Basing this on Nick Rowe's post on Keynes' General Theory (Chapter 3). It appears to be set up as an information transfer market
with real wages W/P, real output Y and employed labor force L so that
WP=dYdL=1κYL
from the basic information transfer model. Looking at constant Y=Y0, we have
where ⟨L⟩ is the expected value of the labor supply. This is Rowe's second equation ("classical labor demand"), written as W/P=MPL(L) and subsequently W/P=1/L. Ours is technically W/P=c/L where c is a constant. Solving the differential equation, we obtain
where ref refers to reference values of the variables L and Y. This is analogous to Rowe's first equation ("classical production function"), written as Y=f(L) and subsequently Y=logL. Ours is technically Y=c1logL+c2 where the ci's are constants.
Rowe's third equation W/P=MRS(L,Y) follows from looking at constant L=L0 so that we have (solving the differential equation again):
ΔL≡L−Lref=κL0log⟨Y⟩Yref
where we can eliminate ⟨Y⟩ to produce (after some re-arranging)
logWP=logYrefκL0+ΔLκL0
Rowe's form is W/P=Y/(1−L), but I'm not entirely sure what the 1 is supposed to mean (full employment?); however we basically obtain this if we take the log of both sides
and expand around L=Lref so that
logWP=logY+log11−Lref+L−Lref1−Lref+⋯
logWP≃logY+logc+cΔL
Of course, W/P:Y→L is a terrible model (Keynes' disagreed with at least part of it) ... here's this model compared to data:
A much more successful market would be P:W→L:
Where the equations that define the supply and demand curves are (respectively):
logP=−logκL0Wref+ΔLκL0
logP=logW0κLref−κΔWW0
You would plot them as functions of ΔL or ΔW so that P∼exp(1−ΔW) or P∼exp(ΔL−1) and they look like this:
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