economics 404 please help tutors

Question 2. Consider an exchange economy E((u',@)_ ) with / agents and Z goods, where the agents' utility functions a' are continuous and increasing in all components, and the endowments of are such that E,@ > 0 . We want to provide a characterization of the Pareto optimal allocations as allocations where the benefits from trade have been exhausted. To do this, we choose a bundle of goods ge Rl , g #0 , and we define the benefit function of each agent (in units of the bundle g ) as follows: if x's R* is a consumption vector for agent i and if v 2u'(0) is a utility level, the benefit b'(x', v) of x' over v is defined by b'(x', v) = max(Be Ru'(x' -Bg)2v} (a) Interpret b' (x' , v') and give a geometric representation of it in a two-good economy. Show that u'(x') 2 v implies b'(x', v') 20, with strict inequality if u'(x') > v. (b) Consider a feasible allocation x= (x'....,X' )ER" (i.e. such that Ex's End ), and let v =u' (x') . Suppose that there is a feasible allocation & such that Show that the allocation x is not Pareto optimal. (c) Illustrate the result of (b) in the Edgeworth box, taking a case where b' (i , v) > 0, b' (x , v) 0 is a parameter. Our consumer treats E, (as well as pi, pz and w) parametrically. The maximization of her utility function subject to the constraints (1) and (2) yields her market demand functions , (P,, Pz, W, E_). j=1,2 for goods 1 and 2, as well as her desired amount of the public good E , ( P. . P z , W, E. ) . 1(a). Write the maximization problem the solution of which is (i, (P,, P2, W. E_,).x, (P,, P2. W. E_,), E(P,, Pz, w, E_,)), and call it Problem PG. Write its Kuhn-Tucker conditions. Assuming that the solution is interior and unique, and that constraints (1) and (2) are satisfied with equality, write a system of equations, not involving the multipliers, the solution of which defines (i, ( P, , P2 , W. E_, ). X, ( P, . P2 , W. E_,), E(P,, P2, W. E_,)). (Just write the system of equations: there is no need to work towards its solution.) 1(b). For the rest of this part we consider the utility function 1: RxR : u(x . x2. E) = x, + la,a.) .B[ ]. where (d,, d ) > >0 and B= BE2 bEEl bzz bre is a symmetric, positive definite matrix. We are interested in the comparative statics expressed by the partial derivatives and DE . Again, assume that the aE_, solutions are interior and the constraints satisfied with equality. University of California, Davis Date: June 23, 2008 Department of Economics Time: 5 hours Microeconomics Reading Time: 20 minutes 1(b) (i). Specialize to this utility function the system of equations obtained in 1(a). 1(b) (ii). Verbally interpret the partial derivatives dry and - aE , as well as their signs. Compute ar, aE - and aE aE_ 1(b) (iii). Let bze 2 0. Find the signs of - and of - DE dE_ aE_ 1(b) (iv). Let bze