This paper examines the theoretical implications of quantity-discounted transportation rates on output effect of discriminatory f.o.b. mill pricing. Assume that the plant location of a monopoly is predetermined and demand curves are linear at two separate markets. It shows that total output under either discriminatory f.o.b. mill pricing or simple f.o.b. mill pricing remains the same when transportation rates are constant or linear. It further shows that total output will be greater under discriminatory f.o.b. mill pricing than under simple f.o.b. mill pricing when the transportation rate curve is convex in the market with less elastic demand and the transportation rate curve is concave in the market with more elastic demand. This indicates that the quantity-discounted transportation rates have an important influence on the output effect of spatial price discrimination.discriminatory f.o.b. pricing, quantity-discounted transportation rates, output effect

Yeung-nan Shieh

English

Research Papers in Economics

     
 
 
  
  
Volume 31, Issue 2 
  
The Implications of Quantity-Discounted Transportation Rates on Output 
Effect of Discriminatory F.O.B. Pricing 
  
 
 
Yeung-nan Shieh  
Department of Economics, San Jose State University 
Abstract 
This paper examines the theoretical implications of quantity-discounted transportation rates on output effect of 
discriminatory f.o.b. mill pricing. Assume that the plant location of a monopoly is predetermined and demand curves 
are linear at two separate markets. It shows that total output under either discriminatory f.o.b. mill pricing or simple 
f.o.b. mill pricing remains the same when transportation rates are constant or linear. It further shows that total output 
will be greater under discriminatory f.o.b. mill pricing than under simple f.o.b. mill pricing when the transportation rate 
curve is convex in the market with less elastic demand and the transportation rate curve is concave in the market with 
more elastic demand. This indicates that the quantity-discounted transportation rates have an important influence on 
the output effect of spatial price discrimination.
I thank Professors H. Hwang,C.C. Mai and an anonymous referee for valuable discussions and comments. Responsibility for any remaining 
errors is mine. 
Citation: Yeung-nan Shieh, (2011) ''The Implications of Quantity-Discounted Transportation Rates on Output Effect of Discriminatory F.O.B. 
Pricing'', Economics Bulletin, Vol. 31 no.2 pp. 1288-1294. 
Submitted: Feb 09 2011.   Published: April 27, 2011. 
 
     Economics Bulletin, 2011, Vol. 31 no.2 pp. 1288-1294
1 
 
1. Introduction 
 
A. C. Pigou (1920), in his famous book The Economics of Welfare, showed graphically that 
output under discriminating monopoly is identical to the simple monopoly output if the demand 
curves in two separate markets are linear.  Later, Robinson (1933) confirmed Pigou’s proposition 
mathematically.  Robinson further pointed out that the division into sub-markets might be 
dictated by geographical barrier.  However, Pigou and Robinson’s analysis is based on the 
traditional non-spatial setting in which transport costs and location decisions are insignificant 
and negligible.  Recently, Hwang & Mai (1990) (henceforth HM) extended Pigou and 
Robinson’s analysis to a bounded linear spatial setting.  Under the assumptions that (1) a spatial 
monopolistic firm locates its plant on a line in between two different markets; (2) the firm sells a 
commodity and charges consumers at two separate markets an f.o.b. mill price or different f.o.b. 
mill prices at plant location; (3) demand curves are linear, HM obtained the following interesting 
and important proposition. 
 
HM: Output under discriminatory f.o.b. mill pricing is identical to that under simple f.o.b. mill 
pricing if the plant location of a monopoly is predetermined and transportation rates are 
constant (HM 1990, p. 572). 
 
