Stephanie owns a small warehouse that is insured for $$200,000$ under a commercial property insurance policy. The policy contains an $80$ percent coinsurance clause. The warehouse sustained a $$50,000$ loss because of a fire in a storage area. The replacement cost of the warehouse at the time of loss is $$500,000.$

a$.$ What is the insurer's liability, if any, for this loss? Show your calculations.

b$.$ Assume that Stephanie carried $$500,000$ of property insurance on the warehous at the time of loss. If the amount of loss is $$10,000,$ how much will she collect?

c$.$ Explain the theory or rationale of coinsurance in a property insurance contract.