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1. Consider an Lgood private ownership economy in which every consumer has a strictly positive endowment vector e3; 3:} U in RL and a utility function u,- : Rf; } R with the following properties: {a} u:- is strictly qussiconcave; and (b) the preferences determined by u:- are locally nonsatiated. In addition? every consumer entirely owns its own rm and every rm has exactly one owner. The rms have the same convex production set Y E EL, which has the following property: (c)er:>terforallteR+. A feasible allocation in this economy is one in which each consumer 3' gets a consumption vector :33: E Bi: and its rm chooses production vector y; E Y such that Ella}: e; pg] E D. a. Give a mathematical denition of strict quasiconcavity for a function in. b. Use the notation above to dene a competitive [Walrasian] equilibrium {without disposal} for this economy. c. Interpret property {c}. d. What can be said about the prot of rm i {owned by consumer i} in a competitive equilibrium of this economy? Be as specic as possible and show that your answer is correct. e. We call a competitive equilibrium in this economy self-sufcient if each consumer 2' gets a consumption vector II\2. There are two types of workers of equal numbers. The total number of workers is normalized to be 1. A worker is either a high productivity type [IE3 = 2} or a low productivity type {5' = 1]. Each worker chooses a task of diiculty e1 [1 E e E 2. The government cannot observe the type of worker or the difculty of the task a worker performs but it can observe the income of a worker. A worker of type I9 working in a type a task receives income at: = HU + e]. The government needs to raise a tax revenue of 1 to meet its scal needs but seeks to raise it in a welfare maximizing manner. The utility function of a type I? who chooses a task of difculty e and pays income tax t is aeywjt] = a: g t, where a! = H + e}. The government maximizes W = 211.; + ah subject to constraints. 1 2 a. Suppose that the government is restricted in its choice of income tax schedule to an aine income tax schedule t = a + Inc where [l :1: l:- <: which task a would each type choose formulate the problems and solve them. b. given results in part welfare function of government problem choosing optimal afne income tax schedule to raise revenue find marginal rate t. c. nowj suppose instead that oers menu for two types from: th those with we are taxed t3. herej is intended e by government. if i worker indifferent between assume chooses raised maximizing criterion w all relevant constraints. since rah="Eh" eh it eh. similarly l a. lconsider consumer m who purchases commodities :1: y at market prices pg: u respectively. however addition incurring direct cost associated such expenditures also required pay transaction fee .5 value its purchases. budget equation consumer. how does this differ from stande contrast was endowed bundle goods : my>2} [1 rather than with income. In this case, the transaction cost is incurred as a result of changes from w and is paid in units of :r, the numeraire. {That is, the transaction cost is incurred whether the consumer is a net buyer or a net seller of I.) Such costs are increasing in the size of the transaction. [i] Graphically depict the consumer's budget constraint which takes into consideration the transaction cost. Also, depict a typical consumer's equilibrium. {ii} Is it possible that a consumer would be a net buyer of :1: without the transaction cost, but would choose to be a net seller of :1: with the cost? {iii} Suppose the transaction cost is [1.5% of the amount of a: traded. Write an expression for the consumer's budget constraint and draw the consumer's budget set. c. Discuss the difference between the formulations in parts a and b. d. Finally, consider an exchange economy with two consumers, 1 and 2, both of whom are as described in part b. Let Lu" 2 {wi,w,) denote the endowment of consumer i. [i] Assume that the consumers incur the cost of their own transactions. Set up the social planner's problem for determining the Pareto eELcient allocations. {ii} Derive and interpret] explain the {necessary} first order conditions for eiciency at a solution to the planner's problem where consumer I is a net buyer of :r