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QUESTION: sir, plz help in solving my assignment.. as im doing distance learning in mba..no one is there to guide me ... i searched in net also.. i didnt get any clue to solve this.. my question is..
Calculate point elasticity of demand for demand function Q=10-2p for decrease in price from Rs 3 to Rs 2 ..
i know the formula.. but dont how to apply the price (decrease from rs.3 t0 2).. even in my book also it not availabe.. so plzzzz guide me..
thanking u
ashwini

Thank you for being so straightforward in seeking my help for getting a handle on application of elasticity of demand. I do understand your problem, and this can happen to any intelligent and perceptive student if adequate intelligible information is not available in the ordinary textbooks. What I find is that most textbooks of intermediate level do not address this problem, and books that take this problem up may have too high standard with stiff mathematical language. The onus therefore lies on the teacher to explain this to students like you, but you probably do not have one. Don’t worry. This is a very simple problem, and I would like to guide you step by step how you can come to grips with it.

First, however, I would like to make it clear to you that the formula of point elasticity that you know of could be misread by beginners. That is why I would like to write a bit about it at the outset.

1.   FORMULA FOR POINT ELASTICITY
Elasticity of demand basically measures the responsiveness of the quantity of a commodity demanded to a change in its price. This responsiveness is measured in percentages. That is to say, elasticity in fact shows a “value in number” (which is usually negative in the case of normal good as is implied in your demand function, but it could as well be positive in the case of Giffen goods or Veblen goods). This “value” we seek works out as:

Percent change in quantity demanded divided by percent change in price.

As a student of business, you are sure aware that percent change of anything is measured by taking the “initial value” and the “final value” of the thing considered, then taking the difference [“final value” –“initial value], and then dividing this difference by “initial value,” and finally multiplying what you get by 100. So, if Q=quantity demanded and P=rice, then:

Percent change in Q ={[“final value of Q”–“initial value of Q”]/”initial value of Q”}100... [1]
Percent change in P ={[“final value of P”–“initial value of P”]/”initial value of P”}100... [2]

Elasticity = (1)/(2) ...          [3]

When you divide (1) by (2), the “100” in the numerator and the “100” in the denominator cancel out, and so you can write (1) and (2) without “100.”

2.   CAVEAT:  Most textbooks may not give a clear picture about why you need to take the AVERAGES for  “initial value of Q” and “initial value of P.” This is simply because, in a straight-line demand curve, such as your Q=10–2P, slope is not the same as elasticity. The slope is constant but the elasticity is varying. If you don’t get it, forget about it. Just take it that we need to take “average initial value of Q” and “average initial value of P” for the denominators in (1) and (2) only, leaving numerators unchanged.

How do you calculate the two “average” numerators? These are actually average of two values:

“average initial value of Q” = [“initial value of Q” + “final value of Q”]/2.....   (4)
“average initial value of P” = [“initial value of P” + “final value of P”]/2.....   (5)

Substitute (4) and (5) into the numerators, and get:

Elasticity =  (6a)/(6b),.....          (6)

In (6), the numerator (6a) and the denominator (6b) are given as:
Numerator ={[“final value of Q”–“initial value of Q”]/[(”initial+final values of Q”)/2]} (6a)
Denominator={[“final value of P”–“initial value of P”]/[(”initial+final values of P”)/2]}(6b)

3.   EXAMPLE: TAKE THE DEMAND FUNCTION:  Q = 10–2P
Given the demand function Q = 10–2P, and the statement that P decreases from Rs. 3 to Rs. 2, we first calculate the “initial Q” and the “final Q” on the basis of “initial P”=3 and
“final P” = 2.

“initial Q” = 10–2x3 = 4
“final Q” = 10–2x2 = 6

Hence,

“final Q” –“initial Q” = 6–4 = 2          (7a)
and
“final P” –“initial P” = 2–3 = –1          (7b)

We then calculate “average initial value of Q” and “average initial value of P,” applying the equations (6a) and (6b) given under rubric 3 (CAVEAT),as under:

“average initial value of Q” = “initial + final values of Q”)/2 = (4+6)/2 = 5         (8)
“average initial value of P” = “initial + final values of P”)/2 = (3+2)/2 = 2.5      (9)

4.   CALCULATION OF ELASTICITY OF DEMAND FOR Q = 10–2P WHEN P DECREASES FROM RS. 3 TO RS. 1
Put the values given in (7a), 7(b), (8), and (9) [which are, respectively, 2, –1, 5, and 2.5] into the equation (3), which is given by (1) and (2), and you get the result for point elasticity E.

E ={[“final value of Q”–“initial value of Q”]/[(”initial + final values of Q”)/2]}
divided by
{[“final value of P”–“initial value of P”]/[(”initial + final values of P”)/2]}
E = (6–4)/ [(4+6)/2] divided by (2–3)/[(3+2)/2]
E = (2/5)/(-1/2.5) = –1
Your result is: point elasticity = negative unity.

