Important information about Pareto efficiency
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Suppose that there are two consumers which we call A and B and two goods which we cal x and y.
(a) Suppose consumer A has utility function
u_a(x,y)=4log(x)+log(y)
Consumer B has utility function
u_b(x,y)=4sqrt(x)+sqrt(y)
The aggregate endowments in good and y are e_x=10 and e_y=10.
Draw the set of all Pareto efficient allocations in the Edgeworth box. Derive this set (the contract curve) anaytically
(b) As it turns out the nice analytic characterization you found above just works, because u_a and u_b are strictly concave and differentiable. Assume now that
u_a(x,y)=x+3y and u_b(x,y)=2x+4y
There functions are concave but not strictly concave
The aggregate endowments are still e_x=10 and e_y=0. Draw A's and B's indifference curves in an Edgeworth box
How does the set of all Pareto efficient allocations look like? Hint: Do not attempt to derive this set analytically. If you have the right indifference curves in your Edgeworth box you should able to figure this out.
(c) Assume now that agents have identical prederences
u_a(x,y) =u_b(x,y)=sqrt(x)=sqrt(y)
Assume that aggregate endowments are e_x>e_y>0. Use the analytical characterization of Pareto efficiency to show that for every Pareto efficient allocations (X_a,Y_a,X_b,Y_b) we must have x_a>=y_a and x_b>=y_b
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