100
PIERPAOLO GIANNOCCOLO
To solve the asymmetric case we must have a parametric solution. Assume a quadratic utility function U(we — Zf) = [we - Zf)2
Then, we can rewrite the solutions given by the maximization
e _ n + k(we - zf)2 - k(uf - zf)2
/ +-
1	2k(we - Zf)
n + k{uf - zf)2 - k(uf - zf)2] - (a + |n)ri
zf
i-ri
r2 ~ /V - Zf)2 + - Zf)2 2k[we - Zf )
^ a +
Zf [r2 - k{uf - Zf)2 + k{we - Zf)2l - (a + f s2)s2 Zf +-L-—-J-= + bs2
For simplicity we assume these value for the parameters: b = 2; a = 1; we=10;k=l
Solving these four equations we obtain the following two best responses:
(1)	-39X3 + 884/c2 + (59 + 59/ + 420y)x + y4 + 398/ + 40y - 1 = 0
(2)	-39/ + (884 + 2c)/ + (59 - 29a + 59/c2 + 420x)y + x4+ (398 - a)/ + (40 + 20c)x -1=0
Where x = Zf y = Zf
Appendix (4) Overlapping generation model Autarky solution
Max : R,
with Rt = + r,_i] Zmax - (ya + ^t-^j ~s,{a + ^st
FOC(st) : tmax-a-bst = 0 FOC(j,_i) : Zmax - a - = 0
^max _
Then the solution is: s =-7-
b
From the (43) the solution is: s = —-^—- with s e (0,1) and
St = st-1 = r