Later, this result has been confirmed and extended by Cheng & Shieh (1994), Shieh & Leung 
(2005) and Tan (2001).  This indicates that Pigou-Robinson’s proposition carries over to the 
spatial economy when the plant location is predetermined and transportation rates are constant.   
However, as is well known in transportation economics, discount for quantity shipped is 
prevalent among various modes of transportation, cf. Fair and Williams (1975, pp. 320-321 and 
325).  Meanwhile, Miller and Jensen (1978), Shieh and Mai (1984), Gilly, Shieh and William 
(1989), and others have examined the theoretical implication of quantity-discounted 
transportation rates on the location decision of the monopolistic firm.  It would be interesting and 
important to examine the impact of quantity-discounted transportation rates on the output effect 
of spatial price discrimination. 
    The purpose of this paper is to fill this gap.   Assume that the plant location is predetermined.  
By utilizing the mean-value theorem suggested by Shih, Mai and Liu (1988) and Varian (1989, 
p.632), we show that total output under discriminatory f.o.b. mill pricing will be greater or less 
than that under simple f.o.b. mill pricing if the transportation rates function is non-linear.  This 
result suggests that the gist of the well-known Pigou-Robinson proposition need not hold in the 
spatial economy with given plant location unless the transportation rates are constant or linear. 
 
2.  The basic model 
 
Our analysis is based on the familiar linear spatial model with the following assumptions. 
(A1)  A monopolistic firm locates its plant along a line of length s between market 1 and market 
2 as illustrated in Figure 1.  Let x and (s – x) be the distances of the plant from market 1 and 
market 2.  
 
 
 
 
1289Economics Bulletin, 2011, Vol. 31 no.2 pp. 1288-1294
2 
 
                                                               s 
                        
                      1                                                                                          2 
                       
                                              x                         
                                                                          
0  K 
                           Figure 1.   A Location Line. 
(A2) The firm sells a commodity to markets 1 and 2, and charges consumers a uniform f.o.b. mill 
price or different f.o.b. mill prices at plant location.  Demand functions at market 1 and market 2 
are 
 
            q1 = α1 - β1p1 = α1 - β1[m1 + t1(q1)x]                                                                                (1) 
            q2 =  α2 - β2p2 = α2 - β2[m2 + t2(q2)(s-x)]                                                                          (2) 
                                                           
where q1 and q2 are quantity demanded, p1 and p2 are the delivered prices, m1 and m2 are mill 
prices quoted at market 1 and market 2, t1(q1) and t2(q2) are transportation rates, t1(q1)x and 
t2(q2)(s-x) are the transport cost of per unit output, α1, α2, β1, β2 are positive parameters.  Based 
on (1) and (2), the inverse demand functions can be written as: 
 
           m1 = a1 - b1q1 - t1(q1)x, a1 = (α1/β1), b1 = (1/β1)                                                                 (3) 
           m2 = a2 - b2q2 - t2(q2)(s-x), a2 = (α2/β2), b2 = (1 /β2)                                                          (4) 
      
(A3)  The cost function, for simplicity, is specified as: 
 
           C = cq                                                                                                                                 (5) 
                                                                                       
where q = q1 + q2 and c is the constant marginal cost.             
(A4) The objective of the monopolist is to find the profit-maximizing output. 
    It should be noted that the inclusion of quantity discounted in the transportation rates 
constitutes a major departure from the conventional model.  That is, instead of assuming constant 
transportation rates, we now permit these rates to vary with quantity shipped. 
    With this set of assumptions, the profit function of the monopolist in market 1 and 2 can be 
specified as: 
 
         π1 = (m1 – c)q1,            π2 = (m2 – c)q2                                                                                    (6) 
  
where q1 and q2 are choice variables. 
    In the case of discriminatory monopoly, the firm charges different f.o.b. mill prices to 
consumers at different locations, thus q1 and q2 are independent variables, and will adjust to 
maximize profits. If the profit functions are strictly concave, i.e., d
2π1/dq1
2 < 0 and d
2π2/dq2
2 < 0, 
the first order conditions are 
1290Economics Bulletin, 2011, Vol. 31 no.2 pp. 1288-1294
3 
 
 
        π1’(q1*) = m1(q1*) – e1q1* – c = 0,   π2’(q2*) = m2(q2*) – e2q2* – c = 0                              (7) 
 
where m1(q1*) = a1 – b1q1* - t1(q1*)x, m2(q2*) = a2 – b2q2* - t2(q2*)(s- x), e1 = b1 + t1’(q1*)x > 0 
and e2 =  b2 + t2’(q2*)(s – x) > 0. 
    Solving (7), we obtain 
 
        q1* = [m1(q1*)  – c]/e1,     q2* = [m2(q2*) – c]/e2                                                                   (8)  
 
where q1* and q2* are the optimal quantity sold in market 1 and 2, respectively.   Total output 
under discriminatory f.o.b. mill pricing would be 
 
