Question

Assume that a firm earns revenue according to R(Q)=a+bQ+cQ^2. Marginal revenue is equivalent to the first derivative of the revenue function. For what values of a,b,c is marginal revenue decreasing for all values of Q?

Answer 1

Answer

Here R = a + bQ + cQ² where R = Revenue

Marginal revenue(MR) = d(TR)/dQ = 0 + b + 2*cQ

=> MR = b + 2cQ ---------------Marginal Revenue

Marginal Revenue is decreasing if d(MR)/dQ < 0 , Marginal Revenue is increasing if d(MR)/dQ > 0 and MR is constant if d(MR)/dQ = 0.

Hence, we want Marginal Revenue to be increasing and hence Marginal Revenue is decreasing when d(MR)/dQ < 0

Now, MR = b + 2cQ => d(MR)/dQ = 2c .

Thus d(MR)/dQ < 0 => 2c < 0 => c < 0.

Hence Marginal revenue is decreasing when c < 0 , whatever be the a and b.

Hence Marginal revenue is decreasing when c < 0 and and b can be any number