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Supplement%209 - On Preference Relations Recall that[see...

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On Preference Relations Recall that [see Handout #9 ] that a basic role is played by MRS de°ned as the absolute value of the slope of an indi/erence curve. Consider the following optimization problem of the consumer [in what follows p x > 0 ; p y > 0 and m > 0 ] : max f u ( x; y ) g subject to the constraint p x ° x + p y ° y = m , and x ± 0 ; y ± 0 To look for an optimal solution ( x ° ; y ° ) with x ° > 0 ; y ° > 0 , we can use the budget equation to get x = m ² p y ° y p x (1) and, upon substitution get the problem max n u ° [ m ± p y ² y ] p x ; y ±o on 0 < y < m p yy This is a maximization problem with a single variable y and if an optimal y ° exists; the following °rst order condition must hold [using the "relevant rules from calculus"]: u x ° [ ² p y =p x ] + u y = 0 or, u x =u y = p x =p y or, ( MRS ) u = p x =p y (2) Given a speci°c functional form of u we may be able to solve for ( x ° ; y ° ) as in the Cobb-Douglas case. Now, we interpret u ( x; y ) > u ( b x; b y ) as " ( x; y ) is preferred to ( b x; b y ) " : In terms of the usual diagram u ( x; y ) > u ( b x; b y ) means ( x; y ) is on a "higher"
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