Historically, most of the diamond mines in the world have been controlled by a few companies and governments. Through clever marketing by diamond producers, many consumers have furthermore become convinced that “diamonds are a girl’s best friend” because “diamonds are forever.” In fact, the claim is that the only way to show true love is to give a diamond engagement ring that costs the equivalent of 3months of salary. (We will refer to this throughout the exercise as “the claim.”)
A: For purposes of this question, assume that diamonds are only used for engagement rings, that there is no secondary market for engagement rings and that the diamond industry acts as a single monopoly.
(a) Let x be the size of diamonds (in karats). Draw a demand curve for x (with the price per karat on the vertical axis)—and make the shape of this demand curve roughly consistent with the claim at the beginning of the question.
(b) If this claim is true, what is the price elasticity of demand for diamonds?
(c) What price per karat would be consistent with the diamond monopoly maximizing its revenues (assuming the claim accurately characterizes demand)?
(d)What price is consistent with profit maximization?
(e) How large would the diamonds in engagement rings be if the marketing campaign to convince us of the claim at the beginning of the question was fully successful and if the diamond industry really has monopoly power?
(f) True or False: By observing the actual size of diamonds in engagement rings, we can conclude that either the market campaign has not yet fully succeeded or the diamond industry is not really a monopoly.
B: Suppose that demand for diamond size is x = (A/p)(1/(1−β)) .
(a)What value must β take in order for the claim to be correct?
(b) How much revenue will the diamond monopoly earn if the claim holds? Does this depend on what price it sets?
(c) Derive the marginal revenue function (assuming the claim holds). Assuming MC > 0, does MR every cross MC?
(d) If MC = 0, how large a diamond size per engagement ring is consistent with profit maximization (assuming the claim holds)?
(e) Suppose the diamond monopoly has recurring fixed costs that are sufficiently high to cause its profits to be zero. If marginal costs were zero, what would be the relationship between the demand curve and the average cost curve?
(f) Suppose β= 0.5 and MC = x. What is the profit maximizing diamond size now?
(g) What if instead β = −1?