2008 Oxford Business &Economics Conference Program
ISBN : 978-0-9742114-7-3
A LOCATION-PRODUCTION MODEL
IN HETEROGENEOUS SPACE
Anthony Paul Andrews
Global Trade Research Institute
Governors State University
Telephone: 708.534.4058.
Telefax: 708 534 8457
I am grateful to my dissertation advisor, Masa Fujita, for invaluable comments and
suggestions. The usual disclaimer applies.
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A LOCATION-PRODUCTION MODEL
IN HETEROGENEOUS SPACE
Abstract:
The development of industrial clusters and locational economies is important in the
organization of spatial production patterns. Just how these patterns emerge and are
developed can be understood by firm location decisions. This paper examines how a
firm chooses its optimal location when spatially heterogeneous labor markets are
either perfectly or imperfectly competitive. It argues that the input combination,
technical efficiency, and transport costs are determining factors in the firm’s location,
and provides comparative static differences between perfectly competitive and
imperfectly competitive labor markets. The framework is a firm location in
heterogeneous space under competitive and constant monopsony labor markets.
JEL Classification: J42, D4
Key Words: Labor Market Structure, Monopsony, Location Theory
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Introduction
Firm location has been an important discussion in urban literature for some time and back in vogue
with respect to the determination of cluster and agglomeration economies (Fujita, et. al. 1999,
Hanson and Xiant, 2004).
However, most analyses on the determination of clusters and
agglomeration economies have been with respect to firms locating in input markets characterized by
pure competition. For example, Isard (1956) made the first significant contribution by establishing
that a firm’s optimal location required that the price ratio for inputs must equal their technical rates of
technical substitution. Sakishita (1967), using a one-dimensional Weber model, extended the theory
by showing that a firm’s optimal location is never an intermediate location.
This study incorporates differentiated inputs of labor and capital and introduces imperfect
competition in the labor market. This allows the examination of how firms are affected by spatial
variations in under differentiated
labor market structures.
It provides insight to labor market
restructuring across space (urban competitive and periphery monopsony). I provide theoretical
between urban competitive and periphery monopsonistic labor markets
which links the firms
location decision in the context of spatial heterogeneous location configurations. The model is
essentially a search equilibrium model for a firm with urban and peripheral labor market
opportunities under perfectly and imperfectly labor market structures,.
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The Location Model of the Firm
Assume that firms locate along a representative ray extending from the urban center (UC) of a
metropolitan area to periphery counties as in Figure 1. Distance from the UC, along this ray, is
denoted by x. In the UC, where x is equal to zero, it is assumed that a highly developed urban
structure characterized by competitive labor markets exists. In addition, it is assumed that, as you
move away from the UC, the degree of competition of the labor market declines.
[Figure 2.1 Here]
Next, assume that the firm’s production function is given as
Y  F ( k , L )  A La K b
(2.1)
where Y is output, L is labor, and K, capital. The constants, a and b are positive parameters with both
a and b being less than one. It is also assumed that the market for the firm’s output is at the UC and
its price, Py , at the UC is given and constant. Furthermore, the firm is assumed to have a fixed target
output, Y , which is treated as a parameter representing the firm’s size. The firm’s production
function, (2.1), then meets the standard conditions of positive marginal products and diminishing
returns for each input.
The firm’s total cost relation at each location x is given by
C  P( L, x ) L  Pk K  txY
(2.2)
where P(L,x) represents the wage rate faced by the firm when its employment is L at distance x; Pk ,
the price of capital which is assumed to be a constant independent of distance from the UC; and t, the
transport cost per unit of output per unit distance, which is positive.
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The wage function faced by the firm is given by
P( L, x )     ( x )   ( x ) L
(2.3)
where  is the reservation wage which is a positive constant;  ( x ) , a parameter which measures the
degree of competitiveness in the labor market at each distance x, and  ( x ) , a parameter which
measures the rate of change in the wage with respect to labor demanded by the firm at a given
distance x. In equation (2.3),  ( x ) represents the slope of the wage gradient in terms of distance x
from the UC when the firm’s demand for labor is close to zero.1 Also,  ( x ) can be specified by a
negative exponential function given by
 ( x )   0 e x
(2.4)
where 0 and  are positive constants. The slope of the wage gradient, , provides information about
the relative degree of competition or market structure between spatially separated local labor markets
(Madden, 1988). The wage gradient over the location space of the firm is show in Figure 2.2. By
definition,
 ( x )    ( x )  0
(2.5a)
 ( x )  2 ( x )  0
(2.5b)
[Figure 2.1: Location Space of the Firm Here]
A unique feature of the mode is its ability to incorporate both perfectly and imperfectly competitive
labor market structures in its specification. This is made possible by the third time in (2.3) which
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reflects the size of the labor pool at each location, x. The following conditions are assumed to hold
regarding (x):
 (x)  0
x.
(2.6)
which implies that the wage elasticity, with respect to labor demand, is nonnegative. Further it is
assumed that (x) is nondecreasing in x so that if (x)>0, the labor market at x is not perfectly
competitive. Figure 2.3 describes the location space of the firm with respect to assumptions (2.4)
through (2.6), where (x) is assumed to be positive for all x>o. It is also assumed that at x = 0, (0) =
0. Hence, the firm’s inverse labor supply function, P(L,0) =  + (0), is perfect elastic which
indicates that the representative firm can purchase as much labor as it wants at the prevailing wage
rate,  + (0).
Suppose again on Figure 2.3, that the firm chooses a location outside the UC at, say, x *. The inverse
labor supply function at x* in now P(L,x*) =  + (x*) + (x*)L, which has slope (x*) with respect to
labor demand L. Thus, firms choosing locations outside the urban center, x>0, face an upward
sloping inverse supply curve of labor with slope (x) > 0.
[Figure 2:3: Spatial Labor Supply Here]
The size of the labor pool available to the firm depends on the distance from the UC. For example, as
the firm moves further from the UC, the inverse labor supply function shifts downward as depicted in
Figure 2.4. A parallel shift implies constant monopsony power over space with respect to the inverse
labor supply curve. At location x1, P (L, x1) gives the firm’s inverse supply function. At location x2
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the firms inverse supply function is given by P(L,x2). Thus, the slope and shape of the inverse supply
function reflects the size of the labor pool available at a particular location. Further, the firm is
assumed to search for the location minimizes its total costs subject to the output constraint.
To determine the optimal input combination at each location x, form the Lagrangian function:
  P( L, x ) L  Pk K  txY  (Y  A La K b )
(2.7)
where  is the Lagrangian multiplier. The first-order conditions for cost minimization are
 
