<p>After many emails, my professor conceded to posting a solution to this problem, the funny thing is that he does not use the confusing $\mathcal{L}$ equation he wrote previously in class.</p>

<p>Solution c.) We have the Planner's problem $$\max_{c^{A},c^{B}}\{\mathbb{E}[u^{A}(c^{A})] + \eta \mathbb{E}[u^{B}(c^{B})]$$ such that the resource constraints 
\begin{align*}
c_1^{A} + c_1^{B} &amp;= e_1^{A}+e_1^{B} = 12\\
c_2^{A} + c_2^{B} &amp;= e_2^{A} + e_2^{B} = 8
\end{align*}
$\eta$ is the Negishi weights - which we will deal with later. Thus we have,
$$\max_{c_1^{A},c_2^{A}}\{\left[ \frac{1}{2}\frac{(c_1^{A})^{1- \gamma^{A} -1}}{1 - \gamma^{A}} + \frac{1}{2}\frac{(c_2^{A})^{1 - \gamma^A}-1}{1-\gamma^{A}}\right] + \eta\left[\frac{1}{2}((12-c_1^{A})-1) + \frac{1}{2}((8- c_2^{A}) - 1)\right]\}$$
We have the FOC: 
\begin{align*}
&amp;c_1^{A}: \frac{1}{2}(1 - \gamma^{A})(c_1^{A})^{-\gamma^{A}} - \frac{1}{2}\eta = 0\\
&amp;c_2^{A}: \frac{1}{2}(1 - \gamma^{A})(c_2^{A})^{-\gamma^{A}} - \frac{1}{2}\eta = 0\\
\end{align*}
We see that $c_1^{A} = c_2^{A}$. In order to solve the problem for $\eta$, the solution must also satisfy the "lifetime budget constraint" of each agent. Since there are only $2$ agents, it is sufficient to solve the problem for just $1$ agent:
$$\Rightarrow \sum_{s=1}^{2}u^{i\prime}(c_s^{i})c_s^{i} = \sum_{s}u^{i\prime}(c_{s}^{i})e_{s}^{i} , \  i = A , B$$
The first sum is the marginal value of consumption and the second sum is the marginal value of endowments. The lifetime budget constraints implies for $A: u^{A\prime}(c_1^{A})c_1^{A} + u^{A\prime}(c_2^{A}) = u^{\prime}(c_1^{A})e_1^{A} + u^{\prime}(c_2^{A})e_2^{A}$ but since $c_1^{A} = c_2^{A}$, $u^{A\prime}(c_1^{A}) = u^{A\prime}(c_2^{A})$. So we get $$c_1^{A} + c_2^{A} = e_1^{A} + e_2^{A} = 8 \Rightarrow c_1^{A} = c_2^{A} = 4 \Rightarrow c_1^{B} = 12 - c_1^{A} = 8 \ \text{and} \ c_2^{B} = 8 - c_1^{A} = 4$$
Note from the FOC's, we have $$\eta = (1-\gamma^{A})(c_1^{A})^{-\gamma^{A}})$$ So if we specify $\gamma^{A}$, we could solve for the specific value of $\eta$. In our example we don't need to do this since $\gamma^{B} = 0$ is a degenerate case.</p>

<p>Solution d.) From the FOC's in part (a) we have 
\begin{align*}
6 + \theta_1^{B} - \theta_2^{B} &amp;= c_1^{B} = 8\\
6 + \theta_1^{B} + 4\theta_2^{B} = c_2^{B} = 4
\end{align*}
Thus we have $\theta_1^{B} = 6/5$ and $\theta_2^{B} = -4/5$. Since $\theta_{s}^{A} + \theta_{s}^{B} = 0$, $s= 1,2$ (zero net supply) We get $$\theta_1^{A} = -6/5 \ \text{and} \ \theta_2^{A} = -4/5$$</p>

