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ajaz_eco_204_2012_2013_chapter_12_Long_Run_CMP_PP

Envelope theorem the marginal profit due to expanding

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Unformatted text preview: l solution: ( )( ) ( ) ( ) ( ) A 1% increase in aluminum prices reduces the average rational smelter’s loss by 9.3% or from a loss of (\$14,373) to a loss of (\$13,036). Envelope Theorem: the marginal profit due to expanding capacity by 1% The Lagrangian was: () [ [ Differentiating the Lagrangian with respect to capacity: This is the impact on profits from a 1 unit increase in capacity. To find the impact due to a 1% increase in capacity we have: 52 ECO 204 Chapter 12: Practice Problems & Solutions for The Long Run Cost Minimization Problem in ECO 204 (this version 2012-2013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. Evaluate at optimal solution: ( ) ( ) A 1% increase in capacity reduces the average rational smelter’s loss by 2.92% or from a loss of (\$14,373) to a loss of (\$13,953.31). Envelope Theorem: the marginal profit due to raising the minimum output by 1% The Lagrangian was: [ () We had required that . Re-writing this constraint as () Differentiating the Lagrangian with respect to [ we have: [ [ : This is the impact on profits from a 1 unit increase in minimum output. To find the impact due to a 1% increase in minimum output we have: 53 ECO 204 Chapter 12: Practice Problems & Solutions for The Long Run Cost Minimization Problem in ECO 204 (this version 2012-2013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. There is no impact on optimal profits from raising the minimum output requirement by 1%. Why? Because the optimal solution is for the smelter to produce well above zero, so that the minimum output constraint does not bind – as such, there is no value in relaxing the constraint. Envelope Theorem: the marginal profit of 1% higher fixed cost The Lagrangian was: [ () Differentiating the Lagrangian with respect to [ : This is the impact on profits from a \$1 increase in TFC. To find the impact due to a 1% increase in TFC we have: Evaluate at optimal solution: ( ) (...
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