 P ( L, x ) L  P( L, x )   a A La 1K b  0
L

 Pk   b A La K b1  0
K

 Y  A La K b  0
K
( 2.8)
( 2.9)
( 210
. )
where P (L,x) = d P(L,x)/d L . Moving the third term of (2.8) and the second term of (2.9) to the
right hand side (rhs) and (2.8) by 2.9) gives
P ( L, x ) L  P( L, x )
Pk

a K
b L
(2.11)
Y  A La K b
(2.12)
for a cost minimum, the ratio of marginal costs equals the ratio of marginal products.
Given this general specification, the model is capable of studying bot perfectly and imperfectly
competitive labor markets.
[Figure 2.4: Labor Supply Curves at x=0, x1, and x2 Here]
2.3
Case I: Perfectly Competitive Labor Markets
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The labor market at each location x is assumed to consist of a continuum of undifferentiated workers
that can be obtained, by the firm, at the current wage. This implies that the firm’s inverse supply
curve of labor is perfectly elastic; that is,
 ( x )  0  x.
(2.13)
Hence, the inverse supply curve under perfect competition is given by
P( L, x )     ( x )  P( x ) .
(2.14)
Substituting (2.14) into (2.11) yields the stand first-order conditions under perfect
P( x )
Pk

a K
b L
(2.15)
Y  A La K b
(2.16)
competition. These conditions are depicted in Figure 2.5. The demand functions for labor and
capital at each location x can be obtained by solving (2.15) and (2.16) for L and K, to get:
b
b
b
 1 1
 a b
L( x)  Y b A b (a / b) PK  P( x) a b  G1P( x) a b


