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At time $(t-1)$, the investor buys some risk free bond, $B_{t-1}$ and some risky asset $X_{t-1}$ at price $P_{t-1}$, such that the budget constraint holds, i.e. $$W_{t-1} = B_{t-1} + P_{t-1}X_{t-1}$$ At period $t$, one unit of risk free bond pays off one unit, so $B_{t-1}$ units of risk free bonds pays off $B_{t-1}$ units of wealth at $t$. For a risky ...
By definition, a direct mechanism is a mechanism that asks all agents for their types and then produces some outcome. Formally, it is a mechanism $\langle M,g\rangle$ in which $M_i$ is wlog equal to $i$'s type space, usually denoted by $\Theta_i$. In your example, there is no such restriction. However, for any indirect mechanism there exists an equivalent ...