Shieh, Yeung-Nan

Yeh, C.N.

English

San Jose State University

San Jose State University
SJSU ScholarWorks
Faculty Publications Economics
1-1-2004
Transportation Rates, Monopsony Power and the
Location Decision of the Firm
Yeung-Nan Shieh
San Jose State University, yeung-nan.shieh@sjsu.edu
C.N. Yeh
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Part of the Economics Commons
This Article is brought to you for free and open access by the Economics at SJSU ScholarWorks. It has been accepted for inclusion in Faculty
Publications by an authorized administrator of SJSU ScholarWorks. For more information, please contact scholarworks@sjsu.edu.
Recommended Citation
Yeung-Nan Shieh and C.N. Yeh. "Transportation Rates, Monopsony Power and the Location Decision of the Firm" Economics Bulletin
(2004): 1-7.
Transportation rates, monopsony power and the location 
decision of the firm 
Yeung−Nan Shieh Chiou−Nan Yeh 
San Jose State University Alabama State University 
Abstract 
This paper investigates the impact of monopsony power on the location decision of the firm. 
It shows that if the transportation rates are a function of distance only and the production 
function is homogeneous of degree one, the optimum location is independent of monopsony 
power. However, if the transportation rates are a function of quantity shipped and distance 
traveled, a homogeneous production of degree one does not ensure independence between the 
optimum location and monopsony power. This result is significantly different from the 
well−known Mai−Suwanakul−Yeh proposition in the constant transportation case. 
We thank Professors D. Cheng, D. Olson, C. C. Mai and the associate editor for very useful comments and suggestions. All 
remaining errors are ours. 
Citation: Shieh, Yeung−Nan and Chiou−Nan Yeh, (2004) "Transportation rates, monopsony power and the location decision of 
the firm." Economics Bulletin, Vol. 18, No. 1 pp. 1−7 
Submitted: September 24, 2003. Accepted: January 13, 2004. 
URL: http://www.economicsbulletin.com/2004/volume18/EB−03R30003A.pdf 
 
       
 
  
         
   
   
    
 
    
  
        
   
 
       
   
 
  
  
  
 
    
   
 
     
    
       
     
     
 
 
 
         
  
 
 
    
        
     
  
1. Introduction 
In recent years a large number of studies have attempted to integrate location theory
with neoclassical production theory.  Most of these studies, following the pioneering 
work of Moses (1958), assume that the input markets are perfectly competitive and show
that the optimum location of the firm is independent of the demand function if the
production function is homogenous of degree one (see Martinich and Hurter 1990).  Little 
attention was devoted to the monopsony case.  Recently Mai, Suwanskul and Yeh (1993)
(henceforth MSY) incorporated the monopsony market structure into the one-output,
two-input Weber-Moses triangle and attempted to fill this gap.  Assume that (i) the
production function is homogenous of degree N; (ii) transportation rates are constant; (iii)
the firm is a price taker in the output market and one input market, but is a monopsonist
in another input market. They examined the impact of monopsony power on the location
decision of the firm and obtained the following interesting and important proposition.
MSY.  The optimum location of the firm is independent of monopsony power if the 
production function is homogeneous of degree one. (MSY 1993, propositions 1 and 2).
This result crucially depends upon the constant transportation rates assumption.  However,
it is well known that in transportation economics, discounts for quantity shipped and 
distance traveled are prevalent among various modes of transportation (see Fair and
Williams 1975, Miller and Jensen 1978, Shieh and Mai 1984, Stahl 1987). It would be
interesting and important to examine the impact of monopsony power on the location 
decision of the firm if transportation rates are a function of quantity shipped and distance
traveled.
The purpose of this paper is to incorporate quantity and distance discounts as key
variables into the transportation rate functions and to examine the impact of monopsony
power on the location decision.  It will be shown that the location decision of the firm is
not independent of the monopsony power if the production function is homogeneous of
degree one.  This result indicates that MSY's proposition in general fails to apply.
2. The monopsonistic location model
In order to investigate the connections between transportation rates, monopsony power
and location decisions, we use a model defined by a number of basic assumptions that
will characterize much of our analysis.
(A1) A firm employs two transportable inputs (L and K) located at A and B to produce a
single output (Q) which is sold in the output market C. The location triangle in
Figure 1 illustrates the location problem of the firm.  In Figure 1, the distances a
and b and the angle π/2 > α > 0 are given.
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 C
 
