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/s 2 )(ds/dc) MC = AC - (ckd/s 2 )(ds/dc) AC (1 – E sc ), where E sc = Elasticity Since (ds/dc)  0, we have congestion as c .

MC = AC - (ckd/s 2 )(ds/dc) AC (1 – E sc ), c $ AC - (AC)(E sc ) Let’s look at demand D. Optimal toll = MC - AC = - (ckd/s 2 )(ds/dc) = -AC*E sc 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 U 1 > 0, U 2 > 0, U 3 < 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: U 2 /U 1 = (-U 3 /U 1 ) T + p. Then: x = x (y - h, T, p). We can show that:  x/  T = -(U 3 /U 1 )  x/  p = v  x/  p, where v = -U 3 /U 1. v = valuation of commuting time.

Road capacity s  travel time function: T (x, nx)  0  T/  s < 0  2 T/  s 2 > 0  t/  nx > 0  2 T/  (nx) 2 > 0.  2 T/  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.

Mgl benefitMgl cost s $ 1 Weighted difference between price and 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 or Demand is completely insensitive to investment p* capacity s p s first best s second best

Optimum construction If p < p*, s f leads to too much s. s s calls for a relative reduction in investment. This will  congestion,  “price,” thus  demand that has been artificially induced by under-pricing p* capacity s p s first best s second best popo soso 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 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 Production of Y decreases production of X If we maximize U (X, Y) we get:

Planning Optimum If we maximize U (X, Y) we get: If we maximize U (X, Y) w.r.t. L x and K x, we get: Does a market get us there?

Market Optimum Does a market get us there? If firms maximize conventionally, we get:

So? Society’s optimum Market optimum Since F Y < 0, p Y /p X 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. Profits for the firm producing X are: Reduced by paying to pollute

Coase Theorem We know that q = -p X F Y   1 gets to Y

Coase Theorem We know that q = -p X F Y

Change the ownership - X owns We know that q = -p X F Y / 

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 = T x + T y q Y’s supply (if Y holds) X’s demand (if Y holds) -p x F Y P y -r/G K = P y -w/G L T*

If X holds, Y pays this muchIf Y holds, X pays this much But, with transactions costs  T = T x + T y q Y’s supply (if Y holds) X’s demand (if Y holds) -p x F Y P y -r/G K = P y -w/G L T* Y gets this much qq X gets this much qq The equilibria are not the same!