For question a , How to get S^Dt, give some ideas, Thank you

Domestic Stock and International Stock. There are two different \"types\" of stock that the representative consumer can buy: \"domestic\" stock and \"international\" stock. Denote by 123' the representative consumer's holdings ofdomestic stock at the beginning of period rand by \"51 the representative consumer's holdings of international stock at the beginning of period t. Similarly, let S,\" and S,' denote, respectively, the nominal price of domestic and international stock in period t, and D,\" and D,' denote, respectively, the per-share nominal dividend that domestic and international stock pay in period t. The period-t budget constraint of the domestic representative consumer is thus RC; +5505 +3311? = Y. + (S: + Dilafs + (S? +13.\" ) , in which all of the other notation is standard: Y, denotes nominal income (over which the domestic consumer has no control) in period t, c, is real units of consumption, and If is the nominal price of each unit of consumption. Also as usual, the lifetime utility of the consumer starting from period t onwards is u(c,) + u(c,+,)+ 2u(c,+,)+ 3u(c,+,)+.. . , in which ,6 5 {0,1} is the usual measure of consumer impatience. A restriction that is that is known to everyone (i.e., is common knowledge) at the beginning of period ris that international stock will pay zero dividends through (and including) period 1+6 and will begin paying dividends in period 1+7. Starting in period it, the nominal dividend based on a unit of international stock will be ldentlcal to the nominal dividend based on a unit of domestic stock. More precisely, the dividend payment schedule that is known to everyone is D,' =D" =D' =D' r+l [+2 [+3 = 0 5 I _ D r _ D r _ D r _ o r _ D DI+T 'Drnvns Dr+85Dr+9 Dr+D!---5Dr+47 _D.r+47!D.r+-IS D,+,,s,..... =DIL4=DI =0: .r+5 [+6 Regarding domestic stock, it is known to all parties that D,\" > 0 V 7 2t - that is, nominal dividends based on domestic stocks will always be strictly positive