Lecture 2 - Economics 101A (Lecture 2) Stefano DellaVigna...

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Economics 101A (Lecture 2) Stefano DellaVigna September 1, 2009
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Outline 1. Optimization with 1 variable 2. Multivariate optimization 3. Comparative Statics 4. Implicit function theorem
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1 Optimization with 1 variable Nicholson, Ch.2, pp. 20-23 (20-24, 9th Ed) Example. Function y = x 2 What is the maximum? Maximum is at 0 General method?
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Sure! Use derivatives Derivative is slope of the function at a point: ∂f ( x ) ∂x =l im h 0 f ( x + h ) f ( x ) h Necessary condition for maximum x is ( x ) =0 (1) Try with y = x 2 . ( x ) == 0 = x =
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Does this guarantee a maximum? No! Consider the function y = x 3 ∂f ( x ) ∂x == 0 = x = Plot y = x 3 .
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Su cient condition for a (local) maximum : ∂f ( x ) ∂x =0 and 2 f ( x ) 2 x ¯ ¯ ¯ ¯ ¯ x < 0 (2) Proof: At a maximum, f ( x + h ) f ( x ) < 0 for all h . Taylor Rule: f ( x + h ) f ( x )= ( x ) h + 1 2 2 f ( x ) 2 x h 2 + higher order terms. Notice: ( x ) . f ( x + h ) f ( x ) < 0 for all h = 2 f ( x ) 2 x h 2 < 0= 2 f ( x ) 2 x < 0 Careful: Maximum may not exist: y =exp( x )
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Tricky examples: Minimum. y = x 2 No maximum. y =exp( x ) for x ( −∞ , + ) Corner solution. y = x for x [0 , 1]
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2 Multivariate optimization Nicholson, Ch.2, pp. 23-30 (24-32, 9th Ed) Function from R n to R : y = f ( x 1 ,x 2 ,...,x n ) Partial derivative with respect to x i : ∂f ( x 1 , ..., x n ) ∂x i = lim h 0 f ( x 1 , ..., x i + h, . ..x n ) f ( x 1 , ..., x i ,...x n ) h Slope along dimension i Total di f erential: df = ( x )
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Lecture 2 - Economics 101A (Lecture 2) Stefano DellaVigna...

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