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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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 Fall '08
 cramton
 Economics, Optimization, Constraint satisfaction

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