        Q* = q1* + q2* = [(m1(q1*) – c)/e1] + [(m2(q2*) – c)/e2]                                                      (9) 
 
     Under simple f.o.b. mill pricing, the profit function can be specified as: 
 
        π = m1q1 + m2q2 – c(q1 + q2)                                                                                               (10) 
 
The monopolist maximizes total profit by charging a uniform price m
o = m1 = m2 in markets 1 
and 2.  The first order condition is 
 
        q1
o - {[m1(q1
o) – c]/e1} + q2
o - {[m2(q2
o) – c]/e2} = 0                                                         (11) 
 
where m1(q1
o) = a1 – b1q1
o - t1(q1
o)x, m2(q2
o) = a2 – b2q2
o - t2(q2
o)(s- x), e1 = b1 + t1’(q1
o)x > 0 and 
e2 =  b2 + t2’(q2
o)(s – x) > 0.   It should be noted that q1
o and q2
o are the quantity sold in market 1 
and 2 respectively under uniform price m
o.   From (11), we at once obtain total output under 
simple f.o.b. mill pricing as: 
 
         Q
o = q1
o + q2
o = {[m1(q1
o) – c]/e1} + {[m2(q2
o) – c]/e2}                                                   (12) 
 
The effect of discriminatory f.o.b. mill pricing on total output can be obtained by comparing (9) 
and (12).  Subtracting (9) from (12), we obtain 
 
       Q
o – Q
* = ({[(m1(q1
o) – c)]/e1} - {[m1(q1
*) – c]/e1})                                                           (13) 
                            + ({[(m2(q2
o) – c)]/e2} -{[m2(q2
*) – c]/e2}) 
 
Next, via the mean-value theorem, cf. Chiang (1984, pp. 261-62), Shih, Mai and Liu (1988, p. 
151) and Varian (1989, p. 632), we obtain 
 
     {[m1(q1
o) – c]/e1} - {[m1(q1
*) – c]/e1}  
                    = (q1
o - q1
*)(- {[m1(q1
e) – c]t1”(q1
e)x/e1
2} – 1)                                                       (14) 
      
     {[(m2(q2
o) – c)]/e2} -{[m2(q2
*) – c]/e2} 
                   = (q2
o - q2
*)(- {[m2(q2
e) – c]t2”(q2
e)(s-x)/e2
2} – 1)                                                   (15) 
 
1291Economics Bulletin, 2011, Vol. 31 no.2 pp. 1288-1294
4 
 
where q1
e in an output level between q1
o and q1* and q2
e is an output level between q2
o and q2*.  
Substituting (14) and (15) into (13), we obtain 
 
     Q
o – Q* = (1/2)[(q1
* - q1
o)({[m1(q1
e) – c]t1”(q1
e)x/e1
2} 
                                    + (q2
* - q2
o)({[m2(q2
e) – c]t2”(q2
e)(s-x)/e2
2}                                          (16) 
 
 
where [m1(q1
e)  –  c] > 0 and [m2(q2
e)  –  c] > 0.  It is easy to see that the output effect of 
discriminatory f.o.b. mill pricing crucially depends upon the signs of (q1
* - q1
o), (q2
* – q2
o), t1” 
and t2”.  This completes the model that constitutes our basic analytical framework. 
 
3. Effect of spatial price discrimination 
 
We are in a position to examine the impact of discriminatory f.o.b. mill pricing on total output.  
Since signs of (q1
* - q1
o), (q2
* – q2
o), t1” and t2” can not a priori be determined, the output effect 
of discriminatory f.o.b.  mill pricing is indeterminate. To pursue our investigation further, we 
assume that the demands elasticity in market 2 is greater than that in market 1, i.e., η1 <   η2.  
Hence, we have (i) m1* > m1
o
 and q1* < q1
o, (ii) m2* < m2
o and q2* > q2
o.  With this in mind, we 
consider two cases: (1) transportation rates are constant or linear with respect to quantity 
shipped; (2) transportation rates are non-linear with respect to quantity shipped. 
 
