Post 8: Solving Quadratic and Other Higher Degree Polynomials

Before solving a quadratic or other higher degree polynomials, a clear distinction should be made between the two terms, equation and function. Let us consider the difference between a quadratic equation and a quadratic function. A quadratic function, say, f(P), has the general format as:

                                       

There is an infinite number of solution for the above equation, as an infinite number of ordered pairs (P, f(P)) can be written for the above equation.

But if we equate the above f(P) = 0, the situation fundamentally changes. Since the variable f(P) is now assigned a 0 value, the result is a quadratic equation in the single variable P.

                           

Now that f(P) is restricted to 0 value, only a select number of P values satisfy and qualify as its solution values, namely, those solution values at which the parabola is figure above intersects the horizontal axis - on which f(P) is 0. The solution P values are often referred to as the roots of the quadratic equation f(P) = 0.

In general, given a quadratic equation of the form:

                            
there are two roots, which can be obtained using the formula,

                          
                                    
For , we would get , which gives us identical repeated roots. For , we get the find the square root of the -ve numbers, which is not possible in real number system. So for those cases, no real-valued root exists.

Higher Degree Polynomials
For higher degree polynomials, one useful method that might work is that of factoring the function. Given a 3rd-degree polynomial equation, factoring results in three terms of the form (x-root) yields 3 roots. Generally, an nth-degree polynomial equation should a total of n roots. Second, and more important in root search is the relationship between the roots and the constant term: since the constant term must be the product of the three roots, each root must be a divisor of the constant term. This relationship can be formalized in the following theorem:

Theorem I Given the polynomial eqaution
                      
                         

where all coefficients are integers, and the coefficient of is unity, if there exists integer roots, then each of them must be a divisor of .

Theorem II Sometimes because the coefficients of the highest degree term is not unity, we cannot apply Theorem I and rather we resort on to a more general theorem:

                        

if there exists a rational root r/s, where r and s are integers without a common divisor except unity, then r is a divisor of , and s is a divisor of .

Note: An example is given below to show how this can be applied to a concrete polynomial function to derive its roots.

Theorem III Given a polynomial 
        
                       

if the coefficients add up to 0, then x=1 is a root of the equation.


Example - Applying Theorem II to solve the quartic equation:

                        .

With , the only possible values for the numerator r in r/s are the set of divisors {1, -1, 2, -2, 3, -3, 4, -4, 6, -6, 12, -12}. And, with , the only possible values for s are the set of divisors {1, -1, 2, -2}. Taking each element in the r set in turn, and dividing it by each element in the s set, respectively, we find r/s can only assume the values

                    1, -1, 1/2, -1/2, 2, -2, 3, -3, 3/2, -3/2, 4, -4, 6, -6, 12, -12

Among these candidates for the roots, many fail to satisfy the given equation. Also, since we are solving a quartic equation, we can expect at most 4 of the listed r/s values to qualify as the roots. The four successful candidates turn out to be 1/2, 2. -2, and -3. According to factoring principle, we can thus write the quartic equation as

                   (x - 1/2) (x-2) (x+2) (x+3) = 0
            =>   (2x - 1) (x-2) (x+2) (x+3) = 0

Post 7: Example of a Liner Model - Partial Market Equilibrium

The analytical procedure of solving an economic model is applied to what is known as static or equilibrium analysis. An equilibrium is a collection of selected interrelated variables so adjusted to one another that no inherent tendency to change prevails in the model which they constitute.

Constructing the Model
In a static equilibrium analysis, the standard problem is that of finding the set of values of the endogenous variables which will satisfy the equilibrium analysis of the model.

a) Identifying the variables: We will consider one commodity model and so it is necessary to include only three variables in the model: the quantity demanded (Qd), the quantity supplied (Qs), and its price (P).

b) Equilibrium condition of the identified variables: Having chosen the variables, our next order of business is to specify an equilibrium condition. The standard assumption is that equilibrium occurs in the market if and only if the excess demand is zero (i.e., Qd - Qs = 0), that is, if and only if the market is cleared. But this raises the question of how Qd and Qs themselves are determined. To answer this, we assume that Qd is a decreasing linear function of P (as P increases, Qd decreases). On the other hand, Qs is postulated to be an increasing linear function of P, with the condition that no quantity is supplied unless the price exceeds a particular positive level.

So the model will contain one equilibrium condition (Qd = Qs) plus two behavioral equations (behavior of Qd and Qs) which govern the demand and supply sides of the market.




