The mathematical problem involves maximizing a utility function subject to constraints. Let’s break it down step by step.

Problem Formulation:

We aim to maximize the utility:

$\max \, u(c_t^y) + u(c_{t+1}^o) + v\left(\frac{m_t}{p_t}\right)$

Subject to the constraints:

1. Budget Constraint at Time $t$: $w_t = c_t^y + \frac{m_t}{p_t} + s_{t+1}$ where $w_t$ is total wealth, $c_t^y$ is consumption at time $t$, $m_t$ represents money holdings, $p_t$ is the price level, and $s_{t+1}$ is savings.

2. Budget Constraint at Time $t+1$: $c_{t+1}^o = s_{t+1}(1 + r_{t+1}) + \frac{m_t}{p_{t+1}}$ where $c_{t+1}^o$ is consumption at time $t+1$, $r_{t+1}$ is the interest rate at $t+1$, and savings $s_{t+1}$ grow with the interest rate.

Approach:

1. Lagrangian Formulation: We incorporate the constraints into the utility function using Lagrange multipliers $\lambda_t$ and $\lambda_{t+1}$.

   $\mathcal{L} = u(c_t^y) + u(c_{t+1}^o) + v\left(\frac{m_t}{p_t}\right)$

   + \lambda_{t+1} \left(c_{t+1}^o - s_{t+1}(1 + r_{t+1}) - \frac{m_t}{p_{t+1}}\right)$$

2. First-Order Conditions (FOC):

   Differentiate $\mathcal{L}$ with respect to $c_t^y$, $c_{t+1}^o$, $m_t$, and $s_{t+1}$, and set the derivatives to zero.

   $\frac{\partial \mathcal{L}}{\partial c_t^y} = 0:$ $u'(c_t^y) = \lambda_t$

   $\frac{\partial \mathcal{L}}{\partial c_{t+1}^o} = 0:$ $u'(c_{t+1}^o) = \lambda_{t+1}$

   $\frac{\partial \mathcal{L}}{\partial m_t} = 0:$ $v'\left(\frac{m_t}{p_t}\right)\frac{1}{p_t} = \lambda_t \frac{1}{p_t} - \lambda_{t+1} \frac{1}{p_{t+1}}$

   $\frac{\partial \mathcal{L}}{\partial s_{t+1}} = 0:$ $\lambda_t = \lambda_{t+1}(1 + r_{t+1})$

3. Intertemporal Euler Equation:

   Combine the FOCs to derive relationships between $c_t^y$, $c_{t+1}^o$, and the marginal utility of money. The main equation typically involves balancing the marginal utilities over time and across money holdings.

   • From the FOCs: $u'(c_t^y) = (1 + r_{t+1}) u'(c_{t+1}^o)$

4. Money Demand Condition:

   The FOC with respect to $m_t$ links the marginal utility of real money balances to consumption and intertemporal preferences.

   $v'\left(\frac{m_t}{p_t}\right) = \frac{\lambda_t}{p_t} - \frac{\lambda_{t+1}}{p_{t+1}}$

Solution Strategy:

1. Use the budget constraints to express $c_t^y$, $c_{t+1}^o$, and $m_t$ in terms of exogenous variables ($w_t, p_t, p_{t+1}, r_{t+1}$).
2. Solve the Euler equation and money demand equation for $c_t^y$ and $m_t/p_t$.
3. Substitute back into the constraints to determine savings $s_{t+1}$.

Would you like to proceed with a specific functional form for $u(\cdot)$ or $v(\cdot)$ (e.g., CRRA utility)?

Follow-Up Questions:

1. What functional forms should we assume for $u(\cdot)$ and $v(\cdot)$?
2. Should we compute numerical solutions with given values for $w_t$, $r_{t+1}$, $p_t$, and $p_{t+1}$?
3. Do you want the optimal paths for consumption and savings derived explicitly?
4. Is there a specific interpretation or economic insight you are looking for in the solution?
5. Should we graph the results or perform sensitivity analysis?

Tip:

When solving intertemporal optimization problems, always confirm the transversality conditions to ensure the solution is dynamically consistent.