5.   CALCULATION WITH FORMULA GIVEN BY CALCULUS AND STATISTICS
You can also calculate elasticity for a linear demand function as given above using the ordinary formula given by calculus as:
E = (dQ/dP)(P/Q)
Which, actually, is what we derived in equation (3) above: (dQ/Q)/(dP/P).
In your demand function the slope is -2, i.e., dQ/dP = -2. And P/Q given by "averages" are 5/2.5. You then get the same result:

E = (dQ/dP)(P/Q)= (-2)[5/2.5] = -1

This formula e= (dQ/Q)/(dP/P), with a little bit more advanced math, can also be usefully expressed as

E = d(logQ)/d(log P),

which is used in econometric estimation when a series of data on prices and quantities are given. There you will require normal equations of regression functions. With more advanced techniques you would require matrix algebra.For now I believe you find yourself in good shape.

6.   FINAL SAY
I hope, Ashwini, this gives you the answer you need. Please do not hesitate to ask for any other help.
I wish you best of success.

---------- FOLLOW-UP ----------

QUESTION: hello sir,

thanking u,
ashwini

Calculation of Point E
Hi Ashwini,
The calculus method is quite simple. The only thing is that you have to know elementary differential calculus. If you know calculus, then go ahead. Otherwise you have to first learn the elements of differential calculus, which should not take more than a couple of weeks for a beginner (for an intelligent student like you, it would take even less).

Assuming that you have some knowledge of calculus, here is what you have to do.

1. First, construct your demand function. You may take an exponential demand function which would require use of logarythms. Let us, for the sake of simplicity, take a straight-line demand function that you initially asked a solution for:

Q= 10 - 2P

2. Second, you differentiate the demand function with respect to price P. The result that you get is called the derivative of Q with respect to P, or simply the slope of the demand function.

3. If you differentiate your demand function Q=10-2P with respect to P, you get the value for your slope = -2.

4. Next you multiply the slope [=-2] by the fraction Pave/Qave, and you arrive at the value for point elasticity, which of course comes to -1.
Please have a look at the attachment for a clearer view with diagram.

5. Please do not forget that slope is the same at each and every point of the straight-line demand curve, but elastcity varies because of the multipicand (Pave/Qave) changing. Please also remember that, if you have a downwad-sloping demand curve, then e=0 at the Q-intercept simply because (Pave/Qave)= 0, and you know it well P is at origin equal to zero. As you move upward to the northwest direction (someting like from Kolkata to Mumbai), (Pave/Qave) goes on increasing from 0 and reaches 1 when you are at the midpoint of the curve. This means at any point between the Q-intercept and the midpoint of the demand curve, elasticity < 1, i.e., the good is inelastic. Again, as you move still further upward, elasticity goes on increasing from 1 up, up, and up, till it reaches infinity at P-intercept. Hence you get elasticit >1 (the good is elastic) at any point between the midpoint of the demand curve and the P-intercept (at this extreme P at P-intercept, nobody buys the good and Q=0, the denominator is 0, which simply means the multiplicand (P/Q)= infinity). At midpoint, of course elasticity = 1. I have also given this in a diagram in the attachment for your better understanding.

5. Of course, for nonlinear demand function, you first have to convert P and Q into logarythm of  P and logarythm of Q, set the function in terms of logarythms [normally in econometrics natural logarythms (ln) are used for convenience and efficiency, which means logarythm to base e= an irrational number between 2 and 3.
Also please have a look at the attachment.

I hope, Ashwini, the attachment will give all the answer you need, if your calculus background is okay. Best of luck.
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Economics

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#### Eklimur Raza

##### Expertise

It appears some students in this website are confused about elasticity of demand and the slope of the demand curve when they are trying to figure out why rectangular hyperbola comes up in case of unitary demand curve. First, they don't know that RH can be depicted in a positive quadrant of price,quantity plane. Secondly, they make the mistake that the slope of RH is constant at -1. Two points could help them: first, e=1 at each and every point of the RH, because the tangent at any point shows lower segment=upper segment (another geometric definition of e); yet slopes at different points,dQ/dP, are different; second, e is not slope but [(Slope)(P/Q)]in absolute terms. Caveat: only if we measure (log P) along the horizontal axis and (log Q) up the vertical axis, can we then say slope equals elasticity --in which case RH on P,Q plane is transformed into a straight-line demand curve [with slope= -tan 45 deg] on (log Q),(logP) plane, and e= -d(log Q)/d(log P). [By the way, logs are not used in college textbooks --although that is helpful in econometric estimation of elasticity viewed as an exponent of P, when demand equation is transformed into log-linear form.] I have not found the geometrical explanation I have given in any textbook followed in undergraduate and college classes in Canada (including the book followed in a university where I taught for a short time and in the book followed in George Brown College, Toronto, where I teach.

##### Experience

About 11 years' teaching economics and business studies, and also English, history and elementary French.Practical experience in a development bank, working with international donor agencies like the World Bank and the ADB. Experience in free-lance journalism, including Canada's "National Post."

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I teach micro- and macroeconomics at George Brown College (continuing education), Toronto, ON, Canada.

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Many articles and editorials, on different subjects, in English newspapers. Recently an applied Major Research Paper, based on a synthesis of the Solow growth model and the Lewis two-sector model, has be accepted by Ryerson University, Toronto. Professors Thomas Barbiero and Eric Cam, Ryerson University, accepted the paper.

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