Translated into mathematical statements, the model can be written as,

Qd = Qs
Qd = a -bP               (a, b > 0)
Qs = -c + dP            (c, d > 0)

For the demand function (Qd), the vertical intercept is at a and its slope is -b, which is -ve and downward sloping. The supply function (Qs) has the slope d being +ve as it is upward sloping, and intercept is at -c. By intercepting the supply curve at a -ve value in y axis, we made sure that the provision stated earlier, that supply will not be forthcoming unless the price is positive (P1 in this case), is satisfied.

Solution by Elimination of Variables
We can set Q = Qd = Qs and rewrite the model equivalently as:

Q = a - bP
Q = -c + dP

=> a -bP = c +dP
=> P* = (a+c)/(b+d)

So the solution value of P, P*, is expressed entirely in terms of the parameters, which represent given data for the model.

To find the value of equilibrium quantity, Q* (= Qd* = Qs*) that correspond to the value P*, simply substitute the demand or supply equation with P* value, and we get,

Q* = a -b * (a+c)/(b+d)
=> Q* = (ad-bc)/(b+d )

The positivity of Q* requires requires that the numerator (ad -bc) be positive as well. Hence, to be economically meaningful (i.e., to have Q* > 0), the present model should contain the additional restriction that ad > bc.

This can rewritten in Set notation as:

D = {(P, Q) | Q = a - bP}
S = {(P, Q) | Q = -c + dP}
D ∩ S = (P*, Q*).

The market equilibrium is unique at (P*, Q*)

Post 6: Steps for solving a mathematical economic model

A mathematical economic model will consist of a system of equations, which may be definitional, behavioral, or in the nature of equilibrium condition. The behavioral equations are usually in the form of functions, which may be linear, or nonlinear, numerical or parametric, and with one or many independent variable(s).

In solving an analytical problem, the first step is to select the appropriate variables - exogenous (variables which are assumed to be determined by forces external to the model) as well as endogenous (variables whose solution we seek from the model) - for inclusion in the model. Next, we must translate into equations the set of chosen analytical assumptions regarding the human (e.g., aggregate consumption pattern in relation to national income), institutional, technological (e.g., production function), legal (e.g., tax structure), and other behavioral aspects of the environment affecting the working of the variables. Only then can we attempt to derive a set of conclusions through relevant mathematical operations and manipulations and to give them appropriate economic interpretations.

The next post (Post 7) on partial market equilibrium would give an example of these steps and how they are used to solve an economic model.

Post 5: Types of Functions

The expression y=f(x) is a general statement to the effect that a mapping is possible between x and y variable. Now let us consider several specific types of function, each representing a rule of mapping.

Constant Function:
A function whose range consists of only one element is called a constant function. As an example, we cite the function:
                                            y = f(x) = 7

In coordinate plane, such a function appear as a horizontal straight line.

Polynomial Function:
The word polynomial means "multiterm", and a polynomial function of a single variable x has the general form:
                                           

Depending on the value of integer n, we have several subclasses of polynomial:
* Constant Function
* Linear Function - As shown below, gives the vertical intercept and gives the slope of the curve. If > 0, the slope of the line would be +ve, else -ve.

                                            

* Quadratic Function - A quadratic function plots as a parabola - a curve with a single built-in bump, as shown below. The curve below implies a +ve ; in the case of < 0, the curve will open the other way, displaying a hill rather than a valley.

* Cubic Function - The graph of a cubic function will, in general, manifest two wiggles, as shown below.



                    

and so forth.

Rational Functions:

A function such as:
                                  
in which, y is expressed as a ratio of 2 polynomials in the variable x, is known as a rational function. A special rational function that has  interesting application in economics is the function:
                                     y = a / x    or   xy = a
which plots as a rectangular hyperbola, as shown below.
                
                                  

This function may be used to represent that special demand curve - with price P and quantity Q on the two axes - for which the total expenditure PQ is constant at all level of price.

The rectangular hyperbola drawn from xy = a never meets the axes, even if it is extended indefinitely upward and to the right. Rather the curve approaches the axes asymptotically: as y becomes large, the curve will come ever closer to the y axis but never actually reach it, and similarly for x axis.

Nonalgebraic Functions:
Exponential functions, such as , in which the independent variable appears in the exponent, are nonalegbraic. The closely related logarithmic functions, such as , are also nonalgebraic. Other types of nonalgebraic functions are trigonometric (or circular) functions.