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

$$\frac{W}{P} : Y \rightarrow L$$

with real wages $W/P$, real output $Y$ and employed labor force $L$ so that

$$\frac{W}{P} = \frac{dY}{dL} = \frac{1}{\kappa} \; \frac{Y}{L}$$

from the basic information transfer model. Looking at constant  $Y = Y_{0}$, we have

$$\frac{W}{P} = \frac{1}{\kappa} \; \frac{Y_{0}}{\langle L\rangle}$$

where $\langle L\rangle$ 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

$$\Delta Y \equiv Y-Y_{ref} = \frac{Y_{0}}{\kappa}\log \frac{\langle L\rangle}{L_{ref}}$$

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 $Y = \log L$. Ours is technically $Y = c_{1} \log L + c_{2}$ where the $c_{i}$'s are constants.

Rowe's third equation $W/P = MRS(L, Y)$ follows from looking at constant $L = L_{0}$ so that we have (solving the differential equation again):

$$\frac{W}{P} = \frac{1}{\kappa} \; \frac{\langle Y\rangle}{L_{0}}$$
$$\Delta L \equiv L-L_{ref} = \kappa L_{0} \log \frac{\langle Y\rangle}{Y_{ref}}$$

where we can eliminate $\langle Y\rangle$ to produce (after some re-arranging)

$$\log \frac{W}{P} = \log \frac{Y_{ref}}{\kappa L_{0}} + \frac{\Delta L}{\kappa L_{0}}$$

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

$$\log \frac{W}{P} = \log Y + \log \frac{1}{1-L}$$

and expand around $L = L_{ref}$ so that

$$\log \frac{W}{P} = \log Y + \log \frac{1}{1-L_{ref}} + \frac{L - L_{ref}}{1-L_{ref}} + \cdots$$
$$\log \frac{W}{P} \simeq \log Y + \log c + c \Delta L$$

Of course, $W/P : Y \rightarrow L$ 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 $P : W \rightarrow L$:

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

$$\log P = - \log \frac{\kappa L_{0}}{W_{ref}} + \frac{\Delta L}{\kappa L_{0}}$$

$$\log P = \log \frac{W_{0}}{\kappa L_{ref}} - \frac{\kappa \Delta W}{W_{0}}$$

You would plot them as functions of $\Delta L$ or $\Delta W$ so that $P \sim \exp(1-\Delta W)$ or $P \sim \exp(\Delta L-1)$ and they look like this:

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