Ramsey pricing of regulated firm services is often considered impractical because of stringent informational requirements. However in cases where demands for regulated firm services are independent and a marginal cost pricing competitive fringe exists, the simple inverse elasticity rule for Ramsey pricing has been shown to apply. This paper extends this result to show that only limited demand elasticity data for the regulated firm is required to apply the same rule, more generally, when zero profits and constant prices exist in the competitive sector in the long run.

William C. Miller

English

Research Papers in Economics

Ramsey pricing with long run competition 
William C. Miller
Pioneer Analytics LLC
Abstract
Ramsey pricing of regulated firm services is often considered impractical because of
stringent informational requirements. However in cases where demands for regulated firm
services are independent and a marginal cost pricing competitive fringe exists, the simple
inverse elasticity rule for Ramsey pricing has been shown to apply. This paper extends this
result to show that only limited demand elasticity data for the regulated firm is required to
apply the same rule, more generally, when zero profits and constant prices exist in the
competitive sector in the long run.
Citation: Miller, William C., (2007) "Ramsey pricing with long run competition." Economics Bulletin, Vol. 12, No. 34 pp. 1-5
Submitted: October 4, 2007.  Accepted: December 14, 2007.
URL: http://economicsbulletin.vanderbilt.edu/2007/volume12/EB-07L50006A.pdf 1
1. Introduction 
 
  It has been well established in the economic literature that information requirements for 
Ramsey or second best optimal pricing for regulated firms can be substantial except in the 
simplest cases.  This is especially true in mixed oligopoly situations where a regulated firm 
competes against one or more unregulated firms offering substitute or complementary products 
or services.  In particular, several authors have shown that above marginal cost pricing in the  
competing sector requires own and cross price demand elasticity data in order to establish 
Ramsey rates different than would have existed in the absence of such competition.  
 
  Sherman (1979) analyzed Postal Service Ramsey rates when rival firms price at or above 
marginal cost.  In a more general context, Ware and Winter (1986) established similar results for 
the above marginal cost case at a Nash equilibrium solution involving Bertrand pricing.  In both 
scenarios, cross price demand elasticities are needed to estimate Ramsey numbers higher (lower) 
than would have existed without competition when products are substitutes (complements).  
More recently, Prieger (1996) offered a Ramsey solution under Stackelberg price leadership 
assumptions which also require the same cross price demand elasticities and a new “strategic” 
price elasticity based on the competitive firm’s price reaction function.   
 
  By contrast, both Braeutigam (1979) and Sherman show that the simple inverse elasticity 
rule applies when a competitive fringe prices at marginal cost.  In this situation, the distinction 
Prieger makes between “myopic” Ramsey pricing by the regulator and other forms of welfare 
improving Ramsey rates vanishes.  Hence with a marginal cost pricing fringe, the regulator needs 
only cost and own price demand elasticity data for the regulated firm in order to establish 
globally optimal Ramsey prices.   
 
  However marginal cost pricing is only a sufficient condition for limiting the required 
demand data.  This note establishes a more general condition showing that the same inverse 
elasticity rule can apply with above marginal cost pricing when rivals are Cournot competitors 
and zero long run profits prevail.  Cournot competition and zero profits might be considered 
appropriate where there is relatively little product differentiation within the competitive sector 
and there are a large number of rivals caused by weak scale effects relative to demand.   
However the regulated firm’s product is differentiated from rivals as a whole, possibly because 
of a quality or reliability distinction.   
 
The particular model presented assumes that the regulated firm offers a non-competitive 
and competitive product where the firm acts as a Stackelberg price leader.  Rival firms take any 
changes in the competitive product rate as given when adjusting their own output according to 
Cournot assumptions.   Free entry and exit conditions exist and therefore a zero profit 
equilibrium is re-established, once new firms enter or existing firms exit as necessary.   Within 
this framework, I demonstrate that deviations from the simple inverse elasticity rule for Ramsey 
rates require long run changes to the competitive sector rate.  I further show that long run price 
changes are absent with linear demand and therefore the inverse elasticity rule applies in this 
instance.  Therefore situations where limited (regulated firm only) data requirements for Ramsey 
pricing are considered appropriate might be broader than once thought.   
  2
 
2.  The Model  
 
  I assume a two stage game where the regulator first sets prices for the regulated firm’s 
two products subject to a break-even constraint, and then the price and the number of firms in the 
competitive sector adjusts re-establishing equilibrium.  Each rival has the same cost structure and 
therefore identical (zero) profits given the common rate charged.   
 
