Universitat Pompeu Fabra, 1996/1997

Due April 23, 1997.

**1.** Let R(x)= {y∈**R**_{+}^{2} | yRx}

Prove the following statement: If a preference relation is complete and transitive, then forall x,y: R(x) is a subset of R(y) or R(y) is a subset of R(x).

**2.** Let ? be a complete and transitive relation. Prove
that:

**2.1**The induced strict relation P is irreflexive, i.e., it is never true that xPx.**2.2**The induced indifference relation I is transitive.**2.3**xPy and yPz imply xPz.

**3.** Consider the following preference relation on **R**_{+}^{2}:
(x_{1},x_{2})R(y_{1},y_{2}) iff
x_{1}>y_{1} or x_{1}=y_{1} and
x_{2}≥y_{2}.

Is this preference relation complete? transitive? strictly monotonic? Draw the set R((1,2)) i.e., the set of bundles weakly preferred to (1,2). What bundles are strictly preferred to (1,2)? What is the indifference curve that goes through (1,2)? Is there a bundle on the diagonal that is indifferent to (1,2) Is this preference relation continuous?

**4.** 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.

Draw the Edgeworth box of this economy (for total quantities of your choice). Find the set of Pareto optimal allocations. Also, draw the set of possible utility pairs.