More on Externalities
© Allen C. Goodman 2002
Transportation
Consider a roadway of distance d.
Services c cars per hour, at speed s. Travel
time for the entire highway is d/s.
Assume that value of driver time and costs
equal k per hour, so that cost per completed
trip = kd/s.
This is a version of average cost per car.
Cost per completed trip = kd/s
Total cost = c*AC = ckd/s
Marginal cost =
dTC/dc = kd/s - (ckd/s2)(ds/dc)
MC = AC - (ckd/s2)(ds/dc)
AC (1 – Esc), where Esc = Elasticity
Since (ds/dc)  0, we have congestion as c .
MC = AC - (ckd/s2)(ds/dc)
AC (1 – Esc),
Let’s look at demand D.
Optimal toll = MC - AC =
$
- (ckd/s2)(ds/dc)
= -AC*Esc
Note: In this model, you
STILL have congestion,
even with the optimal
toll..
- (AC)(Esc)
AC
Toll
D
c
What happens to highway investment when
pricing isn’t optimal?
Let U = U (z, x, Tx)
(1)
z = other expenditure
x = travel
Tx = time devoted to travel; T = time/hour
U1 > 0, U2 > 0, U3 < 0.
Budget constraint:
y = h + z + px
(2)
h is a lump sum tax to finance road construction, so:
L = U (z, x, Tx) +  (y - h - z- px)
*Wheaton, BJE (1978)
L = U (z, x, Tx) +  (y - h - z- px)
In eq’m:
U2/U1 = (-U3/U1) T + p.
Then:
x = x (y - h, T, p).
We can show that:
 x/  T = -(U3/U1)  x/  p = v  x/ p, where v = -U3/U1.
v = valuation of commuting time.
Road capacity s  travel time function:
T (x, nx)  0
 T/  s < 0
 2T/  s2 > 0
 t/  nx > 0
 2T/  (nx)2 > 0.
 2T/  nx  s < 0.
For congestion, assume that travel time T depends only on ratio
of volume nx to capacity s, or T = T (nx/s) = T [n(x/s)]
Yields:
 T/ s = -( T/ x) (x/s).
 T/( s/s) = -[ T/( x/x) ].
Finally, assume that:
(dx/ds)(s/x) <1. 1%  in s  less than 1%  in travel x.
Society’s optimum?
Optimize:
U{z(h, p, s), x (h, p, s), x(h, p, s) T [s, ns (h, p, s)]} (9)
with respect to s.
Balanced budget constraint:
nh + npx = s
(10)
h + px - s/n = 0.
Optimize:
U{z(h, p, s), x (h, p, s), x(h, p, s) T [s, ns (h, p, s)]}
with respect to s.
Balanced budget constraint:
nh + npx = s
h + px - s/n = 0.
With p given, we get:
-nxv  T/ s + (dx/ds) (np - nxv  T/ x) = 1.
If we optimize with respect to s and p, we get:
p = xv  T/ x  -nxv  T/ s = 1.
(9)
(10)
T dx
T
 nxv
 (np  nxv )  1
s ds
x
Mgl benefit
Mgl cost
dx
T
 (np  nxv )
ds
x
T
 nxv
s
Weighted
difference
between price
and social costs
$
1
s
Optimum construction
sfirst best
capacity s
With p given, we get:
-nxv T/ s +(dx/ds) (np - nxv  T/ x) = 1.
If we optimize with respect to s and p,
we get:
p = xv  t/ x -nxv  T/ s = 1.
Solutions will be same ONLY if:
We pick exactly the right price or
Demand is completely insensitive to
investment
ssecond best
p*
p
Optimum construction
sfirst best
capacity s
If p < p*, sf leads to too much s.
ss calls for a relative reduction in
investment. This will  congestion, 
“price,” thus  demand that has been
artificially induced by under-pricing
so
ssecond best
We have had user fees but they certainly
can’t be characterized as optimal.
po p*
p
Estimates of congestion tolls
• Example – For San
Francisco Bay area,
Pozdena (1988) estimates
that congestion tax would
be 0.65 per mile on
central urban highways
• $0.21 per mile on
suburban highways
• Off-peak of $0.03 to
$0.05 per mile.
• For reference, at that
time, the cost of driving
was estimated as between
$0.20 and $0.25 per mile
Trip cost
Peak demand
Social cost
Off-peak demand
Peak tax
Private cost
Nonpeak tax
volume
Estimates of congestion tolls
• Bay area is more
congested than most
metropolitan areas 
taxes may be lower
elsewhere.
• Consumer responses
–
–
–
–
–
Trip cost
Peak demand
Social cost
Off-peak demand
Peak tax
Carpools
Switch to mass transit
Switch to off-peak travel
Alternative routes
Combining trips
Private cost
Nonpeak tax
volume
Coase Theorem
The output mix of an economy is identical, irrespective of the
assignment of property rights, as long as there are zero
transactions costs.
Does this mean that we don’t have to do pollution taxes, that
the market will take care of things?
Let’s analyze.
Externalities and the Coase Theorem
+ + X  F ( Lx , K x , Y )
Y  G ( Ly , K y )
Production of Y decreases
production of X.
L  Lx  Ly
K  Kx  K y
If we maximize U (X, Y) we get:
U [ F ( Lx , K x , Y ), G ( Ly , K y )]
U [ F ( Lx , K x , G ( L  Lx , K  K x )), G ( L  Lx , K  K x )]
Planning Optimum
If we maximize U (X, Y) we get:
U [ F ( Lx , K x , Y ), G ( Ly , K y )]
U [ F ( Lx , K x , G ( L  Lx , K  K x )), G ( L  Lx , K  K x )]
If we maximize U (X, Y) w.r.t. Lx and Kx, we get:
UY
FL
FL

