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Problem on competitive equilibrium economy

The economy consists of an equal number of smokers (S-types) and asthma sufferers (A-types). Good 1 is cigarettes, good 2 is “other stuff.” S-types have the utility function:

xS1 + xS2

where xS1 is the S-type’s consumption of cigarettes and xS2 is the S-type’s consumption of other stuff. A-types have the utility function

xA2 -  2¯xS1

where ¯ xS1 is the per capita consumption of cigarettes of S-types.

The initial endowments are as follows: S-types have one unit of both goods and A-types have 2 units good 2 and no endowment of good 1.

(i) Is there an efficient consumption plan in which S-types consume cigarettes?
(ii) Find a competitive equilibrium of this economy. Show that it is not efficient.

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i) Consumer 1 has perfect substitute type of utility function .Now they will consume cigarettes only when P1<P2. Now P2 being 1 (numeraire).The consumption plan in which person 1 consumes cigarettes is the one where p1<1

ii) This is the externality case type of utility function for person 2. In which we solve the competitive equilibrium normally without externality and then tell that it is inefficient

Now U1= x1+x2 (1,1)
U2=  x2 (1,0)

Budget constraint for person 1:

P1x1 +p2x2= p1(1) + p2(1)

Put p2=1

P1x1 + x2= p1 + 1

Also for person 2 we have

P1x1 +X2= P1

Now we know person 1 has perfect substitutes requirement and peson 2 demands just good 2

So prices should be such that P1<p2  so that person 1 demands only good 1
So Putting x2*=0

We get:
P1X1= p1+1
X1* =1/p1-1

Now X1 in economy= 2

So, 1/p1-1 =2 P1= 3/2 >1 so it contradicts our assumption and we have to take p1=p2=1

Now put P1=p2=1 in budget constraint we get:

X1+x2=1
X1+ x2=1

From both the budget constraints this means any combination that satisfies this requirement will be competitive equilibrium. These are not efficient because there is an externality case involved plus there is no equality between MRS.

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