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Question 2 Consider a smooth exchange economy with I consumers and L 2 2 goods, {], (ut, wilies). but suppose that, unlike in class, for some reason the agents in this economy cannot exchange the commodities directly. Instead, some institutional arrangement forces them to trade in bundles of commodities. There are N bundles of commodities that they can trade, n = 1,.... N. Bundle n is a vector be = (bp. .... bp) 0 units of commodity &. Denote by qn > 0 the price of bundle n, and let q = (qu...., qN) be the vector of bundle prices. Each individual can buy and sell these bundles at the given prices. Let y), denote the number of units of bundle n bought by individual i, with the convention that this number is negative if the individual is actually selling the bundle. Denote by yi = (y;...., y;] the individual's bundle demand. This demand results in consumption of commodities xi = wi+ [nyib where wi E R. denotes the individual's endowment. Her budget constraint is that 4 . 4' = [nyhq" so, which means that she can only afford positive expenditure in some bundles if she raises enough liquidity from the sales of other bundles. A competitive equilibrium in bundles is a pair (q, y), where q is a vector of bundle prices and y = (y'..... y') a profile of bundle demands such that i. each individual is individually rational: for each i, bundle y solves max {ul(wi+ [mynb" ) : q.950 ); ii. all markets clear: Eyi = 0. An allocation of commodities is (still) a profile x = (x], .... x ) of consumption bundles, such that Lixi = [; wi. Allocation of commodities x is said to be first best if there does not exist an alternative allocation * such that ui (8) > ui(xi ) for all individuals, with strict inequality for some. It is said to be second best if there does not exist a profile of bundle demands y = (y', .... y' ) such that Zi yi =0 andQuestion 3 Consider a standard production economy where at least one of the agents has strictly monotone preferences. Recall that an allocation is a pair (x, y) = (x'.....x.y'.....y'je( R4 ]' x (RL)I such that y' e Y' for each j, and ): xi = ); wi + _; y'; and that allocation (x, y) is Pareto efficient if there does not exist another allocation (x, y) such that u' ($1) > ui (xi ] for all consumers, with strict inequality for some. A profile of production plans y = (y'...., y ) is feasible ify e Y for each j; a feasible profile of production plans is technically efficient if there does not exist an alternative feasible plan y such that ); gi > >; y'. Also, given a profile y of production plans, a profile x = (x',..., x ] of consumption bundles is feasible if Ex' = [iw' + );y'. Finally, feasible profile x is allocationy efficient, given y, if there does not exist an alternative profile & that is also feasible given y and such that ui (& ) 2 ui (x' ) for all consumers, with strict inequality for some. Given these definitions: 1. Argue that if (x, y) is Pareto efficient, then profile y is technically efficient (since one agent has strictly monotone preferences). 2. Argue that if (x, y) is Pareto efficient, then profile x is feasible and allocationy efficient given y. 3. In what follows you will argue that, even together, technical and allocation efficiency don't suffice to guarantee Pareto efficiency. Suppose that there are two commodities, two individuals and one firm. Both individuals have smooth utility functions, while the technology of the firm is Y = {(y1. 42) ER, xR ly.