Suppose that potential used-car buyers value high quality used cars at \(€ 8,000\) and low-quality used cars at \(€ 3,000\). Owners of high-quality used cars have a reservation price of \(€ 6,500\), while the reservation price for owners of low-quality used cars is \(€ 2,000\). Everyone is risk neutral. The share of current owners who have low-quality cars is \(\theta\).

a. For what values of \(\theta\) do all the potential sellers sell their used cars?

b. If each potential buyer incurs a transaction cost to purchase a used car (the value of the time spent searching) of \(t=€ 500\), for what values of \(\theta\) do all the potential sellers sell their used cars?

c. If \(\theta\) equals the maximum value calculated in part a, would all potential sellers sell their used cars when transaction costs equal \(€ 500\) per person? Would any of the potential sellers sell?