Natural monopoly solutions

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Regulation of natural monopoly Question 1. In a natural monopoly, a firm serves two groups of customers with different demand profiles. • For group 1, the demand curve is: p1=10-q1 • For group 2, the demand curve is: p2=8-8q2 The total cost function for the utility is TC(q1,q2)=100+2q1+2q2 The price set by the monopolist are p1=5 and p2=4. Do these prices meet the second best Ramsey solution? Question 1 Ramsey Boiteux pricing model solution: pk −ck pk = λ 1+λ 1 εk Note that λ is the same for all the product/consumers To assess if the prices set by the monopolist meet the second best Ramsey solution, it is sufficient to compute the elasticity of demand functions and then replace the values in the Ramsey Boiteux pricing model solution. Question 1 Ramsey Boiteux pricing model solution: pk −ck pk = λ 1+λ 1 εk 𝜀 = | 𝑑𝑞 𝑞 𝑑𝑝 𝑝 |= | 𝑝 𝑞 𝑑𝑞 𝑑𝑝| 𝜀1=| 𝑝1 𝑞1 𝑑𝑞 𝑑𝑝| =| 5 5 (−1)|=1 𝜀2= | 4 1/2 (−1/8)|=1 c1=c2=2 𝑝1 q1 = 10 − 𝑝1 from the demand curve (𝑖. 𝑒. 𝑡ℎ𝑒 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝑇𝐶 𝑖𝑛 𝑞1 𝑎𝑛𝑑 𝑞2) Question 1 Ramsey Boiteux pricing model solution: pk −ck pk = λ 1+λ 1 εk Data: p1=5, p2=4, 𝜺1= 𝜺2=1, c1=c2=2 1) 5 −2 5 = λ 1+λ 1 1 → 3 5= λ 1+λ 2) 4 −2 4 = λ 1+λ 1 1 → 1 2= λ 1+λ λ is different in the two expressions→ the prices do not meet the Ramsey Boiteux pricing model solution