Consider a world of complete information with two agents X and Y and two time periods 1 and 2.
Person X only lives in second period.
Person Y lives in 1st and 2nd periods both.
X and Y are each endowed with and exogenous income I which can be allocated between consumption in both periods.
$s_Y$ = Y’s saving and $0\le s_Y \le I$
$c_{1Y} $ and $c_{2Y} $ are the agent Y’s first and second period consumptions respectively.
The A gent X’s preferences are altruistic for Y. After observing the saving of Y, $sY$,X determine how much his endowment $tX$ to transfer to Y for $t_x\in [0,I]$
$c_{2X}$ the agent X’s consumption in period 2.
Utility functions Y and X are respectively
$$V_Y= ln(C_{1Y}) + bln(C_{2Y})$$
$$V_X=ln(C_{2X})+a *V_Y$$
where a is positive and $a*b\ge 1$ and $b\in (0,1)$
What is the su game perfect equilibrium levels for $t_X$ and $s_Y$?