hw3 - ECON 2100: Advanced microeconomic theory I Problem...

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ECON 2100: Advanced microeconomic theory I Problem Set 3 - suggested solutions Prepared by David Klinowski September 19, 2011 Problem 1 Construct a preference relation on R that is not continuous, but admits a utility representation. Solution Let u ( x )= 1 x< 0 2 x =0 1 x> 0 represent x ° y ⇐⇒ x< 0 or x> 0 and y 0 or x = y =0 ° is not continuous because the upper contour set is not closed. To see this, take x = 1 . Its upper contour set is ° ( 1) = ( −∞ , 0) , which is not a closed set. Problem 2 Prove that if u represents ° ,then : (1) ° is weakly monotone if and only if u is nondecreasing; ° is strictly monotone if and only if u is strictly increasing. (2) ° is convex if and only if u is quasiconcave; ° is strictly convex if and only if u is strictly quasiconcave. (3) ° is quasi-linear if and only if it admits a quasi-linear representation. (4) a continuous ° is homothetic if and only if it is represented by a utility function that is homogeneous of degree one. Solution (1) <if> Suppose u is nondecreasing and x y . u ( x ) u ( y ) since u is nondecreasing x ° y by u representation
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<only if> Suppose ° is weakly monotone and x y . x ° y by weak monotonicity u ( x ) u ( y ) by u representation <if> Suppose u is strictly increasing and x y , x ° = y . u ( x ) >u ( y ) since u is strictly increasing x ± y by u representation <only if> Suppose ° is strictly monotone and x y , x ° = y . x ± y by strict monotonicity u ( x ) >u ( y ) by u representation (2) <if> Suppose u is quasiconcave and u ( x ) u ( y ) . t [0 , 1] u ( tx +(1 t ) y ) u ( y ) by quasiconcavity of u t [0 , 1] tx +(1 t ) y ° y by u representation <only if> Suppose ° is convex and x ° y . t [0 , 1] tx +(1 t ) y ° y by convexity of ° t [0 , 1] u ( tx +(1 t ) y ) u ( y ) by u representation <if> Suppose u is strictly quasiconcave and u ( x ) u ( y ) , x ° = y . t [0 , 1] u ( tx +(1 t ) y ) >u ( y ) by strict quasiconcavity of u t [0 , 1] tx +(1 t ) y ± y by u representation <only if> Suppose ° is convex and x ° y , x ° = y . t [0 , 1] tx +(1 t ) y ± y by strict convexity of ° t [0 , 1] u ( tx +(1 t ) y ) >u ( y ) by u representation (3) <if> Suppose ° admits a quasi-linear representation of the form u ( x, m )= v ( x )+ m .
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hw3 - ECON 2100: Advanced microeconomic theory I Problem...

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