What are the both relationship and inverse relationship between a dependent and an independent variable?

What is an Inverse Relationship?

An inverse relationship is one which is the reverse of another or one in which when one variable factor increases, another decreases. The English term inverse is derived from a Latin word that means “turn upside down”; or opposite in some way. It the sort of relationship that appears in many disciplines, including mathematics, economics and finance. A few examples from each of these areas will illustrate how inverse relationships occur and operate. Let us begin with mathematics.

Inverse Relationship Examples

All the examples of inverse relationships one is likely to encounter involve the reversal or opposite of an association that might be expected. They entail a link between two variables, where either (i) the dependent and independent variables swap roles, i.e., the dependent variable becomes the independent variable and vice versa; or the dependent variable decreases (increases) as the independent variable increases (decreases).

Inverse Relationships in Math

In math, we often come across pairs of variables that are linked in some way. A set of such variables might appear like this: {(-5, -6) (-3, -2) (0, 4) (2, 8)}, where the values that occur first represent one variable and the values in second position represent another variable. In many instances, the values representing the first variable may be described as the x-values; those representing the second variable, as y-values. The link between the two variables may depend on some causal relationship or they may have been paired randomly. Regardless, by virtue of being paired, the x and y values in each pair, and by extension, the two variables which they represent are now in a relationship. That relationship may be described by a rule that takes the values of the first variable (x-values) and tells us the corresponding values of the second variable (y-values). Just as legitimately, the relationship may be described by a rule that takes the values of the second variable (y-values) and tells us the corresponding values of the first variable (x-values). Such rules in mathematics are known as functions.

A mathematical function is simply a rule that describes the relationship between ordered pairs, going either from x-values to y-values, in which case it is written y = f(x) or from y-values to x-values and written x = f(y) or y = f-1(x). The set of values of the variable in brackets is called the domain, while the set of values of the other variable is known as the range. Thus, in y = f(x), the x-values are the domain, while the y-values are the range. Sometimes, a function is described as a machine that takes input – the x-values – and delivers output – the y-values.

As with any rule, its outcome must be unambiguous. It’s a poor rule that gives one result today and another tomorrow. Accordingly, in f = (x), any x-value must result in only one y-value and all x-values must have a result. Hence, for any set of ordered pairs, there will be two rules, with one being the inverse of the other, i.e., the second rule would have described a function that is the inverse of the first rule. And the second function would bear an inverse relationship to the first function.

When One Goes Up, the Other Goes Down

However, an inverse relationship may also exist between the x and y variables rather than the functions. In such cases, an inverse relationship is the opposite of a direct relationship, where in y = f(x), y increases as x increases or in x = f(y), x increases as y increases. In an inverse relationship, given by y = f(x), y would decrease as x increases.

These relationships can be illustrated graphically.

Inverse Relationships in Economics

Demand and supply curves are shown below.

There are many instances of inverse relationships in economics. The one most frequent encountered is the price-demand relationship, where quantity demanded falls (rises) as price increases (decreases). This relationship is widely known as the law of demand.

The demand curve shows the quantity demanded of a good at different price levels. Note that demand is not the same thing as quantity demanded. Demand for a good depends on many factors: the price of the good, the price of other goods, the level of income and wealth, individual preferences, etc. The demand curve above shows the quantities of the good demanded at different price levels, when the other factors are held constant.

The inverse relationship between the price of something and the quantity demanded of it depends on two influences. First, a reduction in price of a product means more of it can be purchased for the same expenditure as before. Second, the lower price of one product increases real income, since less money is required to purchase the product, even though money income remains the same. The rise in real income means that more of all goods, including the one whose price has been reduced, can be purchased.

By contrast, the supply curve illustrates a direct relationship. When prices go up, existing suppliers will try to sell more, while new suppliers will be encouraged to enter the market. As a result, the quantity supplied of the product will increase as prices rise.

Inverse Relationships in Finance

The connection between interest rates and bond prices is an inverse relationship. Bond prices fall as interest rates go up and rise as interest rates go down. This occurs because a bond is a fixed income financial instrument. When a bond is issued, its face value, which is the amount of money, typically $1,000, the bond was issued to raise, is set. In addition, the bond will carry a coupon rate, which determines the fixed coupon payment. Thus a 10% coupon rate means that the $1,000 bond will pay $100 annually.

If a $1,000 bond of similar risk is issued that has a coupon rate of 12%, the 10% bonds will fall in value, because they pay only $100 annually, when the new bonds are paying $120. The price of the old bonds will fall until their $100 per annum payout equals 12%, i.e., $100/0.12 = $833.33. This inverse relationship between bond prices and interest rates can be plotted on a graph, as above.

A Last Word

Bear in mind that the term inverse relationship is used to describe two types of association. In mathematics, it refers to a function that uses the range of another function as its domain. The second function is then the inverse of the first. It may also refer to the association between two variables, where the value of one variable decreases (increases) as the value of the other variable rises (falls).

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