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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. Notice that producing at full capacity maximizes profits in the sense that it minimizes loss. (As a study question, you
should check that operating at full capacity is better than shutting down, confirming the rule that a rational company
incurring losses should shut down when
Now, what is the impact on the average rational smelter’s optimal profits from, holding all else constant, a 1% increase
in: The price of aluminum?
The minimum output?
Fixed cost? By the envelope theorem, the change in the objective (in this case profits) from a small change in a parameter is gotten
by differentiating the Lagrangian with respect to the parameter, evaluated at the initial solution.
Optional note: Recall the Lagrangian was:
[ () [ Any solution to this problem must satisfy the KT conditions:
[ The “product” terms in the KT conditions imply that at the optimum the following terms are zero:
⏟ () ⏟[ ⏟[ This is why differentiating the optimal Lagrangian is equivalent to differentiating optimal profits with respect to the
Envelope Theorem: the marginal profit due to a 1% increase in aluminum price
The Lagrangian was:
() [ [ Differentiating the Lagrangian with respect to aluminum price: 51
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. This is the impact on profits from a $1 increase in aluminum price. To find the impact due to a 1% increase in aluminum
price we have: Evaluate at optima...
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- Fall '14