If the demand function is P = 100 − 4Q

Question

If the demand function is P = 100 − 4Q (a) Find expressions for TR and MR in terms of Q. (b) Find TR and MR when Q=12 (c) Estimate the change in TR brought about by a 0.3-unit increase in output from a current level of 12 units (Hints: Δ(TR) ≈ MR ∗ ΔQ)

Answer 1

a) First we need to recall two things: 1) TR = P*Q and 2) MR is the derivative of TR with respect to Q. With those two things in mind, if we know that P = 100 - 4Q then...

TR = P * Q = (100 - 4Q) * Q = 100Q - 4Q².

Since we know that MR is the first derivative of TR, then...

MR = dTR/dQ = 100 - 4Q (remember that in a perfectly competitive market MC = MR = P)

b) At Q = 12 TR and MR are:

TR = (100 * (12)) - (4 * (12²)) = 624

MR = 100 - (4 * (12)) = 52

c) Using the hint in the question, if ΔTR ≈ MR * ΔQ, then ΔTR ≈ 56 * 0.3 = 16.8