Question

Assume that a monopolist faces a demand curve for its product given by:                                                                                                                                                                              

p=120−1q                                                                                                                                                                                                                                                                                                                           

Further assume that the firm's cost function is:                                                                                                                                                                                                                                                                                                                  

TC=580+11q                                                                                                                                                      

Using calculus and formulas (don't just build a table in a spreadsheet as in the previous lesson) to find a solution, how much output should the firm produce at the optimal price?                                                                                                                                                                                                                                                         

Round the optimal quantity to the nearest hundredth before computing the optimal price, which you should then round to the nearest cent. Note: Non-integer quantities may make sense when each unit of q represents a bundle of many individual items.                                                                                                                  

Answer 1

In order to maximize profit a monopolist produces that quantity at which MR = MC

where MR = Marginal Revenue = d(TR)/dq where TR = Total revenue = p*Q = (120−1q)*q = 120q - q²

=> MR = dTR)/dQ = 120 - 2q.

MC = Marginal cost = d(TC)/dq = 11.

So, MR = MC => 120 - 2q = 11 => q = (120 - 11)/2 = 54.5

thus, firm should produce 54.50 units.

Thus p = 120 - 1q = 120 - 120 - 54.50 = 65.50.

Hence, At the optimal price of 65.50, firm should produce 54.5 units of output