3.1 The constant or linear transportation rates case 
    In this case, transportation rates are constant or linear, t1” = t2” = 0.  Substituting this into (16), 
we obtain 
  
          Q
o – Q* = 0,   as t1” = t2” = 0                                                                                           (17) 
 
Thus, we establish 
 
Proposition 1.  Supposes the plant location of a monopoly is predetermined and transportation 
rates are constant or linear with respect to quantity shipped. Total output remains unchanged 
under discriminatory f.o.b. mill pricing. 
 
In other words, Pigou-Robinson proposition in non-spatial setting and HM proposition in spatial 
setting still hold when transportation rates are constant or linear.  It may be noted that 
Proposition 1 generalizes HM’s result since we do not assume a constant transportation rate. 
 
3.2 The non-linear transportation rates case 
    In this case, transportation rates are non-linear, and t1” ≠ 0 and t2” ≠ 0.  Under assumptions 
that η1 <  η2,  q1* < q1
o, and q2* > q2
o,  from (16), we can obtain 
 
       Q
o – Q* < (>) 0, as t1” > 0 > t2” (t1” < 0 < t2”)                                                                 (18) 
 
Thus, we can conclude that 
 
1292Economics Bulletin, 2011, Vol. 31 no.2 pp. 1288-1294
5 
 
Proposition 2. When the transportation rate curve is convex (t1” > 0) in the market with less 
elastic demand and the transportation rate curve is concave (t2” < 0) in the market with more 
elastic demand, then total output increases under discriminatory f.o.b. mill pricing, and vice 
verse. 
  
In other words, this proposition can be used to predict the direction of output change under 
spatial price discrimination in the case of two groups of transportation rates with opposing 
general curvatures. More importantly, it shows that Pigou-Robinson proposition in the non-
spatial economy and HM proposition the spatial economy need not hold when transportation 
rates depend upon quantity shipped. 
 
4. Conclusion 
 
In this paper, we have explicitly considered quantity  shipped as a key variable in the 
transportation rate function and examined the theoretical implication of this variable on the 
output effect of spatial price discrimination.  In a linear space, HM (1990) and Cheng and Shieh 
(1994) and others considered the case where the demand curves are linear and  demonstrated that 
discriminatory f.o.b. mill pricing does not change total output of a monopoly when the plant 
location is predetermined and transportation rates are independent of quantity shipped.  This 
shows that the well-known Pigou-Robsion proposition can be carried over to the spatial 
economy. 
    When transportation rates depend upon quantity shipped, we demonstrated that HM’s result 
need not hold.  We further showed that, in general, total output will be changed by spatial price 
discrimination unless (i) the transportation rates are constant, or (ii) the transportation rate 
functions are linear. 
    In the case where the transportation rate curve is convex in the market with less elastic 
demand and the transportation rate curve is concave in the market with more elastic demand, we 
showed that discriminatory f.o.b. mill pricing will increase total output.  Our study shows that 
the quantity-discounted transportation rates have an important influence on the output effect of 
spatial price discrimination.   This indicates that all regulatory policies, public and private, which 
aim at increasing total output through rates regulation, should receive careful scrutiny. 
 
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A General Analysis of the Output Effect under ThirdDegree Price Discrimination”

Location and the Theory of Production - A Review,

Location and the Theory of Production:

Price Discrimination”

The Economics of Imperfect Competition.

The Economics of Welfare.

The Output Effect of Price Discrimination in the Weber Triangle”

Transportation Rates and Location of the Firm: A Comparative Statics Result”

The Implications of Quantity-Discounted Transportation Rates on Output Effect of Discriminatory F.O.B. Pricing

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The Implications of Quantity-Discounted Transportation Rates on Output Effect of Discriminatory F.O.B. Pricing

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