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Consider an economy with two types of consumers labeled h and l. Consumers have preferences for current and future consumption given by:
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uh (and!) = lnc;l + ﬁhlnc;I u; (01,02) = lnc; + ,8; Inc; Some consumers are very patient and have a high ,8,“ others are impatient and have a low 6;(ﬁt < 3h). Note that 3 is the weight in the future consumption, or the discount rate. Consumersare equal in their income and there are no taxes (that is, t = t = O). The government does not have to ﬁnance any spending.There are N consumers of each type in this economy. That is, in total there are 2N consumers, N of them are type it and N are type l.I recommend that you make a change of variable to solve this problem: instead of working with the net interest rate (7“), work with the gross interest rate R = 1 + ‘r.