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 U_{1}(x,y)=x+y, and of the second
individual by the utility function U_{2}(x,y)=x+2y.

Let P_{i}(x_{0},y_{0})={(x,y)∈**R**_{+}^{2} | U_{i}(x,y)>U_{i}(x_{0},y_{0})}

Calculate and draw the set P*=P_{1}(1,1)+P_{2}(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
[(x_{1},y_{1}),(x_{2},y_{2})] such that
(2,2) does not belong to T=P_{1}(x_{1},y_{1})+P_{2}(x_{2},y_{2}).
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 U_{1}(x,y)=y-(1-x)^{2} if x≤1;
U_{1}(x,y)=y if x>1. And of the second individual by the
utility function U_{2}(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?

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Nir Dagan / Contact information / Last modified: January 4, 1999.