# 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

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