Notice that the variable inputs required to produce

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Unformatted text preview: a separate constraint). Notice that the variable inputs required to produce the target output (given fixed inputs) are chosen to minimize the total variable cost, not total cost (total fixed cost + total variable cost): ∑( ⏟ ) ( ⏟ ) ⏟ Does it matter that the “objective” doesn’t include the cost of fixed inputs? No, because , total fixed cost, is a constant (it’s the cost of fixed inputs) and therefore independent of the amounts of the variable inputs. As such, the following two variants of the short-run CMP will yield the same answers for the optimal amounts of variable inputs: ( ⏟ ) ⏟ ( ⏟ ) ⏟ Even though these short run CMPs yield the same answers for the optimal variable inputs, in ECO 204 we will work with the short run CMP that has includes the fixed cost in the objective (i.e. CMP ❷ above, reproduced below): ( ⏟ ) ⏟ The firm can solve this CMP in terms of numbers or in terms of parameters. Solving a numerical CMP would give values for the optimal variable inputs from which we would get a dollar value for the minimal cost of producing a specific target output. On the other hand, solving a parametric CMP would give us equations (in terms of parameters) for the optimal inputs from which we would get an equation (in terms of parameters) for the minimal variable, and thus total, cost of producing an arbitrary target output from which we can derive the ECO 100 graphs for average cost, marginal cost, average variable cost, etc. The short run CMP is a constrained “minimization” problem: 4 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. ( ⏟ ) ⏟ From lecture 1 and chapter 1 recall that the ECO 204 constrained optimization methods work for maximization not minimization problems. To transform the “min CMP” into a “max CMP” we multiply the objective (the total cost of inputs) by { } ( ⏟ ) To setup the Lagrangian equation we re-arrange the constraints: { } { ( ⏟ } [ ) ( ⏟ ) ∑ Since we have a combination of equality and inequality constraints we would solve this problem by a combination of the Lagrangian and Kuhn-Tucker methods and obtain solutions for the optimal variable inputs and the Lagrange multipliers. Prior to solving any CMP it’s instructive to an example to see what the optimal values of the Lagrange multipliers will tell us. Consider this general one fixed and one variable input short run CMP (here Price of variable Input and price of fixed input): [ [ ( ) The FOCs and KT conditions for this general 1 variable input CMP are (why do we differentiate with respect to ?): ( [ ( but not ) ) 5 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. Here ( ) ( Marginal product of the variable input, i.e. ) . Now, in ECO 204, we will always find solutions for from which we can compute the o...
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