Sunk Costs $($Exercise $2\mathrm{.}6$ in Luenberger$)$

A young couple has made a deposit of the first month's rent $($equal to $\backslash (\backslash$$ $1,000\backslash ))$ on a $6-$month apartment lease. The deposit is refundable at the end of six months if they stay until the end of the lease.

The next day they find a different apartment that they like just as well, but its monthly rent is only $\backslash (\backslash$$ $900\backslash ).$ And they would again have to put a deposit of $\backslash (\backslash$$ $900\backslash )$ refundable at the end of $6$ months.

They plan to be in the apartment only $6$ months. Should they switch to the new apartment? Assume an $($admittedly unrealistic!$)$ interest rate of $\backslash (12\backslash \%\backslash )$ per month compounded monthly.

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