α
θ
 h
 a  b
 E 
s z
 A  B
  Figure 1. Location Triangle
 (A2) The homogeneous production function of degree N is specified as: 
 Q = f(L, K), fL > 0, fK > 0, fLK = fKL > 0, fLL < 0, fKK < 0  (1) 
 and f L + f K = NQ,  f L + f K = (N-1)f , f L + f K = (N-1)f , f L2 + 2f LK + f K2 = 
L K LL LK L KL KK K LL LK KK
N(N-1)Q. Note that subscripts are used for partial derivatives throughout the paper.
(A3) The transportation rate functions for inputs and output are specified as:
  n = n(s, L),  m = m(z, K),  t = t(h, Q)  (2) 
where n, m and t are transportation rates of L, K, and Q; s, z and h are distances
from the plant to the sources A, B and the output market C.  n s < 0, mz < 0, th < 0,
nL < 0, mK < 0, and  tQ < 0.  By the law of cosines, we can express s and z as:
2 2 1/2 2 2 1/2 
  s = (a  + h  - 2ahcosθ) ,  z = [b  + h  - 2bhcos(α-θ)] (3)
(A4) The prices of inputs and output are evaluated at the firm's location E.  The costs of
purchasing inputs (L and K) at the plant are the prices of inputs at sources (w and r)
plus the full freight costs (ns and mz), i.e., w + ns and r + mz. The price of output at
the plant is the market price (p) minus the freight cost (th), i.e., p - th.
(A5) The firm is a price taker in the output market and the input K market, but has
monopsony power in the input L market (see Yeh, Mai and Shieh 1996).  It faces
an upward sloping input supply curve w(L) with wL > 0.  According to Lerner
(1934), Bronfenbrenner (1971), the degree of monopsony power can be measured
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by the elasticity of input supply, e ≡ (dL/dw)(w/L) = (1/wL)(w/L) with 0 < e < ∞. If
e →∞, the input L market is perfectly competitive and monopsony does not exit.  It
may be noted that the lower is the value of e, the higher is the monopsony power.
(A6) The objective of the firm is to choose the optimum location within the triangle that
maximizes the profit.
It should be noted that the inclusion of (A3) constitutes a major departure from MSY.
That is, instead of assuming constant transportation rates, we now permit these rates to
vary with quantity shipped and distance traveled.
With this set of assumptions, the profit-maximizing location problem can be
formulated as:
 Max π = (p-th)f(L,K) - [w(L) + ns]L - (r + mz)K   (4) 
where L, K, θ and h are choice variables.  The first-order conditions are  
(∂π/∂L) = (p-thu)fL - {w[1+(1/e)] + nsuL} = 0   (5) 

(∂π/∂K) = (p-thu)fK - (r + mzuK) = 0   (6)

 (∂π/∂θ) = -nsθvLL - mzθvKK = 0   (7) 

(∂π/∂h) = -tvQ - nshvLL - mzhvKK = 0   (8) 

where u ≡ 1 + c, uL≡ 1 + cL, uK ≡ 1 + cK, v ≡ 1 + d, vL ≡ 1 + dL, vK ≡ 1 + dK, c ≡ tQ(Q/t) <
0, cL≡  nL(L/n) < 0 and cK≡  mK(K/m) < 0 are the elasticity of transportation rates with
respect to output and inputs; d ≡ th(h/t) < 0, dL ≡ ns(s/n) < 0 and dK ≡ mz (z/m) < 0 are the
elasticity of transportation rates with respect distances h, s and z.  Following Miller and
Jensen (1978), we assume that the elasticity of transportation rates, c, cL, cK, d, dL, and dK 
are constant, and u, uL, uK, v, vL, are vK are positive throughout the paper.  Assume also
that e is a parameter.
We can solve (5) - (8) for L, K, θ, and h in terms of e, r, p, a, b and α if the second-
order conditions are satisfied.  This completes the monopsonistic location model that
comprises the basic analytical framework.
3. The impact of monopsony power on location 
We are now in a position to examine the effects of a change in monopsony power on
the production-location decision.  Applying the standard comparative static procedures to
(5) - (8), we obtain
∂θ/∂e = (E/KD)(F + G)   (9) 

