CESInputDemand

# CESInputDemand - Constant Elasticity of Substitution in...

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Constant Elasticity of Substitution in Applied General Equilibrium The choice of input levels that minimize the cost of production for any set of input prices and a fixed level of production can be expressed as ( 29 min .. i ii x i xw st Y fx = % Limiting for exposition purposes the set of inputs to 3 and focusing on the constant elasticity of substitution (CES) production function yields 123 1 1 2 2 33 ,, 111 1 1 1 2 2 min xxx w x w x wx Y s sss s s s s s ss aaa --- - ++  = ++   Forming the Lagrangian for the optimization problem 1 1 1 2 2 3 3 1 1 2 2 L w x w x w x Y s s s s s l -  =+++ - ++   The first-order conditions for this minimization problem are then 1 1 1 1 1 1 1 1 1 1 2 2 1 1 1 1 1 1 1 1 1 1 2 2 1 2 2 1 0 1 1 11 1 1 0 L w x x Giventhat and L x L w x s s s s s s s s sss s s ss l a s s s s l a - - - - - - ∂- = - + += -- - = - =- - - =-= - = - + = 1 1 1 1 2 2 1 1 2 2 1 1 1 1 3 3 3 1 1 2 2 3 1 1 1 2 2 0 0 0 x L x L Y s s s s s s s l aaaa l a l - - - - - - + = - + = - + The ratio of the first-order condition for the first input to the first-order condition for the second input yields

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AEB 6184 Applied General Equilibrium Notes Professor Charles B. Moss April 12, 2001 2 1 111 1 1 1 1 1 1 1 2 2 33 1 1 1 1 1 2 2 2 1 1 2 2 1 1 12 2 21 11 1 1 2 2 2 1 x xxx w w x wx wxw xx sss s s s s ss s s ss l aaaa l a a aa --- - - - -  ++   = = = ⇒= Simply replacing the price and constant for x 3 yields 1 3 1 31 w w s a a = Substituting these results into the first-order condition with respect to the Lagrange multiplier yields 1 1 1 1 1 3 1 1 1 2 1 2 1 0 ww Y x s s s s s s s s s s s a a a - - - -    - + +=    Focusing on the term inside the brackets 1 1 1 1 1 1 1 3 2 1 23 s s s s s a a a - - - - + + -- First note that 1 s - .
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## This note was uploaded on 11/08/2011 for the course FIN 6646 taught by Professor Moss during the Spring '10 term at University of Florida.

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CESInputDemand - Constant Elasticity of Substitution in...

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