There has been something that has bothered me about the temperature function in the partition function approach last used here [1]: $f(\ell) = \log \ell < 0$ for $\ell < 1$ (or in the original application [2] in terms of money supply $m$). Typically the labor supply $\ell$ is large (millions of people employed), so this isn't a big deal. However it is possible for the "temperature" to go negative, which is a theoretical problem for small $\ell$. In thermodynamics, the analogous function is $f(T) = 1/T$, which is always positive.

Therefore I tried a different function $f(\ell) = \log (\ell + 1)$ (solid) which stays positive and approaches the original function (dashed) for $\ell \gg 1$:

The impact was fairly small on the results of [1] -- the largest difference comes in the ensemble average productivity $\langle p \rangle$ (right/second is from [1], left/first is new calculation):

There was negligible impact on the other results -- the unemployment rate even showed a slight improvement (first is new calculation, second is from [1]):

Overall, a minor impact empirically, but fairly important theoretically.

...

I should note that if $A \rightleftarrows L$ with IT index $p$, we have

$$

A = A_{ref} \left( \frac{L}{L_{ref}} \right)^{p}

$$

If $L \equiv L_{ref} + \ell$, then we can rewrite the previous statement as

$$

A \sim \exp \left( p \log (\ell + 1) \right)

$$

so that the original motivation for the partition function (in [2] above) would tell us that $f(\ell) = \log (\ell + 1)$.

...

**Update 22 September 2016**I should note that if $A \rightleftarrows L$ with IT index $p$, we have

$$

A = A_{ref} \left( \frac{L}{L_{ref}} \right)^{p}

$$

If $L \equiv L_{ref} + \ell$, then we can rewrite the previous statement as

$$

A \sim \exp \left( p \log (\ell + 1) \right)

$$

so that the original motivation for the partition function (in [2] above) would tell us that $f(\ell) = \log (\ell + 1)$.

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