Definition: Subdifferential is a class of slopes. By example -- consider the top half of a stop sign as a
function graphed on the xy-plane. It has well-defined derivatives except at the corners. The subdifferential is made up of only one slope, the derivative, at those points. At the corners there are many 'tangents' which define lines that are everywhere above the stop sign except at the corner.
The slopes of those lines are members of the subdifferential at those points.
In general equilibrium usage, the subdifferential can be a class of prices. It's the set of prices such that expanding the total endowment constraint would not cause buying and selling, because the agents have optimized perfectly with respect to the prices. So if a set of prices is possible for a Walrasian equilibrium, it is in the subdifferential of that alocation.
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