The name is absent



da*

Since c > 0, dhj has the sign of the sum between square brackets in (14). Using the
expression for
σ in (4) and simplifying we find that

∂σ(hj ,hk,hε)    /

dhj        1 1

σ (hj,hk,hε^) у


(mk


2
mj)


σ2 (hj, hk, hε)


ʌ__L_

hj + hε


hk (2hj + hε) (hk + hε) 1


σ(mmk-m⅛y) - 2hj [hj (hj + hε) + hk (hk + hε)]


2hj (hj + hε) [hj (hj + hε) + hk (hk + hε)]


(16)


da* 1        1                  .             1                                /- „ч m1             . η η                    „ ,

dhj has the same sign as the numerator in (16). The partial derivative of the numerator
in (16) with respect to h
j is

2hk (hk + hε)


(mk


2
mj)


σ2 (hj, hk, hε)


(mk mj)2 ∂σ2

+ hk (2hj + hε) (hk + hε) 2 z, T rʒ TTT-
σ2 (hj, hk, hε) ∂hj

<0


2hj (hj + hε) 2hj (2hj + hε),

which is clearly negative. Since limhj0 dhj = and limhj→∞ dhj = 0, it follows from
da*

this that given the other parameters there exists a unique hj > 0 such that dj 0 if and
only if h
j = hj.

Finally,

da*
dhk


exp


(   (mk-mj )

2(hj ,hk,,


)2
,h
ε)


√2πσ (hj,hk,he)

'----------------------v------

>0


Wjh + /

c'' ⅛ ) V
-------zC_


∂σ(hj ,hk ,hε)


∂hk


σ (hj, hk, hε)

SZ

>0


)(■

S


(mk


2
mj)


σ2 (hj, hk, hε)


.     (17)


From (17) it is easy to see that daj > 0 if and only if (mk mj)2 < σ2 (hj, hk, hε). Since
lim
hk→0 σ2 = and σ2 (hj , hk, hε) is strictly decreasing in hk (see (4)), there exists a
unique threshold
hk such that dhj 0 if and only if hk = hk. Observe that hk > hj. ■

Lemma 1


(Lemma A1) limhj→ da* =

Proof.

daj*

limh, o -г;—,
hj0 dhj


First note that both limhj0 σ = and limhj0 dh- = ∙ To determine
we first simplify the limit of (14) as follows:

daj
lim
h
j0 dhj


limhj0 exp -


(mk-mj )2 "I W  hε

2σ2(hj ,hk,hε) J W Wj hj +hε


lim
hj0


∂σ(hj ,hk,hε)
dhj


c' (⅛')


Wj


σ2 (hj, hk, hε)


lim 1

hj 0


(mk


2
mj)


σ2 (hj, hk, hε)


ʌ 1-    (

- lim
hj0


(hj +hε)σ(hj,hk,hε


2∏C^c" (α*)


lim
hj 0


∂σ(hj ,hk,hε)
dhj


σ2 (hj, hk, hε)


41




More intriguing information

1. From music student to professional: the process of transition
2. Does South Africa Have the Potential and Capacity to Grow at 7 Per Cent?: A Labour Market Perspective
3. The name is absent
4. The name is absent
5. SOME ISSUES IN LAND TENURE, OWNERSHIP AND CONTROL IN DISPERSED VS. CONCENTRATED AGRICULTURE
6. Labour Market Institutions and the Personal Distribution of Income in the OECD
7. The name is absent
8. The name is absent
9. Running head: CHILDREN'S ATTRIBUTIONS OF BELIEFS
10. THE DIGITAL DIVIDE: COMPUTER USE, BASIC SKILLS AND EMPLOYMENT