27) Consider (x^2+y^2)<=12 as the PPF, now agent 1 maximizes his utility where the PPF is tangent to the IC's of 1...by using this u will get x1=sqrt(8) and y1=2...clearly since both are <=3 so x1<=3 and y<=3 is also satisfied..thus for agent 2 it will be (3-sqrt(8),1)..hence option d.
28) In this allocation since agent 1 gets to choose his maximization bundle first so clearly when a=1, the slope of the PPF is tangent to 1's IC but at a=1, slope of IC of agent 2=1 while the slope of the PPC and agent 1's IC=sqrt(2)..so clearly this point is not Pareto efficient because the agent's IC's are not tangent at that allocation...even if there were no bounds on A's consumption the slope would still differ and would not be equal to 2..isnt it so..??this is because since A strictly consumes less than 3 for both x1 nd y1 the slope would be sqrt (2) anyhow..and cannot be equal to 1(given value of a)..so it should be option b..isnt it so..??kindly let me know where I am missing..??
29) To be in Pareto efficient (at the point obtained in Q1) the slope of IC(1)=slope of IC(2)=slope of PPF=sqrt(2)...so a=sqrt(2).
However plz check the answers..!!!!
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