Lecture 19: Common property resources Economics 336 Economics 336 (Toronto) Lecture 19: Common property resources 1 / 19
Introduction Common property resource: A resource for which no agent has full property rights for selling exploitation. A widespread negative externality, leads to overexploitation and the tragedy of the commons Classic example: an open-access fishery. Anyone who wishes to may fish. Each fisher who enters makes it harder for others to catch fish. The model: catch per fisher is A(N) when N enter. decreasing marginal returns: A (N) < 0. Each fisher s opportunity cost (boat, lost wages) of entering is B > 0. Economics 336 (Toronto) Lecture 19: Common property resources 2 / 19
Equilibrium and efficiency If there is no limit on entry to the fishery, entry continues as long as A(N) > B, and equilibrium number of fishers satisfies A(N f ) = B. Efficient allocation maximizes social surplus: implying first-order condition Since A < 0, it follows N < N f. max N[A(N) B] A(N ) = B N A (N ) Interpretation: NA (N) is the external cost imposed on others by entry each new entrant imposes on the other N fishers. Implication: limit access to fishery to internalize this congestion cost. e.g. charge a license fee Economics 336 (Toronto) Lecture 19: Common property resources 3 / 19
Common property resource: graphical approach total catch NB T(N)=NA(N) N* f N N Economics 336 (Toronto) Lecture 19: Common property resources 4 / 19
A modern application: Traffic congestion In modern times, most agricultural and natural resources are privately owned, which should eliminate the tragedy of the commons. In modern, urban societies, most people still deal with a common property resource every day: roads. Because road tolls are rare in Canada, roads are a common property resource. The phenomenon of overexploitation is obvious: traffic congestion. The costs of congestion are potentially large. In 2006, the average worker in Toronto CMA commuted 9.4 km to work; it is twice that in outlying suburbs like Pickering and Ajax. So the average worker spends 60-90 minutes a day commuting. You can think of this as tax on work time of 12-20 per cent. About half that time is traffic delays. We will study why congestion costs are inefficiently large, and potential solutions like building more roads or public transit. Economics 336 (Toronto) Lecture 19: Common property resources 5 / 19
Congestion externalities: An example Consider a simple example, based on Arnott and Small (1994). A suburb is inhabited by 1000 commuters who must travel downtown during the peak hour. The suburb and city are connected by two roads. Number of drivers on route i is N i where N 1 + N 2 = 1000. Travel time in minutes on Route 1 is T 1 (N 1 ) = 10 + 1 100 N 1 and on Route 2 is always 15 minutes: T 2 (N 2 ) = 15 So Route 1 is an expressway that is faster at low traffic volumes but which is subject to congestion. Route 2 is a local road system, or perhaps public transit. Economics 336 (Toronto) Lecture 19: Common property resources 6 / 19
Equilibrium commuting patterns Suppose that commuters choose route to minimize travel time. In equilibrium, how many commuters ˆN 1 choose the expressway? Route 1 is preferred by everyone when T 1 (N 1 ) < T 2 (N 2 ), so in equilibrium T 1 ( ˆN 1 ) = 10 + 1 100 ˆN 1 = 15 = T 2 or ˆN 1 = 500 so ˆN 2 = 500 too. Average travel time is 15 minutes on both routes, so total travel time is 15,000 minutes. Economics 336 (Toronto) Lecture 19: Common property resources 7 / 19
Efficient commuting patterns Is this the best we can do? Consider a central planner that assigns cars to min N 1 N 1 T 1 (N 1 ) + (1000 N 1 )T 2 The FOC is or or T 1 (N 1 ) + N 1 T 1 (N 1 ) = T 2 10 + 2 100 N 1 = 15 N 1 = 250 So, at the efficient allocation, T 1 = 12.5, T 2 = 15, and total travel time is 15, 000 250 2.5 = 14, 375 Economics 336 (Toronto) Lecture 19: Common property resources 8 / 19
What s going on here? Notice the difference between the equilibrium condition T 1 ( ˆN 1 ) = T 2 and the social optimum T 1 (N 1 ) + N 1 T 1 (N 1 ) = T 2 Acting independently, drivers on the expressway consider only the private cost of their choice, which is T 1. But the social planner recognizes that each marginal driver raises travel time for each other expressway driver by T 1, and there are N 1 other drivers. So the marginal external cost of an expressway driver is N 1 T 1 (N 1). Economics 336 (Toronto) Lecture 19: Common property resources 9 / 19