<p>Solution e.) We know that $$p = Dq \Rightarrow \begin{pmatrix}
p_1\\
p_2
\end{pmatrix} = \begin{pmatrix} 
1 &amp; 1\\
-1 &amp; 4\\
\end{pmatrix}\begin{pmatrix}
q_1\\
q_2
\end{pmatrix}$$
Then 
\begin{align*}
p_1 &amp;= q_1 + q_2\\
p_2 &amp;= -q_1 + 4q_2
\end{align*}
We know that $q_1 = q_2$ but we do not know the sum because there was no period zero consumption in this model from which to price the bond. Let's choose $q_1 + q_2 = 1$ then $q_1 = q_2 = 1/2$ so $p_1 = 1$ and $p_2 = 3/2$. </p>

CC BY-SA 3.0
2017-03-05T16:15:53.440Z
2017-03-05T16:15:53.440Z
Wolfy
<blockquote>
  <p>Consider a two period, single good, $2$ agent model. Time beings in perios $0$ in a known state (state $0$) but in period $1$ the world may find itself in any one of two states $s = 1,2$ with probabilities $\pi_s = 1/2$ for each state. Both of the consumers agree on the probabilities.</p>
  
  <p>Each consumer as a constant relative risk aversion utility function with utility index $u^{i}(c_{s}^{i}) = \frac{(c^{i}_{s})^{1-\gamma^i}-1}{1-\gamma^i}$, $i = A,B$. Specifically, we assume that $\gamma^{A} &gt; 0$ and $\gamma^{B} = 0$.</p>
  
  <p>In addition to the consumption good there are $J=2$ financial securities with period $1$ payoffs $D$ (a $J\times S$ matrix) given by $$D = \begin{pmatrix}
1 &amp; 0\\
-1 &amp; 4\\
\end{pmatrix}$$
  Thus, the first security pays off $1$ in both states and the second security pays off $-1$ in state $1$ and $4$ in state $2$. There is a spot market for these securities in period $0$ at prices $p_j &gt; 0$. Each security is in zero net supply so that if one consumer is a buyer then the other consumer must be the seller. There are no short-sell constraints on the securities. </p>
  
  <p>Consumers have no period zero endowment and do not consume in the first period so that $e_0^{i} = c_{0}^{i} = 0$, $i = A,B$.</p>
  
  <p>The consumers are not allowed to short-sell the consumption good. I.e., we impost the restriction that $c_{s}^{i}\geq 0$,$\forall i = A,B$ and $s = 1,2$. There is no production in this economy so each agent is exogenously endowed with period $1$ endowments of: $e^{A} = (6,2)$ and $e^{B} = (6,6)$ in states $s = 1$, and $2$, for each agent respectively.</p>
  
  <p>a.) Carefully write out the second consumer's maximization problem including the relevant budget constraints. Let $\lambda_{s}^{B}$ be the Lagrange multipliers on the budget constraints. Carefully write out the first-order conditions for this consumer including the inequality and complementary slackness conditions.  </p>
  
  <p>b.) Solve for the contract curve for this economy.</p>
  
  <p>c.) Verify that the Arrow security prices for this economy are $q_1 = q_2$ and that the optimal allocation of goods is $c^{A} = (4,4)$ and $c^{B} = (8,4)$.</p>
  
  <p>d.) Find the consumers' security portfolios $\theta^{A}$ and $\theta^{B}$.</p>
  
  <p>e.) Normalize the Arrow security prices so that $q_1 = q_2$ and find the security prices $p_1$ and $p_2$.</p>
</blockquote>

