# Inverse Supply Function

The inverse supply function is the supply function solved for the price, meaning the price depends on the quantities supplied. This is the function that is graphed on the supply curve because the graph axes are inverted. By convention, the dependent variable is on the x-axis (quantity), and the independent variable is on the y-axis (price).

## Direct Supply Function vs Inverse Supply Function

The direct supply function is expressed in functional form as follows:

The direct supply function is expressed as:

\[ Q_s = Q_s(p) \]

Where:

- \( Q_s \) represents the quantity supplied of a good or service.
- \( p \) is the price of the good or service.
- \( Q_s(p) \) is a function that shows how the quantity supplied \( Q_s \) varies in response to changes in the price \( p \).

An example of a direct supply function is the following:

\[ p = 20 + 0.1q \]

From the direct supply function, it is possible to solve for the price in terms of the quantities as follows:

We start with the initial equation:

\[ p = 20 + 0.1q \]

We subtract \(20\) from both sides of the equation:

\[ p - 20 = 0.1q \]

We divide both sides of the equation by \(0.1\) to solve for \(q\):

\[ \frac{p - 20}{0.1} = q \]

We simplify the fraction:

\[ q = 10(p - 20) \]

We expand the expression:

\[ q = 10p - 200 \]

Note that in this function, the price depends on the quantity supplied. This is the function that is graphed on the supply curve, meaning the inverse function is graphed rather than the direct one. This is because, by convention, the axes are inverted; note that the independent variable (price) is on the y-axis, and the dependent variable (quantities) is on the x-axis.

In this graph, all other factors besides the price that can affect the quantity supplied are assumed to be constant or non-influential, such as changes in production costs, technological changes, or taxes. Now, from the inverse supply function, it is also possible to find the direct supply function by solving for quantities in terms of price.

We start with the reorganized inverse function:

\[ p = 20 + 0.1q \]

We subtract \(20\) from both sides to begin solving for \(q\):

\[ p - 20 = 0.1q \]

We multiply both sides by 10 to eliminate the decimal coefficient:

\[ 10(p - 20) = q \]

We simplify the expression:

\[ q = 10p - 200 \]

We add \(200\) to both sides to isolate \(q\):

\[ q + 200 = 10p \]

We divide both sides by 10 to solve for \(p\):

\[ p = \frac{q + 200}{10} \]

We simplify the fraction:

\[ p = 20 + 0.1q \]

Therefore, it is possible to switch between the direct supply function and the inverse supply function depending on which one is needed.