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Question 11 Because of superb graduate education at our Department, you have been handpicked by The President to serve as his special assistant to his hotel business that operates informally out of the South Wing of the White House. Your first task is to derive the profit maximizing inputs and output of his hotel business. You quickly learn that this president strictly prefers simple answers. So you consider a Cobb-Douglas production function with labor and capital as inputs. In terms of notation, output q is given by where A, 01, 02, 21, 22 2 0. We denote by 21 and 2, the amount of labor and capital, respectively. a.) Unfortunately, the parameters A, of, and o, are only known to the IRS and the Russians. Lacking the right connections to either of them, you set out to estimate the parameters of the production function. How would you estimate it using OLS? Set up the "econometric model" and state the estimator. b.) After firing up your (illegal) copy of Stata, you estimate parameters A, 1, and a2 and find that A, 61, 62 > 0 with an + 62 0 units of commodity (. Denote by qn > 0 the price of bundle n, and let q = (q1,..., 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 y' = (y;,..., y,) the individual's bundle demand. This demand results in consumption of commodities x' = wi+ [nymb", where wi E R4 denotes the individual's endowment. Her budget constraint is that q . y' = [nynq" 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(w+ [ngnb" ) : q.950}; ii. all markets clear: ), y* = 0. An allocation of commodities is (still) a profile x = (x', ..., x' ) of consumption bundles, such that Ext = _: wi. Allocation of commodities x is said to be first best if there does not exist an alternative allocation & such that u'(x ) > ul(x' ) 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 = (9', ...,9') such that ), 9* =0 and u ( w + [nyhb" ) zu' (x' ) for all individuals, with strict inequality for some. With respect to this model:1. Argue that if allocation x > 0 is first best, then duru(x' ) _dxu" (x' ' ] for all pairs of individuals and commodities. 2. Suppose that by > 0 for all f, with strict inequality for some. Argue that, given that all preferences are strictly increasing, if (q, y) is a competitive equilibrium in bundles, then the resulting allocation of commodities, with x = wi + [nyhb, is second best. 3. You are now going to argue that the previous result cannot be extended to first best. Suppose that L = 3, and there are only two bundles that can be traded: b' = (1, 0,0) and b= = (0, 1, 1). (a) Argue that at any (interior) competitive equilibrium allocation dx , u' (x' ) ( Ox, u' (x' ) + dx, u' (x' ), where A' > 0 is a Lagrange multiplier, for every individual. (b) Argue that at any (interior) competitive equilibrium allocation for all pairs of individuals. (c) Intuitively, argue that the latter condition is insufficient to guarantee that the allocation of commodities is first best