Transcript 3.1 Ramsey Prices - Illinois State University

Ramsey Pricing
Assumptions
• Natural monopoly
– Cost minimization requires one firm in the market
• Strong natural monopoly
– MC pricing crease deficit
• MC with subsidies is not an option
• Barriers to entry
• Product demands are unrelated (we relax this
assumption later in more complex models)
Regulation
• Objective is to maximize social welfare subject
to the firm breaking even
– Per unit charge (constant price)
– Prices across markets can vary
• First best pricing is not an option
• Second best (Ramsey Prices) based on
maximizing welfare subject to an acceptable
level of profit
Ramsey Idea
• We can’t maximize Social Welfare
• Must adopt a price different from MC
• How much should price deviate from MC?
– Raise prices until the marginal welfare losses across
product markets are balanced.
• Prices are raised in inverse proportion to the
absolute value of the demand elasticities in each
market and this minimized the welfare losses
associated with higher prices.
– Elastic, small raise
– Inelastic, large raise
Comparing Elasticities
$
p2(q2)
p1(q1)
p΄
a
f
e
Deadweight loss of market two
d
p1=p2
g
b
c
o
q
q1΄
q2΄
q1=q2
The Math Behind Ramsey
• Consider a multiproduct firm producing outputs q= (q1, … ,qn ), with cost
function given by C(q).
Demands for the n goods are represented by the inverse demand
function, pi(qi), i=1, …, n.
Consumer surplus in the ith market be given by
CS i 

qi
o
pi ( xi )dxi  pi (qi )qi
(3.1)
for i=1, …, n, and profit for the firm is
n
(3.2)
   pi (qi )qi  C (q)
i 1
•
If we maximize a measure of welfare (W), it is the sum of all CSi plus
profit,
W 
CS
i

Max Welfare
W 
CS
i

First Order Condition gives
pi
pi
W
C
 pi (q i ) 
qi  pi (q i ) 
qi  pi (q i ) 
0
qi
qi
qi
qi
Implies
C
pi (q i ) 
qi
Max Welfare s.t. Break even Profit
L  W   (   )
First Order Condition gives
pi C
L
C
 pi (q i ) 
  ( pi (qi )  qi

)0
qi
qi
qi qi
Implies

pi
C 

 pi (q i ) 
1     qi 
qi 
q i

What is Lambda?
Let profit be constrained to     0 , then,
using the envelope theorem, we have
, or roughly
L /   W /   
speaking, if the firm’s profit is decreased by
$1, which will mean a $1 deficit, then welfare
will increase by $λ.
Ramsey Prices
• Rearrange (3.4), we obtain
 p (q )  MC (q)1     q pi(qi )
(3.5)
i
i
i
i
1   
and then dividing both sides by pi (qi ) and
yields
(3.6)
where
pi (qi )  MCi (q)   1

pi (qi )
1   ei
ei  pi (qi ) /qi pi(qi )  0 , the elasticity of demand in market i .
The price in (3.6) is called the Ramsey price in market i. Because (3.6) is
true for all i, the formula states that the percentage deviation of price
from marginal cost in the ith market should be inversely proportional to
the absolute value of demand elasticity in the ith market.
Ramsey Pricing and the Lerner Index
• The Lerner Index is equivalent to the negative
inverse of the price elasticity of demand facing
the firm, when the price chosen is that which
maximizes profits available because of the
existence of market power.
Results and Conclusions
• From (3.6), and for all markets, the percentage deviation of price from
marginal cost, times the price elasticity, sometimes called the Ramsey
number, should be equal to –λ/(1+λ).
• If λ is very small, the Ramsey number approaches zero, implying that
prices will be very close to marginal cost.
– The deficit under efficient pricing is small:
– Reducing profit by $1 and recalculating prices will not increase welfare
• If λ is large, the Ramsey number is close to 1;
– such a situation occurs when an unconstrained monopolist barely
breaks even
– The price-cost deviations are so substantial that large welfare
improvements are possible if a subsidy can be transferred to the firm
to compensate for negative profit.
More Results
• Relatively inelastic demands mean smaller values of |ei| and
higher percentage price deviations from marginal cost.
• Relatively elastic markets, the percentage price deviation will
be smaller,
• Thus, for the ith and jth markets
 pi (qi )  MCi (q)/ pi (qi )
(3.6a)
p (q )  MC (q)/ p (q )
j
j
j
j
j

ej
ei
• Given initial Ramsey prices, if |ei| were to increase, ceteris
paribus, pi and Mci would have to be brought closer together
and /or pj and MCj be moved further apart for the Ramsey
conditions to hold.
Another Interpretation/Result
• By adding  pi  qi pi  MCi  to both sides for markets i and j to
obtain
(3.7)
p j (q j )  MC j (q)
pi (qi )  MCi (q)

MCi (q)  MRi (q) MC j (q)  MR j (q)
Where MRi(q) is the marginal revenue in the ith market.
By (3.7), if we were to increase output by Δqi and Δqj from the
profit-maximizing solution,
– the numerators would be the marginal gain from additional consumption,
– the denominators the marginal loss in profit.
– this marginal-benefit/marginal-loss ratio must be equal across markets at
the point where profit is down to zero.
Comparing Output Market Elasticities
and Ramsey Pricing
$
p2(q2)
p1(q1)
p΄
a
f
e
Deadweight loss of market two
d
p1=p2
g
b
c
o
q
q1΄
q2΄
q1=q2
Comparing Output Market Elasticities
and Ramsey Pricing
• Initially, prices and quantities demanded for both goods are
equal at point d.
• If prices must be increased to eliminate a deficit, which
market should bear more of the burden in terms of a higher
price?
• If prices was raised to p΄, there would be a deadweight loss of
adb in the market one and a loss of ecd in market two. The
former is larger than the latter, indicating that price should
increase more in market two than in market one. This is also
called “what the traffic can bear” or “value-of-service pricing”.