1. Further Equations and Techniques
Chapter 5

2. 5.1 Introduction
Linear functions such as and quadratic functions such as
describe a straight line and a U-shape, respectively.
The cubic functions describe an S-shape: e.g. the figure below is a graph of

3. 5.2 The Cubic Function Definition. The function P defined for all x by
where are constants, and , is called the
general polynomial of degree n with coefficients
Definition. A polynomial of degree 3, i.e is called a cubic polynomial.
Definition. A function of the form is called a cubic function.

4. Example 5.1 Consider a third-degree polynomial function and set .
The resulting function is
Key features of :
sign(y)=sign(x)
The curve reaches very rapidly
y=0 only if x=0, at one point only

5. Example 5.2
Setting and in the third-degree polynomial function results in
Key features of :
When x>0, the graph of lies above the graph of
When x<0, there are two opposing effects: is pushing the curve up, while (negative) is pushing the curve down
The effect of starts dominating for large negative x
There are two turning points
In economics we often deal with opposing effects

6. Example 5.3 Setting and results in . Key features of :
The addition of 15x eliminates the two turning points
When x<0, the downward effect of 15x helps bring the curve down so it never increases in x

7. Example 5.4 Setting and results in . Key features of :
Two turning points
The constant term defines the vertical intercept
Changing the value of the intercept results in a vertical shift of the curve

8. 5.3 Graphical Solution of Cubic Equations
Consider solving the equation
Apply a formula (TOO complicated)
Guess factoring (HOW?)
Use graphical solution
Graphical solution
1) See where the graph crosses the horizontal axis to find the roots: x=-5, x=-2, x=1.
2) This method will not work if we make a mistake with the domain of x: e.g. consider x>2
3) Iterative methods employed by computer programs are based on the graphical solution idea

9. A Cubic Equation with a Single Root

10. Cubic Equations: a Boundary Case
The curve is just touching the horizontal axis at one point x=-3, so we say it is tangent to the axis at that point.
Factoring out this equation results in
x=-3 is called a repeated root

11. General Polynomial Equations
Definition. An equation of the form
is called a polynomial equation of degree n.
It can be shown that a polynomial equation of degree n has n roots, however, not all of these roots are necessarily real.

12. 5.4 Application of the Cubic Function in Economics
Short-run total cost functions are more realistically represented by the cubic functions compared to quadratic or linear ones.
Consider
Costs rise rapidly with output when you produce either too much (q>3.5), or too little (q<1.5)
The range of output 1.5<q<3.5 is where this firm will most likely decide to produce

13. 5.5 The Rectangular Hyperbola
Consider a function
This is a particular case of a rectangular hyperbola.
Key properties of :
y and x vary inversely
In case x=0, y is not defined
The rectangular area defined by the axes and a point on the hyperbola has the same area irrespectively of the position of K

14. 5.6 Limits and Continuity Consider computing for increasingly large
values of x: e.g. x=1000, x=10000, x=
Clearly, the corresponding value of y is going to be getting increasingly smaller.
We say in this case that as x approaches plus infinity, y approaches a limiting value, or limit, of zero.
Notation:

15. Asymptote Consider what happens to when . It is easy to see that
Definition. A curve is said to be asymptotic to the x-axis if
Definition. A curve is said to be asymptotic to the y-axis if
Notation: means “x is approaching zero from the right or from the left.”
The hyperbola has both x- and y-axes as its
horizontal and vertical asymptotes, respectively.

16. Rectangular Hyperbola: General Form
Definition. A function of the form where a,b,c are parameters, is called a
general-form rectangular hyperbola.
Key features of :
The line y=3 is a horizontal asymptote since
The line x=2 is a vertical asymptote since
The parameter c is equal to the area defined by the asymptotes and ANY point on the rectangular hyperbola

17. 5.7 Rectangular Hyperbola: Application in Economics
Consider a demand function described by a rectangular hyperbola:
Key features of :
Consumers are never satiated with the good:
Consumers buy the good no matter how expensive it is: as ,
Total expenditure on that good is constant regardless of the price:

18. Rectangular Hyperbola: an Illustration

19. Variant of the Basic Version
Satiation property can be included by making the demand curve intersect the q-axis.
Total expenditure is no longer constant:
“No matter what the price” property can be modified by making the hyperbola intersect the p-axis as well.

