# himaginary's blog

## 2011年6月12日日曜日

### Analogy between Gibbs free energy and excess demand for money

In thermal physics, Gibbs free energy is defined as follows:
G ≡ U - τσ + pV

where U is the internal energy, τ is the temperature, σ is the entropy, p is pressure, V is volume.

Therefore,
dG = dU - τdσ - σdτ + pdV + Vdp

Now: "Consider a system in thermal contact with a heat reservoir R1 at temperature τ and in mechanical contact with a pressure reservoir R2 that maintains the pressure p, but cannot exchange heat." (Kittel p.262)

In this system S, dτ=0 and dp=0, so
dGS ＝ dUS - τdσS + pdVS

In equilibrium, this equals zero for each set of (τ,p).

In monetary economics, excess demand for money can be derived from equation of exchange as follows:
G ≡ PY - MV

where P is the price, Y is the output, M is the money, V is the velocity of money.

Therefore,
dG = dPY + PdY - dMV - MdV

Now, consider a mediating system in financial contact with a central bank R1 that maintains the money M and in transactional contact with a real economy R2 that maintains the output Y.

In this system S, dM=0 and dY=0, so
dGS ＝ YdPS - MdVS

In equilibrium, this equals zero for each set of (M,Y). This analogy is the idea behind the previous post.

From this analogy, you can derive the monetary economics version of Clausius-Clapeyron relation. Note that in this analogy p(pressure) corresponds to Y(output), V(volume) corresponds to P(price), τ(temperature) corresponds to M(money), and σ(entropy) corresponds to V(velocity of money).

## 自己紹介

This blog is some thoughts on economics by a Japanese non-economist. Translated from my Japanese blog.