 FY 
 FY
U X GL
GL
Does a market get us there?
(*)
Market Optimum
UY
FL
FK

 FY 
 FY
U X GL
GK
(*)
Does a market get us there?
If firms maximize conventionally, we get:
p X FL  pY GL  w
p X FK  pY GK  r
FL G L w


FK GK r
FL
FK
pY


G L GK p X
So?
UY
FL
FK

 FY 
 FY
U X GL
GK
UY
pY
FL
FK



UX
p X G L GK
(*)
(**)
Society’s optimum
Market optimum
Since FY < 0, pY/pX is too low by that factor. Y is
underpriced.
Coase Theorem
The output mix of an economy is identical, irrespective of the
assignment of property rights, as long as there are zero
transactions costs.
Suppose that the firm producing Y owns the right to use water
for pollution (e.g. waste disposal). For a price q, it will sell
these rights to producers of X.
Reduced by
Profits for the firm producing X are:
paying to pollute
Y Y T
 X  p X F ( LX , K X , Y )  wLX  rK X  q (Y  Y ) (***)
 X
 p X FY  q  0
Y
Coase Theorem
Y Y T
 X  p X F ( LX , K X , Y )  wLX  rK X  q (Y  Y ) (***)
 X
 p X FY  q  0
Y
Y  pY G ( LY , KY )  wLY  rKY  q (Y  Y )
 Y
 ( pY  q )G L  w  0
  1 gets to Y
LY
 Y
 ( pY  q )GK  r  0
KY
We know that q = -pXFY
Coase Theorem
 X
 p X FY  q  0
Y
 Y
 ( pY  q ) G L  w  0
LY
 Y
 ( pY  q ) GK  r  0
K Y
FL
FK
pY
 FY 
 FY 
GL
GK
pX
If   1, this looks like (*)
We know that q = -pXFY
Change the ownership - X owns
 X  p X F ( LX , K X , Y )  wLX  rK X  qY (****)
 X
 p X FY  q  0
Y
Y  pY G ( LY , KY )  wLY  rKY  qY
Y
 ( pY  q )G L  w  0
LY
Y
 ( pY  q )GK  r  0
KY
We know that q = -pXFY/
If X owns
FL FY FK FY
pY




G L  GK 
pX
If   1, this looks like (*)
If Y owns
FL
FK
pY
 FY 
 FY 
GL
GK
pX
If   1, this looks like (*)
If  = 1
We are at a Pareto optimum
We are at same P O.
If  is close to 1
We may be Pareto superior
We are not necessarily at
same place.
Where we are depends on
ownership of prop. rights.
Remarks
• These are efficiency arguments.
• Clearly, equity depends on who owns the
rights.
• We are looking at one-consumer economy.
If firm owners have different utility
functions, the price-output mixes may differ
depending on who has property rights.
Graphically
X’s demand (if Y holds)
Y’s supply (if Y holds)
Py -r/GK = Py -w/GL
-pxFY
q
If Y holds, X pays this much
T = Tx + Ty
If X holds, Y pays this much
T*
But, with transactions costs 
X’s demand (if Y holds)
Y’s supply (if Y holds)
Py -r/GK = Py -w/GL
-pxFY
q
q
q
If Y holds, X pays this much
If X holds, Y pays this much
Y gets this much
T = Tx + Ty
X gets this much
T*
The equilibria are not the same!