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Unformatted text preview: Constrained Optimization Professor Erkut Ozbay Economics 300 Constraints Limited budget Limited revenue Limited hours to work . Scarce resources create constraints Constrained optimization Includes an objective function and constraints Choose variables (x 1 ,x 2 ) to maximize (or minimize) an objective function f(x 1 ,x 2 ) subject to constraints Consumption problem You have $6.00 to spend on a lunch of soup and salad Restaurant sells both soup and salad by weight An ounce of soup (S) is $0.25 An ounce of salad (V) costs $0.50 How many ounces of each will you purchase? Budget constraint Budget = $6 Price of Soup (P S )= 0.25 Price of Salad (P V )= 0.50 If you spend all on soup, then you can buy 6/.25 =24 ounces of soup If you spend all on salad, then you can buy 6/.5 = 12 ounces of salad Budget constraint Budget = P S S + P V V Budget = 6 Price of Soup (P S )= 0.25 Price of Salad (P V )= 0.50 6 = 0.25 S + 0.5 V Objective Utility you derive from consumption of soup and salad is 1 1 ( , ) ln( ) ln( ) 2 2 U S V S V = + Choose (S,V) to maximize Subject to 1 1 ( , ) ln( ) ln( ) 2 2 U S V S V = + 6 0.25 0.5 S V = + Substitution method Start with the constraint 6 0.25 0.5 6 0.5 0.25 1 1 6 2 4 24 2 S V V S V S S V = + = = =  Substitution method S = 24 2V Substitute into the utility function 1 1 ( , ) ln( ) ln( ) 2 2 1 1 ( ) ln(24 2 ) ln( ) 2 2 U S V S V u V V V = + =  + Now the problem is a univariate maximization Choose V to maximize 1 1 ( ) ln(24 2 ) ln( ) 2 2 u V V V =  + FOC We know that S = 24 2V, so 1 1 ( ) ln(24 2 ) ln( ) 2 2 u V V V = + * 2 1 2(24 2 ) 2 2 24 2 6 du dV V V V V V = + = =  = * 12 S = SOC Hence, maximum 2 2 2 2 1 1 (24 2 ) 2 2 1 (24 2 ) 2 du dV V V d u dV V V = + =  < Constrained optimization with a budget line and indifference curves Slope of indifference curve / 1/ 2 / 1/ 2 V S MU dS U V V S dV U S MU S V =  =  =  =  1 1 ( , ) ln( ) ln( ) 2 2 U S V S V = + Slope of budget line .5 2 .25 V S P dS dV P =  =  =  At optimum: V V S S MU P MU P = Exercise 2 2 2 4 subject to 8 y x xz z x z = + + + = 2 2 2 4 subject to 8 y x xz z x z = + + + = 2 2 2 2 2 8 8 , 2 (8 ) 4(8 ) y = 16 2 4(8 ) x z z x Then y x x x x x x x x + = = = + + + + 2 2 2 * * y = 16 2 4(8 ) 2 16 4 8(8 ) 16 2 64 8 8 8 x x x x dy x x x dx x x x z x z + + = + = + = = = = 2 2 2 16 4 8(8 ) 2 4 8 6 0 . dy x x x dx d y dx Min = +    =  + = Exercise 1/ 2 1/ 2 10 40 subject to x 2 y x z z = + = 1/ 2 1/ 2 10 40 subject to x 2 y x z z = + = 1/2 1/2 1/2 1/2 x 2 2 x 4 , 10 40 160 10 z z z x Then y x z y x x = = = = + = + 2 2 2 * * 1 6 0 1 0 1 6 0 1 0 0 1 6 0 1 0 1 6 4 o r 4 4 ( 4,1) o r ( 4 , 1) y x x d y d x x x x x x z x = + =  = = = = =  =   2...
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This note was uploaded on 02/03/2012 for the course ECON 300 taught by Professor Cramton during the Fall '08 term at Maryland.
 Fall '08
 cramton
 Economics

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