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ch11-constrained-optimization (1)

ch11-constrained-optimization (1) - Constrained...

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Constrained Optimization Professor Erkut Ozbay Economics 300
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Constraints Limited budget Limited revenue Limited hours to work …. Scarce resources create constraints
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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
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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?
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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
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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
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Objective Utility you derive from consumption of soup and salad is 1 1 ( , ) ln( ) ln( ) 2 2 U S V S V = +
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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 = +
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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 = + - = - = = -
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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 = + = - +
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Now the problem is a univariate maximization Choose V to maximize 1 1 ( ) ln(24 2 ) ln( ) 2 2 u V V V = - +
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FOC We know that S = 24 – 2V, so 1 1 ( ) ln(24 2 ) ln( ) 2 2 u V V V = - + * 2 1 0 2(24 2 ) 2 2 24 2 6 du dV V V V V V - = + = - = - = * 12 S =
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SOC Hence, maximum 2 2 2 2 1 1 (24 2 ) 2 2 1 0 (24 2 ) 2 du dV V V d u dV V V - = + - - = - < -
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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 =
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Exercise 2 2 2 4 subject to 8 y x xz z x z = + + + =
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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 + = = - = + - + - + - + -
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2 2 2 * * y = 16 2 4(8 ) 2 16 4 8(8 ) 0 16 2 64 8 0 8 8 0 x x x x dy x x x dx x x x z x z + - + - = + - - - = - - + = = = - =
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2 2 2 16 4 8(8 ) 2 4 8 6 0 . dy x x x dx d y dx Min = + - - - = - + =
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Exercise 1/2 1/2 10 40 subject to x 2 y x z z = + =
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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 = = = = + = +
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