Assume that the economy consists of two infinitely lived consumers named $i=1,2.$ There is one nonstorable consumption good. Consumer i consumes $c_t^i$ at time $t.$ Consumer $i\text{'}$s preferences over consumption streams are represented by

$E_0{\displaystyle \underset{t=0}{\overset{}{}}}{}^{t}log(c_t^i),in(0,1)$

The endowments are decided each period by a toss of a fair coin. If the outcome of the coin toss is heads $(H),$ consumer $1$ receives one unit of the consumption good, and consumer $2$ receives nothing. If the outcome is tails $(T),$ consumer $2$ receives one unit of the consumption good, and consumer $1$ receives nothing.

$($i$)$ Let $s_t=\{H,T\}$ denote the state of the economy at date $t.q_t^0(s_t)$ denotes the time $0$ price of an Arrow$-$Debreu security that pays one unit of consumption in date $t$ conditional on state $s_t,$ and $c_t^i(s_t)$ denotes consumer $i^{\text{'}}$ s consumption at date $t$ in state $s_t.$ Define an Arrow$-$Debreu competitive equilibrium.

$($ii$)$ Compute the equilibrium under the assumption that Arrow$-$Debreu securities are traded in time $0$ after the realization $s_0=H.$ Explain your answer