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PS3Solution

PS3Solution - Economics 310 Microeconomic Theory A...

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Approach Spring 2007 Solution to Problem Set 3 Question 1: demand, the Hicksian demand and expenditure function The Marshallian demand is a result of utility maximization subject to the budget constraint. The Hicksian demand can be found by minimizing the ex- penditure subject to the utility being equal to some prespeci±ed level u: From duality of the utility maximization problem and the expenditure minimization problem we know that the optimal consumption levels of x and y have to satisfy the same FOCs for both problems. This can be seen graphically. No matter whether we want to maximize the utility or minimize the expenditure, we have to ±nd a tangency point between an indi/erence curve and a budget line. After ±nding the relationship that x and y have to satisfy at the tangency point, we need to plug this formula into the budget constraint for a given income (if we are utility maximizing) or into the utility formula for a given utility level (if we are expenditure minimizing). Utility maximization Expenditure minimization (a) U ( x;y ) = x The consumer receives utility only from x ; therefore, she will spend all her money on it, whether she is utility maximizing or expenditure minimizing. Hence, the Marshallian demand is ( x m ;y m ) = m p x ; 0 ± : To achieve the utility level u; the consumer has to buy x = u units of good x; which implies that the Hicksian demand is ( x h ;y h ) = ( u; 0) : 1
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The expenditure function is e ( p x ;p y ;u ) = p x u (b) U ( x;y ) = xy x + y (hint: this equals 1 1 x + 1 y ) Lagrangian for the utility maximization problem is L = xy x + y ( p x x + p y y m ) : The FOC L x = y 2 ( x + y ) 2 x = 0 ; L y = x 2 ( x + y ) 2 y = 0 ; imply y 2 ( x + y ) 2 p x = x 2 ( x + y ) 2 p y ) y 2 p y = x 2 p x : (1) Plugging this into the budget constraint, we get p x x + p y r p x p y x = m ) x m = m p x + p p y p x ;y m = m p p x p y + p y To obtain the Hicksian demand we need to plug the optimality condition (1) into the utility constraint: u = xy x + y ) u = x q p x p y x x + q p x p y x = x p p x p p y + p p x ; x h = u p p y + p p x p p x ;y h = u p p y + p p x p p y : The expenditure function is e ( p x ;p y ;u ) = p x x h + p y y h = u p p y + p p x ± 2 : (c) U ( x;y ) = min ( x;y ) This consumer wants to consume the goods in a 1 : 1 proportion. She does not receive any additional utility from a good that is consumed in excess. Therefore, it is optimal to set x = y: Plugging this in the budget constraint, we get p x x + p y y = ( p x + p y ) x = m ) x m = y m = m p x + p y : 2
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The cheapest way to achieve utility u is to consume the goods in the same proportion, hence x = y; which results in the utility min ( x;y = x ) = x: There- fore
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PS3Solution - Economics 310 Microeconomic Theory A...

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