Quadratic Equations Chapter 4 NonLinear Economic Relationships Utility
- Slides: 19
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 To find equilibrium, we must have and an inverse supply , 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 • 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? is called the root of function
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 function will also be the roots of 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 • 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 • Substituting, we obtain the general formula: and
Notes on the General Formula • The formula for finding the roots of any quadratic equation of the form is 1. If , there are no (real) roots: the parabola never intersects or “touches” the horizontal axis. 2. 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
Back to Supply and Demand • Coming back to our supply and demand model, we wanted to solve • In this case, for the general formula coefficients are: , the values of the • 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
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