The approximation, Landsburg says, is good “assuming that your contributions are small relative to the initial endowments”. Here’s the thing: why? Suppose Δx/X, Δy/Y and Δz/Z are indeed very small—what then? Why does it follow that the linear approximation works? There’s no explanation, and if you think this is because it’s immediately obvious—well, it isn’t. It may sound plausible, but the math isn’t there. We need to go deeper.
What shapes for U(X,Y,Z) could make the linear approximation not work? It would have to be a curve that had sudden local changes. It would be kinked or fractal. That would be surprising. If U(X,Y,Z) is continuous, smooth, monotonic, and its first and second derivatives are monotonic, I can’t imagine how the linear approximation could fail.
If U(X,Y,Z) is continuous, smooth, monotonic, and its first and second derivatives are monotonic, I can’t imagine how the linear approximation could fail.
There’s an example later in the post, with mixed derivatives. Everything could be smooth and monotonic including all derivatives. Basically think of U(X,Y,Z) as containing a 100XY component.
What shapes for U(X,Y,Z) could make the linear approximation not work? It would have to be a curve that had sudden local changes. It would be kinked or fractal. That would be surprising. If U(X,Y,Z) is continuous, smooth, monotonic, and its first and second derivatives are monotonic, I can’t imagine how the linear approximation could fail.
There’s an example later in the post, with mixed derivatives. Everything could be smooth and monotonic including all derivatives. Basically think of U(X,Y,Z) as containing a 100XY component.