<p>The problem can be solved using two stage budgetting. In stage 1 total income <span class="math-container">$m = \sum_t p_{at} c_{at} + \sum_t p_{bt} c_{bt}$</span> is allocated across periods. In stage two the optimal expenditure <span class="math-container">$E_t$</span> in period <span class="math-container">$t$</span> is divided between <span class="math-container">$c_{at}$</span> and <span class="math-container">$c_{bt}$</span>.</p>
<p>This second stage problem can be written as:
<span class="math-container">$$
\max \left((c_{at})^\rho + (c_{bt})^\rho\right)^{\frac{1}{\rho}} \text{ subject to } p_{at} c_{at} + p_{bt} c_{bt} = E_t
$$</span>
This gives:
<span class="math-container">$$
((c_{at})^\rho + (c_{bt})^\rho)^{\frac{1 - \rho}{\rho}} (c_{at})^{\rho - 1} = \lambda p_{at},\\
((c_{at})^\rho + (c_{bt})^\rho)^{\frac{1 - \rho}{\rho}} (c_{bt})^{\rho - 1} = \lambda p_{bt},\\
$$</span>
Then:
<span class="math-container">$$
c_{at} = c_{bt}\left(\frac{p_{at}}{p_{bt}}\right)^{\frac{1}{\rho - 1}}
$$</span>
This gives:
<span class="math-container">$$
p_{at} c_{at} = p_{bt} c_{bt} \left(\frac{p_{at}}{p_{bt}}\right)^{\frac{\rho}{\rho - 1}}
$$</span>
If we substitute into the budget constraint, we obtain:
<span class="math-container">$$
p_{bt} c_{bt}\left(1 + \left(\frac{p_{at}}{p_{bt}}\right)^{\frac{\rho}{\rho - 1}}\right) = E_t,\\
\to c_{bt}\left((p_{bt})^{\frac{\rho}{\rho - 1}} + (p_{at})^{\frac{\rho}{\rho - 1}}\right) = p_{bt}^{\frac{1}{\rho - 1}} E_t
$$</span>
Similarly:
<span class="math-container">$$
c_{at}\left((p_{bt})^{\frac{\rho}{\rho - 1}} + (p_{at})^{\frac{\rho}{\rho - 1}}\right) = p_{at}^{\frac{1}{\rho - 1}} E_t
$$</span>
Then:
<span class="math-container">$$
(c_{at})^\rho + (c_{bt})^\rho = E_t^\rho \left((p_{at})^\frac{\rho}{\rho - 1} + (p_{at})^\frac{\rho}{\rho - 1}\right)^{1 - \rho}
$$</span>
Then:
<span class="math-container">$$
\left((c_{at})^\rho + (c_{bt})^\rho\right)^{\frac{1}{\rho}} = E_t \left((p_{at})^\frac{\rho}{\rho - 1} + (p_{at})^\frac{\rho}{\rho - 1}\right)^{\frac{1 - \rho}{\rho}}
$$</span>
Now define the period <span class="math-container">$t$</span> price index:
<span class="math-container">$$
(P_t)^{-1} = \left((p_{at})^\frac{\rho}{\rho - 1} + (p_{at})^\frac{\rho}{\rho - 1}\right)^{\frac{1 - \rho}{\rho}}
$$</span>
Given this optimal withing period allocation, we can solve the first stage problem:
<span class="math-container">$$
\max \sum_t \beta^t (P_t)^{-1} E_t \text{ s.t. } \sum_t E_t = \sum_{t} p_{at} e_{at} + \sum_t p_{bt}e_{bt}
$$</span>
This is an optimization problem with perfect subsitutes. So all income will be allocated to the period where <span class="math-container">$\beta^t (P_t)^{-1}$</span> is highest.</p>

