Re: General Equilibrium Question

Posted by Amit Goyal on May 16, 2013; 3:42pm
URL: http://discussion-forum.276.s1.nabble.com/General-Equilibrium-Question-tp7580742p7580834.html

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