Many economics professors choose to wear protective helmets to keep their colleagues from stealing their research ideas.  The trouble is that these helmets contain some exotic and highly toxic substances that need to be disposed of properly when the helmet is retired.  To assure that the helmets are properly dismantled, there is a subsidy for proper disposal. Making a new protective helmet costs $200.  Proper disposal, where all the toxic stuff is correctly handled and processed, costs $800.  However, there is an underground trade in improper disposal of helmets, and the going price for this is $500. Into what range must the subsidy for proper disposal fall in order to provide sufficient incentive for proper disposal but not so much that economists start buying the helmets just to dispose of them and collect the subsidy, which they would totally do?  Truth be told, most economists don’t wear the helmets for most of their careers.  Their ideas are usually only good at the start of their careers.  After 15 years or so, most of their ideas get stale and nobody is interested in reading or publishing their research, even when they work really, really hard on it.  Anyway, imagine that upon purchasing a helmet an economist pays a deposit of $700.  Calculate the present value of the $700 if it will be returned at the end of 15 years if the future is discounted using a rate of 5%.  Please show your calculations.