(2.17)
a
a
a
 1 1
 a b
K ( x)  Y a A a (a / b) PK1  P( x) a b  G2 P( x) a b


(2.18)
[Figure 2:5: Determination of the Optimal Input Combination Here]
By assumption (2.4), the wage rate, P(x) , decreases as x increases and the expansion path shifts
downward and to the right as depicted in Future 2.6. Thus, as x increases in distance from x1 to x2,
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L(x) increases from L(x1) to L(x2) in Panel 3, while K(x) decreases from K(x1) to K(x2) in Panel 1.
That is, as the firm moves farther from the UC, it substitutes labor at lower wages for capital to
produce the given output level (see Panel 2). Analytically, this can be demonstrated (2.4), (2.19) ,
and 2.20, as
L ( x ) 
b
( a  b)
K ( x )  
G1  ( x ) P( x )
a
( a  b)

( a 2b )
( a b )
G2  ( x ) P( x )

 0
b
( a b )
 0
(2.19)
(2.20)
[Figure 2.6: Changes in the Capital-labor Ratio Over Space Here]
2.3.1 Analysis of Spatial Cost
A spatial cost curve is defined as the minimum total cost of producing a fixed quantity of output as a
function of the location of the production site (Zeigler, 1986). Let {L(x), K(x0} be the optimal input
combination at x and from (2.2), (2.17) and (2.18), the total cost function is given as
b
a




 a b 
 a b 
C ( x)  P( x) G1P( x)
  Pk G2 P( x)
  txY




or in implicit form, as
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(2.21a)
2008 Oxford Business &Economics Conference Program
C( x ) 
P( x ) L( x ) 
ISBN : 978-0-9742114-7-3
Pk K ( x )  txY
(2.21b)
Applying the Envelope Theorem to (2.21b) gives
C ( x ) 
P( x ) L( x ) 
tY
.
(2.22)
.
(2.23)
From (2.14) , where P ( x )   ( x ) , and hence
C ( x 0   ( x ) L( x )  tY
With further computation the second derivative is given by
C( x )   ( x ) L( x )   ( x ) L( x ) .
(2.24)
Substituting (2.5) and (2.17) into (2.23) at each x gives
C ( x )  G1
  ( x)
  ( x)
b
a b
 tY
(2.25)
Additionally, substituting (2.5), (2.17), and (2.19) into (2.24) gives
C ( x) 
G1  2  ( x) 
b
 ( x) 
1 
0
b
a

b
(



(
x
))


a

b
   ( x) 
(2.26)
Since (2.26) holds for every x = 0, it is possible to conclude the following:
Lemma 2.1
When the labor market is perfectly competitive, the spatial cost
C(x), is strictly convex.
curve,
Based on Lemma 2.1, Figure 2.7 presents the possible shapes of the spatial cost curve, assuming no
land costs. In Figure 2.7(a), the top panel, the spatial cost curve achieves the minimum at a positive
distance from the UC (i.e., x* > 0) whereas, in Figure 2.7(b), the minimum occurs at the urban center
(i.e., x* = 0). In order to understand when (a) and (b) occur, over from (2.4) and (2.25), observe that
C ( 0)  G1
C (  ) 
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
 
tY
b
a b
 tY
(2.27)
 0
(2.28)
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From (2.27), we have that
a)
b)
C (0)  0 iff
 G  
(   )   1 
 tY 
C (0)  0 iff
 G  
(   )   1 
 tY 
a b
b
a b
b
(2.29)
(2.30)
Hence, if we form the form the function
 G  
Z  1 
 tY 
a b
a
(2.31)
where
  Y
1( a b )
b
1
Ab (a / b) Pk (   )
a b
b
t

