Mock exam 2, Question 9

[Eleanor Cawston]

The question asks what is the marginal revenue on sale of the 30th unit of a good when its demand function is Q = 100 - P.
I said that P = 100 - Q
So TR = 100 Q - Q^2
MR = dTR / dQ = 100 - 2Q
and so MR(30) = 100 - 2*30 = 40

But the answer is 41, by taking the difference of TR at Q = 29 and Q = 30.

Why doesn't the differentiating approach I took work? Does it ever work? Have I completely misunderstood what MR is/how it works?
Thanks,
Eleanor

 

[Richie Holway]

Hi Eleanor,

Good question - the inconsistency comes down to a discrete vs continuous approach. Differentiating total revenue gives us a continuous function for marginal revenue, and so evaluating it at Q=30 considers only an infinitesimally small change in Q, not a change of 1 whole unit (it isn't possible to sell a fraction of a unit). In other words, using a continuous approach only gives us an approximate answer here, where we have enough information to calculate it accurately.

So whenever we are given enough information to calculate the total revenue both before and after the final unit, this is the approach we should use. April 2017, Q28 provides another example of where this is the method to use.

Your method of working out the total revenue and then marginal revenue function can still be useful for other scenarios though, for example for if we are given a demand function and then asked to determine the profit-maximising output level, in which case we would equate the derived marginal revenue function to the marginal cost and then solve for Q.

I hope this helps,
Richie

 

[Eleanor Cawston]

Thanks Richie. That is helpful.