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Understanding the Implicit Function Theorem in Economic Models

This educational piece discusses the concept of the Implicit Function Theorem in economics, particularly in determining optimal precautions and punitive damages. It explores how this theorem can be applied to make predictions in economic models.

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Understanding the Implicit Function Theorem in Economic Models

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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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