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Equation 23 · Part 5 · When Generation Becomes Free, Attention Becomes the Scarce Good

derivative

∂2c(s,θ)∂s ∂θ  <  0,\frac{\partial^{2} c(s,\theta)}{\partial s \, \partial \theta} \;<\; 0,
derivative

What this part means

This notation tracks how one quantity changes with another. The indicated variable tells which change is being measured.

Its job in the formula

This notation tracks how one quantity changes with another. The indicated variable tells which change is being measured.

The passage around this formula

Spence’s contribution was to explain how information can cross the asymmetry without direct verification. An agent takes a costly action — in his original case, education — and the action separates types only under a specific condition: the marginal cost of the signal must be lower for the type the signal is supposed to identify. In the standard notation, with signal level s and type θ\theta increasing in ability, the single-crossing condition is ∂2c(s,θ)∂s ∂θ  <  0\frac{\partial^{2} c(s,\theta)}{\partial s \, \partial \theta} \;<\; 0. and it is this inequality, not the content of the signal, that does the work [ 3 ] . Spence’s model famously does not require the signalling activity to be productive at all. It requires only that it be differentially costly.

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Learn the underlying idea

A derivative describes how quickly one quantity changes as another changes. It is the slope of a curve at a particular point.

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Sources cited in the surrounding passage

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