# Supply Function

The supply function shows the relationship between the quantity supplied of a good and the factors that affect it. Many variables can influence the quantity supplied, such as variations in production costs, technological improvements, and taxes, but prices play a central role.

## Linear Supply Function

An example of a linear supply function is the following:

The supply function is expressed as:

\[ q = 10p - 200 \]

Where:

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

In this example, the quantity supplied depends solely on the price, meaning that all other factors that can affect the quantity supplied are held constant. The positive sign of 10p represents the positive relationship between price and quantity supplied; a higher price leads to a higher quantity supplied and vice versa. Using this supply function, if we input any given price, we get the quantity supplied at that price:

The calculation of the quantity supplied when the price is 60 is:

\[ q = 10p - 200 \]

Substituting \( p = 60 \):

\[ q = 10(60) - 200 \]

\[ q = 600 - 200 \]

\[ q = 400 \]

Therefore, the quantity supplied \( q \) when the price \( p \) is 60 is 400.

However, it is not necessary to assume that all other factors apart from the price are constant; they can be included in the function. Let’s add, for example, costs, the technology used in the production process, and taxes. Of course, many other factors can affect supply, but everything not included in the equation is assumed to either not affect supply or remain constant:

The supply function considering production costs, technology, and government taxes is expressed as:

\[ q = 10p - 200 - 1C + 2A - 3T \]

Where:

- \( q \) represents the quantity supplied of a good or service.
- \( p \) is the price of the good or service.
- \( C \) are the production costs.
- \( A \) is the technology used in production.
- \( T \) are the taxes.
- \( -1 \) is the coefficient indicating the effect of production costs \( C \) on the quantity supplied.
- \( 2 \) is the coefficient indicating the effect of technology \( A \) on the quantity supplied.
- \( -3 \) is the coefficient indicating the effect of government taxes \( T \) on the quantity supplied.

Note the signs of each term. The negative sign in production costs indicates that higher production costs result in a lower quantity supplied. The same situation is shown by the negative sign of taxes. On the contrary, technology has a positive sign because improvements in technology increase the quantity supplied.

## Generalized Supply Function

In the previous examples, the functions have a linear form, but this is not necessarily the case. Supply functions can take other forms, such as logarithmic or multiplicative. Therefore, we can express the function that depends only on the price in a general form:

The 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 varies in response to changes in the price \( p \).

We can also express a general form for the function that depends on several factors:

The supply function considering production costs, technology, and government taxes is expressed as:

\[ Q_s = Q_s(p, C, A, T) \]

Where:

- \( Q_s \) represents the quantity supplied of a good or service.
- \( p \) is the price of the good or service.
- \( C \) are the production costs.
- \( A \) is the technology used in production.
- \( T \) are the taxes.

In these cases, we are not expressing a specific functional form, and therefore, the supply function can be expressed in various functional forms.