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8/14/2019 Transport Lecture14
http://slidepdf.com/reader/full/transport-lecture14 1/23
Transportation Economics:
Pricing I
8/14/2019 Transport Lecture14
http://slidepdf.com/reader/full/transport-lecture14 2/23
Outline:
• 1) Perfect Competition
– many small firms, accept “market price”
• 2) Monopoly
– one big firm, chooses the market price
• 3) Social Optimum
– what would the gov’t provide?
8/14/2019 Transport Lecture14
http://slidepdf.com/reader/full/transport-lecture14 3/23
(I) Perfect competition
• Many firms, each with costs C(q), with
C’>0, C’’>0. Each accepts market price
p as given. To maximize profits,
• or, Price = Marginal costs
• Graph this as,
)q(C p)q(C pqmaxq
′=⇒−
8/14/2019 Transport Lecture14
http://slidepdf.com/reader/full/transport-lecture14 4/23
Price = Marginal Cost
• The marginal cost curve is the “supply curve” for an
individual firm:
qq0
P0
P
C′ (q)
8/14/2019 Transport Lecture14
http://slidepdf.com/reader/full/transport-lecture14 5/23
Supply = Demand
• Adding up the firm’s supply horizontally, we will get
total market supply, and equilibrium po, q
o:
qq0
p0
P
P(x)
Supply
8/14/2019 Transport Lecture14
http://slidepdf.com/reader/full/transport-lecture14 6/23
(II) Monopoly
• Single firm, with costs C(q), where C’>0, C’’><0. It
recognizes that when it sells more, the price will
fall .
• How does revenue change with q?
• Marginal revenue = Price - drop in revenue
• from price fall
q)q( p)q( pdq/]q)q( p[d ′+=
8/14/2019 Transport Lecture14
http://slidepdf.com/reader/full/transport-lecture14 7/23
Marginal revenue
• We can write marginal revenue as,
• MR =
• =
• =
• where is the elasticity
• So MR > 0 if only if E > 1.
]q)q( p)q( p[ ′+
)]q( p/q)q( p1)[q( p ′+)]E/1(1)[q( p −
q)q( p/ pE ′−=
8/14/2019 Transport Lecture14
http://slidepdf.com/reader/full/transport-lecture14 8/23
Marginal revenue, E.g. 1
• With linear demand, P=α −β q
• MR =
•= (α −β q) − β q
• = (α −2β q)
• so that MR is also linear, and is twice as steep as
demand
• = (α / β q)-1
]q)q( p)q( p[ ′+
q)q( p/ pE ′−=
8/14/2019 Transport Lecture14
http://slidepdf.com/reader/full/transport-lecture14 9/23
Marginal revenue, graph
• p
P(q)
q
MR
8/14/2019 Transport Lecture14
http://slidepdf.com/reader/full/transport-lecture14 10/23
Monopoly problem:
• To maximize profits:
• i.e., Marginal revenue = Marginal costs
• Graph this as:
)q(c)]E/1(1)[q( por
)q(c]q)q( p)q( p[)q(cq)q( pmaxq
′=−
′=′+⇒−
8/14/2019 Transport Lecture14
http://slidepdf.com/reader/full/transport-lecture14 11/23
Marginal revenue = Marginal Cost
• The intersection of MR and MC is the “point” of
optimal supply, pm, q
m:
qm
pm
p c′ (q)
P(q)
q
MR
8/14/2019 Transport Lecture14
http://slidepdf.com/reader/full/transport-lecture14 12/23
Consumer Surplus (review)
•
xq
P
P
CS
P(x)
8/14/2019 Transport Lecture14
http://slidepdf.com/reader/full/transport-lecture14 13/23
(III) Social Optimum:
• Measure social welfare=CS - Costs
• To maximize social welfare:
• i.e., Price = Marginal costs
• Graph this as:
)q(c p)q(cdx)x(Pmaxq
0q′=⇒−∫
8/14/2019 Transport Lecture14
http://slidepdf.com/reader/full/transport-lecture14 14/23
Price = Marginal Cost
• The intersection of P(q) and MC is the point of
optimal social welfare, supply, ps, q
s:
qm qs
pm
ps
p
c′ (q)
P(q)
q
MR
8/14/2019 Transport Lecture14
http://slidepdf.com/reader/full/transport-lecture14 15/23
Public Policy
• How to ensure that a monopoly charges the
social optimum, ps, q
s?
• 1) encourage entry and competition – (e.g., telephones, Microsoft antitrust case)
• 2) establish psat a price ceiling
– (e.g. utility and telephone companies)• 3) Have the government be the provider
– (e.g. roads, transit, etc.)
8/14/2019 Transport Lecture14
http://slidepdf.com/reader/full/transport-lecture14 16/23
Policy Problem 1:
• If a company, or the government, charge
price=marginal cost, but there are increasing
returns to scale, then:
• p = MC < AC
• so, Revenue = p q < Costs = AC q
• Either the company, or the government, is
making losses! So public policy in the form of
subsidies are needed.
••
8/14/2019 Transport Lecture14
http://slidepdf.com/reader/full/transport-lecture14 17/23
Policy Problem 2:
• It may be that private costs and benefits
differ from social costs and benefits:
• e.g. 1) pollution - has a extra social costthat private firm might ignore
• So policy is need for this externality
• e.g. 2) waiting time in transit - a social costthat a private firm might ignore?
8/14/2019 Transport Lecture14
http://slidepdf.com/reader/full/transport-lecture14 18/23
Example: Transit Authority
• Final output: q = total passengers on buses
per peak hour. Produced with:
• vehicles per peak hour V, with cost cp
• waiting time, valued at
• Suppose waiting time =1/2V. Then total costs
of buses and waiting are,
V2
qvC ,VcC
WT
W pB ==
WTv
8/14/2019 Transport Lecture14
http://slidepdf.com/reader/full/transport-lecture14 19/23
The gov’t transit authority’s problem:
• Choose V to
• subject to: N=bus capacity
• Question:
• would a private firm running the transit also
take into account consumer’s waiting time?• Answer:
• yes, to some extent.
NVq ≤
WB CCmin +
8/14/2019 Transport Lecture14
http://slidepdf.com/reader/full/transport-lecture14 20/23
Private pricing for transit
• Suppose that the private transit charges price
“p” per bus trip
• But then the “full price” for consumers equals
• pf = p + waiting time
• where is the value that consumers
put on waiting time
)V2/v( p WT+=
WTv
8/14/2019 Transport Lecture14
http://slidepdf.com/reader/full/transport-lecture14 21/23
Revenue for transit
• Write “full price” as a inverse demand pf (q)
• With price p per bus trip, total revenue is,
• p q
•
• where is
the total waiting time
)]V2/v( p[q WTf −=
)V2/qv(C WTW =
)V2/qv()q(qp WT
f
−=
8/14/2019 Transport Lecture14
http://slidepdf.com/reader/full/transport-lecture14 22/23
The private transit authority’s problem:
• Choose V and q, to:
• subject to:
• So that given optimal q*, then V is chosen to :
• differentiate w.r.t number of buses, VWB CCmin +
NVq ≤
BWf B CCq)q( pC pqmax −−=−
8/14/2019 Transport Lecture14
http://slidepdf.com/reader/full/transport-lecture14 23/23
The gov’t transit authority’s problem:
• Choose V and q, to:
• subject to:
• So that given optimal q**, then V is chosento :
WB CCmin +
NVq ≤
BW
q
0 f CCdx)x( pmax −−
∫