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Note4 - Utility Maximization Budget Constraint Let px =...

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Utility Maximization Budget Constraint Let p x = price of good x; p y = price of good y; and I = income. The budget constraint is given by p x x + p y y = I: ° What is the shape of the budget constraint? ° What is the slope of the budget constraint? ° What will be the shape of the budget constraint if p x depends on x (e.g., p x = px ) ? ° In a barter economy, what will be the budget constraint if the endowments are x 0 and y 0 ? Utility Maximization max x;y U ( x; y ) subject to p x x + p y y = I: To the consumer, income and prices are exogenous variables. The solution to the maxi- mization problem may be (i) interior, (ii) at a corner, or (iii) non-unique. Method 1: Lagrangian Method Form the Lagrangian L = U ( x; y ) + ° ( I ° p x x ° p y y ) ; where ° is called the Lagrangain multiplier. If the maximization problem has an interior solution, then the following three °rst-order conditions must be satis°ed: @ L @x = @U ( x; y ) @x ° °p x = 0 ; (13) @ L @y = @U ( x; y ) @y ° °p y = 0 ; (14) @ L = I ° p x x ° p y y = 0 : (15) 14
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Equations (13) and (14) imply that @U ( x;y ) @x p x = ° = @U ( x;y ) @y p y ; (16) or MRS = U x U y = p x p y : (17) Note: A more general set of °rst-order conditions is @ L @x = @U ( x; y ) @x ° °p x ± 0 : (18) If the inequality (18) is strict, then x = 0 : If x > 0 ; then (13) must hold.
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