Home > Maths, Simple algebra > The most basic stuff – supply & demand analysis

## The most basic stuff – supply & demand analysis

Chances are the first type of thing you will do in microeconomics will be along the lines of supply and demand functions. Basically these are functions of quantity in terms of price, ie they tell you how much of a good consumers are willing to buy depending on the price, and they tell you how much producers are willing to supply, depending on the price.

For example you might get a demand function of $Q_D = 320 - 5P$ and a supply function of $Q_S = 50 + 4P$. You can rearrange those equations to put them in terms of P, to get the inverse demand and supply functions. $P = 64 - 0.2Q_D$ and $P = 0.25Q_S - 12.5$

This is telling you that if the price was, say 20, then the quantity demanded will be 220 and the quantity supplied will be 130. There will be an excess of demand over supply, so this will drive up the price – consumers are willing to pay more to get hold of this scarce good, so that higher price will encourage producers to produce more. When we reach a price that brings the quantity supplied in line with the quantity demanded, then we have the equilibrium price.

We can find this just by using simultaneous equations: set $Q_D = Q_S = Q$ so $Q = 320 - 5P = 50 + 4P \Rightarrow 270 = 9P \Rightarrow P= 30$. So now we have our equilibrium price, of 30. We can see then how much quantity is produced at this price by subbing 30 into our demand and supply functions, and in both cases the quantity will come out as 170. So our equilbrium price is 30 and our equilibrium quantity is 170.

Now we can look at the effects of a tax. The government can place a specific tax on the good, which means you pay a tax per unit of output (this is like petrol tax, you pay by unit consumed not as a proportion of the tax). Typically it is the producer that has to pay the tax to the government, ie they have to declare how many units of the good they have sold and pay the tax to the government based on that. So this will affect the producer’s supply function.

Lets say the government puts a specific tax, t, of 5 per unit, so every unit sold means the producer has to pay a tax of 5 to the government. This means that the price the consumer pays and the price the producer gets are no longer the same – the consumer is paying more than the producer ends up with, because he has to pay a portion of it in tax. $P_D -t = P_S \Rightarrow P_D = P_S + t$

So go back to those inverse demand functions: $P_D = 64 - 0.2Q_D$ and $P_S = 0.25Q_S - 12.5$. When we are in equilibrium, so the quantity demanded equals quantity supplied, we can say that $P_D = 64 - 0.2Q$ and $P_S = 0.25Q - 12.5$ and we know that $P_D = P_S + t$ so with t=5, those equations imply that $64 - 0.2Q = 0.25Q - 7.5$.

Solving this for the equilibrium quantity: $64 - 0.2Q = 0.25Q - 7.5 \Rightarrow 71.5 = 0.45Q \Rightarrow Q = 158.889$

Now putting this back into the inverse demand and supply equations gives us:
$P_D = 64 - 0.2(158.889)= 32.222$ and $P_S = 0.25(158.889) - 12.5 = 27.222$. The price the consumer pays, is effectively the ‘market price’, which is 32.222, but the price which the producer actually receives, is 27.222. The gap between them is 5. This is because on each unit that the consumer pays 32.222 for, 5 goes to the government in tax, and 27.222 goes to the supplier.

So the effect of this specific tax, is to increase the market price from 30 to 32.222, and decrease the quantity from 170 to 158.889.

The producing firm’s revenues are equal to price multiplied by quantity, so in the equilibrium without the tax, they made total revenue of $TR = PQ = 30(170) = 5100$ and after the tax they made total revenue of $TR = PQ = 27.222(158.889) = 4325.308$. Meanwhile in the equilibrium which has the tax on it, the government is taking a tax of 5 on all of those 158.889 units sold, so it gets revenue of $GR = tQ = 5(158.889) = 794.444$