## Mathematics in Business and Economics

### Equilibrium

Equilibrium is a term relating to a 'state of rest', a situation where there is no tendency to change. In economics, equilibrium is an important concept. Equilibrium analysis enables us to look at what factors might bring about change and what the possible consequences of those changes might be. Remember, that models are used in economics to help us to analyse and understand how things in reality might work. Equilibrium analysis is one aspect of that process in that we can look at cause and effect and assess the possible impact of such changes.

For the purposes of this resource we are going to look at market equilibrium. Market equilibrium occurs where the amount consumers wish to purchase at a particular price is the same as the amount producers are willing to offer for sale at that price. It is the point at which there is no incentive for producers or consumers to change their behaviour. Graphically, the equilibrium price and output are found where the demand curve intersects (crosses) the supply curve.

Mathematically, what we are looking to find is the point where the quantity demanded (Qd) is equal to the quantity supplied (Qs). Let's take our example from above:

Assume the demand is Qd = 150 - 5P and that supply is given by Qs = 90 + 10P. What we now have is a task that involves understanding how to do simultaneous equations.

In equilibrium we know that Qs = Qd. Remember that Qs = 90 + 10P and that Qd = 150 - 5P. Given that we know that an equation means that whatever is on the left hand side must be the same as that on the right hand side we can re-write our simultaneous equation as follows:

90 + 10P = 150 - 5P

We can now go about collecting all the like terms onto each side (by doing the same to both sides) and solving the equation to find P. Explanation 1 shows the long route and Explanation 2 the route you might normally see in a textbook.

#### Explanation 1

(90 - 90) + (10P + 5P) = (150 - 90) - (5P + 5P)

We have added 5P to both sides and taken away 90 from both sides. This gives us:
15P = 60

Now divide both sides by 15 to get P on its own.

15P / 15 = 60 / 15
The 15P term will now cancel down. How many times does 15 go into 15P? Once.
60 / 15 = 4
P = 4

#### Explanation 2

10P + 5P = 150 - 90
15P = 60
P = 4

We now know the equilibrium price is 4 so we can substitute this into the equations to get the Qd and Qs.

Qd = 150 - 5P
Qd = 150 - 5(4)
Qd = 150 - 20

Qd = 130

Doing the same thing to the supply:
Qs = 90 + 10P
Qs = 90 + 10(4)
Qs = 90 + 40

Qs = 130

So, the equilibrium price is 4 and the equilibrium quantity bought and sold is 130.

Other examples of where this technique might be used include finding equilibrium national income in a two sector economy (for example, if just consumption and investment are considered), in IS/LM analysis, in finding the break even point, in prodcution calculations and many other areas.

Sometimes, you will also see demand and supply equations written differently; don't panic. The principles are exactly the same. Take the examples below:

P = 900 - 0.5Q and P = 300 + 0.25Q

Which is the equation for the demand curve and which for the supply curve?

Remember, the relationship between demand and price is inverse so the negative sign in the first equation tells you it is the equation for the demand curve. There is a positive relationship, however, between price and quantity supplied. The + sign thus tells you that the second equation is the one for the supply curve.

Let's take these equations and find the equilibrium:

In equilibrium, the same price equates the demand and supply, so:
900 - 0.5Q = 300 + 0.25Q (Collect all the like terms together)

900 - 300 = 0.25Q + 0.5Q
600 = 0.75Q
Q = 800

If the equilibrium quantity (Q) = 800 then we can now find the equilibrium price through substituting Q into the two equations:

P = 900 - 0.5 (800)
P = 900 - 400
P = 500

For supply (as a check)
P = 300 + 0.25 (800)
P = 300 + 200
P = 500

#### Some examples to work through:

1. Q = 3P + 2, Qs = 2 - P
2. 3P + 7Q = 10, 4P - Q = 3
3. 5P + 10Q = 10, 2P - Q = 1
4. 2P + Q = 7, 3P + Q = 10
5. 5P + 3Y = 7, 4P - 5Y = 3
6. 4P + 6Q = -13, 3P - 5Q = 14
7. 6P + 3Q = 1, 4P - 2Q = 2
8. 3P + 5Q = 12, 6P - 4Q = 3
9. 6P + 3Q = 2P + Q = 1
10. 2P + Q = 7, 4P + Q = 11
11. 5P + 7Q = 12, 6P - 3Q = 3
12. 2P + 3Z = 2, 6P - 12Z = 13