∂h/∂e = (E/KD)(H + I) (10)
 
where 
E ≡ (w/e2) > 0   (11)
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  F = (p-thu)(N-1)fK(πθLπhh -πθhπhL) 
+ tQvuK[N(u/uK)-1](πθLπKh-πKLπθh)  (12)
  G = nvLLuK[(uL/uK)-1][sh(πθLπKh-πKLπθh)+sθ(πKLπhh-πKhπhL)] 
+ [thfKNu(u-1)+mzKuK(uK-1)](πθhπhL-πθLπhh)   (13)

  H = (p-thu)(N-1)fK(πθθπhL-πθLπhθ) 

+ tQvuK[N(u/uK)-1](πKLπθθ-πKθπθL)  (14)

  I = nvLLuK[(uL/uK)-1][sθ(πKθπhL-πKLπhθ)+sh(πKLπθθ-πKθπθL)] 

+ [thfKNu(u-1)+mzKuK(uK-1)](πθLπhθ-πθθπhL) (15)
and D > 0 by the second-order conditions.  The expressions of πij and eij (i,j = L, K, θ, h)
would be: πLL = (p-thu)fLL - wL[1+(1/e)] - eLL, πLK = πKL = (p-thu)fLK - eLK, πLθ = πθL = -
ns  = πhL = - (tfLvu + nshvLuL), πKK = (p-thu)fKK - eKK, πKθ = πθK = - mz vKuK,θvLuL, πLh θ
πKh = πhK = - tfKvu - mzhvKuK, πθθ = - (nsθθvLL + mzθθvKK) - eθθ, πθh = πhθ = - (nsθhvLL +
2
mzθhvKK) - e  = - (nshhvLL + mzhhvKK) - ehh,  eLL ≡ (t/Q)hfL cu + (n/L)scLuL, eLK ≡θh, πhh
2 2 2
(t/Q)hfLfKcu, eKK≡ (t/Q)hfK cu + (m/K)zcKuK, e ≡ (n/s)s dLvLL + (m/z)z dKvKK, e ≡θθ θ θ θh 
2 2
(n/s)sθshdLvLL + (m/z)zθzhdKvKK, ehh ≡ (t/h)dvQ + (n/s)sh dLvLL + (m/z)sh dKvKK. Note
that (i) if the transportation rates are independent of quantity shipped, c = cL = cK = 0, u = 
uL = uK = 1, eLL = eLK = eKK = 0; (ii) if the transportation rates are independent of distance
traveled, d = dL = dK = 0, v = vL = vK = 1 and eθθ = eθh = ehh = 0.  It is of interest to note
that, to derive (12)-(15), we use the first-order conditions, - mzθvKK = nsθvLL, - mzhvKK 
= tvQ + nshvLL, and the property of homogeneous function of degree N, fKK + fLL = NQ.
It is clear that in (9) and (10) the signs of F, G, H and I can not be determined a priori. 
Thus, we can conclude that the impact of monopsony power on the optimum location is
ambiguous.
Next, we turn to the case in which the production function is homogeneous of degree
one, i.e., N = 1.  Assuming that the elasticity of transportation rates with respect to
quantity shipped and distance traveled are constant, we examine three specific situations:
(1) transportation rates are constant; (2) transportation rates are a function of distance
traveled; (3) transportation rates are a function of quantity shipped and distance traveled.
3.1. Transportation rates are constant
In this case, v = vL = vK = u = uL = uK = 1, and then G = I = 0.  Thus, (9) and (10) can 
be rewritten as:
∂θ/∂e = (E/KD)(N-1)[(p-th)fK(πθLπhh-πθhπhL) + tQ(πθLπKh-πKLπθh)]   (9a)
∂h/∂e = (E/KD)(N-1)[(p-th)fK(πθθπhL-πθLπhθ) + tQ(πKLπθθ-πθLπKθ)]  (10a) 
It is clear that ∂θ/∂e = 0 and ∂h/∂e = 0 if N = 1.  In other words, if the production function
is homogeneous of degree one, the optimum location of the monopsonistic firm is
independent of monopsony power.  This is MSY's case.
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3.2. Transportation rates are a function of distance 
  In this case, 1 > v > 0, 1 > vL > 0, 1 > vK > 0, u = uK = uL = 1,  and then G = I = 0.  The 
expressions in (9) and (10) become
∂θ/∂e = (E/KD)(N-1)(p-th)fK(πθLπhh-πθhπhL) + tQv(πθLπKh-πθhπKL) (9b)
∂h/∂e = (E/KD)(N-1)(p-th)fK(πθθπhL-πθLπhθ) + tQv(πKLπθθ-πθLπKθ)  (10b) 
It is easy to see that ∂θ/∂e = 0 and ∂h/∂e = 0 if N = 1.  In other words, if the production
function is homogeneous of degree one and the transportation rates are a function of
distance only, the optimum location of the monopsonistic firm is independent of
monopsony power.
This result generalizes MSY's proposition since we have not assumed constant
transportation rates. It is well known that in the Weberian location theory (see Weber 
1929, p. 78), the optimum location is found by considering the relative strength of three
forces: the market pull and two material pulls.  Each force is comprised of the quantity
and marginal transportation cost components. If N = 1 and the transportation rates are a
function of distance only, a change in monopsony power will not change the mix of
output and inputs, and the relative transportation costs.  Therefore, the relative pulls of
the output and the inputs are unaffected, and the optimum location is independent of
monopsony power.
3.3. Transportation rates are a function of quantity and distance 
  In this case, 1 > u > 0, 1 > uK > 0, 1 > uL > 0, 1 > v > 0, 1 > vK > 0 and 1 > vL > 0.  If N
= 1, the expressions in (9) and (10) become
∂θ/∂e = (E/DK){tQvuK[N(u/uK)-1](πθLπKh-πKLπθh)+G} (9c)
∂h/∂e = (E/DK){tQvuK[N(u/uK)-1](πKLπθθ-πKθπθL)+I}   (10c) 
Since the signs of ∂θ/∂e and ∂h/∂e are not equal to zero, we can conclude that if the
transportation rates are a function of quantity shipped and distance traveled, a
homogeneous production function of degree one is not sufficient to ensure that the 
optimum location is independent of monopsony power.
This result indicates that MSY's proposition does not hold.  If N = 1 and transportation
rates are a function of quantity shipped, a change in monopsony power will not change
the mix of output and input, but will change the relative transportation costs.  Therefore,
the relative pulls of the market and the input are affected, and the optimum location is not
independent of monopsony power.
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 4. Conclusions
 
