Recursive economy in two islands There are two economies A and B with infinitely many identical households. Since there are infinitely many identical households in each economy, they are all price taker. Maeasure of agents in A and B are equal. Agent preference is a follows
$$\sum_{t=0}^{\infty} \beta^t u(c_t)$$
The only difference between countries A and B is the production technology for $i=A,B$ $$y=z^if(k)$$
The final gods produced are identical for both technologies and capital depreciates fully. Let $z=(z^A,z^B)$ and $z$ follows chain state Markov process of order one and denote the probability that $z’$ occurs conditional in z $\Gamma_{zz’}$.
Part a assume that there is no trade between two island and write down the recursive problem of one type with complete market (that is, with state contingent claim $b_{z’}$
Part b assume that both capital and goods are perfectly mobile across islands without any cost. That is, households can rent their capital to firms in any country depending on the rate of return they  get. So, write down the recursive problem of one type with complete market (that is, with state contingent claim $b_{z’}$
Part c let us assume that technology is linear in capital $f(k)=k$. There is restriction on the movement of capital but not on goods. Assume that there is state contingent claim trade across islands. How can we interpret the welfare loss of this restriction on capital compared to a world without one.
Please can you share hints of this question for each part. I will improve my answer and then I will share it here.
I don’t want you to solve for whole questions. I have some points on the parts which I want you to explain. Please discuss this question with me.
In part a,
The question states there is no trade. Therefore, will I omit the state contingent trade?
That is, the bellman equation is
$$V^i(z^i, K^i, a^i)=max \{ u(c^i)+\beta \sum_{z’} \Gamma_{zz’} V^i(z’^i, K’^i, a’^i)$$
Such that
$$c^i+a’^i= w^i+r^i a^i$$
Let us look at the part b. However, what happens (what changes) if we consider both capital and goods are perfectly mobile across islands without any cos in the bellman equation?
Does I include the state contingent claim?
$$V^i(z^i, K^i, a^i, b^i)=\max \{ u(c^i)+\beta \sum_{z’} \Gamma_{zz’} V^i(z’^i, K’^i, a’^i, b_z’^i)\}$$
$$c^i+a’^i + \sum q_z b_z’^i= w^i+r^i a^i +b^i$$
And I have no idea on part c.
Note that $w$ is wage, $r$ is rate of return , $x’$ is the one forward state of the variable. That $x_t=x$ and $x_{t+1}=x’$
 A: Honestly, the question seems a bit incomplete. Im guessing that it is based on your lectures. There are a few gaps which Ill do my best to fill in.
Can agents in $A$ trade financial assets with agents in $B$? Im assuming no in (a), yes in (b), and yes in (c).

The final gods produced are identical for both technologies and capital depreciates fully.

Im guessing that this means $y_t=c_t+k_{t+1}$
Each HH BC should be
$$r_t\cdot k_t=c_t+k_{t+1}$$
Therefore, the value function will be
$$V(k,K'|z)=\max\{u(c)+\beta\sum_{z'}\Gamma_{zz'}V(k',K'|z')\}$$
$$r\cdot k=c+k'$$
This is very similar to what you have, there doesnt seem to be any labor so not sure where $w$ is coming from.
For part (b) the first thing you have to do is keep track of both variables in $A$ and $B$. So it will be
$$V^i(k^i,K^A,K^B,b^i|z^A,z^B)$$
Looking at the HH BC again. The fact that capital is mobile means that the HH can sell their goods to either $A$ firms or $B$ firm
$$r_t^A\cdot k_t^A+r_t^B(k_t-k_t^A)+b_t=c_t+k_{t+1}+\sum_{z'}q_{t+1}(z')b_{t+1}(z')$$
Where $k_t^A$ is within $0$ and $k_t$.
$$V^i(k,b,\overline{K}|\overline{z})=\max\{u(c)+\beta\sum_{z'}\Gamma_{zz'}V(k',b'(\overline{z'},\overline{K}_z'|\overline{z}')\}$$
$$r^A\cdot k^A+r^B(k-k^A)+b=c+k'+\sum_{z'}q_(\overline{z}'|\overline{z})b'(\overline{z}')$$
For part (c) it would be the same without the movement on capital
$$V^i(k,b,\overline{K}|\overline{z})=\max\{u(c)+\beta\sum_{z'}\Gamma_{zz'}V(k',b'(\overline{z'}),\overline{K}_z'|\overline{z}')\}$$
$$r^i\cdot k+b=c+k'+\sum_{z'}q_(\overline{z}'|\overline{z})b'(\overline{z}')$$
Its fairly similar to what you have but a few notes you are going to have to keep track of all variables in both islands when they can trade.
I havent solved anything but I can tell you what is going on.

*

*If they can move capital between the island then capital is going to flow where $z^i$ is highest. in the case of $f(k)=k$ then all the capital will go to the most productive island. I think that in this case then there will be no trade in the financial market $b=0$ always. Im not 100 percent sure, but I think in this case there will just be aggregate shocks so the agents cant insure each other.

*The contingent claims are used to insure the island against bad shocks. When capital flow is shut down then you will find an island with high current productivity giving consumption to an island with low productivity.

