Quadratic Equations Chapter 4

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

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?

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?

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

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.

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

Parabolic Shapes

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

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:

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.

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?

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:

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

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

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

Total Cost Function

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

Monopoly