Question 672249
Here's the graph: 

see image 1.
<A href="http://imageshack.us/photo/my-images/832/grouchoimage1.jpg/">W3C Web site</A> 


First note that with no law against smoking, Groucho will smoke {{{12}}} cigars a day, since smoking any less is costly to him. It is evident from the graph that the efficient level of smoking equals {{{6}}} cigars.
 
Alternately, since the efficient outcome will occur where {{{MC=MB}}}, you can solve the two equations for {{{x}}}: 

{{{MC=x}}} 

{{{MB=12-x}}}
 
{{{MC=MB}}} => {{{1x=12-x}}} => {{{2x=12}}} =>  {{{x=6}}} 


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 {{{6}}} (not {{{12}}}) and for the second they are {{{5.5}}}, not {{{11}}}. 

This new {{{MB}}} schedule changes the graph to look like:

see image 2.
<A href="http://imageshack.us/photo/my-images/850/grouchoimage2.jpg/">W3C Web site</A> 



Now, the efficient level of cigars reduced is {{{4}}}, with {{{8}}} smoked. Algebraically, the new {{{MB}}} schedule is {{{MB=6-x/2}}}, so now: 

{{{MC=MB}}} => {{{1x=6-x/2}}} => {{{3x/2=6}}} =>{{{3x=12}}} => {{{x=4}}}

3. We know that with machine rental, {{{4}}} cigars will be reduced. Suppose 
Groucho has the right to smoke, and so the status quo has him smoking {{{12}}} a day. 

Harpo has three choices: 

a. {{{not}}}{{{ rent}}} the{{{ machine}}} and not relocate Groucho,{{{but}}}{{{ pay}}} Groucho {{{not}}}{{{ to}}}{{{ smoke}}}
 
b. {{{rent}}} the machine and {{{then}}} pay Groucho not to smoke
 
c. {{{pay}}} to {{{relocate}}} Groucho 

Which would he choose? 

Not renting the machine requires Harpo tocompensate Groucho for the {{{MC }}}
of reducing cigars: {{{1+2+3+4+5+6= 2dollars_in_costs}}}. Harpo’s benefits are 
{{{12+11+10+9+8+7=57}}} dollars, so net benefits are {{{57-21=36}}} dollars.

Machine rental costs ${{{10}}}, plus Harpo has to pay Groucho at least his {{{MC}}} of reduction for the first four cigarettes.  The costs of machine rental are thus:
 
{{{10+1+2+3+4=20}}} 

Relative to the status quo of {{{12}}} cigars smoked, the benefits of machine 
rental have two components. 

First, Brittany gets the full benefits of seeing {{{4}}} cigarettes reduced:

{{{12+11+10+9=42}}} 

Second, she gets the {{{50}}}% improvement in air quality for the remaining {{{ 8}}} cigars that are smoked:
 
{{{4+3.5+3+2.5+2+1.5+1+.5=18}}} dollars 

So, net benefits are {{{42+18-20= 40}}} dollars

Finally, relocation costs ${{{40}}}, and brings Harpo the total area under the 
{{{MB }}} 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:
 
${{{(12+11+10+9+8+7+6+5+4+3+2+1)-40 =38}}}  

So the {{{efficient}}}{{{ choice}}} is machine {{{rental}}}.
 
4. Rather than be banished, Groucho would rent the machine, then pay Harpo 
to allow him to stayand smoke {{{8}}} cigars. If Groucho rents the machine, he would pay the following costs: 

${{{10}}} for machine rental, plus

${{{1+2+3+4=10}}} in nicotine 

withdrawals on {{{4}}} cigars given up, plus ${{{4+3.5+3+2.5+2+1.5+1+.5=18}}} in bribes to Harpo to allow him to smoke {{{8}}} cigarettes 

The total is ${{{38}}}, less than the ${{{40}}} required to relocate. 

So rather than banish Groucho,Harpo could come out ahead by demanding a little more than the ${{{18}}} just necessary for him to allow Groucho to smoke his {{{8}}} 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.