CES utility maximization two goods two period

Question

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!

Answer 1

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.