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Equation 23 · Part 8 · When Generation Becomes Free, Attention Becomes the Scarce Good

Denominator: partial s partial θ

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

What this part means

The complete quantity below the fraction bar; it must be nonzero for this division.

Its job in the formula

partial s partial θ occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

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 fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

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Sources cited in the surrounding passage

These citations provide research context; check each source for the exact claim it supports.