<p>Solution a.) The utility maximization problem for $B$ is \begin{align*}
\max_{c^{B},\theta}\mathbb{E}\left[u(c^{B})\right] = \frac{1}{2}(c_1^{B} - 1) + \frac{1}{2}(c_2^{B} - 1) \ \ \text{s.t.} \ \ &amp;p_1\theta_{1}^{B} + p_2\theta_{2}^{B} = 0, s = 0\\
&amp;c_1^{B}\leq 6 + \theta_{1}^{B} - \theta_{2}^{B}, s = 1\\
&amp;c_{2}^{B}\leq 6 + \theta_{1}^{B} + 4\theta_{2}^{B}, s =2; c_1^{B}\geq 0, c_2^{B}\geq 0
\end{align*}
We have the Lagrangian, $$\mathcal{L}(c^{B},\theta^{B},\lambda^{B}) = \frac{1}{2}(c_1^{B} - 1) + \frac{1}{2}(c_2^{B} - 1) + \lambda_{0}^{B}(0- p_1 \theta_1^{B} - p_2 \theta_2^{B}) + \lambda_{1}^{B}(6 + \theta_1^{B} - \theta_2^{B} - c_1^{B}) + \lambda_2^{B}(6 + \theta_{1}^{B} + 4\theta_2^{B} - c_2^{B})$$
The First-order conditions are:
\begin{align*}
&amp;\lambda_{0}^{B}: p_1\theta_1^{B} + p_2 \theta_{2}^{B} = 0; \lambda_{0}^{B}\geq 0\\
&amp;\lambda_1^{B}: c_1^{B}\leq 6 + \theta_1^{B} - \theta_2^{B};\lambda_1^{B}\geq 0, \lambda_1^{B}(6+\theta_1^{B} - \theta_2^{B} - c_1^{B}) = 0\\
&amp;\lambda_2^{B}: c_2{B} \leq 6+\theta_{1}^{B} + 4\theta_2^{B}, \lambda_2^{B}\geq 0, \lambda_2^{B}(6+\theta_{1}^{B} + 4\theta_2^{B} - c_2^{B}) = 0\\
&amp;c_1^{B}: \frac{1}{2}-\lambda_1^{B}\leq 0; c_1^{B}\geq 0, c_1^{B}\left(\frac{1}{2} - \lambda_{1}^{B}\right) = 0\\
&amp;c_2^{B}: \frac{1}{2}-\lambda_2^{B}\leq 0; c_2^{B}\geq 0, c_2^{B}\left(\frac{1}{2} - \lambda_{2}^{B}\right) = 0\\
&amp;\theta_1^{B}: - \lambda_0^{B}p_1 + \lambda_1^{B} + \lambda_2^{B} = 0\\
&amp;\theta_2^{B}: - \lambda_0^{B}p_2 - \lambda_1^{B} + 4\lambda_2^{B} = 0
\end{align*}</p>

<p>Solution b.) We have $$c_1^{A} + c_2^{B} = 12 \ \ \text{and} \ \ c_2^{A} + c_2^{B} = 8$$
We want to find the set of pareto optimal points (contact curve). We are looking for $$MRS^{A} = MRS^{B}$$
Note that $u^{i} = \pi_1 u(c_1^{i}) + \pi_2 u(c_2^{i})$ Thus, $$MRS^{i} = \frac{\pi_1 u^{i \prime}(c_1^{i})}{\pi_2 u^{i\prime}(c_2^{i})}$$
$$MRS^{A} = \left(\frac{c_1^{A}}{c_2^{A}}\right)^{-\gamma_A} = 1 = MRS^{B}$$ $c_1^{A} = c_2^{A}$ is the equation of the contract curve.</p>

<p>Attempted solution c.) This is where I am stuck, I know that we have to solve for the Planner's problem: $$\mathcal{L} = \sum_{i=A,B} \eta^{i}u^{i}(c_1^{i}) + \lambda_1\left(\sum_{i}e_1^{i} - \sum_{i}c_1^{i}\right) + \lambda_2\left(\sum_{i}e_2^{i} - \sum_{i}c_2^{i}\right)$$
but I am not sure how to proceed any further. Any suggestions would be greatly appreciated. This is my first time solving a planners problem and I am a Mathematics grad student so do not have much exposure to economics.  </p>

