Macro

\fassigns to EL ]. PE's beliefs are common knowledge between M and PE as is the feet that M hows the true value of 9. Thus we have s situation of ineomplete information. (d) Using the Harsanyi transformation sketch the extensive form of the corresponding imperfect-information game. Make sure that information sets are clearly drawn. [You can simplify the stretch by replacing the Coumot duopolv interaetions with terminal nodes and sssoeiating with them the eonesponding cqrlihrium prots] {e} For the game of part (d) show that under the following parameter restrictions there is no pine-strategy separating weak sequential equilibrium {that is, there is no pure-strategy.P equilibrium where the two types of M make different investment ehoioes): l l 19,, I i . 40 2t} 15 {1] With the parameter values of part (e), and assuming that the players are risk neutral, for What values of p is there a pooling weak sequential equilibrium where both types of!\" choose it and, observing this, PE stays out? 1 Question 1 Consider an economy with I consumers and L goods. For each consumer i E {1, .._, I}, the consumption set is HE. Her utility function is given by L- eif-:1 = Z ekeirt, J:=1 where h or CI for i; E {l1._.,L}. As usual we denote 1' = (111...,I'g}. Consider price vectors [p11._.1p5} and wealth levels {11:1, ...,wr]l for which the solution to the utility maximisation problem is interior for every consumer i E {1, _..1 I}. a.) Derive the Walrasian demand Function for good 3' by consumer i. [Be careful with your calculations. Doublecheck! It is easy to make mistakes]I b.) iWhat is the slope of consumer i's Engel curve for good j at [33113]? c.) Find a condition as general as possible on parameters op, .57.\" \"i E {l,...,f}, F; E {l,_..1L} guaranteeing the existence of a positive representative consumer. Do we need restrictions on parameters oh k E {1, .._,L}'? d.) Consider now the special case with just a single consumer and two goods. The consumer's utility function is given by -H:=1 ns EIIJIg] = o1e one Derive the wealth-expansion path for a given price vector {1311333}. e.) In problem d.)1 when does the wealthexpansion path intersect the Ilaxis and when does it intersect the rig-axis? Question 2 It is intuitive to think that the presence of more agents in the economy "shrinks" its core, since there are more coalitions that can object a given allocation. You will understand in this question why this is indeed the case." Fix a standard, two-person exchange economy & = {(u', w'), (u', w')). Define its replica as the four-person exchange economy & = [(u', w'), (u,w), (u',w'),(u',will, where (u', wi) = (u', w) and (u', wi ) = (u], w3). 1. Argue that if (p, x', x?) is a competitive equilibrium for &, then (p, x), x], x], x* ) with x3 = x and x = x', is an equilibrium for 82. 2. Argue that if both utility functions are strictly quasi-concave, and (p, x], x], x', x' ) is a competitive equilibrium for &', then, x1 = x3 and x? = x'. 3. Argue that if both utility functions are strictly quasi-concave, and (x], x], x], x* ) is in the core of & , then, x = x and x] = x'. 4. Argue that if both utility functions are monotone and strictly quasi-concave, and (p, x', x?) is a competitive equilibrium for &, then (x], x3, x', x?) is in the core of &?. 5. Suppose that u (x) = u(x) = xx], wi = (1,0) and we= (0, 1). Argue that allocation ((0, 0), (1, 1)) is in the core of &, yet allocation ((0, 0), (1, 1), (0,0), (1, 1)) is not in the core of 82. 6. Use these results to argue, informally, that the replication of agents does not affect the set of equilibrium allocations of the economy but shrinks its core