How the authors derived these equations ?

[goldiecr12]

I am doing a paper on savings optimization and I would like to understand how the authors of the paper managed to solve the optimization problem in the following two periods. I would like to understand each step.

 

[Deleted member 4993]

I am doing a paper on savings optimization and I would like to understand how the authors of the paper managed to solve the optimization problem in the following two periods. I would like to understand each step.
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View attachment 27150

EXACTLY where are you stuck?

 

[JeffM]

I have looked at this on and off during the day. I am having trouble with the first equation.

If all of the savings are placed in the tax-sheltered option (meaning phi is 1), then utility in period 1 is

[MATH]u_1( y - s + \tau_1s) = u_1(y - (1 - \phi \tau_1)s).[/MATH]
And utility in the second period is

[MATH]u_2( \psi y + Rs - \tau_2Rs) = u_2( \psi y + (1 - \phi \tau_2) Rs).[/MATH]
Looks good if phi = 1.

But if none of the savings are placed in the tax-sheltered option (meaning phi is 0), then utility in the first period is still

[MATH]u_1(y - s) = u_1(y - (1 - \phi \tau_1)s[/MATH], but utility in the second period is

[MATH]u_2( \psi y + Rs - \tau_2rs ) > u_2( \psi y + Rs - phi \tau_2rs) > u_2(\psi y + (1 - \phi \tau_2)Rs)[/MATH].

Where does equation 1 come from?