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),

c

$ AC

- (AC)(Esc)

Let’s look at demand D.

Optimal toll = MC - AC =

- (ckd/s2)(ds/dc)

= -AC*Esc

D

Toll

Note: In this model, you STILL have congestion, even with the optimal toll..

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)]} (9)

with respect to s.

Balanced budget constraint:

nh + npx = s (10)

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.

1)(

x

Tnxvnp

ds

dx

s

Tnxv

Mgl benefit Mgl cost

s

$

1

s

Tnxv

)(

x

Tnxvnp

ds

dx

Weighteddifferencebetween priceand social costs

Optimum construction

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 orDemand is completely insensitive to investment

p*

capa

city

s

p

sfirst best

ssecond best

Optimum construction

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

p*

capa

city

s

p

sfirst best

ssecond best

po

so

We have had user fees but they certainly can’t be characterized as optimal.

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

volume

Trip cost Peak demand

Off-peak demand

Social cost

Private cost

Peak tax

Nonpeak tax

Estimates of congestion tolls• Bay area is more

congested than most metropolitan areas taxes may be lower elsewhere.

• Consumer responses– Carpools– Switch to mass transit– Switch to off-peak travel– Alternative routes– Combining trips

volume

Trip cost Peak demand

Off-peak demand

Social cost

Private cost

Peak tax

Nonpeak tax

Coase TheoremThe 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 L K Y

Y G L K

L L L

K K K

x x

y y

x y

x y

( , , )

( , )Production of Y decreasesproduction of X.

+ + -

If we maximize U (X, Y) we get:

U F L K Y G L K

U F L K G L L K K G L L K K

x x y y

x x x x x x

[ ( , , ), ( , )]

[ ( , , ( , )), ( , )]

Planning Optimum

U F L K Y G L K

U F L K G L L K K G L L K K

x x y y

x x x x x x

[ ( , , ), ( , )]

[ ( , , ( , )), ( , )]

If we maximize U (X, Y) we get:

If we maximize U (X, Y) w.r.t. Lx and Kx, we get:

U

U

F

GF

F

GFY

X

L

LY

L

LY (*)

Does a market get us there?

Market Optimum

U

U

F

GF

F

GFY

X

L

LY

K

KY (*)

Does a market get us there?

If firms maximize conventionally, we get:

p F p G w

p F p G rX L Y L

X K Y K

F

F

G

G

w

rL

K

L

K

F

G

F

G

p

pL

L

K

K

Y

X

So?

U

U

F

GF

F

GFY

X

L

LY

K

KY (*)

U

U

p

p

F

G

F

GY

X

Y

X

L

L

K

K

(**)

Society’s optimum

Market optimum

Since FY < 0, pY/pX is too low by that factor. Y is underpriced.

Coase TheoremThe 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.

Profits for the firm producing X are:

Y Y T

p F L K Y wL rK q Y Y

p F q

X X X X X X

X Y

( , , ) ( ) (***)

YX 0

Reduced by paying to pollute

Coase Theorem

Y Y Y Y Y Y

Y L

Y K

p G L K wL rK q Y Y

Lp q G w

Kp q G r

( , ) ( )

( )

( )

Y

Y

Y

Y

0

0

Y Y T

p F L K Y wL rK q Y Y

p F q

X X X X X X

X Y

( , , ) ( ) (***)

YX 0

We know that q = -pXFY

1 gets to Y

Coase Theorem

Y

Y

Y

Y

Lp q G w

Kp q G r

Y L

Y K

( )

( )

0

0

X

Y p F qX Y 0

We know that q = -pXFY

F

GF

F

GF

p

pL

LY

K

KY

Y

X

If 1, this looks like (*)

Change the ownership - X owns

Y Y Y Y Y Y

Y L

Y K

p G L K wL rK qY

Lp q G w

Kp q G r

( , )

( )

( )

Y

Y

Y

Y

0

0

X X X X X X

X Y

p F L K Y wL rK qY

p F q

( , , ) (****)

YX 0

We know that q = -pXFY/

If Y owns

If 1, this looks like (*)

F

GF

F

GF

p

pL

LY

K

KY

Y

X

If X owns

If 1, this looks like (*)

F

G

F F

G

F p

pL

L

Y K

K

Y Y

X

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.

If X holds, Y pays this muchIf Y holds, X pays this much

Graphically

T = Tx + Ty

q

Y’s supply (if Y holds)X’s demand (if Y holds)

-pxFY Py -r/GK = Py -w/GL

T*

If X holds, Y pays this muchIf Y holds, X pays this much

But, with transactions costs

T = Tx + Ty

q

Y’s supply (if Y holds)X’s demand (if Y holds)

-pxFY Py -r/GK = Py -w/GL

T*

Y gets this muchq

X gets this muchq

The equilibria are not the same!