## From Basic to Involved Mathematics

### Equations and Graphs

There are a number of basic equations used in economics and business. An understanding of what they mean and what they tell us is an important starting point before looking at more complex equations. The key equations you might come across in the first year of a degree course might be:

• Demand functions
• Supply functions
• Production functions
• Cost functions
• Revenue functions
• Consumption functions
• Aggregate Demand functions
• Aggregate Supply functions

To deal with these functions you need to understand what the equation you are being given is telling you and how it relates to the graphical representation that you might be more familiar with from textbooks. Let us look at one example:

Qd = 80 - 3P

This equation tells us that the level of demand is dependent on the price charged. The inclusion of the numbers gives us more specific details about the nature of that relationship. The '80' is the vertical intercept - the point at which the demand curve cuts the y axis. The minus sign tells us that there is a negative relationship between the level of demand and price - so if price goes up, demand will fall and vice versa. The '3P' element tells us that for every 1 unit change in price, demand will rise/fall by 3 units. The bigger the coefficient of P the stronger the relationship is - which may give a clue to the degree of price elasticity because this number will tell you the rate of change of demand when price changes.

We can now work out values of Qd given different price levels. If a business was able to analyse its sales data it may be able to come up with a function of this type that it could factor in to its decision making processes. If the price were 10, let's say, then the demand level would be 80 - 3(10). It is at such times when you have to remember the basics we have covered so far - multiplying positive and negative numbers and the order in which operations are carried out. Here the answer is 50.

#### Graphing Linear Functions

Let's take an example. Assume we have a demand function implied by the equation Qd = 150 - 5P. Assume some values for P ranging from 0 to 30 and put the information into a spreadsheet such as Excel. Excel has a nasty habit of not recognising that in economics, the dependent variable (in this case Qd) is on the horizontal (x) axis and that price (the independent variable) is on the vertical (y) axis. In pure maths, the y axis is the dependent variable and the x axis the independent variable. To get round this problem follow the steps outlined below.

Plot the data as normal, for example:

 Qd Price 0 30 25 25 50 20 75 15 100 10 125 5 150 0
• In the Chart Wizard, select the line option for your graph.
• In the next step, click on the 'series' tab.
• At the bottom where it says 'Category (X) axis labels:' click on the right hand icon - it will shrink to just that box.
• Highlight the Qd series on your table (just the figures not the 'Qd' cell) the cells appear in the shrunken box.
• Click on the right hand icon again to get back to the 'Source Data' dialogue box.
• Highlight Qd in the series box and click 'Remove'.
• When you have your finished graph, right click on the 'X axis' and in the Format Axis dialogue box, uncheck the option 'Value (Y) axis crosses between categories'.

You should end up with a graph that looks something like the one below:

You will notice from the graph that the demand curve cuts the horizontal axis at a value of 150 - this is called the 'horizontal intercept'. Where demand cuts the vertical axis at 30 this is called the 'vertical intercept'.

In the equation, the coefficient of P is -5. If we look at a change in price and a change in demand as a result we can see that for every 5 unit change in price, demand will rise or fall by 25 units. The slope of this graph is therefore 25 ÷ 5 = 5. Given that we know demand is a negative function, we know that if price rises (or falls), demand will fall (or rise) as a result - an inverse relationship. Looking carefully at the equation, we can see that the coefficient of P is indeed -5. This coefficient therefore tells us the slope of the graph.

The diagram below highlights this principle.