condition still holds:

$$2\frac12 K2^{-\frac12}=1$$

which implies K2=1. We also know that the agent would like to borrow but it cannot. Therefore she will choose $C1=Y1=\frac12$. So D=1 (the necessary amount to finance investment only), and C2=2-1=1.