Constrained Optimization

Constrained Optimization - Constrained Optimization...

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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 (x1,x2) to maximize (or minimize) an objective function f(x1,x2) 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 (PS)= 0.25 Price of Salad (PV)= 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 = PS S + PV V Budget = 6 Price of Soup (PS)= 0.25 Price of Salad (PV)= 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 .2 5 0.5 6 0 .5 0 .2 5 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 = - +...
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Constrained Optimization - Constrained Optimization...

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