20. 5.8 The Circle and the Ellipse
Consider an equation
This equation is describing an implicit function y(x).
In general, implicit functions cannot always be made explicit, i.e. put in the form y=f(x).
However, in this case we can do it:

21. A Circle with Specific Center
Consider a circle of radius r with the center at a specific point (a,b).
In this case the circle will be described by the following equation:
Example Setting a=1, b=2, r=3 results in

22. Ellipse The circle equation can be modified to describe an ellipse:
Definition. A curve described by the equation
where -2<a<2, is called an ellipse.

23. 5.9 Circle and Ellipse: Application in Economics
Consider an economy that in the short run has fixed stocks of labor, capital, and other production factors. This economy produces only two goods, X and Y.
Definition. A curve Y(X) where Y is the maximum amount of good Y that can be produced given a specific amount of good X, is called a production possibilities curve.
Production possibilities curves are often conveniently described by an ellipse:
Points on the PPC are called efficient production plans
Production plans inside the area bounded by the PPC are called feasible production plans

24. 5.10 Inequalities
The following rules are important regarding the inequalities:
Rule 5.1 When multiplying both sides of an inequality by a negative constant, the direction of the inequality is reversed:
Rule 5.2 When inverting an inequality, the following rule applies:
Raising both sides of an inequality into power does not obey any simple rule and should be examined on a case-by-case basis.

25. 5.11 Graphical Solution of Inequalities
Example Find values of x for which
Rearranging the inequality to read , we reduce the problem to finding those intervals of x where the parabola’s branches lie above the x-axis:
Coefficient with is positive, so the branches are looking upwards
The equation has two real roots: x=-2 and x=1
As a result, the parabola crosses the x-axis at the roots, and the left-hand-side of the inequality will be positive to the left and to the right of the roots.

26. Strong and Weak Inequality
Definition. An inequality when the right-hand side can never be equal to the left-hand side is called a strong inequality.
Definition. An inequality when the right-hand side can be equal to the left-hand side is called a weak inequality.
Strong inequality example:
Weak inequality example:

27. 5.12 Inequalities: Consumption Function
Consider the following relationship between income Y and consumption :
where a and b are some positive parameters.
Definition. An increase in consumption due to one additional dollar (won, euro ...) is called marginal propensity to consume, denoted as MPC.
It is straightforward to verify that in case of the linear consumption function above the MPC is equal to parameter a.
Definition. The average amount of money spent on one unit of consumption is called average propensity to consume, denoted as APC.
The definition above implies

28. APC and MPC Let us show that APC is always strictly greater than MPC.
Indeed, by definition of APC,
The graph of APC is a rectangular hyperbola
The MPC line is a horizontal asymptote to APC
Consider the difference between MPC and APC:
which implies that MPC is always strictly less than APC.

29. Consumer Budget Constraint
Consider a consumer whose (monthly) income is equal to B.
If consumer buys X units of good X at a price , his expenditure on good X will be equal to
In the same fashion, this consumer will spend on good Y.
We assume this consumer’s total expenditure never exceeds his income B:
Definition. The weak inequality of the form above is called consumer budget constraint.
Rational consumers spend all of their income, so
In case consumers are not rational, they spend less than their incomes:

30. Budget Constraint: Graphical Representation
Consider the following budget:
Rearranging, we obtain:
Definition. The line is called a budget line.
Definition. The vertical intercept of a budget line is called real income in terms of good Y (in case good Y is on the vertical axis).
Definition. The ratio is called the
opportunity cost of X in terms of Y.