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Company K monopolies water supply in a region. Its cost for producing q units of water is C = F + CO whereF > 0 is a ﬁxed cost and c > 0 is a variable cost per unit. The inverse demand function of this market P01) = 2 — q. The inversedemand function intersects the K’s average cost curve AC(q) at least once asotherwise K will never want to operate. Formally, F and c have to satisfy4Fs(2—c)2andc < 2. a. Calculate the average cost function and the marginal cost function. b. Use average cost function to show whether the production exhibits increasing, constant, or decreasing returns to scale?In fact, this market is natural monopoly. A market industry or industry is a natural monopoly if costs are minimized by concentrating production in a single firm. Mathematically, it means the following. For any N 2-2 2 and q = 2?; q,, the cost function satisﬁes the following condition (Kg) 5 2‘2; q,- . If a single firm produces (1 units, it incurs cost of C (q). If N ﬁrms in total produce q units (each firm ‘3′, produces q,- units and EL q,- = q, the total cost they incur is 2:11 C (qi). The condition above says that the market is a natural monopoly if the cost of a single firm is lower than the total costs of N ﬁrms for producing the same level of total output. 0. Prove that the cost function C = F + CQ leads to the natural monopoly.
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