Let the typical individual maximizes lifetime utility function be:

where

Flow budget constraint is given by:

Question: Find the growth rate of expenditure

Solution so far:

Now, I look ahead in the paper, and I see

First, I set up the Hamiltonian for this problem as follows:

I assume that , consumption is the same as expenditure here.

So I have

By the Pontryagin's Maximum Principle, we have the following conditions:

i. and ii.

Now, by i., I know that .

So I then have

Taking the derivative with respect to time, I have:

by ii., I have

So I guess somehow , the interest rate?

In the attachment I have a copy of the paper that this equation is on, they are on page 4 of the pdf file.

Thank you very much!!!