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1. Consider an economy with two consumers, 1 and 2. Each consumer consumes only grapes and wine and can use grapes as an input to produce wine. Grapes used as input cannot be consumed directly as grapes. Consumer 1 produces 1 unit of wine for each unit of grapes it uses as input. Consumer 2 produces 2 units of wine for each unit of grapes it uses as input. Consumer I initially owns 1 unit of grapes and no wine. Consumer 2 initially owns 1 unit of grapes and 2 units of wine. Consumer 1 gets utility H1 {531, ml} 2 lm] from consuming gl 3 [l units of grapes and an 3 [l units of wine. Consumer 2 gets utility ugg, mg} = 5:2 [mg1) from consuming 5:2 3 [I units of grapes and '[I'J'g 3} 1 units of wine, and gets utility 32(5'3, mg) 2 [l otherwise. a. Give a verbal explanation of the way the consumers' preferences differ from each other. b. Find the optimal production and consumption of each consumer separately, assuming that the consumers do not trade with each other. c. Consider the private ownership economy consisting of the two consumers, each owning a competitive rm that uses that consumers winemaking technology. Prove that in a com petitive [Walrasian] equilibrium neither good can have a price of [L d. Find every competitive equilibrium for this economy, letting grapes be numeraire. How many such equilibria are there? Compare the utility levels of the two consumers to what they receive in part b, when they do not trade. e. Find a Pareto efcient [Pareto optimal} allocation for this economy that is not a com petitive equilibrium allocation and that gives positive utility to consumer 1. {Try to nd an allocation that is easy to characterize with little calculation.) Describe how this allocation differs from a competitive equilibrium allocation. f. Is there any Pareto efcient allocation in which both rms produce positive quantities of wine? Is every competitive equilibrium allocation in this economy Pareto efcient? g. In a competitive equilibrium in this economy, the consumer with the more productive technology supplies more input and the consumer who has more need of output for con sumption gets more of the output. Is this property [more productive agents supply more input and agents more in need of a good get to consume more of it} a characteristic of all competitive equilibria in general equilibrium models? Explain and justify your answer by f. Is there any Pareto eicient allocation in which both rms produce positive quantities of wine":I Is every competitive equilibrium allocation in this economy Pareto eicient? g. In a competitive equilibrium in this economy} the consumer with the more productive technology supplies more input and the consumer who has more need of output for con sumption gets more of the output. Is this property [more productive agents supply more input and agents more in need of a good get to consume more of it) a characteristic of all competitive equilibria in general equilibrium models? Explain and justify your answer by checking whether the property continues to hold in the economy above no matter what the consumersT initial endowments are. 2. There are two types of consumer who have differing tastes. A rm faces a consumer. It is known that the probability of the consumer being of type H is % and being type L is %. The H type has utility function U3 2 Fi t and the L type has utility function UL = t? where q is the quality of a particular good the rm sells and t is the monetary payment for the good. The rm can choose any level of quality 9' 3: U at a cost of 4:; dollars. There is no xed cost. The reservation utility of both types is zero. a. Suppose the rm can identify the type of the consumer before offering a two part tariff (T? p}; i.e.J t = T + pg. 1What are the optimal two part tariffs and what is the corresponding earante {before the rm learns the type of the consumer) expected prot?I b. From now onJ assume that the rm cannot identify the types directly. What is the expected prot maximising twopart tariff {Typ} for the rm'? 'What is the corresponding 1 expected profit? c. Suppose now that the firm can offer any (non-linear) contract. What is an optimal con- tract and what is the corresponding expected profit of the firm? Compare your answer with your answer in part a. d. Now, suppose there are two consumers whose types are drawn independently from the identical distribution in the above. The consumers know their types but not the other con- sumer's type. The producer knows only their distributions. Suppose the producer produces q and auctions it off to the two consumers according to the ascending (English) auction. What is the expected revenue of the firm? What is the optimal q and corresponding profit of the firm? 3. The following is a three period model of a financial contract involving a lender (bank) and a borrower (firm). In the first period, the firm either builds a new facility at cost c > 0 or it does not. If the firm builds the new facility, its subsequent production function in value terms becomes 4VL; otherwise it is 2VZ. Here, L is the amount of loan used in the production. In the second period, the bank offers a contract (L, t) to the firm. A contract consists of L (2 0) the amount of loan and t (2 0), the gross amount the borrower has to repay. Assume that the firm does not default. In the third period, the firm either accepts the contract or rejects the offer. If a contract (L, t) is offered and accepted, the bank receives the payoff of t - RL, where R > 1, while the payoff to the firm is 4VL - t - c if it built the facility in the first period and 2vZ -t, if it did not. If the firm rejects the contract, the payoff to the bank is zero and the payoff to the firm is -c if the firm built the facility in the first period and is zero otherwise. Assume that the firm accepts the contract if accepting and rejecting the contract result in the same payoffs. Both the bank and the firm are risk neutral. Assume that # > c. a. Suppose the bank knows whether the firm built the new facility or not in the first period when it offers a contract. Draw the game tree of this game. b. Find all sequential equilibria of the game in (a). c. Suppose from now on that the firm's possible investment in the first period is in software and the bank does not know whether or not the firm made the investment when it offers the