In Section 14.3, I carry out several examples where the producers’ or consumers’ surplus is calculated. I want to give you a few more examples including some of the examples worked out by students in class.
Let’s take a look at producers’ surplus. To get a good idea of this concept, let’s visualize what area on a supply or demand graph represents. In the graph below, we have a supply function $latex \displaystyle S(Q)=0.9Q$. The supply and demand are in equilibrium when 100 units are produced at 90 dollars per unit.
On this graph heights are in dollars per unit and widths are in units. This means the units on any area will be
$latex \displaystyle \frac{\text{dollars}}{\text{unit}}\cdot \text{units}=\text{dollars}$
The area under the supply curve is
This is the amount of money a supplier would be willing to receive if each of the units yielded revenue according to the prices on the supply curve from 0 to 100 units.
However, if the market is in equilibrium all 100 units with earn 90 dollars per unit yielding
Since the market is in equilibrium, the supplier actually receives a higher price per units giving an additional 4500 dollars in revenue. This extra amount is called the producers’ surplus.
The consumer would be willing to pay more than the equilibrium price. The amount they save by paying the equilibrium price is called the consumers’ surplus.
Problem 1 Suppose the supply curve for a particular product is
$latex \displaystyle S(Q)={{Q}^{{\scriptstyle{}^{5/2};}}}+2{{Q}^{{\scriptstyle{}^{3/2};}}}+50$
and that the equilibrium quantity is Q = 16. Find the producers’ surplus.
First find the equilibrium price (black). Then find the area under the supply curve (red) and the area under the equilibrium price (green). The difference between these amounts (blue) is the producers’ surplus.
Notice that the producers’ surplus is the area between the equilibrium price and the the supply curve. We can compute the surplus by computing this area.
This gives rise to the formula often quoted for the producers’ surplus,
$latex \displaystyle \text{Producers } \text{Surplus}=\int\limits_{0}^{{{Q}_{e}}}{ \left( {{P}_{e}}-S\left( Q \right) \right)} dQ$
A similar formula exists for the consumers’ surplus and is essentially the area between the demand curve and the equilibrium price,
$latex \displaystyle \text{Consumers } \text{Surplus}=\int\limits_{0}^{{{Q}_{e}}}{ \left( D(Q)-{{P}_{e}} \right) dQ}$
Problem 2 Suppose the demand and supply curves for a product are
$latex \displaystyle D(Q)=900-20Q-{{Q}^{2}}$
$latex \displaystyle S(Q)={{Q}^{2}}+10Q$
Find the producers’ and consumers’ surplus.
Start by finding the equilibrium point (black). Then find the producers’ surplus (blue).
The consumers’ surplus is
