# CES utility maximization two goods two period

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!

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.