CC BY-SA 3.0
2017-03-03T03:10:00.307Z
Why do you think you need to solve the planner's problem to solve (c)? I think you need to show that the proposed prices and allocations are consistent with equilibrium. That is, the allocations are feasible and, given the prices, solve each consumer's utility maximisation problem.
Theoretical Economist
CC BY-SA 3.0
2017-03-03T03:11:14.190Z
Well my professor said that we should solve it, since similar problems ask to solve the Planner's problem
Wolfy
CC BY-SA 3.0
2017-03-03T03:17:42.247Z
I don't see how the planner's problem should give you the state (Arrow security) prices, but that might just be my having forgotten how to solve GE models with securities markets. In any case -- why are you not able to solve the constrained optimisation as you did with the second consumer's problem?
Theoretical Economist
CC BY-SA 3.0
2017-03-03T03:19:49.557Z
@TheoreticalEconomist Well my professor provided the Planner's problem equation without solving it, and he sort of hand-waved through how to solve it. I may know how to solve it (without actually knowing) but I have never seen an example of a worked out solution so I am a bit daunted by the equation.
Wolfy
CC BY-SA 3.0
2017-03-03T03:24:50.243Z
It does look daunting; I'll give you that. However, solving the social planner's problem will just boil down to finding the first-order conditions and finding the solution to those. I'm sure you can work that out. At the very least, that's one way of doing it.
Theoretical Economist
CC BY-SA 3.0
2017-03-03T03:27:02.707Z
@TheoreticalEconomist I think I see what you mean. To clarify, are you saying the I can take the first-order conditions I found and finding the solutions to those? or, are there new first-order conditions from just part c.) alone?
Wolfy
CC BY-SA 3.0
2017-03-05T15:35:26.730Z
@TheoreticalEconomist My professor posted a solution due to many emails asking for a detailed solution of this. Funny thing is that his $\mathcal{L}$ that I presented above he does not use and instead sets the Planner's problem in a less daunting way. Just wondering if you would be interested in me posting that solution to this question.
Wolfy
CC BY-SA 3.0
2017-03-05T15:40:01.027Z
I'm guessing your professor maximised one consumer's utility subject to a minimum utility constraint on the other, which is equivalent. In any case, you should post the solution -- I would be interested in reading it, and this question can be cleared from the unanswered queue.
Theoretical Economist
CC BY-SA 3.0
2017-03-05T15:40:57.087Z
@TheoreticalEconomist Ok, should I post an answer to my own question or just re-edit the question with the solution to c, d and e?
Wolfy
CC BY-SA 3.0
2017-03-05T15:41:40.727Z
You should post an answer to your own question. You should also be able to accept your own answer as well.
Theoretical Economist
CC BY-SA 3.0
2017-03-05T15:42:04.670Z
@TheoreticalEconomist Ok will do momentarily
Wolfy
CC BY-SA 3.0
2017-03-05T16:16:19.037Z
@TheoreticalEconomist posted the solution
Wolfy
<mathematical-economics><financial-economics>
Arrow-Debreu Pricing, Planner's Problem
Wolfy
# Arrow-Debreu Pricing, Planner's Problem

(Asked by: username_0 on 2017-03-03)

> Consider a two period, single good, $2$ agent model. Time beings in perios $0$ in a known state (state $0$) but in period $1$ the world may find itself in any one of two states $s = 1,2$ with probabilities $\pi\_s = 1/2$ for each state. Both of the consumers agree on the probabilities.
>
> Each consumer as a constant relative risk aversion utility function with utility index $u^{i}(c\_{s}^{i}) = \frac{(c^{i}\_{s})^{1-\gamma^i}-1}{1-\gamma^i}$, $i = A,B$. Specifically, we assume that $\gamma^{A} > 0$ and $\gamma^{B} = 0$.
>
> In addition to the consumption good there are $J=2$ financial securities with period $1$ payoffs $D$ (a $J\times S$ matrix) given by $$D = \begin{pmatrix}
> 1 & 0\\
> -1 & 4\\
> \end{pmatrix}$$
> Thus, the first security pays off $1$ in both states and the second security pays off $-1$ in state $1$ and $4$ in state $2$. There is a spot market for these securities in period $0$ at prices $p\_j > 0$. Each security is in zero net supply so that if one consumer is a buyer then the other consumer must be the seller. There are no short-sell constraints on the securities.
>
> Consumers have no period zero endowment and do not consume in the first period so that $e\_0^{i} = c\_{0}^{i} = 0$, $i = A,B$.
>
> The consumers are not allowed to short-sell the consumption good. I.e., we impost the restriction that $c\_{s}^{i}\geq 0$,$\forall i = A,B$ and $s = 1,2$. There is no production in this economy so each agent is exogenously endowed with period $1$ endowments of: $e^{A} = (6,2)$ and $e^{B} = (6,6)$ in states $s = 1$, and $2$, for each agent respectively.
>
> a.) Carefully write out the second consumer's maximization problem including the relevant budget constraints. Let $\lambda\_{s}^{B}$ be the Lagrange multipliers on the budget constraints. Carefully write out the first-order conditions for this consumer including the inequality and complementary slackness conditions.
>
> b.) Solve for the contract curve for this economy.
>
> c.) Verify that the Arrow security prices for this economy are $q\_1 = q\_2$ and that the optimal allocation of goods is $c^{A} = (4,4)$ and $c^{B} = (8,4)$.
>
> d.) Find the consumers' security portfolios $\theta^{A}$ and $\theta^{B}$.
>
> e.) Normalize the Arrow security prices so that $q\_1 = q\_2$ and find the security prices $p\_1$ and $p\_2$.