First, let demand for the non-competitive product, the regulated firm’s competitive 
product and the competitive sector’s product be determined by the functions D
1(p1), D
2(p2, pr) 
and D
r(p2, pr) where p1, p2 and pr are the corresponding rates.  The usual own price marginal 
conditions D
1
p1 < 0, D
2
p2 < 0  and D
r
pr < 0 and product substitutability conditions D
2
pr  = D
r
p2 > 0  
apply.  Then the competitive sector long term profit function with n rivals can be written as: 
 
π
 r(p2, pr, n) = D
r(p2, pr)(pr - ur) - nFr = 0, 
 
where ur and Fr are each rival’s constant unit and fixed cost, respectively.   The Cournot  
solution also requires the first order condition: 
 
D
r/n + D
r
pr(pr - ur) = 0. 
 
Therefore assuming a non-zero Jacobian for these two expressions, the functions pr(p2) and n(p2) 
can be established using the implicit function rule.  Similarly, the regulated firm’s profit function 
can be specified as: 
 
B
d(p1, p2, pr) = D
1(p1)(p1 - u1) + D
2(p2, pr)(p2 - u2) - Fd,  
 
where u1 and u2 are the constant unit costs for products one and two and Fd is the firm’s fixed 
cost.  Finally substituting pr(p2) and n(p2) into the profit functions, total welfare can be shown as 
dependent only on the two rates for the regulated firm:  
 
W(p1, p2, pr(p2), n(p2)) = S(p1, p2, pr(p2)) + B
d(p1, p2, pr(p2)) + π
 r(p2, pr(p2), n(p2)),      
 
where S() is consumer surplus with zero wealth effects assumed. 
 
  The regulator’s task is to maximize W subject toB
d(p1, p2, pr(p2)) = 0.  Therefore writing 
the Lagrangian: 
 
‹ = W(p1, p2, pr(p2), n(p2)) + 8[B
d(p1, p2, pr(p2))], 
 
the first order welfare maximizing conditions with respect to p1 and p2 can then be shown as:  
 
     M‹/Mp1 = - D
1 + (1 + 8)B
d
p1 = 0 
 
                  = 8D
1 + (1 + 8)(p1 - u1)dD
1/dp1 = 0,               3
 
 and  
 
M‹/Mp2 = - D
2 - D
r(dpr/dp2) + B
r
p2 + B
r
pr(dpr/dp2) + B
r
n(dn/dp2) + 
 
              (1 + 8)(B
d
p2 + B
d
pr(dpr/dp2)) = 0      
    
     = 8D
2 + (1 + 8)[(p2 - u2)(D
2
p2 + D
2
prdpr/dp2)] - D
r(dpr/dp2) = 0,                
 
sinceB
r
p2 + B
r
pr(dpr/dp2) + B
r
n(dn/dp2) = 0.  The two first order conditions can then be 
transformed to derive the optimal price mark-ups on the regulated products.  First note that the 
total effect on product two demand from a change in p2 is the sum of the direct effect and the 
indirect effect transmitted through an induced change in pr: dD
2/dp2 = D
2
p2 + D
2
prdpr/dp2.  The 
following expressions corresponding to the above can then be derived by manipulation:   
 
               e1(p1 - u1)/p1 = -8/(1 + 8)                (1) 
 
     e 2(p2 - u2)/p2  = -8/(1 + 8) + (Rr/R2)z/(1 + 8),                    (2)      
 
where e1 = (dD
1/dp1)(p1/D
1), e2 = (dD
2/dp2)(p2/D
2), z = (dpr/dp2)p2/pr and Rr/R2 = D
rpr/D
2p2. 
Finally subtracting (1) from (2) gives:  
 
e2(p2 - u2)/p2 - e1(p1 - u1)/p1 = (Rr/R2)z/(1 + 8).              (3) 
 
The two terms on the LHS are the Ramsey numbers for the regulated products.  This last 
expression shows that if long term profits are zero in the competitive sector, then the regulator 
deviates from the simple inverse elasticity rule e2(p2 - u2)/p2 = e1(p1 - u1)/p1 only when p2 affects 
pr or z … 0.  Otherwise, the Ramsey number for the substitutable product should be set higher 
(lower) in absolute value than the number for the non-substitutable product if z is negative 
(positive).  However it is important to note that if z = 0, then the total effect on product two 
demand from an own price change reduces to the direct effect only or dD
2/dp2 = D
2
p2.  This 
implies that the regulator needs to use only the regulated firm’s own price demand elasticities to 
determine optimal rates for p1 and p2 according to the simple inverse elasticity rule.     
 
