Having seen that the core shrinks to the set of competitive equlibria, it can be asked how quickly does this convergence take place. Debreu (1975) showed under certain conditions that the maximum distance between the core for the r-fold economy (Cr) and the set of competitive equilibria (WEA) shrinks at least as fast as the inverse of the number of consumers. Debreu's conditions are met in the case of Exercise 5.32 so it can be used to illustrate his result.
The notion of distance here is the maximum distance between points in two sets. So if X={1} and Y={[0,3]} then d(X,Y)=2. I am going to measure distance between two points on the contract curve by their separation on the x1-axis.
Debreu's result can be stated as showing that there exists a constant K such that r.d(Cr,WEA)<=K.
A diagram helps to follow what is going on.
(e) Show the WEA is (5,5). By Theorem 5.6 this is in the core for all r.
(f) Consider a point (z,z) on the contract curve such that 4<=z<5. Note that a line drawn between 1's initial endowment (8,2) and (z,z) must intersect 1's indifference curve through (z,z) twice. Form the coalition of r type 1 consumers and allocate them g=1/r(8,2)+(1-1/r)(z,z) and (r-1) type 2 consumers and allocate them (10-z,10-z). Check this is feasible.
(g) Now choose r such that u1(g)>u1(z,z) and deduce that for a given z with 4<=z<5 if r>(8-z)(z-2)/2z(5-z) then z is not in the core. Use this result to show part (d) of the question.
(h) For 5 < z <=6 show in a similar way that z is not in the core for r > (8-z)(z-2)/2(10-z)(z-5).
(i) Deduce that for 4 <= z <=5 and r <= (8-z)(z-2)/2z(5-z) part of the core is in the section of the contract curve running from (z,z) to (5,5).
(j) Deduce that for 5 < z <= 6 and r <= (8-z)(z-2) / 2(10-z)(z-5) the other part of the core is in the section of the contract curve running from (5,5) to (z,z).
(k) Deduce that r.d(Cr,WEA) < r.modulus(z-5)< 3/2 to get Debreu's result!
G. Debreu (1975), 'The rate of convergence of the core of an economy', Journal of Mathematical Economics, 2, 1-8.