Misallocation and Productivity
Margarida Duarte and Diego Restuccia
University of Toronto
Basic references are: Restuccia and Rogerson (2008) and Hsieh and Klenow (2009).
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Misallocation and Productivity
Basic model. One good is produced each period. The production unit is an establishment.
An establishment is a decreasing returns to scale technology that for simplicity we assume
it requires only labor input:
y = s1−γ lγ ,
where s is the productivity of the establishment and l the labor input. We assume there
is only two types of establishments sL and sH . We also assume there is a fixed number of
establishments of each type, N = NL + NH .
There is a representative household who owns the establishments, has one unit of labor
that rents to firms and standard preferences over consumption.
We assume competitive markets, firms rent labor from households and sell goods to
consumers.
Problem of establishments. An establishment of type i maximizes profits by choosing the
labor demand subject to the competitive market wage:
πi = max si1−γ liγ − wli .
li
1
The first order condition of this problem is:
γs1−γ
liγ−1 = w,
i
which implies
li =
γ 1/(1−γ)
w
si .
Note that labor demand by the firm is increasing in s and decreasing in the wage rate.
Substituting the labor demand into output and profits yields,
y i = si
γ γ/(1−γ)
w
πi = (1 − γ)yi
which are also increasing in si and decreasing in w.
Labor market clearing requires that labor demand be equal to labor supply which is equal
to 1,
X
li Ni = 1.
i
Definition A competitive equilibrium is a wage w and allocations c, li , yi , πi such that:
i) Given w and Π, c = w + Π. ii) Given w, li solves the firms problem for each type of
establishment i. iii) Market clear:
X
li Ni = 1,
i
2
X
yi Ni = c.
i
We can substitute the labor demands into the labor market clearing condition to solve
for w as:
γ 1/(1−γ)
w
(sL NL + sH NH ) = 1.
Hence,
w = γ(sL NL + sH NH )1−γ .
Given w, solve for the labor demands of each type li , profits, output, consumption and
aggregate output. In particular, labor demands are given by:
li = P
si
i si Ni
and aggregate output is given by
Y = (sL NL + sH NH )1−γ .
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Efficient Allocation
We can show that the equilibrium allocation without distortions is efficient. To do so, note
that a benevolent planner would maximize welfare of the consumers by maximizing output.
So the problem of the planner is to allocate labor across establishments to maximize output
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subject to the labor adding up to 1,
max Ye =
X
si1−γ liγ Ni ,
i
subject to
P
i li N1
= 1. The first order conditions of this problem give
sH
lH
=
,
lL
sL
and substituting on the labor constraint implies
si
.
i si Ni
li = P
Hence, the efficient labor allocations coincide with the equilibrium allocations and aggregate
output is the same. We now consider distortions to equilibrium allocations that would put
a wedge between the efficient allocation and the equilibrium allocations.
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Distorted Economy
Assume that taxes are levied on the output (revenue) of each type of establishment. For
simplicity assume only high productivity establishments are taxed. Tax revenues are rebated
back to consumers as a lump sum transfer T .
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The problem of the H type establishment now changes to:
γ
πH = max(1 − τ )s1−γ
H lH − wlH ,
lH
which implies
lH =
(1 − τ )γ
w
1/(1−γ)
si .
Using the resource constraint for labor, we obtain the equilibrium wage:
w = γ(sL NL + (1 − τ )1/(1−γ) sH NH )1−γ .
Given w, solve for the labor demands of each type li , profits, output, consumption and
aggregate output. In particular, labor demands are given by:
lL =
sL
,
(sL NL + (1 − τ )1/(1−γ) sH NH )
lH =
(1 − τ )1/(1−γ) sH
,
(sL NL + (1 − τ )1/(1−γ) sH NH )
and
and aggregate output is given by
γ
1−γ γ
Y = NL s1−γ
L lL + NH sH lH ,
Y =
(sL NL + (1 − τ )γ/(1−γ) sH NH )
.
(sL NL + (1 − τ )1/(1−γ) sH NH )γ
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Note that in the distorted equilibrium compared to the equilibrium without distortions,
wages are lower, labor demand by L establishments is higher, labor demand by H establishments is lower, and aggregate output is lower.
There is nothing special about the tax being on the high productivity establishments,
any tax/subsidy combination that distorts the optimal size of establishments will reduce
aggregate output in this setting.
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Evidence
In the efficient allocation as well as in the equilibrium without distortions, note that the
marginal product of labor is equalized across all establishments. Since H establishments
are more productive, this is accomplished by this establishments being larger, hiring more
labor to reduce the marginal product to the one of the L establishments. In the distorted
economy, the marginal product of labor is not equalized, only the value of marginal products
is equalized, with the wedge being the tax on high establishments. This creates a reallocation
across establishments that distorts their size and reduces measured productivity.
If we had data on the productivity of establishments, and their demands of factors, we
could inquire whether marginal products are equalized across establishments in the data.
This is what Hsieh and Klenow (2009) do using micro data for the United States, China and
India. They find that if factors were reallocated in China and India to equalize marginal
products (to the same extent than the US) then aggregate TFP will increase in those countries by 40-60%.
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Discuss evidence for agricultural productivity in Malawi from Restuccia and SantaeulaliaLlopis (2014).
References
[1] Hsieh, Chang-Tai, and Peter J. Klenow. “Misallocation and Manufacturing TFP in
China and India.” The Quarterly Journal of Economics 124, no. 4 (2009): 1403-1448.
[2] Restuccia, Diego, and Richard Rogerson. “Policy distortions and aggregate productivity
with heterogeneous establishments.” Review of Economic Dynamics 11, no. 4 (2008):
707-720.
[3] Restuccia, Diego, and Raul Santaeulalia-Llopis. “Land Misallocation and Productivity.”
Unpublished manuscript, University of Toronto (2014).
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