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Monday, March 17, 2014

The informaton transfer version of Nick Rowe's version of Keynes' General Theory


Basing this on Nick Rowe's post on Keynes' General Theory (Chapter 3). It appears to be set up as an information transfer market

WP:YL


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

WP=1κY0L


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

ΔYYYref=Y0κlogLLref


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):

WP=1κYL0

ΔLLLref=κL0logYYref


where we can eliminate Y to produce (after some re-arranging)

logWP=logYrefκL0+ΔLκL0


Rowe's form is W/P=Y/(1L), 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

logWP=logY+log11L


and expand around L=Lref so that

logWP=logY+log11Lref+LLref1Lref+

logWPlogY+logc+cΔL


Of course, W/P:YL 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:WL:



Where the equations that define the supply and demand curves are (respectively):


logP=logκL0Wref+ΔLκL0


logP=logW0κLrefκΔWW0


You would plot them as functions of ΔL or ΔW so that Pexp(1ΔW) or Pexp(ΔL1) and they look like this:




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