Endowment economy

Question

Consider an economy populated by two types of infinitely lived consumers, odd and even. There is mass one of each type of consumer. There is a single good in the economy. The economy starts at $t = 0$. Odd agents have an endowment sequence $$e^0=\{3,1,3,1...\}$$ and even agents have $$e^0=\{1,3,1,3...\}$$ The preferences of both types of agents are described by the utility function $$U ( c ) =\sum\beta log\ c_t$$.

I need to compute the price and allocations. What I've done so far:

$$\max\sum\beta log\ c_t \ s.t \sum p_tc_t^i=\sum p_te_t^i$$ and market clear $c_t^1+c_t^2\leq e_t$

$$L=\sum\beta log\ c_t-\lambda^i(\sum p_tc_t^i-\sum p_te_t^i)$$ and gotten the focs: $\frac{\beta^t}{c-t^i}=\lambda^ip_t$. Guessing that $p_t = \beta_t$ I get that the consumption allocations of both agents are constant. I denote them by $c^1$ and $c^2$. Using this information in the budget constraints, I find: $$c^i\sum\beta^t=\sum p_te_t^i$$. Here is where I'm stuck. I know that we are supposed to use the geometric sequence but I am not sure. Is $c^1=\frac{1}{1+\beta}?$

Answer 1

Let's take for granted that $\sum_t p_t e_t^i$ is finite.

The first order conditions give: $$ \frac{\beta^t}{c_t^i} = \lambda^i p_t $$ This gives: $$ c_t^i = \frac{1}{\lambda^i}\frac{\beta^t}{p_t} \tag{1} $$ If we stubstitute into the budget constraint, we obtain: $$ \begin{align*} &\sum_t \frac{1}{\lambda^i} \beta^t = \sum_t p_t e_t^i,\\ \to &\frac{1}{\lambda^i} \sum_t \beta^t = \sum_t p_t e_t^i,\\ \to &\frac{1}{\lambda^i} \frac{\beta}{1 - \beta} = \sum_t p_t e_t^i,\\ \to &\frac{1}{\lambda^i} = \frac{(1-\beta)}{\beta}\sum_t p_t e_t^i. \end{align*} $$ Substituting back into $(1)$ gives for time period $v$: $$ c_v^i = \frac{(1-\beta)}{\beta} \frac{\beta^v}{p_v} \sum_t p_t e_t^i \tag{2} $$

Equilibrium on the market at every period $v$ gives: $$ \begin{align*} &c_v^1 + c_v^2 = 4,\\ \to &\frac{(1 - \beta)}{\beta} \frac{\beta^v}{p_v} \sum_t p_t (e_t^1 + e^t_2) = 4,\\ \to &\frac{(1 - \beta)}{\beta} \frac{\beta^v}{p_v} \sum_t p_t = 1 \end{align*} $$ This must hold for every $v$, so $p_v = \beta^v$ is a solution here.

Then we can compute the consumption of individual 1 in period $v$ as follows: $$ \begin{align*} c_v^1 &= \frac{(1-\beta)}{\beta}\sum_t p_t e_t^i,\\ &= \frac{(1-\beta)}{\beta}\left[\sum_{t \text{ odd}} \beta^t 3 + \sum_{t \text{ even}} \beta^t \right],\\ &= \frac{(1-\beta)}{\beta} \left[ \frac{3 \beta}{1 - \beta^2} + \frac{\beta^2}{1 - \beta^2}\right],\\ &= \frac{(1 - \beta)}{\beta} \frac{ 3 \beta + \beta^2}{1 - \beta^2},\\ &= \frac{(1 - \beta)}{\beta} \frac{(3 + \beta) \beta}{(1-\beta)(1+\beta)},\\ &= \frac{3 + \beta}{1 + \beta}. \end{align*} $$

We have that $c_v^2$ will be equal to $\dfrac{1 + 3 \beta}{1 + \beta}$.