ch11-constrained-optimization

ch11-constrained-optimization - Constrained Optimization...

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onstrained Optimization 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 alad salad • Restaurant sells both soup and salad by weight n ounce of soup (S) is $0 25 – An ounce of soup (S) is $0.25 – An ounce of salad (V) costs $0.50 ow many ounces of each will you purchase? • How many ounces of each will you purchase?
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Budget constraint Budget = $6 rice of Soup (P =025 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 unces of salad 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 11 (, ) l n () l n ( ) 22 USV S V
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• Choose (S,V) to maximize 11 ( , ) ln( ) 22 US V S V  Subject to 6 0.25 0.5 SV
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Substitution method • Start with the constraint 6 0.25 0.5 0 5 0 2 5 SV S   60 . 50 .25 11 6 4 VS 24 2 
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Substitution method • S = 24 – 2V • Substitute into the utility function 1 11 ( , ) ln( ) 22 ) l n ( 2 4 2) l n () USV S V V V V uV 
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• Now the problem is a univariate maximization • Choose V to maximize 11 ) l n ( 2 4 2) l n () V V V 22 uV
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11 ) l n ( 2 4 2) l n () V V V  •F O C 22 uV 21 0 2(24 2 ) 2 2 4 2 du dV V V V  e know that S = 24 V so * 6 VV V * We know that S 24 2V, so 12 S
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•S O C 11 du  2 2 2 (24 2 ) 2 21 0 dV V V  • Hence, maximum 22 (24 2 ) 2 d VV V
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Constrained optimization with a budget e and indifference curves line and indifference curves lope of indifference curve 11 ( , ) ln( ) 22 USV S V  Slope of indifference curve /1 / 2 / 2 V S MU dS U V V S dV U S MU S V   Slope of budget line .5 2 .25 V P dS dV P S At optimum: VV SS M UP M
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Exercise 22 4 x x z z  24 bject to 8 yx z subject to xz 
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22 4 x x z z  24 subject to 8 yx xz 8 x   8 , ( 8 ) 4 ( 8 ) zx Then x x x x 2 2( y = 16 2 4(8 ) xx x x  
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22 2 = 16 2 4(8 ) x x x y ) 21 6 48 ( 8 ) 0 xx dy x    16 2 64 8 0 dx  * * 8 0 x 80 z xz 
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21 6 4 8 ( 8 ) dy x xx dx  2 248 6 0   2 in . Min
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Exercise 10 40 yxz  1/2 subject to x 2 z
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/2 1/2 10 40 yxz  subject to x 2 z x2 z 2 x z 4 z x , 10 Then 160 10 yx x
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160 10 y x  0 x dy  2 2 16 d xx x  2 ** 4 o r 4 x  4 (4,1) or ( 4, 1) z x
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160 dy 2 2 10 0 dx x y  23 2 320 0 x y * At 4, (4) 0 (4,1) minimum 4 x  2 * 4, ( 4) 0 ( 4, 1) maximum (4 ) d y x      
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Constrained optimization Choose to maximize (or to minimize) 2 (, ) f xx 12 ) x x subject to ) x x c ( , gx
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ch11-constrained-optimization - Constrained Optimization...

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