Subsidy in supply and demand model

The aim of this post is to analyze what happens to the quantity of goods produced and the market equilibrium price when the government provides a subsidy to suppliers.

We wish to find out what the equilibrium price and quantity are compared with what would happen if there was not a subsidy.

Before we can consider the implications that the subsidy has on the market, we first need to analyze the market free from interference.

Let’s consider an example where the demand curve is specified as follows:

Q_D = 10 – P

The supply curve before the subsidy has been implemented is defined as:

Q_s = P

In this case we know that the market equilibrium is here supply equals demand. This is the same as saying that the quantity demanded (Q_D) and quantity supplied (Q_s). This implies:

10 – P = P

10 = 2P

P = 5

We can now find the quantity that is consumed/produced in equilibrium by substituting our equilibrium price back into either the supply or demand function. It can trivially be seen that

Q = 5

By substituting the price back into the Q_s (quantity supplied) equation.

Introducing a subsidy

Now consider the case the subsidy (s) = 2. In this case for every unit the supplies provide, they get the subsidy as well as the price. Therefore, we can now write our quantity supply equation becomes:

Q_s = P + S

Q_s = P + 2

The market equilibrium in this case can be solved in the similar manner as it was above:

10 – P = P + 2

P = 4

Q = 6

It might be of interest to see the overall size of the subsidy paid by the government. Recall, that the government pays the subsidy for each unit sold. Therefore, the overall amount paid by the government is:

S * Q

2*6

12

Therefore the total amount paid by the government is $12. There are a few things worth noting

• The price decreases by less than the size of the subsidy
• The subsidy leads to an increase in quantity being produced

It is also very important to make sure that the subsidy is incorporated into the supply equation correctly. Suppose instead that Q_s = 2P then when we incorporate the subsidy into the supply curve that:

Q_s = 2(P+S)

Q_s = 2P+ 2S

Make sure that you do not just add the subsidy on to the end of the supply equation such that:

Q_s = 2P + S

The intuition is that the subsidy is essentially just part of the price and that we could easily define a new price which includes the subsidy as: P* = (P + S) and put that price into the supply equation.