Problem on competitive equilibrium economy

Question

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:

x^S₁ + x^S₂

where x^S₁ is the S-type’s consumption of cigarettes and x^S₂ is the S-type’s consumption of other stuff. A-types have the utility function

x^A₂ -  2¯x^S₁

where ¯ x^S₁ 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.

Answer 1

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/p_1-1

Now X₁ in economy= 2

So, 1/p_1-1 =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.