Thermodynamics and Economics
Expressing Microeconomics as a Heat Engine
By Mark Ciotola
First published on February 15, 2019
An Analogy
The firm is at the heart of microeconomics. For a firm to survive, it must make a gross profit. Several of the concepts of firms and microeconomics are analogous to those concerning heat engines.
For example, thermal energy removed from the hot reservoir \(Q_h\) is analogous to revenue (R). Thermal energy placed into the cold reservoir \(Q_c\) is analogous to cost (C).
We can continue the analogy further. For each successive unit of product, Marginal Revenue MR would be \(dT_h\). Marginal Cost MC would be \(dT_c\).
Transforming The Analogy Into Calculations
Then, Marginal Profit MP would be:
\(dP = dT_h – dT_c\), or,
\(MP = MR – MC = T_h – T_c\)
Gross Margin can be expressed as:
\(GM = \frac{R – C}{R}\)
Does the form of this equation look familiar? Let us express recall a similar expression for Carnot efficiency:
\(\epsilon = \frac{Th – Tc}{Th} =1 – \frac{Tc}{Th}\)
So we can now express Gross Margin thermodynamically:
\(GM = \epsilon = 1 – \frac{Tc}{Th}\)
We can now express Marginal Profit thermodynamically:
\(P =Q_h~\epsilon = Q_h~\frac{Tc}{Th}\)
In Summary:
The work performed by the heat engine represents profit (P):
\(W = T_h – T_c\)
Likewise, Total Revenues would be \(\sum T_h , \) where \(\sum\) is the summation sign. Total Costs would be:
\(\sum T_c\).
Total Profit would be can be expressed in terms of total work W:
\(\sum W\), or
\(TW = \sum T_h – \sum T_c\).
More Exact Expressions
For those inclined to the greater exactness of calculus, Total Revenues would be \(\int \! T_h \, \mathrm{d}x \) where \(\int\) is the summation sign.
Total Costs would be:
\(\int \! T_c \, \mathrm{d}x \).
Total Profit would be can be expressed in terms of total work W:
\(\int \! W \, \mathrm{d}x \), or
\(TW = \int \! T_h \, \mathrm{d}x – \int \! T_c \, \mathrm{d}x\).