Algebraic Treatment of Market Equilibrium

By Anne Alexander

Recall from your textbook’s discussion that equilibrium in a market occurs where the market it brought into balance. Equilibrium is a place from where there is no tendency for change. The equilibrium price in a market causes quantity demanded to equal quantity supplied – buyers’ and sellers’ plans coincide at that price. There is no excess demand and no excess supply. Using this knowledge, we can find equilibrium in a market with another way of representing supply and demand, actual functional forms for supply and demand curves.

A demand function is a function that represents a demand curve. The demand function shows us the exact relationship between price and quantity demanded. Demand functions are also just shorthand ways of representing both a demand curve and a demand schedule. . When we have a demand function, we can actually plot a demand curve AND find points on the demand schedule.

On the other side of the market "coin," a supply function is a function representing the exact relationship between price and quantity supplied. Supply functions are also shorthand representations – they can be used to find both supply curves and to find points in a supply schedule.

You can see that with the supply and demand functions, we have easy ways of representing sellers’ and buyers’ intentions in a market. These functions are also handy to have for finding equilibrium outcomes in a market. Rather than visually having to scope out exactly where quantity supplied equals quantity demanded on a demand and supply schedule or on a market graph, we can find exact equilibrium outcomes using demand and supply functions and a little bit of algebra. Let’s look a little more deeply into this possibility!

As an example, let’s consider the market for wheat. Measuring quantity in millions of bushels, suppose we have a market demand curve that is given by:

Q_D = 50 – 2P

Now, we can find several points along a demand schedule using this demand curve. The following table shows some points:

We can also see what the graph of this demand curve would look like. We can use the points found above for the demand schedule to sketch the demand curve.

Now suppose that the supply curve for the wheat market is given by:

Q_S = -6 + 12P

(Yes, that’s a negative sign in front of the intercept. This is normal for a supply function – they usually don’t start at the origin point of the graph, but up a bit on the price axis.)

Just like with the demand function, we can find some sample points on a supply schedule to help you visualize what the supply function shows you here. The following table shows some points (you should also check these with your own calculator):

The supply curve will look something like:

Since the scale is off a bit, the curve doesn’t look exactly right – but you get the idea!

Now there is something that you should keep in mind right now. You’re about to learn how easy it is to handle supply and demand functions. In just a minute, you’ll learn how you can find the equilibrium in a market when you have nothing but the demand function and the supply function. This means you’ll be able to find equilibrium even when you don’t feel like finding points on a demand schedule or drawing a supply curve on a graph. But you should always remember that if you start to get a confused, you could always also use the functions to graph supply and demand or find schedules if you feel more at ease learning these concepts visually. The functions are very easy ways of expressing things and contain a lot of information, but you may be more comfortable extracting the information from them and then working with them. That’s OK too – use the method you find more workable.

OK – back to the example. Remember that equilibrium is the place in the market where there is no tendency for change. Buyers’ and sellers’ plans coincide, and therefore quantity demanded equals quantity supplied. Now when we have a supply function and a demand function, we already know quantity demanded and supplied in terms of price. We can therefore use these functions to find what price causes quantity demanded and quantity supplied to "clear," that is where there is no excess demand or excess supply. Recall that the demand function was Q_D = 50 – 2P, and that the supply function was

Q_S = -6 + 12P. Now since equilibrium occurs where Q_D = Q_S, we can solve for the price that does this by setting the functions equal to each other:

Q_D = Q_S Þ 50 – 2P = -6 + 12P

Now we will isolate the "P" term, so we can solve for equilibrium price. We’ll do this step-by-step so you can refresh your memory on algebra manipulations if you need to.

50 – 2P = -6 + 12P

Þ 50 + 6 – 2P = 12P

Þ 56 = 12P + 2P

Þ 56 = 14P

Þ P = $4 per bushel

We now know that the equilibrium price in the market, the price that causes quantity demanded to be equal to quantity supplied, is $4 per bushel of wheat. Now that we know this important piece of information, we can find out what quantity is traded in the wheat market. Notice that since at P = $4, both quantity demanded and quantity supplied are the same simply because of the definition of equilibrium, we can find the quantity traded in the market by using either the demand or the supply function. In this case, I would prefer to use the demand function (it’s easier, and I’m lazy). To find equilibrium quantity, we simply plug the equilibrium price into the demand function and crank the number out.

Q_D = 50 – 2P

Þ Q_D = 50 – 2 ($4)

Þ Q_D = 50 – 8 = 42 million bushels

Therefore in equilibrium, 42 million bushels of wheat trade hands, with the price of each bushel being $4. Any other price will not clear the market – with P = $4, 42 million bushels are demanded, and 42 million bushels are supplied. There will be no tendency for a change from this point unless something happens to the demand function, the supply function, or both.

Just to recap, remember that the steps to solving for equilibrium when you have a demand function and a supply function for a market are:

1. 1. Set Q_D = Q_S
   2. Isolate "P" term on one side. Solve for equilibrium price.
   3. Plug the price found in step 2. Into either the supply or demand function. Solve for equilibrium quantity.

And always remember, if graphing or finding points on schedules are easier for you use these methods. You can easily find graphs and schedules when you have a demand and supply functions.