Utility function assignment

Ann views coffee and sugar as perfect complements. In particular she always takes one cup of coffee with 3/2 teaspoons of sugar. Let x1 denote the cups of coffee and x2 denote teaspoons of sugar that Ann consumes. a) Write down a utility function u(x1 x2) that models Ann’s preferences for coffee and sugar. (1 mark) Is this function unique? If yes explain why. If not provide an example of another utility function v(x1 x2) that expresses the same preferences as u(x1 x2). (1 mark) b) Suppose that one unit of x1 (i.e. one cup of coffee) costs $1.5 and one unit of x2 (i.e. 1 tsp of sugar) costs $0.1. Ann spent $438.89 on coffee and sugar over a period of ten months. During that period how many cups of coffee and how many teaspoons of sugar did she consume? (2 marks) 1 c) Consider bundle A that contains one cup of coffee and no sugar A = (xA1 xA2 ) = (1 0). Provide an economic explanation of the sign (i.e. positive negative zero) of MU1 at bundle A as well as the sign of MU2 at bundle A. (2 marks) d) Is MU1 constant and independent of the values of x1 and x2? If yes prove your answer. If no provide a counterexample. (1 mark) Question 2 (3 marks) Brian considers coffee denoted (x1) and energy drinks denoted (x2) as perfect substitutes. In particular he is always indifferent between having two cups of coffee or one energy drink regardless of his previous consumption of energy drinks or coffee. a) Write down a utility function u(x1 x2) that models Brian’s preferences for coffee and energy drinks. (1 mark) Is this function unique? If yes explain why. If not provide an example of another utility function v(x1 x2) that expresses the same preferences as u(x1 x2). (1 mark) b) Is MU1 constant and independent of the values of x1 and x2? If yes prove your answer. If no provide a counterexample. (1 mark) question 3A monopolist has the cost function c(y) = 100y. Assume that there is only one type of consumers Type A described by the inverse market demand function pA = 10 000 − 2QA where QA denotes market quantity and pA market price. a) Find the total revenues total costs and total profits of the monopolist if she is charging a uniform price. (1 mark) Find consumer’s surplus (0.5 mark) Is there any deadweight 2 loss in this situation? If yes calculate the deadweight loss. If no explain why the outcome is Pareto efficient. (1 mark) b) Find the total revenues total costs and total profits of the monopolist if she can practice First-Degree Price discrimination. (1 mark) Find consumer’s surplus (0.5 mark) Is there any deadweight loss in this situation? If yes calculate the deadweight loss. If no explain why. (1 mark) Now assume that there is an additional group of consumers Type B which are described by the market demand function pB = 5 000 − 2QB. c) Assume that the monopolist is charging a uniform price to the whole market composed now of Type A and Type B consumers. Specifically the market demand is now p(Q) = pA + pB where Q = QA + QB (i.e. total output sold to all consumers). Find the total revenues total costs and total profits of the monopolist. (1 mark) d) Assume that the monopolist can practice First-Degree price discrimination over the whole market composed now of Type A and Type B consumers. Find the total revenues total costs and total profits of the monopolist for each type of consumer (2 marks) What is consumers’ surplus for Type A consumers? (0.5 mark) What about Type B consumers? (0.5 marks)e) Suppose now that the monopolist can practice second-degree price discrimination. Discuss what happens to the monopolist profits for each type of consumer relatively to part d) above. (0.5 marks) How much quantity do Type A consumers buy and how much do Type B consumer buy compared to part d) ? (0.5 marks) What happens to consumers’ surplus for Type A consumers compared to part d)? (0.5 marks) What about consumers surplus for Type B consumers relatively to part d) ? (0.5 marks) Is the outcome Pareto efficient for Type A consumers? (0.5 marks) Is the outcome Pareto efficient for Type B consumers? (0.5 marks)

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