Solution a.) The utility maximization problem for $B$ is \begin{align\*}
\max\_{c^{B},\theta}\mathbb{E}\left[u(c^{B})\right] = \frac{1}{2}(c\_1^{B} - 1) + \frac{1}{2}(c\_2^{B} - 1) \ \ \text{s.t.} \ \ &p\_1\theta\_{1}^{B} + p\_2\theta\_{2}^{B} = 0, s = 0\\
&c\_1^{B}\leq 6 + \theta\_{1}^{B} - \theta\_{2}^{B}, s = 1\\
&c\_{2}^{B}\leq 6 + \theta\_{1}^{B} + 4\theta\_{2}^{B}, s =2; c\_1^{B}\geq 0, c\_2^{B}\geq 0
\end{align\*}
We have the Lagrangian, $$\mathcal{L}(c^{B},\theta^{B},\lambda^{B}) = \frac{1}{2}(c\_1^{B} - 1) + \frac{1}{2}(c\_2^{B} - 1) + \lambda\_{0}^{B}(0- p\_1 \theta\_1^{B} - p\_2 \theta\_2^{B}) + \lambda\_{1}^{B}(6 + \theta\_1^{B} - \theta\_2^{B} - c\_1^{B}) + \lambda\_2^{B}(6 + \theta\_{1}^{B} + 4\theta\_2^{B} - c\_2^{B})$$
The First-order conditions are:
\begin{align\*}
&\lambda\_{0}^{B}: p\_1\theta\_1^{B} + p\_2 \theta\_{2}^{B} = 0; \lambda\_{0}^{B}\geq 0\\
&\lambda\_1^{B}: c\_1^{B}\leq 6 + \theta\_1^{B} - \theta\_2^{B};\lambda\_1^{B}\geq 0, \lambda\_1^{B}(6+\theta\_1^{B} - \theta\_2^{B} - c\_1^{B}) = 0\\
&\lambda\_2^{B}: c\_2{B} \leq 6+\theta\_{1}^{B} + 4\theta\_2^{B}, \lambda\_2^{B}\geq 0, \lambda\_2^{B}(6+\theta\_{1}^{B} + 4\theta\_2^{B} - c\_2^{B}) = 0\\
&c\_1^{B}: \frac{1}{2}-\lambda\_1^{B}\leq 0; c\_1^{B}\geq 0, c\_1^{B}\left(\frac{1}{2} - \lambda\_{1}^{B}\right) = 0\\
&c\_2^{B}: \frac{1}{2}-\lambda\_2^{B}\leq 0; c\_2^{B}\geq 0, c\_2^{B}\left(\frac{1}{2} - \lambda\_{2}^{B}\right) = 0\\
&\theta\_1^{B}: - \lambda\_0^{B}p\_1 + \lambda\_1^{B} + \lambda\_2^{B} = 0\\
&\theta\_2^{B}: - \lambda\_0^{B}p\_2 - \lambda\_1^{B} + 4\lambda\_2^{B} = 0
\end{align\*}

