O-Ring Theory of Development

O-Ring Theory of Development
Jesús Fernández-Villaverde
University of Pennsylvania
October 19, 2014
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October 19, 2014
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O-Ring
Technology I
Consider a …rm using a production process consisting of a given
number n of tasks.
Each task requires 1 worker.
The quality of a worker is q: expected percentage of maximum value
B the product retains.
Probabilities are independent across workers.
Expected production:
Ey = (Πni=1 qi ) nBk α
where k is capital.
Quantity cannot substitute quality.
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O-Ring
Problem of the …rm I
Firms are risk neutral: w.l.g. we can deal with y instead of Ey .
Distribution φ (q ) of workers with wages w (q ).
Capital in the economy k rented at price r .
Problem of the …rm is:
max (Πni=1 qi ) nBk α
n
∑ w ( qi )
rk
i =1
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O-Ring
Equilibrium conditions
FOCs:
Πi 6=i qi nBk α = w 0 (qi )
α (Πni=1 qi ) nBk α
1
=r
Note that:
nBk α > 0
implies assortative matching of workers across …rms (Becker, 1981).
With perfect sorting (and dropping i when there is no confusion):
w 0 (q ) = q n
1
nBk α
r = αq n nBk α
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1
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Aggregate output I
With perfect sorting, production function of the …rm that operates
with productivity qi and capital ki is:
qin nBkiα
Firms are indi¤erent about which level of q to operate.
Then, for two levels of q, the demand for capital should imply that
marginal productivities are equated:
αqin nBkiα
1
= αqjn nBkjα
or
ki =
Jesús Fernández-Villaverde (PENN)
qi
qj
O-Ring Theory
1
n
1 α
kj
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O-Ring
Aggregate output II
By setting qj = 1,
n
ki = qiα 1 k1
and production is:
1 2α
α n
Eyi = qi 1
nBk1α
By market clearing in labor, we will have n1 d φ (q ) …rms operating at
each level q.
By market clearing in capital:
k =
Z 1
0
Jesús Fernández-Villaverde (PENN)
n
1
k
qi1 α k1 d φ (q ) ) k1 = n R
n
1
n
q 1 α d φ (q )
0 i
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Aggregate output III
Then, output per …rm:
1 2α
1 α n
Eyi = qi
0
nB @n R
k
1
0
n
qi1 α d φ (q )
1α
A
Total output (and applying a law of large numbers across …rms of the
same quality):
1α
0
Z 1 1 2α
k
n
A 1 d φ (q )
y =
qi 1 α nB @n R
n
1 1 α
n
0
q d φ (q )
0 i
R 1 11 2αα n
q
d φ (q )
α
= nα B R0 i n
αk
1 1 α
q d φ (q )
0 i
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Interest rate and wages I
We can come back to the FOCs of the …rm:
w 0 (q ) = q n
1
nBk α
r = αq n nBk α
1
or, substituting the second FOC in the …rst one:
0
w (q ) = q
n 1
nB
αq n nB
r
α
1 α
For q = 1,
r = αnBk1α
where k1 =
R1
0
q i1
Jesús Fernández-Villaverde (PENN)
n
k
α 1
n d φ (q )
1
.
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O-Ring
Interest rate and wages II
Then:
w 0 (q ) = q 1
1
αn
1
nBk1α
Integrating:
w (q ) = (1
n
α) q 1 α Bk1α
where the constant of integration must be zero to ensure market
clearing.
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Numerical example I
Two countries with same capital.
Country 1: mass 1 of workers with quality 1.
1
2
Country 2: a two-point distribution q0.5 with
mass.
mass and q1 with
1
2
Output country 1:
y1 = nα Bk
α
Output country 2:
α
α
y2 = n k B
Jesús Fernández-Villaverde (PENN)
1
2
1 α
O-Ring Theory
1
2
1
2
1 2α
1 α n
+1
n
1 α
+1
α
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O-Ring
Numerical example II
Ratio:
y2
=
y1
As n ! ∞,
1
2
1 α
y2
!
y1
1 2α
1 α n
1
2
1
2
1
2
+1
n
1 α
+1
α
1 α
However, with a normal Cobb-Douglas production function:
y cd = k α h1
α
where h1 = 1 and h2 = 0.75 (half the workers as productive as in
country 1 and half only half as productive):
y2cd
= 0.751
y1cd
Jesús Fernández-Villaverde (PENN)
O-Ring Theory
α
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Numerical example III
If α =
1
3
y2
y1
y2cd
y1cd
! 0.63
= 0.8255
Capital country 1:
k1 = nk
Capital country 2:
k1 =
k0.5 =
Jesús Fernández-Villaverde (PENN)
2
1
2
nk
n
1 α
1
2
O-Ring Theory
+1
n
α 1
k1
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O-Ring
Numerical example IV
Interest rate country 1:
r = αnB (nk )α
1
Interest rate country 2:
r = αnB
2
1
2
n
1 α
nk
+1
!α
1
Note that this explains why capital will not ‡ow to country 2 even if
country 2 has less capital to start with.
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Numerical example V
Wages country 1:
w (1) = (1
α) B (nk )α
Wages country 2:
w (1) = (1
w (0.5) = (1
w = (1
Jesús Fernández-Villaverde (PENN)
α) B
2
α) B
α) B
1
2
2
1
2
n
1 α
1
2
n
1 α
nk
+1
n
1 α
nk
+1
O-Ring Theory
!α
2
!α
1
2
1
2
n
1 α
nk
+1
1
2
!α
n
1 α
+1
!
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Empirical regularities
1
Wage and di¤erences in productivity among countries are
large)Clark’s (1987) evidence about textile industries.
2
Firms hire workers of di¤erent skill and produce di¤erent quality
products.
3
There is a positive correlation among the wages of workers in di¤erent
occupations within enterprises.
4
Firms only o¤er jobs to some workers rather than paying all workers
their estimated marginal product.
5
Income distribution is skewed to the right.
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Extension I: endogenous n.
1
Rich countries specialize in complicated products.
2
Firms are larger in rich countries.
3
Firm size and wages are positively correlated.
4
Firms only o¤er jobs to some workers rather than paying all workers
their estimated marginal product.
5
Income distribution is skewed to the right.
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Extension II: endogenous q.
We can think about q as the result of investment on human capital or
e¤ort by the worker.
If q is perfectly observable, the model does not change much.
But, if there is an error (mismatch, randomness, etc.), there are
important consequences.
Possibility of multiple equilibria.
Jesús Fernández-Villaverde (PENN)
O-Ring Theory
October 19, 2014
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