The corporate sector solves:

as the marginal liquidity service (expressed in terms of pledgeable income) in state ш assuming that the available liquidity is L(oj) = Assumption 1 implies that т(ш) > 0 for all ш.

Assume that there exists at least one state of nature in which there is excess liquidity, that is, in which the decision d(oo) is in the interior of the feasible decision set D. This is a mild assumption and is satisfied in all our examples. It implies (see the Appendix for more detail) that pledgeable income is never redistributed to the corporate sector in states of liquidity shortage (t(oo) = 0 if т(ш) > 0), and so the available liquidity in equation (9) is appropriately defined.
Optimization with respect to L*,, at equilibrium market prices, x yields
Like risk premia, liquidity premia are determined by a covariance formula, but this time involving the intertemproal marginal rate of substitution 1 + т(ш) of the corporate sector. An asset’s liquidity premium is high when it delivers income in states in which liquidity has a high value for the corporate sector.
For completeness, we can finally introduce external claims on the corporate sector (shares, bonds, etc.). Let Lj be the date-0 supply of claim j paying 9j(uj)

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