General equilibrium - exchange economy with time and perishable goods
Of course, note that in general, consumer B is willing to increase her consumption in period 1. it is a matter of simply solving the problem in the regular way. i.e, MRS=p and find the $p$ which "clears" the market.

Normalize the price of the current period good to $1$ and let $p$ be the price of the second period good, then the optimum for player $A$ is:
$\max_{x_{A1},x_{A2}} \sqrt{x_{A1}}+\sqrt{x_{A2}}$ s.t $x_{A1}+px_{A2}\le 1$
Clearly the constraint is binding and thus the problem becomes:
$\max_{x_{A1}} \sqrt{x_{A1}}+\sqrt{\frac{1-x_{A1}}{p}}$
Calculate the first order condition, repeat the process for consumer B etc...