#### 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

\(TW = \int \! T_h \, \mathrm{d}x – \int \! T_c \, \mathrm{d}x\).