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ECON 1450 – Professor Berkowitz The Implicit Function Theorem

ECON 1450 – Professor Berkowitz The Implicit Function Theorem. Punitive damages. The defendant is strictly liable, but the penalty of covering damages is enforce only α < 1 of the time Therefore, in choosing x (precaution), the defendant solves

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ECON 1450 – Professor Berkowitz The Implicit Function Theorem

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  1. ECON 1450 – Professor BerkowitzThe Implicit Function Theorem

  2. Punitive damages • The defendant is strictly liable, but the penalty of covering damages is enforce only α < 1 of the time • Therefore, in choosing x (precaution), the defendant solves • Choose x: min x + p(x)αD(x) = min x + αED(x)

  3. Specific example • Suppose α = 1/3, so that • Choose x: min x + αED(x) = min x + (1/3)ED(x) • The socially efficient precaution, x*, holds when the defendant believes that he will always pay when he is in the wrong. Intuitively, then, x(α=1/3) < x*, and more generally x(α<1) < x*. How can we show this?

  4. Proof • Suppose we can solve for x(α), where • x(α=1) = x*. Then, if our intuition is correct, then ∂ x(α)/ ∂α > 0!!! • How do we show this?? • Choose x: min x + αED(x) – we get a FOC: • L(x, α) == 1 + α ∂ED(x)/ ∂x == 0, where • ∂ L(x, α)/∂x = α ∂2ED(x)/ ∂x ∂x > 0 • ∂ L(x, α)/∂α = ∂ED(x)/ ∂x < 0

  5. Implicit Function Theorem!!! • By the implicit function theorem, ∂x/ ∂α = -∂L/∂α/∂L/∂x > 0 !!!!! • This is a very general tool that is used for making prediction in economic models

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