The intertemporal budget constraint expresses the balance between the household's consumption and income over time, taking into account the possibility of borrowing and saving. It can be expressed as:
$$\sum_{t=0}^{\infty} \frac{C_{i,t}}{(1+r)^t} = \sum_{t=0}^{\infty} \frac{Y_{i,t}}{(1+r)^t} + \frac{A_i}{(1+r)^0}$$
where:
- $A_i$ is the initial level of the household i's assets, which can be positive (if the household has savings) or negative (if the household has debt).
- $Y_{i,t}$ is the income of the housold i at time t
- $r$ is the interest rate
To solve the optimization problem of maximizing lifetime utility subject to the intertemporal budget constraint, we can use the method of Lagrange multipliers. The Lagrangian can be written as:
$$\mathcal{L} = \sum_{t=0}^{\infty} e^{ -\rho t}\frac{1}{1-\theta} \left(\frac{C_{i,t}}{C_{t-1}^{\phi}} \right)^{1-\theta} + \lambda \left(\sum_{t=0}^{\infty} \frac{C_{i,t}}{(1+r)^t} - \sum_{t=0}^{\infty} \frac{Y_{i,t}}{(1+r)^t} - \frac{A_i}{(1+r)^0} \right)$$
Taking the first-order conditions with respect to $C_{i,t}$ and $A_i$, we obtain:
$$\frac{\partial \mathcal{L}}{\partial C_{i,t}} = e^{ -\rho t}\frac{1}{1-\theta} \left(\frac{C_{i,t}}{C_{t-1}^{\phi}} \right)^{-\theta} (1-\theta) \frac{1}{C_{t-1}^{\phi}} + \frac{\lambda}{(1+r)^t} = 0$$
$$\frac{\partial \mathcal{L}}{\partial A_i} = -\frac{\lambda}{(1+r)^0} = 0$$
Solving for $\lambda$ and substituting into the first-order condition for consumption, we obtain the Euler equation:
$$\frac{e^{ -\rho t}}{C_{i,t}} = \frac{1+r}{1-\theta} \left(\frac{C_{i,t+1}}{C_{i,t}^{\phi}} \right)^{1-\theta}$$
This equation relates the marginal utility of consumption at time t to the expected marginal utility of consumption at time t+1, discounted by the interest rate. It implies that the household will adjust its consumption over time to equalize the marginal utility of consumption across periods, taking into account the opportunity cost of consuming today (in terms of foregone interest earnings) and the effect of consumption on the reference level of consumption $C_{t-1}$.
To solve for the optimal consumption path, we can use the method of dynamic programming. The household's problem can be written as:
$$V(A_{i,t}) = \max_{C_{i,t}} \left(e^{ -\rho t}\frac{1}{1-\theta} \left(\frac{C_{i,t}}{C_{t-1}^{\phi}} \right)^{1-\theta} + \beta V(A_{i,t+1}) \right)$$
subject to the intertemporal budget constraint. Here, $V(A_{i,t})$ represents the value function, which is the maximum lifetime utility that household i can achieve given its initial level of assets $A_{i,t}$. The parameter $\beta = \frac{1}{1+r}$ is the discount factor, which determines the relative weight placed on future utility compared to present utility.
To solve for the optimal consumption path, we can use the method of backward induction. Starting from the last period T, we can solve for the optimal consumption choice $C_{i,T}$ that maximizes the objective function subject to the intertemporal budget constraint. We can then use this optimal consumption choice to solve for the value function $V(A_{i,T-1})$ in the second-to-last period T-1, by substituting $C_{i,T}$ into the objective function and maximizing with respect to $C_{i,T-1}$. This process is repeated for each period t=1,2,...,T-1, until we arrive at the initial period t=0 and obtain the optimal consumption path $C_{i,0}$, which maximizes the lifetime utility of household i subject to the intertemporal budget constraint.
The optimal consumption path satisfies the Euler equation, which implies that the ratio of marginal utilities of consumption in adjacent periods is equal to the intertemporal discount factor:
$$\frac{MU_{i,t}}{MU_{i,t+1}} = \frac{1}{1+r}$$
where MU_{i,t} is the marginal utility of consumption at time t. This equation implies that the household will adjust its consumption over time to equalize the marginal utility of consumption across periods, taking into account the opportunity cost of consuming today (in terms of foregone interest earnings) and the effect of consumption on the reference level of consumption $C_{t-1}$.
