I think you've got the equation wrong and it should be
y = ax^2 + bx + c
In basic market analysis, we assume that many functions are straight lines. So the demand function is assumed to be a line of the form price P = a - (b times quantity demanded) = a - bQ say
In that case, the total revenue TR from selling a quantity Q will be price-times-quantity as follows
TR = PQ = (a - bQ)Q = aQ - bQ^2 or -bQ^2 + aQ
This is a quadratic equation but without the c-element, which means that it has no value when Q=0 and that is obvious in this case. If no goods are being sold, then TR will be zero also. The values to be attached to a and b determine the shape of the revenue function: how high it rises, how it declines at price falls and also as sales quantity falls.
As an example, let the demand function be P = 100 - 5Q then a = 100 and b = 5
The total revenue is given by TR = -5Q^2 + 100Q
In this example, if Q=0, then TR=0 also.
Also, if Q=20, then TR=0 again. To sell that quantity, the price has had to fall to zero, so TR will be zero. The shape of the TR function here is a parabola with its highest point at Q=10. At that sales quantity, the revenue will be at a maximum.
As a second example, we sometimes assume that total costs functions are linear and of the form
TC = fixed costs + (unit variable cost times quantity produced) = a + bQ
Here a stands for the fixed costs which must be paid even if there is no output of goods. The b-value tells the average cost of producing each item of goods.
So, for example, the TC equation could be TC = 4 + 10Q which means that fixed costs are 4 money units for any volume of output and that each item of output costs 10 to make for sale.
We can combine the TR and TC to give us Profit which is TR minus TC. Let's use the numerical examples above and call Profit with the symbol Pr, then
Pr = TR - TC = [-5Q^2 + 100Q] - [4 + 10Q] = -5Q^2 + 90Q - 4
This also is a quadratic equation, this case clearly of the form y = ax^2 + bx + c.
In this case, the -4 tells us that a loss of 4 money units will be suffered when Q=0.
The -5 and the +90 describe the shape of the graph of this function.
If you draw this graph, you'll find that Pr starts at (0, -4) and reaches a maximum point for Q=9 and then declines to the zero again at about Q=18.
An interesting feature of this example is that maximum revenue occurs at Q=10, but maximum profit at Q=9. That shows the effect of the fixed costs and the upward sloping TC line..
This kind of quadratic analysis occurs frequently in elementary economics because of the assumption about straight line relationships. In real life, functions are not linear but the analysis can be useful, using the simplification.
Is this helpful?
