Diamond Trading company (DTC), a subsidary of De Beers, is the dominant supplier

Question

Diamond Trading company (DTC), a subsidary of De Beers, is the dominant supplier of high- quality diamonds for the wholesale market. Assume they are a monolpoly on the wholesale diamonds. The quantity that DTC chooses to sell thus has a diresct impact on the wholesale price of diamonds. Let the wholesale prie of diamonds (in uhundreds of dollars) be given by the following inverse demand function: P=120-Qdtc. Assume that DTC has a cost of 12 (hundred dollars) per high- quality diamond.

a. Write DTC's profit function in terms of Qdtc, and solve for DTC's profit- maximizing quantity. What will be the wholesale price of diamonds at that quantity? what will DTC's profit be?

Explanation / Answer

We know profit is the function of Revenue and cost.

Revenue = Px Q

                   = (120 –Qdtc)xQdtc

                    = 120Qdtc – Qdtc^2

Cost = Cost per unit x Q

                = 12 x Qdtc

                = 12QDtc

Profit = Revenue – cost

                = 120Qdtc – Qdtc^2 - 12QDtc

                = 108Qdtc – Qdtc^2

For profit maximizing quantity, we need to differentiate Profit function with respect to Quantity, and equate it with 0.

108 – 2Qdtc = 0

Qdtc =54 units

Price will be:

P=120 –Qdtc

   = 120 – 54

   =66

Profit = 108Qdtc – Qdtc^2

            = 108x54 – 54^2

           =5832 -2916

           =2916

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