government. The government now maximizes the welfare function:
πdA +
∕∙n+
AAh
dB
и (F -
∙.K) dAdB^
(22)
with π and v* defined as in (14) and (15), subject to
F(K,Al) (1 - и) - (1 - au) K = О
(23)
F(K,Aft) (1 - и) - (1 - au) K = F (K*,Aft) (1 - t) - (1 - βt) K* - C(24)
The optimality condition with respect to и is:
∂W ['b+ Ah ['b+ A"
-— = Q = Ud' - U')J J [(F - aK)] dAdB + H'J J [u (Fκ - a) Ku] dAdB
+h ' C (u (f ft -aK η 9~a-) dB
h ' £ (u (fl
., ∂Al∖
aKl) — dB (25)
7 du J
The optimality condition with respect to a is:
∂W
da
Zb+ Ah B++ Ah
I [uK] dAdB + H' I [u (Fκ - a) Ka] dAdB
+( h' ι (F‘- aK' > ■)dB - £ h' (u fl - aKl) ) b'
First, it follows from equation (25) that u > О, a = 1, H' = U' cannot be an
optimum, because
dw =£ h' (u (f (AK'■) - k^) d-A) dB = 0 (27)
Second, u has to be greater than zero to satisfy Д^" = 0. This can be explained
as follows: The first term on the RHS of equation (25) is strictly positive, the
three other terms are strictly negative. Therefore, u = 0 and u < 0 are no possible
solutions.
Can u > 0, a = 1, H' > U' be an optimum?
12