Optimal Consumption Path

Macro

How to understand the Optimal Consumption Path? From the ECON4004 Advanced Macroeconomics, we use the Hamiltonian to solve the households’ Consumption decisions. The typical Hamiltonian function $H$ looks like:

$$H=\ln C+\lambda (rK+wL-C)$$

The term next to $\lambda$ is the Asset-accumulation Equation, where

$$\dot{K}=rK+wL-C$$

noted that $\dot{K}=\frac{\delta K}{\delta t}$ is the change in the level of capital with respect to time $t$. When $\dot{K}=0$, then we could consider this as the Steady State.

After taking the first order condition of Hamiltonian, we get:

$$\begin{array}{rl}\frac{\delta H}{\delta C}& =\frac{1}{C}-\lambda =0\\ \frac{\delta H}{\delta K}& =\lambda r=\lambda \rho -\dot{\lambda }\end{array}$$

Since $\frac{1}{C}=\lambda$, we take the log of these two terms, and we get:

$$\ln C=-\ln \lambda$$

we take the derivative of these two terms in respect to $t$:

$$\frac{\delta \ln C}{\delta t}=\frac{1}{C}\frac{\delta C}{\delta t}=\frac{\dot{C}}{C}$$

same for $\delta \lambda$, so

$$\frac{\dot{C}}{C}=-\frac{\dot{\lambda }}{\lambda }$$

Then we could get

$$\frac{\dot{C}}{C}=r-\rho$$

And this is the Optimal Consumption Path. The intuition is that, if $r>\rho$, this means that the return to capital is higher than the discount rate $\rho$. So the optimal behavior is to save more and consume less today.