A young couple has made a nonrefundable deposit of the first month's rent (equal to $1000) on a one-year apartment lease. The next day they find another apartment that they like just as well, but its monthly rent is only $900. They plan to be in the apartment for a year. Should they switch to the new apartment? Assume an (annual) interest rate of 12%.

Question

BlackTom AI · Accepted Answer

To evaluate whether the couple should switch, begin by recognizing that the $1000 deposit on the first month’s rent is a sunk cost once paid, so it should not affect the switch decision.

Option 1 (True): Consider the incremental financial effect of switching. If they stay, their annual rent is 12 payments of $1000, totaling $12,000 for the year. If they switch, their annual rent would be 12 payments of $900, totaling $10,800. The difference is $1,200 in annual savings from moving to the cheaper apartment.

However, since the problem gives an annual interest rate of 12%, we should compare the present value of the savings. The PV of the $1,200 savings over one year at 12% is approximately 1,200 / 1.12 ≈ $1,071.43. This positive PV indicates a net economic benefit to switching when we ignore the already‑paid $1,000 deposit (a sunk cost).

Option 2 (False): This would imply that switching is not worthwhile or that the deposit matters in the decision. The $1,000 already paid is sunk and should not enter the decision; the comparison should be based on the future cash flows and their present value. Ignoring the sunk cost leads to a positive PV of savings, supporting switching.

In summary, the switch yields a positive present value of savings (about $1,071 when discounted at 12%), so the financially rational choice is to switch. The correct reasoning hinges on treating the $1,000 deposit as sunk and evaluating only the incremental rent savings with their time value of money.

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