Calculate the equilibrium price and quantity from math equations

In this post, I will cover how to find the equilibrium quantity and price when given an equation representing the supply and demand curves. Consider the following equation:

Find the equilibrium quantity and price given the inverse demand equation P_D = 10 - 3Q and and the inverse supply function P_S = 2P

find equilibrium price and quantity given math equations

Firstly, let's look at what the inverse demand and supply equations are actually representing. These equations simply represent the relationship between price and quantity in 'maths language'. For example, the supply equation P_S = 2Q says that the price of supply will always be twice the size of the quantity of goods being supplied. Suppose that the quantity supplied was 1 unit. In this case, the price of supply would be P_S = 2*(1) = 2. If the quantity supplied was 2 units, the price of supply would be P_S = 2*(2) = 4.

Constructing a supply and demand schedule

In an early post, we saw how the supply schedule can be used to draw a supply curve. We can actually use these supply and demand equations to construct a supply and demand schedule presented in the table below.

Qty suppliedPrice of supplyQty of demandPrice of demand
12*(1) = 2110 - 3(1) = 7
22*(2) = 4210 - 3(2) = 4
32*(3) = 6310 - 3(3) = 1

From the supply and demand schedule, we can see that the quantities supplied and demanded are the same at the point where both the prices are the same value of $4.

Calculating price and quantity using mathematical equations

Above we mechanically found the equilibrium by finding where the price for supply was the same for demand. However, there is an easier way to do this. When both the prices are the same for supply and demand, the quantity supplied equals the quantity demanded. And since the equations

P_S = 2Q
and
P_D = 10 - 3Q

Both describe the price dependent on quantity, if we set the equations equal, that is the same as setting the prices equal. Therefore, to find the equilibrium quantity, we equate:

2Q = 10 - 3Q

We can solve for Q in this equation by firstly adding 3Q to both sides such that:

 2Q + 3Q = 10 - 3Q + 3Q

which simplifies to

 5Q = 10

Now dividing both sides by 5 gives:

 \frac{5}{5}Q = \frac{10}{5}Q

Giving the equilibrium quantity:

Q = 2

Now we just need to solve for the equilibrium price. We just found the point where the quantity supplied equals quantity demanded. We know at this point the prices must be the same. Therefore, we can use either the inverse demand equation (P_D = 10 - 3Q) or the inverse supply equation (P_S = 2Q) to find the equilibrium price, as:

P_S = 2*(2) = 4

P_D = 10 - 3*(2) = 4

Both equations provide gives us price is $4. We should also see that this approach gave us the same answer as we found when we constructed the supply and demand schedules.

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