In this paper, we have incorporated quantity and distance discounts as key variables
into the transportation rate functions and examined the theoretical implications of these
variables to the effect of monopsony power on the location decision of a monopsonistic
firm.  MSY's study focuses on the constant transportation rates case.  Our work has
generalized the study of MSY in the sense that their results are valid only in some special
cases.
Assuming that the production function is homogeneous of degree one, we show that
the optimum location is independent of monopsony power if the transportation rates are a
function of distance only.  In this case, MSY's proposition holds.  However, if the 
transportation rates are a function of quantity shipped, then a homogeneous production
function of degree one does not ensure that the optimum location is independent of
monopsony power.  This result shows that in general MSY‘s proposition fails to apply.
The upshot of our analysis is that the present of quantity discount in the transportation
rate functions has an extremely important influence on the monopsonistic firm's location 
decision.
References
 Bronfenbrenner, M. (1971) Income Distribution Theory, Aldine Atherton: Chicago.
 Fair, M. L. and Williams, E.W. (1975) Economics of Transportation and Logistics, 
 Business Publication, Chicago.
 Lerner, A. P. (1934) —The Concept of Monopoly and the Measurement of Monopoly
 Power“ Review of Economic Studies 1, 157-175.
 Mai, C. C. Suwanakul, S. and Yeh, C. N. (1993) —Monopsony and the Production-
  Location Decision of The Firm“ Regional Science and Urban Economics 23, 779-790.
  Martinich, J. S. and Hurter, A. P. (1990) —Generalized Comparative Statics for the  
  Production-Location Problem“ Regional Science and Urban Economics 20, 275-294.
  Miller, S. M. and Jensen, D. M. (1978) —Location and the Theory of Production: A
  Review, Summary and Critique of Recent Contributions“ Regional Science and Urban 
  Economics 8, 117-128.
  Moses, L. N. (1958) —Location and Theory of Production“ Quarterly Journal of
  Economics 72, 259-272.
  Shieh, Y. N. and Mai, C. C. (1984) —Location and the Theory of Production: 
  Clarifications and Extensions“ Regional Science and Urban Economics 14, 199-218.
  Stahl, C. (1987) —Theory of Urban Business Location“ in Handbook of Regional and 
  Urban Economics 2 by E.S. Mills, Ed., North-Holland: Amsterdam,750-820.
  Yeh, C. N. Mai, C. C. and Shieh, Y. N. (1996) —Location and the Theory of Production
 under Monopsony“ Papers in Regional Science 75, 433-440.
  Weber, A. (1929) Theory of the Location of Industries, C. Friedrich, trans., University of
  Chicago Press: Chicago.
6



