It multiplies any distortion associated with misallocation but is not itself affected
by the allocation of resources.
Second, the tax wedges affect output through TFP. Therefore, this proposition
illustrates a very important result found elsewhere in the macro literature: the
misallocation of resources at the microeconomic level often shows up as a
reduction in TFP at the macroeconomic level. This result has been emphasized
by Chari et al. (2007), Hsieh and Klenow (2007), and Restuccia and Rogerson
(2007), and also plays a key role in Caselli and Gennaioli (2005) and Lagos
8
The solution for
τ
satisfies
τ
= (1

σ
(1

τ
))
T
θ
+
σ
(1

τ
)
T
ρ
where
T
ρ
≡
R
1
0
τ
i
“
A
i
(1

τ
i
)
Q
ρ
”
ρ
1

ρ
di
. That is,
T
ρ
is a weighted average of the sectorspecific
tax rates, where the weights depend on
ρ
;
T
θ
is defined analogously.
20
CHARLES I. JONES
(2006). Importantly, the tax wedges get multiplied by the intermediate goods
multiplier. We will discuss the effect of these wedges in more detail below.
Finally, a key difference relative to the previous result on the symmetric al
location is that the curvature parameter determining the productivity aggregates
has changed. For example,
ρ
1
−
ρ
replaces the original
ρ
. Notice that if the domain
of
ρ
is
[0
,
∞
)
, the range of
ρ
1
−
ρ
is
[0
,

1)
: there is less complementarity in
determining
Q
ρ
than
S
ρ
.
This result can be illustrated with an example. Suppose
ρ
→ ∞
. In this
case, the symmetric allocation depends on the smallest of the
A
i
, the pure weak
link story.
In contrast, the equilibrium allocation depends on the harmonic
mean of the (tax adjusted) productivities, since
ρ
1
−
ρ
→ 
1
. Disasterously low
productivity in a single variety is fatal in the symmetric allocation, but not in the
equilibrium allocation. Why not?
The reason is that the equilibrium allocation is able to strengthen weak links by
allocating more resources to activities with low productivity. If the transportation
sector has especially low productivity that would otherwise be very costly to the
economy, the equilibrium allocation can put extra physical and human capital
in that sector to help offset its low productivity and prevent this sector from
becoming a bottleneck.
Of course, this must be balanced by the desire to
give this sector a
low
amount of resources in an effort to substitute away from
transportation on the consumption side.
This can be seen in the math:
the
equilibrium solution for allocating capital is
K
i
K
=
1

τ
i
1

τ
bracketleftBigg
(1

σ
(1

τ
))
parenleftbigg
A
i
(1

τ
i
)
Q
θ
parenrightbigg
θ
1

θ
+
σ
(1

τ
)
parenleftbigg
A
i
(1

τ
i
)
Q
ρ
parenrightbigg
ρ
1

ρ
bracketrightBigg
.
Another perspective on the solution is gained by returning to a special case
we considered earlier. Suppose
θ
= 1
,
ρ
→ ∞
, and
σ
= 1
/
2
, and suppose
τ
i
= 0
. In this case,
Q
θ
→
max
A
i
while
Q
ρ
becomes the harmonic mean of
the
A
i
. Total factor productivity is the product of the two. Contrast this with
the same example for the symmetric allocation: there, TFP was the product of
the arithmetic mean and the minimum. Allocating resources optimally shifts
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 Spring '14
 Roe,TerryLee
 Economics, Generalized mean, intermediate goods