Question 672249: Groucho and Harpo work together in the same office. Groucho smokes; Harpo hates smoke.
Groucho currently smokes 12 cigars per day. He faces marginal costs (withdrawal pains) of
reducing smoking equal to $x where x is the number of cigars reduced (not smoked). In other
words, Groucho’s marginal cost of smoke abatement is: MC = x. Harpo receives marginal
benefits (reduced discomfort and improved health) equal to $(12-x) when Groucho reduces
his cigar smoking. In other words, Harpo’s marginal benefit from smoke abatement is:
MB = 12-x . In addition to reducing the number of cigars smoked in the office, it is possible
to rent an air purifier that reduces smoke in the air by 50% for $10 per day. It is also possible
to rent a separate office for Harpo so he does not have to breathe any smoke for $40 per day.
a. Draw a diagram showing the marginal costs and benefits of pollution reduction with the
number of cigars reduced on the horizontal axis. Use this diagram to determine the
efficient number of cigars assuming the air purifier and office rentals are not allowed.
b. Suppose instead that the air purifier is installed. What is the efficient number of cigars
now? (Hint: the new marginal benefit curve is: MB =0.5 (12-x).
c. Suppose that either option (air purifier or office rental) is allowed. What is the most
efficient outcome: air purifier rental, office rental, or neither?
d. What would be the outcome (number of cigars smoked, if any; air purifier or office
rental, if any) if Groucho and Harpo’s boss were to declare the office a non-smoking area
but allow them to negotiate a mutually agreeable solution? What would happen if
Groucho were granted the right to smoke and then negotiations were permitted? (Hint: if
you properly invoke the Coase Theorem you do not need to perform any additional
calculations to answer these two questions).
Here is what I did:
a.
MC = MB
MC = x
MB = 12-x
so: x = 6 cigars
b. 6-0.5x = x
x = 4 cigars
c. Air purifier 50% costs: $10
MB = integral 6 to 0 (12-x) plus integral 12 to 0 (6-0.5x).
Net benefit = 63
but neither.
d. NO IDEA.
Please help!
Answer by MathLover1(20849)   (Show Source):
You can put this solution on YOUR website! Here's the graph:
see image 1.
W3C Web site
First note that with no law against smoking, Groucho will smoke  cigars a day, since smoking any less is costly to him. It is evident from the graph that the efficient level of smoking equals  cigars.
Alternately, since the efficient outcome will occur where  , you can solve the two equations for  :
=>  =>  => 
2. The clean air machine,by reducing the smoke from each cigar, also reduces
the marginal benefits of clean-up by half for each cigar reduced. Thus for the first cigar reduced, the benefits are (not ) and for the second they are  , not  .
This new  schedule changes the graph to look like:
see image 2.
W3C Web site
Now, the efficient level of cigars reduced is  , with  smoked. Algebraically, the new schedule is  , so now:
=>  =>  => => 
3. We know that with machine rental, cigars will be reduced. Suppose
Groucho has the right to smoke, and so the status quo has him smoking a day.
Harpo has three choices:
a.   the and not relocate Groucho,  Groucho  
b.  the machine and  pay Groucho not to smoke
c.  to  Groucho
Which would he choose?
Not renting the machine requires Harpo tocompensate Groucho for the 
of reducing cigars:  . Harpo’s benefits are
 dollars, so net benefits are  dollars.
Machine rental costs $ , plus Harpo has to pay Groucho at least his  of reduction for the first four cigarettes. The costs of machine rental are thus:
Relative to the status quo of cigars smoked, the benefits of machine
rental have two components.
First, Brittany gets the full benefits of seeing cigarettes reduced:
Second, she gets the  % improvement in air quality for the remaining  cigars that are smoked:
 dollars
So, net benefits are  dollars
Finally, relocation costs $ , and brings Harpo the total area under the
 curve in benefits. (Since Groucho can smoke all he wants after relocation, he bears no costs). Therefore, the net benefits to Harpo of paying for relocation are:
$
So the   is machine  .
4. Rather than be banished, Groucho would rent the machine, then pay Harpo
to allow him to stayand smoke cigars. If Groucho rents the machine, he would pay the following costs:
$ for machine rental, plus
$ in nicotine
withdrawals on cigars given up, plus $ in bribes to Harpo to allow him to smoke cigarettes
The total is $ , less than the $ required to relocate.
So rather than banish Groucho,Harpo could come out ahead by demanding a little more than the $ just necessary for him to allow Groucho to smoke his cigars.
This again demonstrates the Coase theorem: in the absence of transaction costs,
efficient outcomes will be achieved regardless of the initial distribution of property
rights.
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