1. Quadratic Equations
Chapter 4

2. Non-Linear Economic Relationships
Utility function
Total cost function
Supply function is typically non-linear
Demand function is typically non-linear
Production possibilities frontier

3. Quadratic Supply and Demand
Consider an inverse demand function and an inverse supply function To find equilibrium, we must have , which means we must solve That transforms into This is a quadratic equation since it contains the square of q. How do we go about solving it?

4. Quadratic Functions and their Roots
From the supply and demand model, we want to solve
Define a quadratic function
Define to be the level of output such that the quadratic function above turns into zero, in other words
The number is called the root of function
It turns out that solving for the equilibrium level of output in our model with quadratic supply and demand boils down to finding the root(S?) of quadratic function
How do we find roots of quadratic functions?

5. Graphical Meaning of a Root
Graphically, to find a root of a function is to find a point where this function’s graph intersects the horizontal axis.
Quadratic function
Roots

6. Graphs of Quadratic Functions
Going back to our demand-supply model, we need to find roots of
It is easy to see that the roots of will also be the roots of function
What would be the shape of a quadratic function?
It turns out that the shape of a quadratic function is a parabola.

7. Quadratic Function Roots
May number to two
May not exist
May be just one root

8. Parabolic Shapes

9. Roots of Quadratic Functions
Roots of a quadratic function
May number to two
May not exist
May be just one root
One way to solve for the equilibrium level of output in our model is to plot the
graph of or and look where parabola
branches intersect the horizontal axis: impractical
Another thing we can do is to factorize the right-hand side of

10. Factorizing Quadratic Expressions
Let us expand the following expression:
Factorizing means to take expression and transform it into Numbers a and b are called factors. How do we do that?
Let us try to factorize Assuming that factors a and b exist (later on that), we know that
That implies the following:

11. Vicious Circle We tried to factorize and came up with
We’re almost there! Let’s try to solve this system. From the first equation, it is clear that Substituting this into the second equation, we obtain
We’re back to a quadratic equation again since !
Factorizing is a nice idea, but we need more tools to solve for the roots of a quadratic equation.
Fortunately, there exists a general formula for finding such roots.

12. Completing the Square Consider equation
What we’re going to do now is called completing the square
Let us first add and subtract to the left-hand side:
It follows that
is a complete square since
What do we do with this?

13. General Formula for Finding Roots of Quadratic Equations
We found that finding roots of is equivalent to finding the roots of
Notice that, if , this equation has no solution: the case where a parabola doesn’t intersect the horizontal axis at all
If it’s positive, we have two possibilities:
In the general case of and
Substituting, we obtain the general formula:

14. Notes on the General Formula
The formula for finding the roots of any quadratic equation of the form
If , there are no (real) roots: the parabola never intersects or “touches” the horizontal axis.
It may happen that In this case there is a single root ,
corresponding to the case when parabola’s vertex is “touching” the x-axis is

15. Back to Supply and Demand
Coming back to our supply and demand model, we wanted to solve
In this case, for the general formula , the values of the coefficients are:
Substituting, we obtain
It seems like we have two roots, but we really have only one since the negative root doesn’t make economic sense. Hence, the equilibrium level of output in our demand-supply model is equal to
Exercise: verify that it’s the same root if we divide the original equation by 2.5

16. Total Cost Function
A firm’s total cost function may assume the following form:
Positive intercept=fixed costs
Realistically, costs rise rapidly as firms increase output

17. Total Cost Function

18. Monopolistic Revenue Function
A monopoly is the only producer in a particular industry
If a monopoly lowers the price, it sells more
Revenue is defined as the product of output and price: R=PQ
We’ll show later that revenue rises as a monopoly expands its output level when Q is small, but then it starts decreasing
We can describe this case with a parabola whose branches are looking down, for example

19. Monopoly