# Fun Algebraic Equations Involving Economics

By Kathleen Knowles, 03 Jul 2020

During my first year in university, I had a roommate who was majoring in economics. Among some of her reasons for choosing this course of study was the simplicity of the mathematics included -- or so she thought. The following weeks of lectures gave her a nasty shock. Mathematics is a crucial and vital part of economics.

## How Economists Use Algebra

Algebra is a set of mathematical expressions containing variables, constants, and operations. Before you can fully understand economics, you need a good understanding of these mathematical concepts. Calculating profits, returns, allocation of resources, investment risks, among other things, require you to know how to identify and calculate variables, and constants. Most economic calculations and measurements contain statistics, algebra, and calculus (differential equations).

Economists assess risks involved with a particular event and find ways to save money using math. Big companies hire economists to determine which course of action is best for their business. The economist conducts a market survey and notes down all the factors in play. These factors, also known as constants and variables, are used to calculate particular outcomes that determine if an action will be profitable or not and what the long-term ramifications may be.

## Three Laws of Algebra

There are three fundamental laws of algebra: commutative, associative, and distributive.

The **commutative law** states that addition or multiplication of two or more numbers is commutative. The order at which they are placed doesn’t matter, i.e. 2+3+4=4+2+3 and 1x2x3=3x2x1

The **associative law** states that the grouping of numbers doesn’t affect the result when adding or subtracting them, i.e. 1+2 + (3+4) = 1+ (2+3) +4 and 2(3x4) = (2x3) x4.

The **distributive law** states that (X+Y) Z=XZ + YZ and X(Y+Z) =XY+XZ, which means multiplying a number separately or as a group yields the same result.

We will use these laws to solve some friendly economic problems that contain algebraic equations.

## Questions

We will first present the questions. A break down of each question is included in the section below.

### Question One

A cosmetic company just started a new face cream product line. The company produces 800 face creams in a week. The company offers "pay on delivery" services. For each customer that orders the face cream, there is a 10% chance that they may cancel the order before the week runs out. This means that there is a 90% chance they won’t. If the company receives 800 orders for the week, how many people are expected to cancel their orders?

### Question Two

A coffee shop has about 10,000 USD worth of assets before depreciation. After assessing the shop, you realize that the cost of repair will be four times the cost of maintenance. The expected repairs and maintenance expense to fixed assets ratio is given at 8% using this formula:

*Repairs and Maintenance Expense to Fixed Assets Ratio = (Cost of Repairs + Cost of Maintenance) / Total Value of Fixed Assets.*

Determine how much money will be needed to put the shop back in working order and if it is worth 10,000 USD.

### Question Three

A company is starting on new product X but has yet to determine the price of the product. They aim to price the product such that their demand will be the same as the number of products manufactured. X is bought based on how much it costs, and a study shows that the demand per month is three times the price subtracted from 21. The quantity that can be manufactured is 4 times the price plus 3. What will be the expected product price?

### Question Four

Let us assume that you are a financial analyst with the following data, and you want to find the return on assets for a company to see if it’s making enough money.

Financial Leverage Index = 3 and the Return on Equity (ROE) is 50% of the total capital. What is the Return on Assets (ROA)?

## Solutions

### Solution One

When someone orders a face cream, they either cancel the delivery or do not. Let’s say that *X* is the number of people that do not cancel their orders. This means that *X* is binomially distributed. *P* is the probability of not canceling, and *n* is the quantity ordered. The probability distribution is:

*P(x;n,p)=(nx)px(1−p)n−x, x=0, 1,...n*

The mean of the distribution is *E(x) = np* and the variance *σ2 = np(1−p)*.

In this problem n=800 and p=0.9.

*E(x) = 800 x 0.9 and σ2 = 800 x 0.9 (1−0.9)*

So, E(X) = 720 and σ = 8.49 and the expected cancels is 80.

### Solution Two

Since Quantity demanded = Quantity supplied, we can say that *21−3P = 3+4P*.

Subtracting 3 from both sides and adding 3P to both sides yields:

*21−3P−3 3+4P−3*

*21−3P 4P*

*21−3P+3P 4P+3P*

*21 7P*

Divide both sides by 7.

*3 P*

This means that for the company to be able to supply what is demanded, they need the price to be at $3 per product.

### Solution Three

The total amount of money required will be the sum of the cost of repairs and the cost of maintenance. Let's assume that the cost of maintenance is *m*. This means that the cost of repairs is *4m*.

*0.8 = (m + 4m) /10,000*

*M+4m = 10,000 x 0.8*

*5m = 8000*

*M = 8000\5*

M = 1,600 and most of repairs is *3 x 1600*=6400

This means the cost of the coffee house is almost the same as how much it will take to renovate it.

### Solution Four

Let’s break down the problem to identify the meaning and value of the different variables.

The formula for the financial leverage index is:

*Financial Leverage Index = Return on Equity (ROE) / Return on Assets (ROA)*

Let's assume that *ROA = x*.

*3 = 56% / X*

*3*X =56%*

*X= 56/3*

*X = 18.7%. *The Return on Assets is 18.7%.

That's a lot of math -- but it is useful math. Economists are filled with valuable knowledge and they get there using algebra.

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