Appendix
Proof that M=0 when all industries have
ui
 :
vi
This is immediate from the definition of M because when ∀ I,
ui

vi
then
ui - λvi = ui - (ui/vi)vi = 0 .
|||
Proof that the maximum value of M=1:
First consider the case of only two industries.
Then
M .
| u 1  v 1 || u 2  v 2 |
u1  u 2
Using the definition of λ and expanding the expression, it is straightforward to show that
u1 - λv1 = -(u2 - λv2)
Then M can be written as
M 
v 1  u 1
u1  u 2
by simply defining industry 1 as the one for which ui < λvi .
Substituting for λ, M is then
u `1  u 2
M 
v `1  v 2
v1  u1
u1  u 2
1
Then
v1
1
v `1  v 2
M

u`
u1  u 2
0
1
So M is maximized by setting u1 = 0. Also,
M

v 2
v1
u1  u 2
( v `1  v 2 ) 2
u1  u 2
1
0
so M is maximized by setting v2 = 0 . Because u1 = 0 and v2 = 0, λ = u2 / v1 so
M 
u2
v1
v `1
u2
1
.
Now, to show this in general for more than two industries, arbitrarily divide all industries
into two parts and call the first part of the partition industry 1 and the second industry 2.
Because there can never be more structural mismatch than that which occurs when all vacancies
are in one industry and all unemployed are in the other industry, ignoring industry divisions
could never increase the amount of mismatch in the economy when at full mismatch. More
concretely, if v2 = 0, then the vacancies for all the industries that make up industry 2 also equal 0.
Similarly, if u1 = 0, then the number of unemployed in each of the industries that make up
industry 1 also equal zero. Thus, the above proof holds for any number of industries.
|||
2
Proof that increasing the number of unemployed by a scalar in each industry leaves s unchanged:
Let uit and vit be unemployment and vacancy rated in industry i at time t.
Suppose ui1 = γui0 ∀ i.
Then
M1

.5 u i 1  1v i 1
i
u
i1
.5 u i 0  1v i 1
i
u
i0
i
u

v
i1
1
0
i1
where
i
u

v
i0
i
i
i0
and
i
Since vi1 = vi0 ∀ i and ui1 = γui0 ∀ i, λ1 = γλ0.
Thus,
M1

.5 u i 0   0 v i 0
i
u
i0
.5 u i 0  0 v i 0
i
u
i0
i
M0
|||
3
4
U Mining
2000
2000
1000
1000
0
0
U Manufacturing
Jan 2001
Jan 2002
Jan 2003
Jan 2004
Jan 2005
Jan 2006
Jan 2007
Jan 2008
Jan 2009
Jan 2010
Jan 2011
Jan 2001
Jan 2002
Jan 2003
Jan 2004
Jan 2005
Jan 2006
Jan 2007
Jan 2008
Jan 2009
Jan 2010
Jan 2011
Jan 2001
Jan 2002
Jan 2003
Jan 2004
Jan 2005
Jan 2006
Jan 2007
Jan 2008
Jan 2009
Jan 2010
Jan 2011
Jan 2001
Jan 2002
Jan 2003
Jan 2004
Jan 2005
Jan 2006
Jan 2007
Jan 2008
Jan 2009
Jan 2010
Jan 2011
Appendix Figure 1: Experienced Unemployed and Vacancies by Industry
Source: CPS & JOLTS data downloaded from www.bls.gov
2000
2000
1000
1000
0
0
V Mining
U Construction
V Manufacturing
U Whole & Retail
5
V Construction
V Whole & Retail
U Transp & Util
Jan 2001
Jan 2002
Jan 2003
Jan 2004
Jan 2005
Jan 2006
Jan 2007
Jan 2008
Jan 2009
Jan 2010
Jan 2011
Jan 2001
Jan 2002
Jan 2003
Jan 2004
Jan 2005
Jan 2006
Jan 2007
Jan 2008
Jan 2009
Jan 2010
Jan 2011
2000
2000
1000
1000
0
0
V Transp & Util
U Information
6
V Information
U Financial Act
U Ed & Hlth Serv
Jan 2001
Jan 2002
Jan 2003
Jan 2004
Jan 2005
Jan 2006
Jan 2007
Jan 2008
Jan 2009
Jan 2010
Jan 2011
Jan 2001
Jan 2002
Jan 2003
Jan 2004
Jan 2005
Jan 2006
Jan 2007
Jan 2008
Jan 2009
Jan 2010
Jan 2011
Jan 2001
Jan 2002
Jan 2003
Jan 2004
Jan 2005
Jan 2006
Jan 2007
Jan 2008
Jan 2009
Jan 2010
Jan 2011
Jan 2001
Jan 2002
Jan 2003
Jan 2004
Jan 2005
Jan 2006
Jan 2007
Jan 2008
Jan 2009
Jan 2010
Jan 2011
Appendix Figure 1: Experienced Unemployed and Vacancies by Industry (continued)
2000
2000
1000
1000
0
0
V Financial Act
U Prof & Bus Serv
2000
2000
1000
1000
0
0
V Ed & Hlth Serv
U Leisure & Hosp
7
V Prof & Bus Serv
V Leisure & Hosp
U Other Services
Jan 2001
Jan 2002
Jan 2003
Jan 2004
Jan 2005
Jan 2006
Jan 2007
Jan 2008
Jan 2009
Jan 2010
Jan 2011
Jan 2001
Jan 2002
Jan 2003
Jan 2004
Jan 2005
Jan 2006
Jan 2007
Jan 2008
Jan 2009
Jan 2010
Jan 2011
2000
2000
1000
1000
0
0
V Other Services
U Government
8
V Government
9
Jan 2011
Jan 2010
Jan 2009
Jan 2008
U Sales
V Sales
Jan 2011
Jan 2010
Jan 2009
Jan 2011
V Services
V Professional
Jan 2008
0
Jan 2005
0
Jan 2010
1000
Jan 2009
1000
Jan 2008
2000
Jan 2007
2000
Jan 2006
3000
Jan 2005
3000
U Services
Jan 2007
U Professional
Jan 2007
V Management
Jan 2006
U Management
Jan 2005
0
Jan 2011
0
Jan 2010
1000
Jan 2009
1000
Jan 2008
2000
Jan 2007
2000
Jan 2006
3000
Jan 2005
3000
Jan 2006
Appendix Figure 2: Experienced Unemployed and Vacancies by Occupation
Source: CPS data (BLS) & HWOL data (the Conference Board)
3000
2000
2000
1000
1000
0
0
U Office & Admin
V Office & Admin
10
U Construction
Jan 2011
Jan 2010
Jan 2009
Jan 2008
Jan 2007
Jan 2006
Jan 2005
Jan 2011
Jan 2010
Jan 2009
Jan 2008
Jan 2007
Jan 2006
Jan 2005
3000
V Construction
V Instal & Maint
U Production
3000
2000
1000
U Transportation
Jan 2011
Jan 2010
Jan 2009
Jan 2008
Jan 2007
Jan 2006
Jan 2005
0
V Transportation
11
Jan 2011
Jan 2010
Jan 2009
Jan 2008
Jan 2007
U Instal & Maint
Jan 2007
0
Jan 2005
0
Jan 2011
1000
Jan 2010
1000
Jan 2009
2000
Jan 2008
2000
Jan 2006
3000
Jan 2005
3000
Jan 2006
Appendix Figure 2: Experienced Unemployed and Vacancies by Occupation (continued)
V Production