Price elasticity of demand and Methods of computing square roots: Difference between pages

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In economics, the '''price elasticity of demand''' ('''PED''' or '''E<sub>d</sub>''') is a measure to show the responsiveness (or elasticity) of the quantity demanded for a good or service to a change in its price, [[ceteris paribus]]. It is also known as the percentage change in quantity demanded for a good or service for a percentage change in the price of the same good or service. In short, the price elasticity of demand is given by the following formula:
There are a numbers of ways to calculate [[square root]]s of [[number]]s, and even more ways to [[Estimation|estimate]] them.


<math display="block">Price\ elasticity\ of\ demand = \frac {Percentage\ change\ in\ quantity\ demanded}{Percentage\ change\ in\ price}</math>
The mathematical operation of finding a root is the opposite operation of [[exponentiation]], and therefore involves a similar but reverse thought process.


As the price and the quantity demanded of a good are inversely related (i.e. an increase in price will always cause a decrease in quantity demanded and vice versa), the sign of PED is always negative. Hence, economists tend to just ignore the sign and compare the absolute values instead.
Firstly, one needs to know how [[Arithmetic precision|precise]] the result is expected to be.  This is because often [[square root]]s are [[Irrational numbers|irrational]].  For example, square root of a nice round [[whole number]] '''28''' is a [[Fraction (mathematics)|fraction]] which in its [[Decimal places|decimal notation]] has [[Infinity|infinite]] length, and therefore it is impossible to express it exactly:<math display="block">\sqrt{28} \approx 5.291502622129181....</math>


In absolute terms, if the percentage change in the price is more than proportionate than the percentage change in the quantity demanded, the elasticity value is greater than one and the demand for the good is price elastic. Likewise, if the percentage change in the price is less than proportionate than the percentage change in the quantity demanded, the elasticity value is smaller than one and the demand for the good is price inelastic.
Moreover, for some [[real number]]s the square root is a [[complex number]]. For example, square root of '''-4''' is a complex number '''2''[[imaginary unit|i]]''''' :<math display="block">\sqrt{-4} = 2 i</math>


In diagrams, the price elasticity of demand is shown by the [[gradient]] of the demand curve. A steeper curve refers to a more price inelastic demand whereas a gentler curve refers to a more price elastic demand.
In many cases there may be multiple valid answers.  For example, square root of '''4''' is '''2''', but '''-2''' is also a valid answer.  One can verify that they are both valid answers by squaring each candidate answer and checking if you obtain '''4''' as the result of verification:<math display="block">2^2 = 2 \times 2 = 4</math>


== Factors affecting price elasticity of demand ==
<math display="block">(-2)^2=(-2) \times (-2) = 4</math>
Factors that affect the price elasticity of demand for a good include:
* '''Availability of substitutes''': When substitutes of a good is more available, consumers would be able to choose between the different types of goods available. Hence, a good with more substitutes available would be more price elastic as an increase in price would cause many consumers to change to one of the substitutes.
* '''Closeness of substitutes''': If the substitutes of a good is very similar to the original good, consumers are able to change between the different types of goods easily as it makes no difference in using either good. For example, the demand for a specific brand of sugar is likely to be price elastic as the taste of sugar is largely the same for different types of brands. Hence, a good with more similar substitutes would be more price elastic than one with less similar substitutes.
* '''Degree of necessity''': When a good has a high degree of necessity (i.e. the good is more of a need than a want), a change in the price would not have much difference in the quantity demanded for the good, thus having a price inelastic demand. For example, the demand for necessities like food would be more price inelastic compared to the demand for luxuries such as jewellery as consumers are more likely to reduce their demand for luxuries than for necessities. However, goods that are addictive (such as cigarettes and drugs) would be considered as necessities by addicts, thus causing them to have price inelastic demands.
* '''Proportion of income''': If a large part of income is used to buy the good, it would have a more price elastic demand. As the percentage change in the price of an expensive good would cause a large change in the absolute cost of buying the good, consumers would thus prefer to find for a cheaper substitute for expensive goods than for cheaper goods. Hence, the demand for expensive goods would tend to be more price elastic.
* '''Time period''': Over time, the demand for goods would tend to become more price elastic. For example, when the prices of oil increases, the quantity demanded for oil would not fall by a lot in the beginning, but in the [[long run]], consumers would tend to change to more fuel efficient cars or cut down their travelling.


{{Stub}}
Please note that calculating a square root is a special case of the problem of [[Nth root|calculating N<sup>th</sup> root]].
[[Category:Economics]]
 
== Calculating ==
Most [[calculator]]s provide a function for calculation of a square root.
 