Transportation Rates, Monopsony Power and the Location Decision of the Firm

SJSU ScholarWorks

San Jose State UniversitySJSU ScholarWorksFaculty Publications Economics1-1-2004Transportation Rates, Monopsony Power and theLocation Decision of the FirmYeung-Nan ShiehSan Jose State University, yeung-nan.shieh@sjsu.eduC.N. YehFollow this and additional works at: https://scholarworks.sjsu.edu/econ_pubPart of the Economics CommonsThis Article is brought to you for free and open access by the Economics at SJSU ScholarWorks. It has been accepted for inclusion in FacultyPublications by an authorized administrator of SJSU ScholarWorks. For more information, please contact scholarworks@sjsu.edu.Recommended CitationYeung-Nan Shieh and C.N. Yeh. "Transportation Rates, Monopsony Power and the Location Decision of the Firm" Economics Bulletin(2004): 1-7.Transportation rates, monopsony power and the location decision of the firm Yeung−Nan Shieh Chiou−Nan Yeh San Jose State University Alabama State University Abstract This paper investigates the impact of monopsony power on the location decision of the firm. It shows that if the transportation rates are a function of distance only and the production function is homogeneous of degree one, the optimum location is independent of monopsony power. However, if the transportation rates are a function of quantity shipped and distance traveled, a homogeneous production of degree one does not ensure independence between the optimum location and monopsony power. This result is significantly different from the well−known Mai−Suwanakul−Yeh proposition in the constant transportation case. We thank Professors D. Cheng, D. Olson, C. C. Mai and the associate editor for very useful comments and suggestions. All remaining errors are ours. Citation: Shieh, Yeung−Nan and Chiou−Nan Yeh, (2004) "Transportation rates, monopsony power and the location decision of the firm." Economics Bulletin, Vol. 18, No. 1 pp. 1−7 Submitted: September 24, 2003. Accepted: January 13, 2004. URL: http://www.economicsbulletin.com/2004/volume18/EB−03R30003A.pdf                                                                                                                                         1. Introduction In recent years a large number of studies have attempted to integrate location theorywith neoclassical production theory.  Most of these studies, following the pioneering work of Moses (1958), assume that the input markets are perfectly competitive and showthat the optimum location of the firm is independent of the demand function if theproduction function is homogenous of degree one (see Martinich and Hurter 1990).  Little attention was devoted to the monopsony case.  Recently Mai, Suwanskul and Yeh (1993)(henceforth MSY) incorporated the monopsony market structure into the one-output,two-input Weber-Moses triangle and attempted to fill this gap.  Assume that (i) theproduction function is homogenous of degree N; (ii) transportation rates are constant; (iii)the firm is a price taker in the output market and one input market, but is a monopsonistin another input market. They examined the impact of monopsony power on the locationdecision of the firm and obtained the following interesting and important proposition.MSY.  The optimum location of the firm is independent of monopsony power if the production function is homogeneous of degree one. (MSY 1993, propositions 1 and 2).This result crucially depends upon the constant transportation rates assumption.  However,it is well known that in transportation economics, discounts for quantity shipped and distance traveled are prevalent among various modes of transportation (see Fair andWilliams 1975, Miller and Jensen 1978, Shieh and Mai 1984, Stahl 1987). It would beinteresting and important to examine the impact of monopsony power on the location decision of the firm if transportation rates are a function of quantity shipped and distancetraveled.The purpose of this paper is to incorporate quantity and distance discounts as keyvariables into the transportation rate functions and to examine the impact of monopsonypower on the location decision.  It will be shown that the location decision of the firm isnot independent of the monopsony power if the production function is homogeneous ofdegree one.  