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( ) ( ) ( ) 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 20122013) 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 . Rewriting 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 20122013) 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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This document was uploaded on 01/19/2014.
 Fall '14

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