{| class="wikitable"
!
!General Steps
!Example
|-
|width="20%" valign="top"|How to calculate a square root using a simple [[calculator]].
|width="40%" valign="top"|
* ''First, make sure the operating space is clear. This is usually accomplished by clicking the '''C''' button a couple of times.''
* Then type the number whose root you are trying to calculate.
* Then press the square root button (<math>\sqrt{}</math>).
* The number you see on the screen is one of the answers.  Remember, that often there are multiple valid answers, as explained above.
|width="40%" valign="top"|
* Press '''C''' button a couple of times.
* Type '''16'''
* Press <math>\sqrt{}</math> button.
* The answer is '''4.'''  ''Keep in mind that '''-4''' is also a valid answer.''
|}
 
== Estimating ==
If the result does not have to be very precise, the following estimation techniques could be helpful:
{| class="wikitable"
!Methodology
!Example
|-
|width="49%" valign="top"|Suppose you need to find [[square root]] of some number <math>N</math>. 
Find some number <math>A</math> such that <math>A^2</math> (that is <math>A</math> squared, or <math>A</math> times <math>A</math>) is [[Approximation|approximately]] [[Equality (mathematics)|equal]] to <math>N</math> ''(but how close? This needs to be expanded)''.
 
Then we can think of <math>A</math> as being [[Approximation|approximately]] a [[square root]] of <math>N</math>.
|width="49%" valign="top"|Suppose we need to estimate the square root of 2.
We know that <math>1^2 = 1</math>, and <math>2^2 = 4</math>.
 
Therefore, one of the answers to <math>\sqrt{2}</math> is somewhere between 1 and 2.
|}
 
==References==
*{{Cite web|url = http://mathworld.wolfram.com/SquareRoot.html | title = Square Root}}
 
 
{{math-stub}}
 
[[Category:Mathematics]]

Latest revision as of 04:59, 30 June 2016

There are a numbers of ways to calculate square roots of numbers, and even more ways to estimate them.

The mathematical operation of finding a root is the opposite operation of exponentiation, and therefore involves a similar but reverse thought process.

Firstly, one needs to know how precise the result is expected to be. This is because often square roots are irrational. For example, square root of a nice round whole number 28 is a fraction which in its decimal notation has infinite length, and therefore it is impossible to express it exactly:[math]\displaystyle{ \sqrt{28} \approx 5.291502622129181.... }[/math]

Moreover, for some real numbers the square root is a complex number. For example, square root of -4 is a complex number 2i :[math]\displaystyle{ \sqrt{-4} = 2 i }[/math]

In many cases there may be multiple valid answers. For example, square root of 4 is 2, but -2 is also a valid answer. One can verify that they are both valid answers by squaring each candidate answer and checking if you obtain 4 as the result of verification:[math]\displaystyle{ 2^2 = 2 \times 2 = 4 }[/math]

[math]\displaystyle{ (-2)^2=(-2) \times (-2) = 4 }[/math]

Please note that calculating a square root is a special case of the problem of calculating Nth root.

Calculating

Most calculators provide a function for calculation of a square root.

General Steps Example
How to calculate a square root using a simple calculator.
  • First, make sure the operating space is clear. This is usually accomplished by clicking the C button a couple of times.
  • Then type the number whose root you are trying to calculate.
  • Then press the square root button ([math]\displaystyle{ \sqrt{} }[/math]).
  • The number you see on the screen is one of the answers. Remember, that often there are multiple valid answers, as explained above.
  • Press C button a couple of times.
  • Type 16
  • Press [math]\displaystyle{ \sqrt{} }[/math] button.
  • The answer is 4. Keep in mind that -4 is also a valid answer.

Estimating

If the result does not have to be very precise, the following estimation techniques could be helpful:

Methodology Example
Suppose you need to find square root of some number [math]\displaystyle{ N }[/math].

Find some number [math]\displaystyle{ A }[/math] such that [math]\displaystyle{ A^2 }[/math] (that is [math]\displaystyle{ A }[/math] squared, or [math]\displaystyle{ A }[/math] times [math]\displaystyle{ A }[/math]) is approximately equal to [math]\displaystyle{ N }[/math] (but how close? This needs to be expanded).

Then we can think of [math]\displaystyle{ A }[/math] as being approximately a square root of [math]\displaystyle{ N }[/math].

Suppose we need to estimate the square root of 2.

We know that [math]\displaystyle{ 1^2 = 1 }[/math], and [math]\displaystyle{ 2^2 = 4 }[/math].

Therefore, one of the answers to [math]\displaystyle{ \sqrt{2} }[/math] is somewhere between 1 and 2.

References


Template:Math-stub

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