This result indicates that MSY's proposition in general fails to apply.2. The monopsonistic location modelIn order to investigate the connections between transportation rates, monopsony powerand location decisions, we use a model defined by a number of basic assumptions thatwill characterize much of our analysis.(A1) A firm employs two transportable inputs (L and K) located at A and B to produce asingle output (Q) which is sold in the output market C. The location triangle inFigure 1 illustrates the location problem of the firm.  In Figure 1, the distances aand b and the angle π/2 > α > 0 are given.1                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            C αθ h a  b E s z A  B  Figure 1. Location Triangle (A2) The homogeneous production function of degree N is specified as:  Q = f(L, K), fL > 0, fK > 0, fLK = fKL > 0, fLL < 0, fKK < 0  (1)  and f L + f K = NQ,  f L + f K = (N-1)f , f L + f K = (N-1)f , f L2 + 2f LK + f K2 = L K LL LK L KL KK K LL LK KKN(N-1)Q. Note that subscripts are used for partial derivatives throughout the paper.(A3) The transportation rate functions for inputs and output are specified as:  n = n(s, L),  m = m(z, K),  t = t(h, Q)  (2) where n, m and t are transportation rates of L, K, and Q; s, z and h are distancesfrom the plant to the sources A, B and the output market C.  n s < 0, mz < 0, th < 0,nL < 0, mK < 0, and  tQ < 0.  By the law of cosines, we can express s and z as:2 2 1/2 2 2 1/2   s = (a  + h  - 2ahcosθ) ,  z = [b  + h  - 2bhcos(α-θ)] (3)(A4) The prices of inputs and output are evaluated at the firm's location E.  The costs ofpurchasing inputs (L and K) at the plant are the prices of inputs at sources (w and r)plus the full freight costs (ns and mz), i.e., w + ns and r + mz. The price of output atthe plant is the market price (p) minus the freight cost (th), i.e., p - th.(A5) The firm is a price taker in the output market and the input K market, but hasmonopsony power in the input L market (see Yeh, Mai and Shieh 1996).  It facesan upward sloping input supply curve w(L) with wL > 0.  According to Lerner(1934), Bronfenbrenner (1971), the degree of monopsony power can be measured2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       by the elasticity of input supply, e ≡ (dL/dw)(w/L) = (1/wL)(w/L) with 0 < e < ∞. Ife →∞, the input L market is perfectly competitive and monopsony does not exit.  Itmay be noted that the lower is the value of e, the higher is the monopsony power.(A6) The objective of the firm is to choose the optimum location within the triangle thatmaximizes the profit.It should be noted that the inclusion of (A3) constitutes a major departure from MSY.That is, instead of assuming constant transportation rates, we now permit these rates tovary with quantity shipped and distance traveled.With this set of assumptions, the profit-maximizing location problem can beformulated as: Max π = (p-th)f(L,K) - [w(L) + ns]L - (r + mz)K   (4) where L, K, θ and h are choice variables.  The first-order conditions are  (∂π/∂L) = (p-thu)fL - {w[1+(1/e)] + nsuL} = 0   (5) (∂π/∂K) = (p-thu)fK - (r + mzuK) = 0   (6) (∂π/∂θ) = -nsθvLL - mzθvKK = 0   (7) (∂π/∂h) = -tvQ - nshvLL - mzhvKK = 0   (8) where u ≡ 1 + c, uL≡ 1 + cL, uK ≡ 1 + cK, v ≡ 1 + d, vL ≡ 1 + dL, vK ≡ 1 + dK, c ≡ tQ(Q/t) <0, cL≡  nL(L/n) < 0 and cK≡  mK(K/m) < 0 are the elasticity of transportation rates withrespect to output and inputs; d ≡ th(h/t) < 0, dL ≡ ns(s/n) < 0 and dK ≡ mz (z/m) < 0 are theelasticity of transportation rates with respect distances h, s and z.  Following Miller andJensen (1978), we assume that the elasticity of transportation rates, c, cL, cK, d, dL, and dK are constant, and u, uL, uK, v, vL, are vK are positive throughout the paper.  Assume alsothat e is a parameter.We can solve (5) - (8) for L, K, θ, and h in terms of e, r, p, a, b and α if the second-order conditions are satisfied.  This completes the monopsonistic location model thatcomprises the basic analytical framework.3. The impact of monopsony power on location We are now in a position to examine the effects of a change in monopsony power onthe production-location decision.  