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Economics/Marginal Revenue Curve Equation vs Manual Marginal Revenue Calculation

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Question
Dear Professor,  

The textbook that I am reading says there is a shortcut to finding marginal revenue (MR) given that the demand curve is a straight line, P=a+bQ. It says the MR curve is a straight line that begins at the same point on the vertical axis (y-intercept) as the demand curve but with twice the slope which is MR=a+2bQ.  

Here are my questions regarding it:
1) How can there be marginal revenue at Q=0 ? How can there be a    
  y-intercept ?  
2) The MR curve equation just does not follow the MR calculated
  manually. (I calculated Total Revenue (TR) from the demand
  curve and I substract every additional TR with the previous one
  to get MR. But it is just not the same. So given a constant    
  Marginal Cost (MC), how do we determine the profit maximization
  price and quantity using MR=MC ? Which method do we use?

Thank you for your time.

Regards,
Justin

Answer
AR, MR and TR
AR, MR and TR  

AC and MC
AC and MC  
Hi Justin,
Thank you for asking me a question on a topic which is a particular favourite of mine. This is in fact the hub of microeconomic analysis. If you have a good grounding, you will find demand, cost, and other topics becoming more and more illuminating. Even though there are technical treatises which demand advanced-level mathematics, you can get a good handle on the topic with only elementary mathematics— algebra, geometry, and a little bit of calculus. Let me get down to your two queries.

MARGINAL REVENUE AT Q=0
First, we define MR as extra or additional revenue because of an extra or additional output. Here one thing must be carefully noted: MR does NOT come from ONLY THE LAST UNIT of Q. The last unit of Q gives rise to TR, and that TR less the TR obtained from the previous Q is MR. Suppose “n” denotes  the current level and “n-1” the previous level, then we define MR:
MR = TR(n) – TR(n-1)  or when we consider continuous function  MR = d(TR)/dQ.
Now the question is, when is Q=0? This means, Q which seller would like to produce at the maximum possible price at which buyers will simply not buy is NOT produced. This is the Q-INTERCEPT. Were he to produce that Q, he would have to charge the highest possible “intercept” price [=AR]. Now, refer to our above definition of MR, the producer’s Q at “intercept price” would be his LAST Q , and any Q precedent to  that he simply had not considered, which means that “TR at an imaginary n-1 time before the TR at n time with intercept price” had been by all means zero. Hence, at the intercept price when TR=max, we also have MR=max. The positive value comes because the AR he is not getting because he is not selling constitutes his MR because “this” MR MINUS ZERO [as there had been no other MR –recall our definition] is exactly “this” AR at Q=0. The following table may make it clear:





Q   P=AR=TR/Q   TR = PQ        MR=[TRn]-[TRn-1]
o    200          0          +200
         
         +180
1    180          180          +160    
         +140
     
2    160          320          +102
     
         +100
3   140          420          +80


 
These above figures are “discrete” figures. As you go on increasing Q, MR will go down “in steps” or giving a “serrated curve.” Look at the image. So, as the serrated curve for the “discrete MRs” shows, you can see MR is not exactly at intercept when Q=0. Then why do we say MR equals the intercept? Yes, it does when we consider a “continuous function” as is given by the above table, and as already explained above. The continuous function is, as you know from calculus, an approximation –we pass the “smooth line of MR” through all these steps.
Given that, it must also be remembered that MR can be NEGATIVE after it attains MAXIMUM level. “Negative MR means that in order to sell additional units, the firm must decrease its price on earlier units so much that its total revenues decline (Samuelson).”  See the image.

MARGINAL REVENUE AS A CONTINUOUS FUNCTION OF Q JUST AS TR IS A CONTINUOUS FUNCTION OF Q
TR= QP  or, as you have mentioned, TR=  Q (a+bQ), which we may write as:

TR = aQ+bQ^2

You have two terms in the right-hand side of the above equation. When you differentiate TR with respect to Q (which is a quadratic here), you get the following. (1) The first term has 1 as exponent to Q, and that 1 comes to the left of a, and we deduct 1 from the exponent 1, which yields 1-1=0; anything raised to the power of zero results in 1. (2) In the second term which involves Q raised to power 2, the exponent “2” moves to the left of b, and in this case we deduct 1 from the exponent 2, which yields 2-1=1. Hence we have:

MR = a+2bQ

Note, here we have “2bQ” and b<0 [because of the downsloping curve], and so the “rate b” is multiplied by 2. That is the reason why MR falls at twice the rate of AR. In any intermediate-level book (such as Taro Yamane’s “Mathematics for Economists”), you will get very good geometric exposition and proof. If you are prepared to take a little more trouble by getting deep into calculus, you can use elasticity of demand formula, manipulate that, and get a proof of this. Let me give you the least mathematical explanation in the image –check it.

THE QUESTION ON PROFIT MAXIMIZATION
Yes, profit is always maximized when MR = MC. Here are the reasons.
Since, in continuous MR and MC functions, the total area under the MC curve and the total area under the MR curve at any particular level of Q give us the TC and the MC, it is at the point when MC=MR that the DIFFERENCE between the area under the MR curve and the area under the MC curve is the MAXIMUM. Just draw two curves –downsloping MR curve and upsloping MC curve, and check it that at intersection the “triangle” gives the maximum area for profit [=sum of all MR – sum of all MC]. Any point of Q to the left of the equilibrium level will reduce the “triangle” and also any point of Q to the right of the equilibrium level will bring about a “negative triangle, meaning loss” which has to be deducted from the “max triangle” if Q is produced beyond the equilibrium level. That’s why the maximum level of profit is attained when

MC=MR at maximum level of profit.

As you can visualise in a diagram, for the producer it is something like walking the tight rope –neither left, nor right.

NO MATTER WHETHER RISING OR CONSTANT MC
You know, MC and AC are also intricately related. Look at the attachment.
MC may at first fall, but then rises, and cuts the AC from below at the lowest point of the AC.
Even if you take constant cost, it is still the same principle of MR=MC for profit maximization

CONCLUSION
This is a very wide topic. So I am giving you only the barest exposure. There are volumes written on these related topics. I would suggest, with this background, go through a good intermediate textbook on microeconomics, and if possible also some mathematical microeconomics textbook (there are many). Mathematical derivation will hone your conception.
I hope, Justin, I have been able to give you the clarification you sought.
Best of luck.  

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Eklimur Raza

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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."

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

Publications
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.

Education/Credentials
Master degree in Interantional Economics and Finance and diploma with honours in Business Administration from Canada.

Awards and Honors
Received First Prize in an inter-university Literary Contest.

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