Solution b.) We have $$c\_1^{A} + c\_2^{B} = 12 \ \ \text{and} \ \ c\_2^{A} + c\_2^{B} = 8$$
We want to find the set of pareto optimal points (contact curve). We are looking for $$MRS^{A} = MRS^{B}$$
Note that $u^{i} = \pi\_1 u(c\_1^{i}) + \pi\_2 u(c\_2^{i})$ Thus, $$MRS^{i} = \frac{\pi\_1 u^{i \prime}(c\_1^{i})}{\pi\_2 u^{i\prime}(c\_2^{i})}$$
$$MRS^{A} = \left(\frac{c\_1^{A}}{c\_2^{A}}\right)^{-\gamma\_A} = 1 = MRS^{B}$$ $c\_1^{A} = c\_2^{A}$ is the equation of the contract curve.

Attempted solution c.) This is where I am stuck, I know that we have to solve for the Planner's problem: $$\mathcal{L} = \sum\_{i=A,B} \eta^{i}u^{i}(c\_1^{i}) + \lambda\_1\left(\sum\_{i}e\_1^{i} - \sum\_{i}c\_1^{i}\right) + \lambda\_2\left(\sum\_{i}e\_2^{i} - \sum\_{i}c\_2^{i}\right)$$
but I am not sure how to proceed any further. Any suggestions would be greatly appreciated. This is my first time solving a planners problem and I am a Mathematics grad student so do not have much exposure to economics.

**Tags:** mathematical-economics, financial-economics

> **Comments:**
> **username_1**: Why do you think you need to solve the planner's problem to solve (c)? I think you need to show that the proposed prices and allocations are consistent with equilibrium. That is, the allocations are feasible and, given the prices, solve each consumer's utility maximisation problem.  
> **username_0**: Well my professor said that we should solve it, since similar problems ask to solve the Planner's problem  
> **username_1**: I don't see how the planner's problem should give you the state (Arrow security) prices, but that might just be my having forgotten how to solve GE models with securities markets. In any case -- why are you not able to solve the constrained optimisation as you did with the second consumer's problem?  
> **username_0**: @username_1 Well my professor provided the Planner's problem equation without solving it, and he sort of hand-waved through how to solve it. I may know how to solve it (without actually knowing) but I have never seen an example of a worked out solution so I am a bit daunted by the equation.  
> **username_1**: It does look daunting; I'll give you that. However, solving the social planner's problem will just boil down to finding the first-order conditions and finding the solution to those. I'm sure you can work that out. At the very least, that's one way of doing it.  
> **username_0**: @username_1 I think I see what you mean. To clarify, are you saying the I can take the first-order conditions I found and finding the solutions to those? or, are there new first-order conditions from just part c.) alone?  
> **username_0**: @username_1 My professor posted a solution due to many emails asking for a detailed solution of this. Funny thing is that his $\mathcal{L}$ that I presented above he does not use and instead sets the Planner's problem in a less daunting way. Just wondering if you would be interested in me posting that solution to this question.  
> **username_1**: I'm guessing your professor maximised one consumer's utility subject to a minimum utility constraint on the other, which is equivalent. In any case, you should post the solution -- I would be interested in reading it, and this question can be cleared from the unanswered queue.  
> **username_0**: @username_1 Ok, should I post an answer to my own question or just re-edit the question with the solution to c, d and e?  
> **username_1**: You should post an answer to your own question. You should also be able to accept your own answer as well.  
> **username_0**: @username_1 Ok will do momentarily  
> **username_0**: @username_1 posted the solution  

---

## 1 answers

### Answer by username_0 on 2017-03-05. Score: 2 | Accepted answer

After many emails, my professor conceded to posting a solution to this problem, the funny thing is that he does not use the confusing $\mathcal{L}$ equation he wrote previously in class.

