1196 JOURNAL OF POLITICAL ECONOMY PROPOSITION 2.Let h(x)= (17) 2p' 2'0 ifx≤0 if x>0, T(+(/2p)r(险) where I()is the gamma function,and M:R-R and U:RR are two Kummer functions described in the Appendix.The function h(x) is positive and increasing in (-0).In addition,h solves equation (16) with h(0)= r(2+(r/2p》r(2)】 Any solution u(x)to equation (16)that is strictly positive and increasing in(-∞，0)must satisfy u(x)=B,h(x)with 8,>0. We shall also need properties of the function h that are summarized in the following lemma. LEMMA 2.For cach x R,h(x)>0,h'(x)>0,h"(x)>0,h"(x)>0, limh(x)=0,and lim'(x)=0. Since g must be positive and increasing in (-0,k'),we know from proposition 2 that B,h(x) for x<k q(x)= (18) x+入 +B,h(-x)-c for x≥. Since g is continuous and continuously differentiable at k', 8,）-太-A,A-k)+c=0, r+入 8h)+B,a(-- r+λ =0. These equations imply that B,= (19) [h'()+(-kI(r+入) and k'satisfies [-c(r+入][h'(h)+h'(-k*〗-h(k)+(-k)=0. (20) Reproduced with permission of the copyright owner.Further reproduction prohibited without permission.Reproduced with permission of the copyright owner. Further reproduction prohibited without permission