Neoclassical Growth Theory

The assets owned at the end of period t are denoted by $a_t$. The period budget constraint for period 1 is $c_1 + a_1 = \omega_1 + (1 + r)a_0$, and for period 2 is $c_2 + a_2 = \omega_2$. The single budget constraint at present value is $c_1 + \frac{c_2}{1 + r} = \frac{\omega_1}{1 + r} + \omega_2.

The law of motion for wealth/assets is of the form $a_t = c_t + (1 + r)a_{t-1}$. The missing restriction in this consumption problem is the transversality condition, which states that $\lim_{t \to \infty} \lambda_t a_t = 0$, where $\lambda_t$ is the shadow price of wealth in period t.

To solve the optimization problem using the Lagrangian method, we construct the Lagrangian function: $L(c_1, c_2, \lambda) = \ln(c_1) + \frac{1}{1+\rho} \ln(c_2) + \lambda \left(c_1 + \frac{c_2}{1 + r} - \frac{\omega_1}{1 + r} - \omega_2\right)$. Then, we take the first-order conditions by differentiating L with respect to $c_1$, $c_2$, and $\lambda$, and setting the derivatives equal to 0.