101A_lec15

# 101A_lec15 - Economics 101A(Lecture 15 Stefano DellaVigna...

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Unformatted text preview: Economics 101A (Lecture 15) Stefano DellaVigna March 12, 2009 Outline 1. Production Function II 2. Returns to Scale 3. Two-step Cost Minimization 4. Cost Minimization: Example 1 Production Function II • Isoquants Q ( y ) = { x | f ( x ) = y } • Set of inputs z required to produce quantity y • Special case. Two inputs: — z 1 = L (labor) — z 2 = K (capital) • Isoquant: f ( L, K ) − y = 0 • Slope of isoquant dK/dL = MRT S • Convex production function if convex isoquants • Reasonable: combine two technologies and do bet- ter! • Mathematically, convex isoquants if d 2 K/d 2 L > • Solution: d 2 K d 2 L = − f 00 L,L f K − 2 f 00 L,K f L + f 00 K,K ³ f L ´ 2 /f K ³ f K ´ 2 • Hence, d 2 K/d 2 L > if f 00 L,K > (inputs are com- plements in production) 2 Returns to Scale • Nicholson, Ch. 9, pp. 302-305 (Ch. 7, pp. 190—193, 9th) • E f ect of increase in labor: f L • Increase of all inputs: f ( t z ) with t scalar, t > 1 • How much does input increase? — Decreasing returns to scale: for all z and t > 1 , f ( t z ) < tf ( z ) — Constant returns to scale: for all z and t > 1 , f ( t z ) = tf ( z ) — Increasing returns to scale: for all z and t > 1 , f ( t z ) > tf ( z ) •...
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## This note was uploaded on 09/05/2009 for the course ECON 101a taught by Professor Staff during the Spring '08 term at Berkeley.

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101A_lec15 - Economics 101A(Lecture 15 Stefano DellaVigna...

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