1. Consider the public good model of the two flatmates and TV with the initial wealths w1 =w2 =...

Question

1. Consider the public good model of the two flatmates and TV with the initial wealths w1 =w2 = 100.(a) Suppose the TV itself is the public good, so that, G = 1 means TV is provided andG = 0 means TV is not provided. The cost of the TV is c = 75 and the utility functionsare such that, for each i = 1, 2: U i(xi, G) = xi if G = 1; and U i(xi, G) = xi/2 ifG = 0. Find the reservation price of each individual. Is it optimal to provide the TV?(b) Instead of (a), suppose the quality of the TV is the public good with the cost function ofTV quality given by C(G) = G2. Also, suppose the utility function of each individuali = 1,2 is: U i(xi, G) = xiG. Show that the optimal amount of the public good isthe same in every Pareto efficient allocation. What is this amount? Will the optimalamount of the public good change if the initial wealths of the two individuals change?2. A monopolist sells a good in two markets. The inverse demand curves in the two marketsare P1(y1) = 75- (y1/2) and P2(y2) = 100- y2, where yi is the quantity in market i = 1, 2.The monopolist’s total cost function is given by C(Y ) = 50+2Y +Y 2, where Y is the totaloutput of the monopolist.(a) Suppose, because it is impossible for anyone other than the monopolist to transport thegood between the two markets, the monopolist is able to price discriminate betweenthe two markets. How much will the profit maximizing monopolist sell in each of thetwo markets and at what prices?(b) Suppose it is now possible for anyone to transport the good from one market to theother at zero transportation cost. Find the profit maximizing monopolist’s optimalprice and the quantities it will sell in the two markets at this price.3. T plc operates the only public transport system in the town of ABC. T plc can only chargea single uniform price of $F per trip. ABC has 1000 identical citizens, who each takesX = 40 - 5F trips per year if the price per trip is $F . T plc has a constant marginal costof $0.4 per trip.(a) If T plc is not allowed to practice two part tariff pricing, find the monopoly price of atrip and the total number of trips it will sell per year.(b) If T plc is allowed to practice two part tariff pricing by charging an annual membershipfee to each citizen in addition to the uniform price $F per trip, find the optimalmembership fee, the uniform price per trip and the number of trips each citizen willbuy per year.(c) Calculate T plc’s annual profits in (a) and (b).

Robert · Accepted Answer

Problem set 3: 
2.  
(a) 
Inverse demand functions in two markets; 
P1(y1) = 75 - (y1/2) 
P2(y2) = 100 - y2 
Cost function: C(Y) = 50 + 2Y + Y^2 
Marginal cost (MC) = 2 + 2Y 
Market 1: 
P1(y1) = 75 - (y1/2) 
Total revenue (TR1) = p1*y1 = 75y1 – 0.5y1^2 
Marginal revenue (MR1) = dTR1/dy1 = 75-y1 
In order to maximize the profit, the firm would equate marginal revenue of this market with 
marginal cost i.e. 
75-y1 = 2 + 2y1 or 
3y1 = 73, implies profit maximizing output in market 1 (y1*) = 24.33 
And profit maximizing price in market 1 (p1*) = 75-(24.33/2) = 62.835 
Market 2: 
P2(y2) = 100 - y2 
Total revenue (TR2) =p2*y2 =  100y2 – y2^2 
Marginal Revenue (MR2) = dTR2/dy2 =  100 – 2y2 
In order to maximize the profit, the firm would equate marginal revenue of this market with 
marginal cost i.e. 
100-2y2 = 2 + 2y2 or 
4y2 = 98, implies profit maximizing output in market 2 (y2*) = 24.5 
And profit maximizing price in market 2 (p2*) = 100-24.5 = 75.5 
(b) 
In this case, we will derive the combined demand for two markets; 
Inverse demand functions in two markets; 
P1(y1) = 75 - (y1/2) , implies y1 = 150 – 2p1 
P2(y2) = 100 - y2, implies y2 = 100-p2 
Combined demand: Y = y1+y2 or 
Y = 150 – 2p1 + 100-p2 or 
Y = 150-2P+100-P                            [p1 = p2 = P] 
So combined demand: Y = 250-3P, or P = 83.33 – (Y/3) 
Total profit (TR) = P*Y = 83.33Y – (Y^2)/3 
Marginal revenue (MR) = dTR/dY = 83.33 – 2Y/3 
At the profit maximizing point, MR = MC or 
83.33 – 2Y/3 = 2 + 2Y or 
81.33 = (8Y/3), implies total output produced by monopolist (Y*) = 30.49875 
And price charged (P*) = 83.33-(30.49875/3) = 73.16375 
Output supplied to market 1: y1 = 150 - 2p1 = 150-(2*73.16375) = 3.6725 
Output supplied to market 2: y2 = 100-73.16375= 26.8363
3. 
(a) 
Uniform price = $F 
1000 consumers and each takes : X = 40-5F trips  
Constant marginal cost = 0.4 per trip 
1000 consumers and each takes : X = 40-5F trips  
So market demand: Xt = 1000X = 40*1000-5*1000F or 
Xt = 40000 – 5000F or 
F = 8 - (Xt/5000),

1. Consider the public good model of the two flatmates and TV with the initial wealths w1 =w2 = 100.(a) Suppose the TV itself is the public good, so that, G = 1 means TV is provided andG = 0 means TV...

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