CC BY-SA 4.0
2021-10-31T12:08:45.090Z
2021-10-31T12:08:45.090Z
tdm
<p>In an Arrow-Debreu economy, there are two periods and N identical agents.
In each period, the agent consumes two goods <span class="math-container">$c_{At}$</span>, <span class="math-container">$c_{Bt}$</span> where <span class="math-container">$ t = 0,1 $</span> and has the endowments <span class="math-container">$(e_{a0},e_{b0},e_{a1},e_{b1})$</span></p>
<p>The agent's utility function is <span class="math-container">$U(c_{a0},c_{b0}) + \beta U(c_{a1},c_{b1}) $</span> where <span class="math-container">$U$</span> follows the CES form: <span class="math-container">$U(c_{at},c_{bt}) = (c_{at}^p + c_{bt}^p)^{1/p}$</span>.</p>
<p>I need to find the competitive equilibrium of the economy.</p>
<p>So as usual I set up the Lagrangian and try to find the FOCs:
<span class="math-container">$ L= (c_{a0}^p + c_{b0}^p)^{1/p} + \beta (c_{a1}^p + c_{b1}^p)^{1/p} + \lambda(p_{a0}c_{a0} + p_{b0}c_{b0} + p_{a1}c_{a1} + p_{b1}c_{b1} - p_{a0}e_{a0} - p_{b0}e_{b0} - p_{a1}e_{a1} - p_{b1}e_{b1}) $</span></p>
<p>After solving for the FOCs, I have:
<span class="math-container">$$(c_{a0})^{p-1}(c_{a0}^p + c_{b0}^p)^{(1-p)/p} = -\lambda p_{0a} $$</span>
<span class="math-container">$$(c_{b0})^{p-1}(c_{a0}^p + c_{b0}^p)^{(1-p)/p} = -\lambda p_{0b}$$</span>
<span class="math-container">$$\beta (c_{a1})^{p-1}(c_{a1}^p + c_{b1}^p)^{(1-p)/p} = -\lambda p_{1a}$$</span>
<span class="math-container">$$\beta(c_{b1})^{p-1}(c_{a1}^p + c_{b1}^p)^{(1-p)/p} = -\lambda p_{1b}$$</span>
<span class="math-container">$$p_{a0}c_{a0} + p_{b0}c_{b0} + p_{a1}c_{a1} + p_{b1}c_{b1} = p_{a0}e_{a0} + p_{b0}e_{b0} + p_{a1}e_{a1} + p_{b1}e_{b1} $$</span>
After dividing the first equation by the second one, and the third by the fourth one I have:
<span class="math-container">$$ c_{0b} = c_{0a}(\frac{p_{0a}}{p_{0b}})^{-1/(p-1)} $$</span>
<span class="math-container">$$ c_{1b} = c_{1a}(\frac{p_{1a}}{p_{1b}})^{-1/(p-1)} $$</span></p>
<p>However, this is not enough to plug in the budget constraint to find the optimal consumption. I have also tried dividing the first FOC by the third one (and then substituting the two relationships I have above) but it has not produced anything meaningful for me either.</p>
<p>Can someone give me some guidance on how to proceed? Thank you!</p>

<utility><competitive-equilibrium>
CES utility maximization two goods two period
sucksatmath
# CES utility maximization two goods two period

(Asked by: username_0 on 2021-10-30)

In an Arrow-Debreu economy, there are two periods and N identical agents.
In each period, the agent consumes two goods $c\_{At}$, $c\_{Bt}$ where $ t = 0,1 $ and has the endowments $(e\_{a0},e\_{b0},e\_{a1},e\_{b1})$

The agent's utility function is $U(c\_{a0},c\_{b0}) + \beta U(c\_{a1},c\_{b1}) $ where $U$ follows the CES form: $U(c\_{at},c\_{bt}) = (c\_{at}^p + c\_{bt}^p)^{1/p}$.

I need to find the competitive equilibrium of the economy.

So as usual I set up the Lagrangian and try to find the FOCs:
$ L= (c\_{a0}^p + c\_{b0}^p)^{1/p} + \beta (c\_{a1}^p + c\_{b1}^p)^{1/p} + \lambda(p\_{a0}c\_{a0} + p\_{b0}c\_{b0} + p\_{a1}c\_{a1} + p\_{b1}c\_{b1} - p\_{a0}e\_{a0} - p\_{b0}e\_{b0} - p\_{a1}e\_{a1} - p\_{b1}e\_{b1}) $

After solving for the FOCs, I have:
$$(c\_{a0})^{p-1}(c\_{a0}^p + c\_{b0}^p)^{(1-p)/p} = -\lambda p\_{0a} $$
$$(c\_{b0})^{p-1}(c\_{a0}^p + c\_{b0}^p)^{(1-p)/p} = -\lambda p\_{0b}$$
$$\beta (c\_{a1})^{p-1}(c\_{a1}^p + c\_{b1}^p)^{(1-p)/p} = -\lambda p\_{1a}$$
$$\beta(c\_{b1})^{p-1}(c\_{a1}^p + c\_{b1}^p)^{(1-p)/p} = -\lambda p\_{1b}$$
$$p\_{a0}c\_{a0} + p\_{b0}c\_{b0} + p\_{a1}c\_{a1} + p\_{b1}c\_{b1} = p\_{a0}e\_{a0} + p\_{b0}e\_{b0} + p\_{a1}e\_{a1} + p\_{b1}e\_{b1} $$
After dividing the first equation by the second one, and the third by the fourth one I have:
$$ c\_{0b} = c\_{0a}(\frac{p\_{0a}}{p\_{0b}})^{-1/(p-1)} $$
$$ c\_{1b} = c\_{1a}(\frac{p\_{1a}}{p\_{1b}})^{-1/(p-1)} $$

