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ajaz_204_2009_lecture_2 - University of Toronto Department...

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University of Toronto Department of Economics ECO 204 2009 2010 Sayed Ajaz Hussain Lecture 2 1 Ajaz Hussain. Department of Economics. University of Toronto (St. George)
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Today Quick review of unconstrained and equality constrained optimization from last time The envelope theorem What is it and when to use it? An application to unconstrained optimization An application to equality constrained optimization Lagrange multiplier λ interpreted Ajaz Hussain. Department of Economics. University of Toronto (St. George) 2
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(Static) Optimization Ajaz Hussain. Department of Economics. University of Toronto (St. George) 3 Optimization “Maximization” or “Minimization” Problems Unconstrained Optimization Equality Constrained Optimization Inequality Constrained Optimization ECO 204 Chapter 1.1 ECO 204 Chapter 1.2
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Last Time: T.O. Airport Parking Garage You’re the parking manager at T.O. airport You’ve identified two segments of customers Ajaz Hussain. Department of Economics. University of Toronto (St. George) 4 Short Term Parkers Demand Curve: P S = 3 Q S /200 Long Term Parkers Demand Curve: P L = 2 Q L /200 Q P 3 2 D S D L
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Road Plan Ahead Ajaz Hussain. Department of Economics. University of Toronto (St. George) 5 Review Unconstrained Revenue Optimization Solution Review Equality Constrained Revenue Optimization Solution Envelope Theorem Application: Envelope Theorem on Unconstrained Problem Application: Envelope Theorem on Equality Constrained Problem
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Last Time: Short Term Parking Unconstrained Optimization For now, focus on short term parking problem only Max R S = P S Q S Max R S = 3Q S Q S 2 /200 dR/dQ S = 0 3 Q S /100 = 0 Q S = 300 P S = $1.5 Ajaz Hussain. Department of Economics. University of Toronto (St. George) 6 R S R S Q S
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Last Time: Short Term Parking Equality Constrained Optimization For now, focus on short term parking problem only Max R S s.t. Q S = 200 Max R S s.t. Q S 200 = 0 Max L = R S ‐ λ [Q S 200] Max L = 3Q S Q S 2 /200 ‐ λ [Q S 200] dL/dQ S = 0 3 Q S /100 ‐ λ = 0 dL/d λ = 0 Q S 200 = 0 Q S = 200 Q S = 200 P S = $2 R S = $400 Ajaz Hussain. Department of Economics. University of Toronto (St. George) 7 FOC with respect to λ always gives the equality constraint
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Equality Constrained Optimization Ajaz Hussain. Department of Economics. University of Toronto (St. George) 8 R S R S Q S 300 $450 200 $400
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Short Term Parking: Interpreting λ Ajaz Hussain. Department of Economics.
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