Two-Period Model with Borrowing Constraint: Optimizing Consumption

How can we optimize consumption in a two-period model with a borrowing constraint?

Given Y1=100, Y2=100, r=0.05, and β=0.95, determine the optimal values of C1 and C2.

If r increases to 0.1 while maintaining Y1=100, Y2=100, and β=0.95, what are the new optimal values of C1 and C2?

How would lifting the borrowing constraint impact the solutions for (1) and (2)?

Solution:

To optimize consumption in a two-period model with a borrowing constraint, we need to follow these steps:

When solving the two-period model with a borrowing constraint, we aim to maximize utility while adhering to the budget constraint. Let's break down the process:

1. Given Y1=100, Y2=100, r=0.05, and β=0.95, we first ignore the borrowing constraint and maximize the utility function: ln(C1) + βln(C2).

2. By setting up the Lagrangian function and solving the first-order conditions, we find the optimal values of C1 and C2 to be approximately C1=204.75 and C2=0.938.

3. If r increases to 0.1, we repeat the optimization process and find the new optimal values of C1 and C2 based on the updated interest rate.

4. When the borrowing constraint is lifted in scenario (3), allowing C1 to exceed Y1, the solutions for optimal consumption in period 1 and period 2 may change. The household can borrow to increase current consumption, potentially leading to adjustments in the allocation of consumption across periods.

Therefore, the lifting of the borrowing constraint can alter the optimal values of C1 and C2, offering the household more flexibility in managing consumption levels in each period.