Applying the standard comparative static procedures to(5) - (8), we obtain∂θ/∂e = (E/KD)(F + G)   (9) ∂h/∂e = (E/KD)(H + I) (10) where E ≡ (w/e2) > 0   (11)3                                                                                                                                                                                                                                                                                                                                     F = (p-thu)(N-1)fK(πθLπhh -πθhπhL) + tQvuK[N(u/uK)-1](πθLπKh-πKLπθh)  (12)  G = nvLLuK[(uL/uK)-1][sh(πθLπKh-πKLπθh)+sθ(πKLπhh-πKhπhL)] + [thfKNu(u-1)+mzKuK(uK-1)](πθhπhL-πθLπhh)   (13)  H = (p-thu)(N-1)fK(πθθπhL-πθLπhθ) + tQvuK[N(u/uK)-1](πKLπθθ-πKθπθL)  (14)  I = nvLLuK[(uL/uK)-1][sθ(πKθπhL-πKLπhθ)+sh(πKLπθθ-πKθπθL)] + [thfKNu(u-1)+mzKuK(uK-1)](πθLπhθ-πθθπhL) (15)and D > 0 by the second-order conditions.  The expressions of πij and eij (i,j = L, K, θ, h)would be: πLL = (p-thu)fLL - wL[1+(1/e)] - eLL, πLK = πKL = (p-thu)fLK - eLK, πLθ = πθL = -ns  = πhL = - (tfLvu + nshvLuL), πKK = (p-thu)fKK - eKK, πKθ = πθK = - mz vKuK,θvLuL, πLh θπKh = πhK = - tfKvu - mzhvKuK, πθθ = - (nsθθvLL + mzθθvKK) - eθθ, πθh = πhθ = - (nsθhvLL +2mzθhvKK) - e  = - (nshhvLL + mzhhvKK) - ehh,  eLL ≡ (t/Q)hfL cu + (n/L)scLuL, eLK ≡θh, πhh2 2 2(t/Q)hfLfKcu, eKK≡ (t/Q)hfK cu + (m/K)zcKuK, e ≡ (n/s)s dLvLL + (m/z)z dKvKK, e ≡θθ θ θ θh 2 2(n/s)sθshdLvLL + (m/z)zθzhdKvKK, ehh ≡ (t/h)dvQ + (n/s)sh dLvLL + (m/z)sh dKvKK. Notethat (i) if the transportation rates are independent of quantity shipped, c = cL = cK = 0, u = uL = uK = 1, eLL = eLK = eKK = 0; (ii) if the transportation rates are independent of distancetraveled, d = dL = dK = 0, v = vL = vK = 1 and eθθ = eθh = ehh = 0.  It is of interest to notethat, to derive (12)-(15), we use the first-order conditions, - mzθvKK = nsθvLL, - mzhvKK = tvQ + nshvLL, and the property of homogeneous function of degree N, fKK + fLL = NQ.It is clear that in (9) and (10) the signs of F, G, H and I can not be determined a priori. Thus, we can conclude that the impact of monopsony power on the optimum location isambiguous.Next, we turn to the case in which the production function is homogeneous of degreeone, i.e., N = 1.  Assuming that the elasticity of transportation rates with respect toquantity shipped and distance traveled are constant, we examine three specific situations:(1) transportation rates are constant; (2) transportation rates are a function of distancetraveled; (3) transportation rates are a function of quantity shipped and distance traveled.3.1. Transportation rates are constantIn this case, v = vL = vK = u = uL = uK = 1, and then G = I = 0.  Thus, (9) and (10) can be rewritten as:∂θ/∂e = (E/KD)(N-1)[(p-th)fK(πθLπhh-πθhπhL) + tQ(πθLπKh-πKLπθh)]   (9a)∂h/∂e = (E/KD)(N-1)[(p-th)fK(πθθπhL-πθLπhθ) + tQ(πKLπθθ-πθLπKθ)]  (10a) It is clear that ∂θ/∂e = 0 and ∂h/∂e = 0 if N = 1.  In other words, if the production functionis homogeneous of degree one, the optimum location of the monopsonistic firm isindependent of monopsony power.  This is MSY's case.4                                                                                                                                                                                                           3.2. Transportation rates are a function of distance   In this case, 1 > v > 0, 1 > vL > 0, 1 > vK > 0, u = uK = uL = 1,  and then G = I = 0.  The expressions in (9) and (10) become∂θ/∂e = (E/KD)(N-1)(p-th)fK(πθLπhh-πθhπhL) + tQv(πθLπKh-πθhπKL) (9b)∂h/∂e = (E/KD)(N-1)(p-th)fK(πθθπhL-πθLπhθ) + tQv(πKLπθθ-πθLπKθ)  (10b) It is easy to see that ∂θ/∂e = 0 and ∂h/∂e = 0 if N = 1.  In other words, if the productionfunction is homogeneous of degree one and the transportation rates are a function ofdistance only, the optimum location of the monopsonistic firm is independent ofmonopsony power.This result generalizes MSY's proposition since we have not assumed constanttransportation rates. It is well known that in the Weberian location theory (see Weber 1929, p. 78), the optimum location is found by considering the relative strength of threeforces: the market pull and two material pulls.  Each force is comprised of the quantityand marginal transportation cost components. If N = 1 and the transportation rates are afunction of distance only, a change in monopsony power will not change the mix ofoutput and inputs, and the relative transportation costs.  