Now if we substitute into the lagrangian equation we

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Unformatted text preview: ptimal cost in terms of parameters as . Now, if we substitute into the Lagrangian equation we get the optimal value of the Lagrangian equation, the “optimized objective of the constrained optimization problem”, where some terms are going to be zero: [⏟ [⏟ ( [ ) { ⏟ } This allows us to establish a few general results about the cost of producing a target output in the short run. Suppose that after we solve this CMP some parameter were to change slightly. Very likely, this’ll lead to new values for the optimal amount of the variable input and therefore a new value for the optimal cost. Without solving the problem again, we can use the envelope theorem to compute the (approximate) change in the optimal cost due to a small change in a parameter. Following the three steps of the envelope theorem we would first write down what we are trying to maximize in terms of parameters. In the case of a one variable and one fixed input CMP this would be (here, the variable input has a lower limit of zero in general could have a non-zero minimum limit): [ [ ( [ ( [ ) ) ( ⏟ ⏟ ⏟ ) [ ( ⏟ ) ⏟ [ Next, we would differentiate the parameterized Lagrangian with respect to the parameter that is changing and then evaluate that derivative at the optimal solution to get the change in the optimal Lagrangian due to the change in the parameter. For example, here are the expressions for the change in the optimal Lagrangian due to, all else constant, a changes in the parameters , respectively: 6 ECO 204 Chapter 13: The Short Run Cost Minimization Problem (this version 2012-2013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. But since solution): { } these derivatives are in fact (after taking account of the change in the optimal ( ) That is, if the price of the variable input rises, total cost must increase by the amount of the initial optimal variable input, or stays the same (why?). This is illustrated below: Fixed Input Initial iso-cost line New iso-cost line Fixed Input Target Output 0 Variable Input ( ) That is, if the price of the fixed input rises, total cost must increase by the amount of the initial optimal fixed input, or stays the same (why?). This is illustrated below: 7 ECO 204 Chapter 13: The Short Run Cost Minimization Problem (this version 2012-2013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. Fixed Input Initial iso-cost line New iso-cost line Fixed Input Target Output 0 Variable Input ( ) That is, if the company uses another unit of the fixed input, then total cost must increase by the price of the fixed input (but cost can’t stay the same). This is illustrated below. Fixed Input Target Output Initial iso-cost line New iso-cost line New Fixed Input 1 0 Initial Fixed Input Variable Input ( A priori, we don’t know the sign of ) ; we know that since . However, one can argue that in fact by noting that is attached to an equality constraint that in theory Marginal cost, which cannot be negative (think about it). If is alway...
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