Econ402_Solutions_Set2

Econ402_Solutions_Set2 - Economics 402, Winter 2009 Problem...

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Economics 402, Winter 2009 Problem Set 2 Suggested Solutions 1 Loci (20 raw points) 1. Assume that the partial derivatives of F with respect to both x and y are positive. Using the same careful and set-by-step reasoning, we did in class: what is the slope of the F(x, y)=0 in (x,y)-space? Here we are told to assume: @F @x > 0 ; @F @y > 0 : The easiest way to derive the loci is graphical. The steps of the graphical procedure are as follows: (i) Start with any arbitrary ( x; y ) ( x 0 ; y 0 ) F ( x 0 ; y 0 ) = 0 (ii) Increase x by some arbitrary amount to x 1 > x 0 : (iii) Given the arbitrary increase in x , ±gure out which direction y must move in order to sat- isfy F ( x 1 ; y 1 ) = 0 : (iv) Connect the dots. (i) x x 0 y y 0 F( x 0 , y 0 )=0 (ii) x x 0 y y 0 F( x 1 , y 0 )>0 x 1 (iii) x y 1 x 0 F( x 1 , y 1 )=0 y x 1 y 0 (iv) x y 1 x 0 y x 1 y 0 We can see that when both partial derivatives are positive the locus of x and y , i.e. the set of points where F ( x; y ) = 0 is satis±ed, is downward-sloping. I started with an arbitrary ( x; y ) pair, 1
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( x 0 ; y 0 ) , satisfying F ( x 0 ; y 0 ) = 0 x to x 1 , and this means that F ( x; y ) = 0 no longer holds at ( x 1 ; y 0 ) : Because the partial derivative of F with respect to x is positive, we know that F ( x 1 ; y 0 ) > 0 . In which direction must we move y in order to reestablish F ( x; y ) = 0 ? Because we need the value of F to go down in order for F ( x; y ) = 0 to hold, and because we know that the partial of F with respect to y is everywhere positive, we evidently must decrease y by some amount to y 1 such that F ( x 1 ; y 1 ) = 0 and y in equilibrium. Note that we cannot say for sure how ±at or steep that relationship is or whether it is a line between the two variables where F ( x; y ) = 0 2.
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This note was uploaded on 03/13/2009 for the course ECON 402 taught by Professor House during the Winter '08 term at University of Michigan.

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Econ402_Solutions_Set2 - Economics 402, Winter 2009 Problem...

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