This problem asks you to compare two streams of discounted utility received at timest= 0,1,2,..., and then to use the result in the analysis of a repeated game. The first stream isS1, the second is S2where

S1=(x,x,x,...,x, x , x , x , x ,...),

S2=(x,x,x,...,x,x+T,xC,xC,xC,...),

T >0,C >0, and thex+Tentry is at the'th component of S2. To be very explict aboutS2, att= 0,1,...,1, the utility received isx, atit isx+T, and at+1 onwards, it isxC. Throughout, the discount factor is denotedand 0< <1.

1. Give the expected discounted utility ofS1. Your answer may not include an infinite summation.

2. Give the expected discounted utility ofS2. Your answer may not include an infinite summation.

3. Show that the expected discounted utilities are equal at=T/(T+C).