However, this is not enough to plug in the budget constraint to find the optimal consumption. I have also tried dividing the first FOC by the third one (and then substituting the two relationships I have above) but it has not produced anything meaningful for me either.

Can someone give me some guidance on how to proceed? Thank you!

**Tags:** utility, competitive-equilibrium

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## 1 answers

### Answer by username_1 on 2021-10-31. Score: 3

The problem can be solved using two stage budgetting. In stage 1 total income $m = \sum\_t p\_{at} c\_{at} + \sum\_t p\_{bt} c\_{bt}$ is allocated across periods. In stage two the optimal expenditure $E\_t$ in period $t$ is divided between $c\_{at}$ and $c\_{bt}$.

This second stage problem can be written as:
$$
\max \left((c\_{at})^\rho + (c\_{bt})^\rho\right)^{\frac{1}{\rho}} \text{ subject to } p\_{at} c\_{at} + p\_{bt} c\_{bt} = E\_t
$$
This gives:
$$
((c\_{at})^\rho + (c\_{bt})^\rho)^{\frac{1 - \rho}{\rho}} (c\_{at})^{\rho - 1} = \lambda p\_{at},\\
((c\_{at})^\rho + (c\_{bt})^\rho)^{\frac{1 - \rho}{\rho}} (c\_{bt})^{\rho - 1} = \lambda p\_{bt},\\
$$
Then:
$$
c\_{at} = c\_{bt}\left(\frac{p\_{at}}{p\_{bt}}\right)^{\frac{1}{\rho - 1}}
$$
This gives:
$$
p\_{at} c\_{at} = p\_{bt} c\_{bt} \left(\frac{p\_{at}}{p\_{bt}}\right)^{\frac{\rho}{\rho - 1}}
$$
If we substitute into the budget constraint, we obtain:
$$
p\_{bt} c\_{bt}\left(1 + \left(\frac{p\_{at}}{p\_{bt}}\right)^{\frac{\rho}{\rho - 1}}\right) = E\_t,\\
\to c\_{bt}\left((p\_{bt})^{\frac{\rho}{\rho - 1}} + (p\_{at})^{\frac{\rho}{\rho - 1}}\right) = p\_{bt}^{\frac{1}{\rho - 1}} E\_t
$$
Similarly:
$$
c\_{at}\left((p\_{bt})^{\frac{\rho}{\rho - 1}} + (p\_{at})^{\frac{\rho}{\rho - 1}}\right) = p\_{at}^{\frac{1}{\rho - 1}} E\_t
$$
Then:
$$
(c\_{at})^\rho + (c\_{bt})^\rho = E\_t^\rho \left((p\_{at})^\frac{\rho}{\rho - 1} + (p\_{at})^\frac{\rho}{\rho - 1}\right)^{1 - \rho}
$$
Then:
$$
\left((c\_{at})^\rho + (c\_{bt})^\rho\right)^{\frac{1}{\rho}} = E\_t \left((p\_{at})^\frac{\rho}{\rho - 1} + (p\_{at})^\frac{\rho}{\rho - 1}\right)^{\frac{1 - \rho}{\rho}}
$$
Now define the period $t$ price index:
$$
(P\_t)^{-1} = \left((p\_{at})^\frac{\rho}{\rho - 1} + (p\_{at})^\frac{\rho}{\rho - 1}\right)^{\frac{1 - \rho}{\rho}}
$$
Given this optimal withing period allocation, we can solve the first stage problem:
$$
\max \sum\_t \beta^t (P\_t)^{-1} E\_t \text{ s.t. } \sum\_t E\_t = \sum\_{t} p\_{at} e\_{at} + \sum\_t p\_{bt}e\_{bt}
$$
This is an optimization problem with perfect subsitutes. So all income will be allocated to the period where $\beta^t (P\_t)^{-1}$ is highest.


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