Solution c.) We have the Planner's problem $$\max\_{c^{A},c^{B}}\{\mathbb{E}[u^{A}(c^{A})] + \eta \mathbb{E}[u^{B}(c^{B})]$$ such that the resource constraints
\begin{align\*}
c\_1^{A} + c\_1^{B} &= e\_1^{A}+e\_1^{B} = 12\\
c\_2^{A} + c\_2^{B} &= e\_2^{A} + e\_2^{B} = 8
\end{align\*}
$\eta$ is the Negishi weights - which we will deal with later. Thus we have,
$$\max\_{c\_1^{A},c\_2^{A}}\{\left[ \frac{1}{2}\frac{(c\_1^{A})^{1- \gamma^{A} -1}}{1 - \gamma^{A}} + \frac{1}{2}\frac{(c\_2^{A})^{1 - \gamma^A}-1}{1-\gamma^{A}}\right] + \eta\left[\frac{1}{2}((12-c\_1^{A})-1) + \frac{1}{2}((8- c\_2^{A}) - 1)\right]\}$$
We have the FOC:
\begin{align\*}
&c\_1^{A}: \frac{1}{2}(1 - \gamma^{A})(c\_1^{A})^{-\gamma^{A}} - \frac{1}{2}\eta = 0\\
&c\_2^{A}: \frac{1}{2}(1 - \gamma^{A})(c\_2^{A})^{-\gamma^{A}} - \frac{1}{2}\eta = 0\\
\end{align\*}
We see that $c\_1^{A} = c\_2^{A}$. In order to solve the problem for $\eta$, the solution must also satisfy the "lifetime budget constraint" of each agent. Since there are only $2$ agents, it is sufficient to solve the problem for just $1$ agent:
$$\Rightarrow \sum\_{s=1}^{2}u^{i\prime}(c\_s^{i})c\_s^{i} = \sum\_{s}u^{i\prime}(c\_{s}^{i})e\_{s}^{i} , \ i = A , B$$
The first sum is the marginal value of consumption and the second sum is the marginal value of endowments. The lifetime budget constraints implies for $A: u^{A\prime}(c\_1^{A})c\_1^{A} + u^{A\prime}(c\_2^{A}) = u^{\prime}(c\_1^{A})e\_1^{A} + u^{\prime}(c\_2^{A})e\_2^{A}$ but since $c\_1^{A} = c\_2^{A}$, $u^{A\prime}(c\_1^{A}) = u^{A\prime}(c\_2^{A})$. So we get $$c\_1^{A} + c\_2^{A} = e\_1^{A} + e\_2^{A} = 8 \Rightarrow c\_1^{A} = c\_2^{A} = 4 \Rightarrow c\_1^{B} = 12 - c\_1^{A} = 8 \ \text{and} \ c\_2^{B} = 8 - c\_1^{A} = 4$$
Note from the FOC's, we have $$\eta = (1-\gamma^{A})(c\_1^{A})^{-\gamma^{A}})$$ So if we specify $\gamma^{A}$, we could solve for the specific value of $\eta$. In our example we don't need to do this since $\gamma^{B} = 0$ is a degenerate case.

Solution d.) From the FOC's in part (a) we have
\begin{align\*}
6 + \theta\_1^{B} - \theta\_2^{B} &= c\_1^{B} = 8\\
6 + \theta\_1^{B} + 4\theta\_2^{B} = c\_2^{B} = 4
\end{align\*}
Thus we have $\theta\_1^{B} = 6/5$ and $\theta\_2^{B} = -4/5$. Since $\theta\_{s}^{A} + \theta\_{s}^{B} = 0$, $s= 1,2$ (zero net supply) We get $$\theta\_1^{A} = -6/5 \ \text{and} \ \theta\_2^{A} = -4/5$$

Solution e.) We know that $$p = Dq \Rightarrow \begin{pmatrix}
p\_1\\
p\_2
\end{pmatrix} = \begin{pmatrix}
1 & 1\\
-1 & 4\\
\end{pmatrix}\begin{pmatrix}
q\_1\\
q\_2
\end{pmatrix}$$
Then
\begin{align\*}
p\_1 &= q\_1 + q\_2\\
p\_2 &= -q\_1 + 4q\_2
\end{align\*}
We know that $q\_1 = q\_2$ but we do not know the sum because there was no period zero consumption in this model from which to price the bond. Let's choose $q\_1 + q\_2 = 1$ then $q\_1 = q\_2 = 1/2$ so $p\_1 = 1$ and $p\_2 = 3/2$.


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