L061113ProductionWEB

L061113ProductionWEB - S.Grant ECON501 3 PRODUCTION THEORY...

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S.Grant ECON501 3. PRODUCTION THEORY Ref: MWG Chapter 5 Productive units — “ f rms” corporations, other legally recognized businesses productive possibilities of individuals or households potential productive units that are never actually organized. “Black box” — able to transform inputs into outputs. 1 S.Grant ECON501 3.1 Production Sets production vector y R L e.g. y = 5 2 6 3 0 R 5 Menu of all possible production vectors constitutes Y , the production set. a) Transformation frontier Y = © y R L | F ( y ) 0 ª F ( y )=0 means y element of boundary of Y . MRT ck y )= ∂F y ) /∂y c ∂F y ) /∂y k Notice ∂F y ) ∂y k dy k dy c + ∂F y ) ∂y c =0 so, dy k dy c = MRT ck y ) 2
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S.Grant ECON501 b) Production function . q = f ( z ) Y = ½ ( z 1 ,... z L 1 ,q ) | q f ( z 1 ,...,z L 1 ) 0 , z c 0 , c =1 ,...L 1 ¾ Holding level of output f xed: MRTS = ∂f z ) /∂z c ∂f z ) /∂z k additional amount of input k that must be used to keep output f xed at ¯ q = f z ) , when amount of input c decreased marginally. 3.2 Properties of Production Sets (see pp 130-155) 1. free disposal 2. non-increasing returns to scale 3. non-decreasing RTS 4. constant RTS 3 S.Grant ECON501 3.3 Pro f t Maximization Pro f t Max. Pblm (PMP) max y Y p.y or max y p.y s.t. F ( y ) 0 Pro f t function π ( p )=max y Y p.y Supply correspondence y ( p )= { y Y | p.y = π ( p ) } Ex. 3.1 Y = © y R 2 | y 1 + y 2 0 , y 1 0 ª π ( p )= ½ 0 if p 2 p 1 if p 2 >p 1 y ( p )= 0 if p 2 p 1 © y R 2 | y 2 = y 1 0 ª if p 2 = p 1 unde f ned if p 2 >p 1 4
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S.Grant ECON501 First order approach (i) Transformation frontier max y p.y s.t. F ( y ) 0 L = p.y λF ( y ) FONC y c : p c = λ ∂F ( y ) ∂y c or in matrix notation p = λ F ( y ) (ii) Production function max z 0 pf ( z ) w.z FONC z c : p ∂f ( z ) ∂z c w c ( = w c ,i f z c > 0 ) or in matrix notation p f ( z ) w and ( p f ( z ) w ) .z =0 5 S.Grant ECON501 Properties of the Pro f tFunct ion Given Y is closed and satis f
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This note was uploaded on 09/04/2010 for the course ECON 501 taught by Professor Grant during the Spring '10 term at Rice.

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L061113ProductionWEB - S.Grant ECON501 3 PRODUCTION THEORY...

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