a b
b
(2.32)
and we can conclude the following:
Theorem 2.1: When the labor market is perfectly competitive for each Y , there exists a
unique optimum location, x*, where
a)
if    < Z, then x* > 0 ; or
b)
if     Z, then x* = 0.
[Figure 2.7 (a) For x* > 0 Here]
[Figure 2.7(b): For x* = 0 Here]
Theorem 2.1(a) represents the cost curve when the cost function achieves the minimum at a positive
distance from the UC, while Theorem 2.1(b) represents the case where the cost function achieves the
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minimum at x = 0. In case (a), at the optimal location, x* > 0, we have c ( x*)  0. Hence, using
(2.22) it follows that
 P( x*) L ( x*)  tY
(2.33)
.
In other words, at the optimal location x*, the marginal labor cost saving per unit distance equals the
marginal transport cost increase per unit distance. Using (2.14) and (2.17), (2.33) can be rewritten as
G1   ( x*)
   ( x*)
b
a b
 tY
(2.34)
or, since  ( x )   e  x , it follows that
  e
  x*
G  
 1 
 tY 
a b
b
  x*
  e 
a b
b
(2.35)
which is the equation defining the optimal location of the firm when there is perfect competition in
the labor market.
2.3.2 Comparative Static Results
It is not possible to examine how changes in the output level, Y , affect the optimal location, x*. In
the following analysis, for convenience, we represent Y simply by Y. Also, recall from Theorem
2.1.a, that x* > 0 iff    < Z. Furthermore, relation (2.35) holds if x* > 0. Substituting G1, given
in (2.19), into (2.35) gives
Y
1( a  b )
b

1
b
( a b )
b
A (a / b) Pk 
t

a b
b

   e  x*  e  x*
(2.36)
Taking natural logarithms of both sides of (2.36) and differentiating with respect to Y and x* yields
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dx *  1  (a  b)   a  b
 e   x* 




dY   b Y  b
   e   x* 
1
(2.37)
Since (a+b)/b > 1 > (  e x* ) / (    e  x* ), it follows that
dx *
dY


 0 as a  b  1 .


Therefore, the effect of output levels on the optimal location depends on the degree of homogeneity
of the production function as summarized in the following theorem:
Theorem 2.2: Suppose     Z , where Z is given by 2.34, then under perfectly
competitive labor markets
 0, if
d x* 
  0, if
dY
  0 if