 
3.  Long Run Impacts on the Competitive Sector       
 
  This section shows that linear demand is sufficient to establish that the competitive sector 
price is unaffected in the long term by regulated rates.  More generally, it also describes the 
condition determining the direction of impact of p2 on pr.  First note that under Cournot 
competition, each rival j perceives its profit function as:      
 
B
r
j = (D
r(p2, pr) - 3 Vi)(pr - ur) - Fr, 
 
where i = 1 to n,  i … j, and the sum of all other firm volumes in the industry 3 Vi is assumed  4
fixed.  Therefore each firm varies pr to achieve:   
 
MB
r
j/Mpr = D
r - 3 Vi + D
r
pr(pr - ur) = 0    
    
or  
 
            D
r/n + D
r
pr(pr - ur) = 0,                 (4) 
 
since all Vi are equal.  Holding all Vi constant, each firm’s second order condition is:  
 
                     2D
r
pr + D
r
prpr(pr - ur) < 0.               (5)  
 
Also equilibrium with costless entry and exit requires the following zero long run profit 
condition:  
 
                (D
r/n)(pr - ur) - Fr = 0.                  (6) 
          
  Marginal effects on pr from changes in p2 are then determined by totally differentiating 
(4) and (6) with respect to p2.  Therefore:  
 
[D
r
pr(1 + 1/n) + D
r
prpr(pr - ur)]dpr/dp2 – [D
r/n
2]dn/dp2 = -D
r
p2/n - D
r
prp2(pr - ur) 
 
[D
r/n + (D
r
pr/n)(pr - ur)]dpr/dp2 – [D
r(pr - ur)/n
2]dn/dp2 = -(D
r
p2/n)(pr - ur). 
 
The determinant of the Jacobian formed by the matrix of bracketed terms on the LHS is then:  
 
Z = -[D
r
pr(1 + 1/n) + D
r
prpr(pr - ur)]D
r(pr - ur)/n
2 + [D
r/n + (D
r
pr/n)(pr - ur)]D
r/n
2 
           
      = -(D
r
pr(pr - ur) - D
r/n + D
r
prpr(pr - ur)
2)D
r/n
2. 
 
Using (4) to substitute for D
r/n then signs Z as: 
 
-[2D
r
pr+ D
r
prpr(pr - ur)](pr - ur)D
r/n
2 > 0, 
 
from (5).    
 
Then by Cramer’s rule, the marginal effect on price is determined directly from:  
 
dpr/dp2 = [(D
r
p2/n + D
r
prp2(pr - ur))D
r(pr - ur)/n
2 - (D
r
p2/n)D
r(pr - ur)/n
2]/Z 
 
            = D
r
prp2D
r(pr - ur)
2/Zn
2,  
 
and substituting for Z:  
 
       dpr/dp2 = -D
r
prp2(pr - ur)/[2D
r
pr+ D
r
prpr(pr - ur)], 
  5
in which case sign[dpr/dp2] = sign[D
r
prp2].  Also note that the effect on competitive firm volume 
is directly evidenced by (6).  In particular, D
r/n decreases, stays the same or increases with 
respect to increases in p2 as dpr/dp2 is positive, zero or negative.  
 
   The key result here is that with linear demand D
r
prp2 = 0 so that dpr/dp2 = 0 and then the 
simple inverse elasticity rule applies in the long run using only own price elasticities for the 
regulated firm.  Therefore the distinction between Prieger’s myopic result and optimal Ramsey 
pricing disappears in this instance.  At a more general level, if dpr/dp2 ≠ 0 then cross price 
demand data is required to set optimal second best rates.  For example if demand elasticities are 
constant, it can be easily verified that dpr/dp2  < 0 and therefore e2(p2 - u2)/p2 – e1(p1 - u1)/p1 < 0 
by (3). In this case, the price mark-up for the substitutable product should be set higher relative 
to the non-substitutable product than implied by the simple inverse elasticity rule.  This last 
result is consistent with general findings from the literature when the number of competitors is 
assumed fixed.     
 
  
4. References  
 
Baumol, W., and D. Bradford (1970) “Optimal Departures from Marginal Cost Pricing” The 
American Economic Review 60, 265-283.  
 
Braeutigam, R. (1984) “Socially Optimal Pricing with Rivalry and Economies of Scale” The 
Rand Journal of Economics 15, 127-134.    
 
Braeutigam, R. (1979) “Optimal Pricing with Intermodal Competition” The American Economic 
Review 69, 38-49.  
 
George, A., and R. Sherman (1979) “Second Best Pricing for the U.S. Postal Service” Southern 
Economic Journal 45, 685-695.  
 
Höffler, F. (2006) “Monopoly Prices versus Ramsey-Boiteux Prices: Are they “Similar,” and: 
Does it Matter?” Journal of Industry, Competition and Trade 6, 27-43.      
 
Prieger, J. (1996) “Ramsey Pricing and Competition: The Consequences of Myopic Regulation” 
Journal of Regulatory Economics 10, 307-321.    
 
Ware, R., and R. Winter (1986) “Public Pricing under Imperfect Competition” International 
Journal of Industrial Organization 4, 87-97.   
 
 
 
 
 
 
 
 

Ramsey pricing with long run competition

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