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Q1:1. (14 total points) Suppose a consumer has a utility functiongiven by U(X,Y)= MIN(X, 3Y). The consumer has $60 to spend(M = $60). The Price of Good Y is Py = $1.a) (10 points) Draw the consumer's Price Consumption Curve forthe following prices for Good X: Px = $1, Px = $2, Px = $3. Foreach set of prices, make sure to accurately draw each budgetconstraint, and also identify the consumer's optimal bundle oneach budget constraint. Make sure your graph is neat, accurate,scaled properly, and fully labeled to receive full credit.b) (4 points) Using a graph, sketch the consumer's demand curvefor Good X. Make sure to identify three points on the demandcurve. Make sure your graph is neat, accurate, scaled properly, andfully labeled to receive full credit.See Answer
Q2:Exercise 2- Attitudes towards risk and fair betsThe following graph represents Mike's marginal utility of income (not the utility but the marginal utility):Marginal UtilityMU(I)IncomeWe know that Mike's income I is greater than $20, but we do not know his exact income. Alejandro offers Mikethe following bet: Alejandro will roll a 6 sided dice (all numbers equally likely). If it is 1 or 2, Mike pays Alejandro$20. Otherwise, if it is 3, 4,5 or 6, Alejandro pays Mike $10.a) Is this a fair bet?b) Will Mike accept or reject the bet? Do your answer depend on Mike's initial wealth? (assuming he canalways pay the $20 if he loses)See Answer
Q3:Question 1: Auctions a. Using the second-price, sealed-bid auction model described in Lecture 1, but assuming buyers' valuations are privately drawn from a uniform distribution over [0, 0], show that: i. The optimal bidding strategy for any buyer is to bid their true valuation and so: ii. Explain how the expected revenue to the seller will increase with u. b. Suppose instead that the seller implements a first-price, sealed-bid auction. i. Show that the bidding strategy b' : = - (1)u, is u, is an equilibrium bidding strategy when there are two bidders in the auction (N = 2). ii. What is the expected revenue in this auction? c. Would a risk-averse seller prefer one auction-type over the other? d. Consider a Position Auction discussed in lectures, with each buyer's valuation known to all other buyers: i. Show that the strategy b³ (as defined in lectures) is a best response to b¹ and, similarly, that b² is a best response to b³ and b. ii. Will a seller necessarily increase revenue by increasing the number of positions available for advertising on a webpage?See Answer
Q4:Question 2: Networks and Platforms There is a mass M = 1 of consumers uniformly distributed along a straight (Hotelling) road of length 1. A single games platform is located at one end of the road and consumer i's "willingness to pay" from playing on the platform is given by: u₂ (di) = 1 - di (1) Here, d; is the distance consumer i must travel to reach the console store. Assume that consumers who do not participate on the platform receive zero utility. The platform charges a price p for the access and faces costs C(x) = x², where is the proportion of consumers using the platform. The platform sets prices to maximise profits. a. Determine the demand curve and inverse demand curve faced by the platform. b. What is the profit-maximising price charged by the store? c. What price would a welfare-maximising social planner choose? d. What is the dead-weight loss associated with the monopoly price? Now assume that a new multi-player game is introduced that allows the platform users to gain utility by interacting with others using the same platform, so the "willingness to pay" of each consumer is now given by: 7 ui (di, 9) = 1 + 1/2 − di - 8 (2)/n7 == u,(di,q) +1/12-di Here, q is the proportion of consumers agent i expects to use the platform. e. Recalculate parts b., c., and d. above for the new situation. f. How did the introduction of the multi-player game impact on the profits of the platform? The firm realises that there is actually a mass N of advertisers willing to advertise on the platform. Each advertiser j expects to receive profit ; (Pa, 9) from advertising, where: (2) Tj (Pasq) = 2q-cj- Pa (3) Here, c; is a random variable drawn from a uniform distribution over [0, 1], pa is the price charged by the console manufacturer to the advertiser, and q is the proportion of consumers expected to use the platform. The platform faces additional costs Ny² from adding advertisers, where y is the proportion of adver- tisers advertising on the console. g. Given your solutions to part e., calculate the price på the platform should charge advertisers. h. Should the platform vary the price p charged to consumers? If so, how? i. Compare the profit-maximising prices charged with the marginal costs for the games platform in the two markets. Comment on your answers. j. What prices would a social planner charge if it were able to set prices on the platform? k. How large is the Spence distortion?See Answer

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