a b 1
a b 1
a b 1
That is, if the firm’s production function has the property of decreasing returns to scale (increasing
returns to scale), the optimal location of the firm is farther from (closer to) the UC as the firm’s
output level increases. If the firm’s production function operates under constant returns to scale, the
firm’s optimal location will not move. In the UC, lower transportation costs and scale economies
would offset higher wage costs.
Comparing changes in parameters  , Pk , and t on the optimal location x*, the results are shown in
Theorem 2.3 below.
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Theorem 2.3: Suppose  +  < . Then under perfectly competitive labor markets,
x* increase as either (1) A decreases, (2) Pk increases, or (3) t
decreases.
These results can be explained intuitively as follows using (2.33). Given the fixed level of output, Y ,
as the efficiency parameter A increases, L(x) decreases at each x. Hence, as depicted in Figure
2.8(a), the optimal location, x*, moves closer towards the urban center as A increases. As the capital
price, Pk , increases, L(x) increases at each x due to substitution. Hence, as depicted in Figure 2.8(b),
x* moves farther from the urban center as Pk increases. Finally, as t increases, the marginal transport
cost, t Y , increases. Hence, x* moves closer to the UC as t increases (see Figure 2.8c).
The comparative statics of Y, A, Pk , and P(x) with respect to the optimal input combination (L(x),
K(x)) at each location are provided in the following theorem:
Theorem 2.4: Under perfectly competitive labor markets, we have that
i.
Y increases  L(x) and K(x) increase;
ii.
A increases  L(x) and K(x) decrease;
iii.
Pk increases  L(x) increases and K(x) decreases; and
iv.
P(x) increases  L(x) decreases and K(x) increases.
These results are intuitively obvious.
[Figure 2.8(a) Impact of Changes in A on x* Here]
[Figure 2.8(b): Impact of PK increase on x* Here]
[Figure 2.8: Impact of Changes of PK on t Here]
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2.4
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Case 2: Constant Monopsony Power Over Space
The structure of local labor markets and labor markets in urban centers depend, to a large extent, on
the composition of economic activity and resources present in each. It is usually the case that
workers in monopolistic labor markets receive wages below their urban counterparts because of the
lower degree of concentration and/or specialization of employment associated with peripheral
markets.
More importantly, the negative wage gradient in Figure 2.2 may be due to (1) the
commuting costs of works or (2) monopsony power in the labor market. If the number of firms in
spatially dispersed locations is small, opportunities for monopsonistic power over workers in the
radius of these labor markets may be exercised (Nakagome, 1988) This implies that the elasticity of
labor supply is less than infinity for firms in periphery labor markets.
Accordingly, the wage
elasticity of labor supply reflects the size of the labor pool at a given location in addition to the
sensitivity of the firm’s wage offer for additional employment.
Under perfect competition, it was assumed that the supply curve of labor was infinitely elastic so that
the labor requirement of the firm could be purchased as the going rate. In this section, we assume
that the firm has monopsonistic power over space for workers in the radius of its local labor market.
Under monopsony, the representative firm employs additional labor only by raising its wage. This
means that the marginal cost of labor increases by more than the wage of the marginal worker. The
existence of a wage differential, between firms in the urban center and the periphery, allows firms in
monopolistic labor markets to pay lower wages than firms in the urban center. To analyze firm
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location in imperfect competition we consider the case of monopsony and, as before, we assume a
modification of generalized inverse supply curve of labor presented in (2.3) and modified to be
P( L, x )     ( x )  L
(2.38)
where we assume that
 (x)  

x.
(2.39)
Assumption (2.39) implies that the rate of change in the wage with respect to changes in the quantity
of labor supplied is the same at every distance from the urban center. For example, in Figure 2.9, if
the firm locates at x1, the wage is    (1) and the corresponding inverse supply function is given by
P(L,1). If, on the other hand, the firm locates at x 2, the wage is    ( 2) and the corresponding
inverse labor supply function is given by P(L,2). However, the slopes of the two functions are the
same; that is,  1 is equal to  2 .
Substituting (2.38) and (2.39) into (2.11), the conditions for optimal choice input combinations at
each x combining with (2.12) it is possible to derive the labor input demand function, which is given
by:
L( x )
a b
b
1

1
Y b A b ( a / b) Pk

   ( x )  2 ( x ) L( x )
.
(2.40)
To derive the graphical solution of (2.40) at each location x, let the rhs of (2.40) be represented in the
following manner:
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 ( L, x ) 
1
b

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1
b
Y A ( a / b ) Pk
;
   ( x )  2 ( x ) L( x )
(2.41)
that is, we partition (2.40) into two functions consisting of its rhs and lhs with both depicted in
Figure . Note that ate each given x, the value of L(x) is determined at the intersection between the
(L,x) and the L((a+b)/b curve.
Based on Figures 2.10 and 2.11, we can readily conclude the
following:
[Figure 2.9: Constant Monopsony Power Over Space Here]
[Figure 2.10: Determination of L9x) and K(x) Here]
[Figure 2.11: Determination of L(x)]
Lemma 2.2:
Under assumption 2.44:
i.
At each x (0  x  ),  a unique optimal input
Combination, (L(x), K(x)) where L(x), K(x) >0.;
ii.
L(x) is increasing in x and K(x) is decreasing in x; and
L(  )  lim L( x )  , and
iii.
x 
K (  )  lim K ( x )   .
x 
The total minimum cost function, under (2.39) is given by
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C( x )     ( x )  L( x ) L( x )  Pk K ( x )  txY .
(2.42)
The following Lemma is introduced :
Lemma 2.3:
Under the labor market with constant monopsony
Power, it is assumed that L( x ) / L( x )   .
(2.43)
Applying the Envelope Theorem to (2.42) generates the following results:
C( x )   ( x ) L( x )  tY
C ( x) 
(2.44)
L( x) 


