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6 In a certain country, all civil servants are recruited on their 25th birthdays. (Birthdays are assumed to occur uniformly over the calendar year). Of those who reach exact age 40, 10% obtain employment in private companies and leave the civil service immediately. All remaining civil servants retire when they reach age 60 or die in service before this age. Civil servants have for many years experienced the mortality of English Life Table No. 12 - Males, and will do so for the indefinite future. The population of civil servants has been stationary for many years, and civil servants are recruited at the rate of 3,000 each year. (i) How many civil servants are in service at any given date? (ii) The government of the country has just decided to reduce the size of the civil service by 10%. This is to be done by immediately lowering the retirement age of civil servants. Find the new retirement age (to the nearest month), assuming that there are no other changes. 7 In a mortality table with a one-year select period, the following relationships apply at age r: (0 0 for k e {1. ..., L}. As usual we denote r' = (r;, ...,;). Consider price vectors (p1, .... pz) and wealth levels (w], ...,w') for which the solution to the utility maximization problem is interior for every consumer i e {1, .... /}. a.) Derive the Walrasian demand function for good j by consumer i. (Be careful with your calculations. Double-check! It is easy to make mistakes.) b.) What is the slope of consumer i's Engel curve for good j at (p, wi)? c.) Find a condition as general as possible on parameters ox, By, i e {1,..., / }, k E {1. ..., L} guaranteeing the existence of a positive representative consumer. Do we need restrictions on parameters ox, ke {1, .... L}? d.) Consider now the special case with just a single consumer and two goods. The consumer's utility function is given by Derive the wealth-expansion path for a given price vector (P1, p2)- e.) In problem d.), when does the wealth-expansion path intersect the r,-axis and when does it intersect the ry-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', wl ), (u', w?)). Define its replica as the four-person exchange economy 82 = ((u', w'), (u', wa), (u', w'), (u', wa)). where (u', w3) = (u', w') and (u', wi ) = (uz, w?). 1. Argue that if (p, x', x?) is a competitive equilibrium for &, then (p, x', x], x3, x' ) with x3 = x' and x* = x', is an equilibrium for &?. 2. Argue that if both utility functions are strictly quasi-concave, and (p, x', x', x', x* ) is a competitive equilibrium for &', then, x' = x3 and x? = x*. 3. Argue that if both utility functions are strictly quasi-concave, and (x], x?, x3, x* ) is in the core of &', then, x' = x3 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) =x'x?, wi = (1, 0) and w2 = (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 &?. 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." 1 We are using the term "shrink" loosely, since the presence of more agents changes the dimension of the allocation space, so comparing the sizes of the cores will require some refinement of the argument. In the limit, one can show that replication ad infinitum reduces the core to just the set of equilibrium allocations