General Equilibrium Question

[Ayushya Kaul]

3. Suppose that a city can be described by an interval [0; 1]. Two citizens, A
and B live in this city at dierent locations; A at 0:2 and B at 0:75. Government
has decided to install a nuclear power plant in this city but is yet to choose
its location. No citizen wants a nuclear power plant in her backyard. Both of
them have the same utility function, u(d) = d, where d denotes the distance
between the nuclear plant and home.
(a) Find the set of feasible allocations.
(b) Find the set of Pareto optimal allocations.
(c) Find the set of utilitarian allocations.
(d) Find the Rawlsian allocation.

[Sumit]

B) set of pareto optimal solutions (0.2,0.75)..

[Ayushya Kaul]

How did you arrive at that answer?

[Ayushya Kaul]

Any idea about this:
2. Consider a bilateral transaction where buyer B has valuation vB > 0 and
seller S has valuation 0. Utility of the agents are as follows. Utility of B is
(vB- pB^1/2) and utility of S is pS, where pB denotes the price that B pays and
pS denotes the price S receives. Note that, pB need not be equal to pS, but of
course pB>= pS.
(a) Find the set of feasible allocations.
(b) Draw a diagram where utility of B is plotted on the horizontal axis and
utility of S is plotted on the vertical axis. Show the set of feasible allocations
in this two-dimensional utility space. This is called the utility possibility set.
(c) Identify Pareto optimal allocations in the utility possibility set.
(d) Identify utilitarian allocations in the utility possibility set.

[Amit Goyal]

(a) Find the set of feasible allocations.
Set of feasible allocations (locations to place the nuclear power plant) is [0, 1], i.e. all locations from 0 to 1 (including 0 and 1).

(b) Find the set of Pareto optimal allocations.
The set of Pareto optimal allocations (locations to place the nuclear power plant) is {0, 1} U (0.4, 0.5), i.e. all locations between 0.4 and 0.5 (not including 0.4 and 0.5) and locations at the extreme 0 and 1.

(c) Find the set of utilitarian allocations.
Utilitarian allocation/location is one that maximizes the sum of utilities (distances) of the two individuals. In this case, it is location 1.

(d) Find the Rawlsian allocation.
Rawlsian allocation/location is one that maximizes the utility (distance) of the worst of the two individuals. In this case, it is location 0.475.

[Amit Goyal]

There is a line in the problem that is incomplete. Complete it first:

"Note that, pB need not be equal to pS, but of
course pB  pS. "

[sakshi]

sir how u arrived at answers of part b and d

[laracroft]

Sir
How did you arrive at the solutions of part b,c,d ??

[sakshi]

sir i think it should be " but ofcourse pB > and equal to pS..

[Amit Goyal]

2. Consider a bilateral transaction where buyer B has valuation vB > 0 and
seller S has valuation 0. Utility of the agents are as follows. Utility of B is
(vB- pB^1/2) and utility of S is pS, where pB denotes the price that B pays and
pS denotes the price S receives. Note that, pB need not be equal to pS, but of
course pB ≥ pS.
(a) Find the set of feasible allocations.
In principle, any price combination in the following set is feasible.
{(pB, pS): pB ≥ pS ≥ 0}
But if we impose individual rationality, buyer will not pay more than the price: vB^2. Thus, we will say that set of feasible price combinations is:
{(pB, pS): vB^2 ≥ pB ≥ pS ≥ 0}

(b) Draw a diagram where utility of B is plotted on the horizontal axis and
utility of S is plotted on the vertical axis. Show the set of feasible allocations
in this two-dimensional utility space. This is called the utility possibility set.  
Utility possibility Set = {(uB, uS)| uB + uS^1/2 ≤ vB, uB ≥ 0, uS ≥ 0}

(c) Identify Pareto optimal allocations in the utility possibility set.
Efficiency calls for pB = pS. Thus,
Utility possibility Frontier = {(uB, uS)| uB + uS^1/2 = vB, uB ≥ 0, uS ≥ 0}

(d) Identify utilitarian allocations in the utility possibility set.
Utilitarian utility pair can now be found easily by maximizing
uB+ uS s.t. uB + uS^1/2 = vB

[Sumit]

thank you so much sir.....could you plz explain part d) Rawlsian allocation more...I fail to catch it...

n for others pareto optimal solution is {0, 1} U (0.4, 0.5) bcoz...
consider interval [0,1] on number line...
First consider point zero..it is pareto optimal bcoz there is no deviation from this point on number line in interval [0,1] which will increase utility(i.e distance) for any one citizen or both citizen without reducing utility of atleast one citizen.....same logic apply for point 1 and interval (0.4,0.5) as well...

now consider interval (0,0.4]...the points lie in this interval is not pareto optimal bcoz....consider any point btw this interval say 0.1 ...In this case citizen A utility=0.1 & B=0.65....now If we deviate to point say 0 from this point then u will see utility of both will increase...so this is not a pareto optimal point..as we are able to increase utility of both the citizen....same logic is apply to all the points in interval (0,0.4] & [0.50,1)...in all the points you will find atleast one point from which utility of one or both citizen increase without reducing other citizen utility..

0.4 n 0.5 is not pareto optimal bcoz if we deviate to point 0 from 0.4 & to 1 from 0.5....you will see there is increase in utility of citizen B from 0.35 to 0.75 in case first(i.e 0.4 to 0) and increase in utility of citizen A from 0.3 to 0.8 in case two (i.e 0.5 to 1)