For a finite-horizon discrete time optimization problem, my textbook provides a condition called the "bequest condition", which I'm not familiar with. Specifically, where the state at time $t$ is denoted $x_t$, the bequest condition states that at the final state, $x_T$, it must hold that $x_T\geq \bar x$. What does this mean?
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It is simply the terminal condition of the model. You do not specify the variable that x represents, but if it were, for example, capital, this would simply state that in the last period (since horizon is finite) there is a condition that states that capital must be greater or equal to some given amount (bequest --> legacy, you might find some sense in that word). Again, you do not specify the model, but normally these types of conditions relate to either a simplifying assumption that allows you to solve models using certain methods (for example undetermined coefficient, iterative solution methods, etc.), or to some logical notion.