Therefore, the relative pulls ofthe output and the inputs are unaffected, and the optimum location is independent ofmonopsony power.3.3. Transportation rates are a function of quantity and distance   In this case, 1 > u > 0, 1 > uK > 0, 1 > uL > 0, 1 > v > 0, 1 > vK > 0 and 1 > vL > 0.  If N= 1, the expressions in (9) and (10) become∂θ/∂e = (E/DK){tQvuK[N(u/uK)-1](πθLπKh-πKLπθh)+G} (9c)∂h/∂e = (E/DK){tQvuK[N(u/uK)-1](πKLπθθ-πKθπθL)+I}   (10c) Since the signs of ∂θ/∂e and ∂h/∂e are not equal to zero, we can conclude that if thetransportation rates are a function of quantity shipped and distance traveled, ahomogeneous production function of degree one is not sufficient to ensure that the optimum location is independent of monopsony power.This result indicates that MSY's proposition does not hold.  If N = 1 and transportationrates are a function of quantity shipped, a change in monopsony power will not changethe mix of output and input, but will change the relative transportation costs.  Therefore,the relative pulls of the market and the input are affected, and the optimum location is notindependent of monopsony power.5                                                                                                                                                                                                                                                                                          4. Conclusions In this paper, we have incorporated quantity and distance discounts as key variablesinto the transportation rate functions and examined the theoretical implications of thesevariables to the effect of monopsony power on the location decision of a monopsonisticfirm.  MSY's study focuses on the constant transportation rates case.  Our work hasgeneralized the study of MSY in the sense that their results are valid only in some specialcases.Assuming that the production function is homogeneous of degree one, we show thatthe optimum location is independent of monopsony power if the transportation rates are afunction of distance only.  In this case, MSY's proposition holds.  However, if the transportation rates are a function of quantity shipped, then a homogeneous productionfunction of degree one does not ensure that the optimum location is independent ofmonopsony power.  This result shows that in general MSY‘s proposition fails to apply.The upshot of our analysis is that the present of quantity discount in the transportationrate functions has an extremely important influence on the monopsonistic firm's location decision.References Bronfenbrenner, M. (1971) Income Distribution Theory, Aldine Atherton: Chicago. Fair, M. L. and Williams, E.W. (1975) Economics of Transportation and Logistics,  Business Publication, Chicago. Lerner, A. P. (1934) —The Concept of Monopoly and the Measurement of Monopoly Power“ Review of Economic Studies 1, 157-175. Mai, C. C. Suwanakul, S. and Yeh, C. N. (1993) —Monopsony and the Production-  Location Decision of The Firm“ Regional Science and Urban Economics 23, 779-790.  Martinich, J. S. and Hurter, A. P. (1990) —Generalized Comparative Statics for the    Production-Location Problem“ Regional Science and Urban Economics 20, 275-294.  Miller, S. M. and Jensen, D. M. (1978) —Location and the Theory of Production: A  Review, Summary and Critique of Recent Contributions“ Regional Science and Urban   Economics 8, 117-128.  Moses, L. N. (1958) —Location and Theory of Production“ Quarterly Journal of  Economics 72, 259-272.  Shieh, Y. N. and Mai, C. C. (1984) —Location and the Theory of Production:   Clarifications and Extensions“ Regional Science and Urban Economics 14, 199-218.  Stahl, C. (1987) —Theory of Urban Business Location“ in Handbook of Regional and   Urban Economics 2 by E.S. Mills, Ed., North-Holland: Amsterdam,750-820.  Yeh, C. N. Mai, C. C. and Shieh, Y. N. (1996) —Location and the Theory of Production under Monopsony“ Papers in Regional Science 75, 433-440.  Weber, A. (1929) Theory of the Location of Industries, C. Friedrich, trans., University of  Chicago Press: Chicago.6

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Transportation Rates, Monopsony Power and the Location Decision of the Firm

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