# Problem set 2 of Welfare economics

Universitat Pompeu Fabra, 1996/1997

Due April 29, 1997.

1. Consider an economy with two individuals. The preferences of one individual (over two commodities) is represented by the utility function U1(x,y)=x+y, and of the second individual by the utility function U2(x,y)=x+2y.

Let Pi(x0,y0)={(x,y)∈R+2 | Ui(x,y)>Ui(x0,y0)}

Calculate and draw the set P*=P1(1,1)+P2(1,1). Is the set P convex? Does the bundle (2,2) belong to P*? Draw the Edgeworth box with total resources (2,2) in explaining your answer.

For the case that the total resources are (2,2), find an allocation [(x1,y1),(x2,y2)] such that (2,2) does not belong to T=P1(x1,y1)+P2(x2,y2). Draw this point in the Edgeworth box, and draw the set T in a separate diagram.

2. Consider an economy with two commodities and two individuals. The preferences of one individual is represented by the utility function U1(x,y)=y-(1-x)2 if x≤1; U1(x,y)=y if x>1. And of the second individual by the utility function U2(x,y)=x+y,

Draw the Edgeworth box when the total resources are (2,1). Is the allocation (1,1) for individual 1, and (1,0) for individual two Pareto efficient? Can it be implemented as a price equilibrium? Why? Are the preferences of both individuals monotone? strictly monotone?