Thermodynamics and Economics

By Mark Ciotola

First published on February 15, 2019

Thermodynamics and Economics

By Mark Ciotola

First published on February 15, 2019

This course concerns expressing microeconomic phenomena in terms of thermodynamic principles.

“Economics is the social science that studies the choices that individuals, business, governments, and entire societies make as they cope with scarcity, and the incentives that influence and reconcile those choices.”[1]

People exist in a physical universe and are themselves comprised of physical matter. Both economics and physics involve physical quantities. Unsurprisingly, it is possible to express many concepts in economics using thermodynamics. This has the benefit that economic models can be constrained using physics, and physics can provide a deeper understanding of economics.

Reference:

1. Arik Levenson (citing Parkin), Economics 001 Principles of Microeconomics

This course concerns expressing microeconomic phenomena in terms of thermodynamic principles.

“Economics is the social science that studies the choices that individuals, business, governments, and entire societies make as they cope with scarcity, and the incentives that influence and reconcile those choices.”[1]

People exist in a physical universe and are themselves comprised of physical matter. Both economics and physics involve physical quantities. Unsurprisingly, it is possible to express many concepts in economics using thermodynamics. This has the benefit that economic models can be constrained using physics, and physics can provide a deeper understanding of economics.

Reference:

1. Arik Levenson (citing Parkin), Economics 001 Principles of Microeconomics

Introduction

Microeconomics concerns individual and firm-level decisions regarding the allocation of resources. There are several important concepts to analyze a firm-level venture. We will just consider a single product firm for the sake of simplicity. Hence it is quite important for decision-making.

At its most basic level, microeconomics considers at what volumes and prices it makes sense for a enterprise to continue producing a good or service.

Revenues, Costs and Profits

General Definitions

Revenue (R) is the amount of money a firm receives in payment for a product sold.

Costs (C) are how much money a firm pays to make a product.

Profit (P) is the amount left over when cost is subtracted from revenue:

$$P = R – C$$

For example, a pizza requires $5 of ingredients, labor and overhead to make. A pizzeria can cell that pizza for$12. The profit would then be $7: $$7 = 12 – 5$$ Definitions for Totals Total Revenue (TR) is the sum of revenues a firm receives in payment for all units of the product sold. Total Cost (TC) is the sum of costs a firm pays to make all units of the product sold. Total Profit (TP) is the sum of profits obtained from all goods sold: $$TP = TR – TC$$ The pizzeria sells 100 pizzas each for$12, resulting in $1,200 in total revenue. The pizzas cost$5 to make, resulting in total costs of $500. Then the total profit would be$700.

$$700 = 1,200 – 500$$

Definitions for Marginal Quantities

However, not all units of a product cost the same to make, nor are they always sold at the same price. So it is often of interest to determine the revenue, cost and profit of each additional unit produced and sold.

Marginal Revenue (MR) is the revenue received for one additional unit of product sold.

Marginal Cost (MC) is the cost paid to produce for one additional unit of product.

Marginal Profit (MP) is the profit obtained when producing one additional unit of product:

$$MP = MR – MC$$

Margins

Businesspeople often talk of Gross Margins (GM), which is:

$$GM = \frac{Revenue – Cost~of~Goods~Sold}{Revenue}$$.

Here, the Cost of Goods Sold (COGS) is simply Costs (C). So Gross Margin becomes:

$$GM = \frac{R – C}{R}$$

Fixed Quantities

Fixed Costs (FC) are the minimum costs required to produce any amounts of units (even zero amount). For example, just to get into business, a pizzeria has to buy an oven.

Other Important Quantities

Entry costs are the minimum cost to start a particular business. Such are closely related to fixed costs.

Opportunity Cost (OC) is the amount of profit given up by not pursuing a course of action.

Risk (R) is cost due to uncertainty.

Interest rates (i) reflect a combination of costs due to risk and sacrificed opportunities. In reality, they can also represent bargaining power differentials.

Example

Imagine you are a celebrity setting up an online retail firm selling tee shirts with your photo for $10. All orders are fulfilled using a third-party online service. They charge you$5 amount for each tee shirt.

Your marginal revenue is $10. Your marginal cost is$3. Your marginal profit is $7. If you sell 100 tee shirts, your total profit would be: $$Total~Profit = Total~Revenues – Total~Costs$$ $$400 = 100~shirts~ (\frac{7}{shirt} – \frac{3}{shirt})$$ What if you had to pay for an up-front set-up fee of$100 to process your photo? This would be a fixed cost, as well as a start-up cost. Even if you sold no tee shirts, you would still incur the $100 just to go into business. If you sold no shirts, you would have a$100 loss instead of a profit.

However, if you once again sold $100 shirts, your profit after fixed costs would be the$400 from above less the $100 start-up fee, for a net profit of$300.

What is the minimum amount of shirts you should expect to sell in order to decide to go into business? You will need to at least break-even, which means to cover the fixed costs plus expected marginal costs. Here, marginal profit will provide $4 per shirt, so it takes 25 shirts to cover the initial$100 investment.

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

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

Introduction

Supply and Demandtheory models at which prices consumers will purchase a product, at what prices will producers sell a product, and the resulting market price for the product. Supply and demand provides a framework for understanding pricing and sales volume. Although real life pricing sometimes deviates from this idealized theory, the Supply and Demand approach can help identify why those deviations exist.

Supply Function

The supplyfunction is the price at which producers will sell a product as a function of quantity sold. According to theory, producers will demand a higher price for each additional unit of product sold, due to increasing marginal costs. Let’s use gold as an example. Each additional unit of gold mined requires gold producers to mine more deeply into the ground (or use lower grade ore), which is more expensive. In summary, suppliers tend to charge more money per additional unit demanded, at least in the short run.

Demand Function

The demandfunction is the price at which consumers will buy a product as a function of quantity purchased. According to theory, consumers will only  pay a lower price for each additional unit of product purchased, due to a decreasing marginal utility. (Utility means the value the the product to consumers). Continuing with gold for an example, consumers only really need so much gold. They might be willing to pay all their savings for a small amount of gold engagement ring. They might still be willing to pay a lot of money per gram for additional gold jewelry. Beyond that, most people do not have a high need or desire for moregold, and would not be willing to pay as much for more of it. In summary, as supply increases, prices consumers are willing to pay tend to fall. Once supply sufficiently increases, producers literally cannot even give away their product for free and price drops to zero.

Determining the Market Price

These functions can be plotted, with the axes being price and quantity. Where these two functions intersect indicates both market price and quantity produced. The plots shown are examples. Actual plots may vary.

Market price and quantity are where supply and demand functions meet to determine market price.

Although some plots may be straight lines, the plots will often form concave curves in real life. This means the the marginal affects are accelerating for supply while decelerating for demand.

Market price and quantity are where supply and demand functions are concave curves.

4. 4. Expressing Supply and Demand Thermodynamically

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.