Solving for market equilibrium with individual consumers and firms

This article will go over the economics of multiple consumers and producers, and adding up their individual supply and demand functions to determine a market supply and a market demand.  These individual supply and demand functions will be aggregated to get market supply and demand mathematically and then market equilibrium will be calculated.  After this, a graph will be produced to visually show the markets interact and equilibrium price and quantity are determined.  The question is:

Suppose that our market consists of three consumers (say consumer 1, 2, and 3) with the individual demand curves q1 = 5 – p, q2 = 10 – 2p, and q3 = 7 - .5p respectively. Suppose that there are 4 identical firms with the marginal cost curves MC(q) = 1 + .25q. Assume no other firms can enter the industry so that we are in the short run in this sense.

a.) Find the equation for and plot the market demand curve. This means simply summing q1 + q2 + q2. This sum is the total demand by all 3 members in our market at price p.

b.) Find the equation for and plot the short run market supply curve for a competitive industry with the above 4 firms. This means we must first solve for q in the MC equation and then sum the q’s across firms. All 4 firms are the same so you just multiply q by 4.

c.) Solve for the short run competitive equilibrium output and price. Diagram the equilibrium and label your diagram.

To solve the first part of this economics problem (a) we need to horizontally sum up each individual’s demand curve.  This means that we need to solve for q (quantity) which thankfully has already been done for us.  When we add together each person’s q, the sum will be market q.  For example, if person 1 buys 4, person 2 buys 3, and person 3 buys 2, then the total purchased in the market will be 9 (4+3+2).  So:

Qd (market demand) = q1 + q2 + q3 = 5 – p + 10 – 2p + 7 - .5p

Qd = 22 – 3.5p  if p < 5

And this makes sense, because if p is 0, then market demand will be 22 (look through the problem entering 0 in for p to see what I mean).  The trick here is to recognize that this equation only holds if p is less than or equal to 5.  This is because consumer 1 and 2 will drop out of the market as soon as the price is equal to 5.  If you look at their individual demand functions and plug in a p of 5, then their individual quantity will be 0 (they are no longer active in the market).  However the third consumer is still in the market.  This means that for p greater than 5 our demand curve will consist only of the third consumer:

Qd = 7 - .5p if p > 5

This simply means that we get a kink in our graphical representation of market demand, which is shown below:

To get our market supply equation, we need to again add up the quantity supplied by each individual firm.  To do this you must rearrange the equations and solve for q (quantity).  Then you can add together each of the individual quantity’s supplied to get market quantity.  The intuition behind this step is that in a perfectly competitive market, marginal cost (MC) is going to equal price (p) in equilibrium).  So when we rearrange the supply functions, we can think of the firm as saying: “I am willing to supply this much if the price is this…” which is valid because the firm is trying to cover their cost.

Since each firm is identical we start with the equation for 1 firm:

MC = 1 +.25q, which means that MC starts at 1, and goes up by .25 as more quantity is produced.

We can change MC to p and solve for q to get:

q = 4p – 4, which shows us that the price has to be higher than 1 for any quantity to be produced.  This makes sense because the minimum marginal cost is 1.  Also, if p is 2 then q will be 4.  If we plug a q of 4 into our first equation we get a MC of 2, so we know we did it right.

Now we need to multiply this individual firm’s quantity by 4, because there are four firms and they are identical.  So our market supply function will be:

Qs = 16p – 16

And graphically this looks like:

Now to find equilibrium we set Qs = Qd, for more info on this process see the post on calculating market equilibrium.  This gives us:

Qs = Qd = 16p-16 = 22 – 3.5p

Add 3.5p and 16 to both sides to get:

19.5p = 38 or p = 1.948

We can confirm that this results in equilibrium by plugging it into both of our Q equations and we will get a value near 15.17 for both of them letting us know that we did it right.  Finally, a graphical representation of this economy in equilibrium is shown below:

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