Double Marginalization Part II

... continued from Double Marginalization Part I

II. Double Marginalization: Retailer and Manufacturer both have monopoly power

Here, the monopolist retailer makes a profit by charging a higher retail price than the price demanded by the monopolist manufacturer:

                        P > P_M

 Profit for the retailer is given by:

                        ∏ _R      = Q_D * (P – P_M)

                                    = (2480 – 174 P) * (P – P_M)

                                    = 2480 P – 2480 P_M - 174 P² + 174 P * P_M

Profit for the retailer is maximized at:

                        ∂∏_R ∕ ∂P = 2480 – 174 * 2 P + 174 P_M

                                or         174 * 2 P = 2480 + 174 P_M

                        or         P          = (2480 / 348 ) + (P_M / 2)

                        or         P          = 7.13 + (P_M / 2)

Substituting this value of P in the demand curve results in:

                        Q_D        = 2480 – 174 P

                                    = 2480 – 174 ( 7.13 + P_M/2)

                                    = 2480 – 1240 - 87 P_M

                                    = 1240 -  87 P_M

So, the profit function of the monopolist manufacturer is

                        ∏_M       = Q_D * P_M

                                    = (1240 - 87 P_M) * P_M

                                    = 1240 P_M - 87 (P_M)²

So, the manufacturer’s profit is maximized at:

                        ∂∏_M ∕ ∂P_{M =} 1240 - 2 * 87 * P_M = 0

                        or 174 P_M = 1240

                        or P_M = 1240 / 174 = $7.13

Thus, the retail price charged by the retailer to the final customer is given by

                        P          = 7.13 + (P_M /2)

                                    = 7.13 + (7.13 / 2)

                                    = $10.70

So, because of double marginalization, the final consumer pays a higher price of $10.70 compared to Case I above, where she had to pay only $7.13.

Now, let’s calculate the total channel profits.

The manufacturer’s profits are given by:

                        ∏_M       = 1240 P_M - 87 (P_M)²

                                                = 1240 * 7.13 – 87 (7.13)²

                                                =  $ 4418.39

The retailer’s profits are given by:

                        ∏ _R      = 2480 P – 2480 P_M - 174 P² + 174 P * P_M

                                                = 2480 * (10.70 – 7.13)

-  174 * (10.70)²

+ (174 * 10.70 * 7.13)

                                    = $2206.97

So, the total channel profits are given by

                        ∏        = ∏_{M +} ∏ _R     

                                    = $ 2206.97 + $ 4418.39

                                    = $ 6625.36

So, because of double marginalization, the total channel profits of $6625.36 are lower compared to Case I above, where the channel profits were $ 8838.76.