L( x ) 
L( x) 
 ( x) 
(2.45)
From (2.45) and Lemma 2.3, it holds that
C( x )  0 
(2.46)
x.
from which it can be concluded that
Lemma 2.4:
Under assumption (2.39), the cost curve is
Strictly convex.
Also, observe from (2.44) that
lim C( x )  tY   lim  ( x ) L( x )
x
(2.47)
xb
and further, solving (2.40) for the numerator and evaluating L(x) at infinity, it follows that
lim  ( x ) L( x )  L( )lim  ( x )  0.
x 
x
Therefore, from (2.47) it can be concluded that
lim
C ( x )  tY  0.
x
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(2.48)
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From Lemma 2.4, the configuration of the spatial cost curve, C(x), can assume two different forms,
as depicted in Figure 2.12. When C ( x ) is strictly less than zero, the optimal location distance of the
firm is positive and outside the urban center. On the other hand, if C ( x ) is nonnegative, the optimal
location is at the urban center.
Evaluating the spatial marginal cost function at the urban center, we have that
C( 0)    L( 0)  tY
(2.49)
Hence,
C ( 0 )  0
iff
tY    L( 0)
(2.50)
C ( 0 )  0
iff
tY    L( 0)
(2.51)
and that (2.40) can be uniquely determined. Thus, it is possible to make the following conclusions:
Theorem 2.5
Under the labor market with constant monopsony power, there exists a unique optimal
location, x*, where
a.
b.
if tY  L( 0), then x* > 0; and
if tY  L( 0), then x* = 0.
The results are not as general as the results of Theorem 2.1 because the functional form of the labor
demand function is much more complex.
3. Comparative Static Results Under Constant Monopsony Power Over Space
It is possible to examine how the optimal location of the firm changes with respect to changes in each
of the parameters. First, we consider changes in the optimal location with respect to the level of
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output followed by each of the respective parameters. The results, with respect to Y, we have from
tY   ( x*) L( x*, Y ) , that
dx *
2bL( x*)
 o as
 ( a  b)  1
dY
   ( x*)  2L( x*)
[Figure 3.1: Spatial Cost Function Under Constant Monopsony
Power Over Space Here]
This implies that the optimal location decreases with increases in output under increasing or constant
returns to scale. If (a + b) is less than one, decreasing returns to scale, the sign of derivative is
inconclusive.
Other comparative static results are presented as the following theorem:
Theorem 2.7: Under the labor market with constant monopsony power,
x* increases as
i.
ii.
iii.
iv.
v.
vi.
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t decreases;
 is indeterminate;
 decreases for <L<1/;
 decreases;
A decreases; and
Pk increases.
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The results, except iii and vi, are similar to those under perfectly competitive labor markets. Result
iii says that the contribution to the wage at distance is inversely related to the percentage change in
labor's contribution. Result iv means that as the labor supply curve becomes steeper at every distance
x, the firm will move closer to the urban center.
4. Summary
The results of the analysis of the optimal location of the firm under perfect and imperfect labor
markets indicate the existence of a unique optimal location and strictly convex spatial cost curves.
The comparative static results between perfect and imperfect competition differ only with expect to
changes in the optimal location and the level of output. Under perfect competitive labor markets,
changes in the optimal location with respect to changes in output depended on the homogeneity of the
production function. Also, the firms moves further from (closer to) the urban center in the case of
decreasing (increasing) returns to scale. In the case of constant returns to scale, the optimal location
of the firm remains unchanged and that regardless of scale effects, the optimal location was unique.
Under imperfect competition, constant monopsony power in periphery labor markets, results
indicated the same scale effect to hold for increasing and constant returns with respect to the scale of
production function and the firm’s optimal location. However, with respect to decreasing returns to
scale the results are inconclusive. Also, the steeper the labor supply curve, the closer the firm’s
location to the urban center. That is, the more inelastic the labor supply function the closer the
optimal location if the firm to the urban center. Finally, under both cases, the comparative static
results differed only in terms of specifications of each model.
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Notes
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1.
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The wage gradient for (2.3) is defined as
P( L, x )
d ( x ) d ( x )


