Energy Engineering - Energy Economics

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: p 1= 10 -q 1 • For group 2, the demand curve is: p 2= 8 -8q 2 The total cost function for the utility is TC(q 1,q 2)= 100+2q 1+2q 2 The price set by the monopolist are p 1=5 and p 2=4 . Do these prices meet the second best Ramsey solution? Question 1 Ramsey Boiteux pricing model solution: p k − c k p k = λ 1 + λ 1 ε k 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: p k − c k p k = λ 1 + λ 1 ε k ������ = | ������ � � ������ � � |= | � � ������ � ������ � | ������ 1= | � 1 � 1 ������ � ������ � | = | 5 5 ( − 1 ) |=1 ������ 2= | 4 1 / 2 ( − 1 / 8 ) |=1 c 1=c 2=2 � 1 q1 = 10 − � 1 from the demand curve (������.�.������ℎ� ���������������������������������������� �� ������������ ������� �1 �������� �2) Question 1 Ramsey Boiteux pricing model solution: p k − c k p k = λ 1 + λ 1 ε k Data: p 1=5, p 2=4, ������ 1= ������ 2=1, c 1=c 2=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 Question 2. A electricity utility serves two groups of customers with different demand profile. • For group 1, the demand curve is: p 1= 100 -q 1 • For group 2, the demand curve is: p 2= 60 -q 2/2 The total cost function for the utility is TC(q 1,q 2)= 1800+20q 1+20q 2 An independent agency wants to meet a breakeven constraint to encourage a firm to produce for both types of consumers. What should be the prices in this case? Question 2. Ramsey Boiteux pricing model solution: p k − ck p k = λ 1 + λ 1 εk Let’s set λ 1 + λ = k . We can re -arrange the solution as (p k − ck)εk p k =k ������ 1= | � � ������ � ������ � |= � 1 100 − � 1 ������ 2= � 2 60 − � 2 If we replace elasticity in the solution above, we obtain � 1− 20 � 1 � 1 100 − � 1 = � 2− 20 � 2 � 2 60 − � 2 If we simplify and rearrange, we obtain: � 1 = 2p 2-20 Question 2. If we now consider the break even constrain: π = p 1q 1 + p 2q 2 −1800−20q 1 −20q 2 =0 or (by replacing q with the inverse demand curve): π =120p 1 − p 1 2 +160p 2 −2p 2 2 −6200 =0 If we substitute the relationship between p 1 and p 2 derived in the previous slide: p 2 2−80p 2 +1500=0 Question 2. If we solve the 2 nd order equation, we obtain 2 (p,q ) pairs : (p 1,q 1)=(40,60); (p 1,q 1)=(80,20); (p 2,q 2)=(30,60); (p 2,q 2)=(50,20); Question 2. If we solve the 2 nd order equation, we obtain 2 (p,q ) pairs : (p 1,q 1)=(40,60); (p 1,q 1)=(80,20); (p 2,q 2)=(30,60); (p 2,q 2)=(50,20); The economic meaningful solution is the one highlighted in green, because the quantity are higher, compared to the second case Question 3. A monopolist faces a demand curve: p=300 -5q . The total costs function is TC(q)=1000+5q 2. Determine the price and quantity at the equilibrium. Quantify the deadweight loss and the monopolist profit. How your answers change if a two -part tariff is introduced and you know that the monopolist has 100 customers? To determine the price the monopolist will charge and the quantity sold : MR=MC Question 3. MR = d p (q)q dq = d(300−5q) q dq MC= d T C (� ) dq = d 1000+5q 2 dq MR=MC 300 -10q=10q → q=15 By replacing q in the demand function → p=225 What about the profit ? Revenues - TC(q)= 225*15 -(1000+5*15 2) = 1250 Question 3. What about the deadweight loss (DWL) ? To compute the DWL, calculate the blue area! x → quantity in pure competition y → marginal cost if q=15 q 15 p 225 DWL x y D S (:MC) Question 3. Quantity in pure competition: p=MC 300 -5q=10q → q=20 Marginal cost if q=15 MC=10q → MC=150 DWL= b∗h 2 = (225−150)∗ (20−15) 2 =187,5 15 p 225 DWL 20 150 D S (:MC) Question 3. What about the 2 part tariff? T= A+p PC *q (where p PC is the price in pure competition and A is equal to FC/n) Data: FC (fixed costs) =1000; n=100 customers; q PC =20 A= 1000 100 =10 p PC =300 -5q=300 -5*20=200 T=10+200q Question 3. Is this 2 part tariff feasible? A*n=FC