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Unformatted text preview: ECO 310, Fall 2007 Midterm Review October 21 1 Constrained Optimization Consider the following maximization problem: max x,y f ( x, y ) subject to g ( x, y ) ≤ c. Define L ( x, y, λ ) = f ( x, y ) + λ ( c − g ( x, y )) , and you will get first-order necessary conditions: L x = f x ( x ∗ , y ∗ ) − λg x ( x ∗ , y ∗ ) = 0 , L y = f y ( x ∗ , y ∗ ) − λg y ( x ∗ , y ∗ ) = 0 . 1.1 Lagrangian L is called a Lagrangian. If ( x ∗ , y ∗ ) is a solution to the original constrained maximization, then you can write FONCs as if ( x ∗ , y ∗ ) maximized L without constraints. Note that it is just an as-if story. 1 You have to be very careful about the direction of the constraint inequality ( g ( x, y ) ≤ c or g ( x, y ) ≥ c ), the sign in front of λ (plus or minus), and the “penalty term” ( λ ( c − g ( x, y )) or λ ( g ( x, y ) − c )), although you are on the right track if you make mistakes an even number of times. 1.2 Multiplier λ is called a Lagrangian multiplier. λ ≥ 0, and λ = 0 if the constraint is not binding ( g ( x ∗ , y ∗ ) < c ). This is called the complementary slackness condition. More symmetrically, we can write λ ≥ , g ( x ∗ , y ∗ ) ≤ c, and at least one of the two holds with equality . Interpretation of λ : penalty per unit for violating the constraint, or how much you can get if you relax the constraint by one unit. This interpretation is formally proved by the next envelope theorem. 1 The as-if story becomes real if f is concave and g is convex. This fact will not be asked in the exam. 1 1.3 Envelope Theorem Consider a maximization problem with parameter α : V ( α ) = max x,y f ( x, y ; α ) subject to g ( x, y ; α ) ≤ . Define L ( x, y, λ ; α ) = f ( x, y ; α ) − λg ( x, y ; α ) . Note that V ( α ) = f ( x ∗ ( α ) , y ∗ ( α ) , ; α ) = L ( x ∗ ( α ) , y ∗ ( α ) , λ ∗ ( α ); α ) , where ( x ∗ ( α ) , y ∗ ( α )) is the optimizer and λ : ( α ) is the multiplier. (By the com- plementary slackness condition, λg = 0 at the optimum) Let’s assume that x ∗ ( α ), y ∗ ( α ), and λ ∗ ( α ) are differentiable with respect to α . Then, take the derivative of V ( α ) with respect to α , and we get V ( α ) = L x dx ∗ dα + L y dy ∗ dα + L λ dλ ∗ dα + L α = L α = f α − λg α because the first three terms in the first line disappear because of the FONCs...
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- Fall '08
- Economics, demand functions, input demand functions, Hicksian