I'm trying to solve a two-period two-good consumption problem. Endowment in the two periods is $w_1$ and $w_2$, the interest rate is $\rho$, and total utility is $$ U(x_1, y_1) + \beta U(x_2, y_2)\ . $$

So far I have that for each $t$, total consumption is given by $$ C_t = x_t p_{xt} + y p_{yt}\ , \qquad \qquad (1) $$ and the budget constraint is $$ C_2 = w_2 + (1 + \rho)(w_1 - C_1)\ . \qquad \qquad (2) $$ So in the end, I need to solve $$ \max_{x_1, y_1, x_2, y_2} U(x_1, y_1) + \beta U(x_2, y_2) $$ subject to (1) and (2).

I'm having trouble getting the optimisation off the ground. If I write down the Lagrangian and take derivatives I don't get anything approximating the Euler equation $$ u'(C_1) = \beta(1 + \rho)u'(C_2) \qquad \qquad (3) $$ which I see everywhere for these types of problems.

My intuition tells me that I should be able to solve this problem in two stages. If I define $u$ to be the indirect utility function $$ u(C) = U(x^*(C, p_{x}, p_{y}), y^*(C, p_{x}, p_{y}))\ , $$ where $x^*$ and $y^*$ are the (still unknown) optimal consumption rules, then I should be able to

1. optimally allocate money over the two periods by solving (3) subject to (2);
2. optimally consume in each period by solving $$ \frac{\partial_x U}{p_{xt}} = \frac{\partial_y U}{p_{yt}} $$ subject to (1).

Is this correct? If so, how can I formalise this argument? If not, what am I missing? References appreciated!

I'm trying to solve a two-period two-good consumption problem. Endowment in the two periods is $w_1$ and $w_2$, the interest rate is $\rho$, and total utility is $$ U(x_1, y_1) + \beta U(x_2, y_2)\ . $$

So far I have that for each $t$, total consumption is given by $$ C_t = x_t p_{xt} + y p_{yt}\ , \qquad \qquad (1) $$ and the budget constraint is $$ C_2 = w_2 + (1 + \rho)(w_1 - C_1)\ . \qquad \qquad (2) $$ So in the end, I need to solve $$ \max_{x_1, y_1, x_2, y_2} U(x_1, y_1) + \beta U(x_2, y_2) $$ subject to (1) and (2).

I'm having trouble getting the optimisation off the ground. If I write down the Lagrangian and take derivatives I don't get anything approximating the Euler equation $$ u'(C_1) = \beta(1 + \rho)u'(C_2) \qquad \qquad (3) $$ which I see everywhere for these types of problems.

My intuition tells me that I should be able to solve this problem in two stages. If I define $u$ to be the indirect utility function $$ u(C) = U(x^*(C, p_{x}, p_{y}), y^*(C, p_{x}, p_{y}))\ , $$ where $x^*$ and $y^*$ are the (still unknown) optimal consumption rules, then I should be able to

1. optimally allocate money over the two periods by solving (3) subject to (2);
2. optimally consume in each period by solving $$ \frac{\partial_x U}{p_{xt}} = \frac{\partial_y U}{p_{yt}} $$ subject to (1).

Is this correct? If so, how can I formalize this argument? If not, what am I missing? References appreciated!