1 2 32 0 32 0 3 30 0 30 0 when approaching

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Unformatted text preview: The firm’s costs (or cost function) is given by the function , 5 80 32 . The firm 30. Of course, we wants to minimize its costs subject to a production constraint, also have the non‐negativity constraints, 0 and 0. We can translate this into one of a maximization problem by looking at , . The firm then chooses and to maximize 5 80 32 , subject to the constraints above. The Lagrangian for this problem is: 5 80 32 30 . Our first‐order conditions are: : 10 80 0 10 80 0. 1. 2. : 32 0 32 0. 3. : 30 0 30 0. When approaching a problem like this, first check out corner solutions for the choice variables ( and ). Suppose 0. Then 80, which violates our K‐T condition that 0, we conclude 32. So we 0. So this can’t apply. Likewise, if we assume can’t have corner solutions. Need the constraint bind? Suppose not. Then 0 (the constraint doesn’t affect the 30 0. But from the second conditions objective function at the margin) and so of 1) and 2) (with 0) we see 8 and 32, which means the constraint would bind. Hence, 0. 2. The Implicit Function Theorem We have on a few occasions made use of a very useful theorem in economics, the implicit function theorem. It is time to point it out explicitly. Suppose we had a nice, function which is continuous and has continuous first partial derivatives, where , 0 at some point ∗ , ∗ in its domain. We’ll keep things somewhat general here as to what , , , and the condition , 0 may represent. ∗∗ If we assume , 0, the implicit function states that there exists a neighborhood of ∗∗ , such that any value of corresponds to a unique value of and hence, we can write as a continuous function of , . Moreover, this function is differentiable. More generally, consider the system...
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