# Expressing Supply and Demand Thermodynamically

#### By Mark Ciotola

First published on February 16, 2019

Let us examine how Supply and Demand can be expressed thermodynamically.

## Early in the Process

Heat engine operating across high temperature difference.

Let us consider a Carnot engine operating upon exhaustible, nonrenewable reservoirs. This is analogous to bridging market supply and demand in order to make a profit. Initially, the temperature difference between the hot and cold reservoirs is quite high. In thermodynamics, we would say the engine operates at a high efficiency. As a business, we would say the engine has a high profit margin.

## A Dynamic Process

Yet the efficiency of each subsequent unit produced decreases. This indicates increasing marginal costs, thus decreased marginal profit (or put another way, the cost increases for each unit of profit produced).

Let us denote thermal energy as $$Q$$. Let us consider the removal of 1 unit of thermal energy from the hot reservoir, $$Q_h$$. Hence, the marginal revenue will be $$Q_h$$. The profit is the work $$W$$performed.

For a Carnot heat engine, work is:

$$W = Q_h (1 – \frac{T_c}{T_h})$$.

Heat engine operating across moderate temperature difference.

Efficiency vs Thermal Energy Removed

Hence profit is decreasing as well for each additional unit produced. So there is a decreasing incentive to produce additional units unless additional units can be sold at additional prices. Hence the supply curve slopes upward as quantity supplied increases.

Rate of work performed versus thermal energy removed

At the same time, as thermal energy flows from the hot reservoir to the cold one, the temperature difference decreases as the hot reservoir cools and the cold reservoir heats up. This is analogous to fulfilling limited market demand.

## Another View

A hot reservoir supplies and a cold reservoir demands. Yet how to express changing “price” of supply and demand?

### What is conserved when a cold reservoir receives heat?

An general, the colder a reservoir is, the greater demand it will have for heat. What does demand mean physically? A higher thermal gradient. But what about the changing slope of a demand curve.

## Market Satisfaction Equilibrium

Heat engine operating across no temperature difference can perform no work.

Eventually the two temperatures become equal, which is analogous to having satisfied all of the customer demand.

Temperature Difference versus Thermal Energy Removed

Hence, the demand curve slopes downward, just as the temperature difference does.

## Market Price Equilibrium

Market price and quantity are where supply and demand curves meet.

Where there are both continuous sources of supply and demand, the market will eventually reach a market equilibrium where these two curves meet. Since such requires a continual flow of goods, this can be said to have reached dynamic equilibrium.