L
x
dx
dx
  ( x )   ( x )
The first term is the slope of the wage gradient as a function of distance and the second term
is the slope of the inverse labor supply curve in distance space.
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Bibliography
Andrews, Anthony P. (1991) The Impact of Labor Market Structure on the Location of the Firm:
Theory and Empirical Testing, Ph.D. Dissertation, (Philadelphia: University of Pennsylvania).
Boal, William and Michael R. Rnsom (1997) “Monopsony in the Labor market” Journal of Economic
Literature, 35(10:86-112.
Erickson, R.E. and M. Wasylenko (1980), “Firm Location and Site Selection in Suburban
Municipalities,” Journal of Urban Economics,8:69-86.
Fujita, Masahisa (1990) Urban Economic Theory (Cambridge: Cambridge University Press).
Hanson, George H. and Chong Xiang (2004) “The Home-Market Effect and Bilateral Trade Patterns”
American Economic Review, 94(4):1108-1129/
Isard, W. (1956) Location and Space-Economy (Cambridge: Cambridge University Press).
Madden, Janice (1985) “Urban Wage Gradients: An Empirical Analysis,” Journal of Urban
Economics, 16:291:301.
Sakahita, N. (1967) “Production Function, Demand Function and the Optimal Location of the Firm,”
Papers of the Regional Science Association, 20:109-22.
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Figure 2.1:
Location Space of the Firm
0 (= UC*)
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Figure 2.1: Location Space of the Firm
   (0)
   ( x)

x
0
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Figure 2:3: Spatial Labor Supply
P(L,0)
P(L,x)
P(L,x*)
   (0)
L
 (0)
   ( x)
0
x
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Figure 2.4: Labor Supply Curves at x=0, x1, and x2.
P( L, x1 )
P ( L, x )
P( L, x2 )
P ( L, 0)
   ( x2 )
   ( x1 )
L
0
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Figure 2:5: Determination of the Optimal Input Combination
K
K
a P( x)
L
b PK
Y  ALa K b
L
0
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Figure 2.6: Changes in the Capital-labor Ratio Over Space
( x1  x2 )
Panel 1
Panel 2
K
K1
K2
K(x1)
K(x)
K(x2)
x
L
L(x1)
L(x2)
x1
x2
L(x)
Panel 3
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Figure 2.7 (a) For x* > 0
C(x)
x*
Figure 2.7(b): For x* = 0
C(x)
x
x=0
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Figure 2.8(a) Impact of Changes in A on x*
 P( x) L( x)
tY  Marginal Transport
Cost
x*
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Figure 2.8(b): Impact of PK increase on x*
 P( x) L( x)
tY 
x* = Pk Increase on x*
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Figure 2.8: Impact of Changes of PK on t
tY
 P( x) L( x)
x* = t - Increase on x*
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Figure 2.9: Constant Monopsony Power Over Space
P(L,0)
P(L,1)
   (0)
P(L,2)
L
   (1)

1
2
   (2)
0
x1
K
x2
1   2
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Figure 2.10: Determination of L9x) and K(x)
K
K
a    ( x)   ( x)
b
PK
Y
L
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Figure 2.11: Determination of L(x)
1
1
 

Y b A b (a / b) PK
L

1
1
Y b A b ( a / b) PK
   ( x)
1
f ( L,  )
1
Y b A b ( a / b) PK
 
f (l , x), 0  x  
f ( L, 0)
L(0)
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L(x)
37
L(  )
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Figure 3.1: Spatial Cost Function Under Constant Monopsony Power Over Space
Panel A
C(x)
x*
tY    (0)
Panel B
C(x)
x
tY   (0)
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