Optimal highway tolls There is a simple way to decentralize the optimum. Suppose that drivers value each minute of time at price w. Charge an expressway toll f = N1 T 1 (N 1 )w = 2.5w Now drivers choose route 1 iff T 1 + f T 2, which leads to the efficient outcome. The toll is a Pigouvian tax on congestion costs. Note that everyone s total cost (time plus tolls) remains 15 minutes. But the government can redistribute revenue to all 1000 commuters equally, making everyone better off. In this example, a Pigouvian toll creates a Pareto improvement. Economics 336 (Toronto) Lecture 19: Common property resources 10 / 19
Road pricing in practice Highway tolls have long been used in some countries, but mainly as a way to pay for physical infrastructure, not for pricing congestion externalities. Recent technological changes have led to the advent of congestion charges on existing road infrastructure in a number of cities. Watch this video Economics 336 (Toronto) Lecture 19: Common property resources 11 / 19
The Pigou-Knight-Downs paradox Suppose that society did not want to use tolls. Can we solve the congestion problem just by building more roads? To model road capacity, suppose that travel time on route 1 is now T 1 (N 1 ) = 10 + 10 N 1 C 1 where capacity C 1 is defined as the level of traffic flow at which average speed drops to half the maximum. T 2 = 15 as before. If C 1 = 1000 then T 1 = 10 + 1 100 N 1 as in the first example. What happens if capacity is increased, say to C 1 = 1500? Since travel time drops on route 1, commuters switch from route 2. As long as C 1 2000, the equilibrium is T 1 ( ˆN 1 ) = 10 + 10 ˆN 1 C 1 = 15 = T 2 Economics 336 (Toronto) Lecture 19: Common property resources 12 / 19
At the new equilibrium, ˆN 1 = 1 2 C 1 as long as C 1 2000, so there is some traffic on route 2. Everyone s travel time remains 15 minutes. The new capacity does not decrease anyone s travel time, until there is enough capacity to handle all traffic. This is the Pigou-Knight-Downs paradox. The problem is that there is latent demand for a free highway. Some potential expressway commuters avoid the highway due to congestion costs, taking other routes (or foregoing low-value trips). Expanding road supply therefore creates its own demand, which in turn eliminates the benefits of additional capacity. Economics 336 (Toronto) Lecture 19: Common property resources 13 / 19
Extensions The phenomenon of latent demand is a key one in transportation planning. Planners now generally believe that adding capacity cannot solve congestion. Other aspects: Incorporating land use effects: In the long run, adding capacity also increases the value of development land in the periphery, which further increases road demand. So roads contribute to urban sprawl. Public transit: We could add a third route to the model, a commuter train. While this would displace some cars, the reduced congestion would cause some route 2 drivers to switch to route 1, making it congested again. Economics 336 (Toronto) Lecture 19: Common property resources 14 / 19
Downs-Thomson paradox Suppose that frequency of public transit (Route 2) is increased when demand increases, because of economies of scale. Say T 1 (N 1 ) = 10 + 10 N 1 C 1 T 2 (N 2 ) = 18 N 2 250 = 14 + N 1 250 Now the equilibrium has ˆN 1 = 1000C 1 2500 C 1 T 1 ( ˆN 10 1 ) = 10 + 2.5 C 1 /1000 Adding road capacity now increases travel time. There is some evidence this happens in reality. When a new commuter train is added, existing bus service is often reduced. Some previous bus passengers switch to driving, which more than offsets reduced road demand due to the train. Economics 336 (Toronto) Lecture 19: Common property resources 15 / 19
Braess paradox Suburb A and destination B are now connected by two routes 1 and 2 each passing through its own congested node (a highway interchange). A C 10+N c /100 20 D 20 10+N d /100 Route 1 Route 2 Travel time is T i (N i ) = 30 + N i /100 on each route. In equilibrium ˆN 1 = ˆN 2 = 500 and ˆT 1 = ˆT 2 = 35. B Economics 336 (Toronto) Lecture 19: Common property resources 16 / 19
Now suppose the two nodes are connected, where the connection takes 2 minutes travel time. This creates a new, potentially faster route 3: A C 10+N c /100 20 Route 3 D 20 10+N d /100 Route 1 Route 2 B Economics 336 (Toronto) Lecture 19: Common property resources 17 / 19
Travel times are T 1 (N) = 30 + N 1 + N 3 100 T 2 (N) = 30 + N 2 + N 3 100 T 3 (N) = 22 + N 1 + N 2 + 2N 3 100 In equilibrium, T 1 = T 2 implies N 1 = N 2 = N, and T 1 = T 3 implies 30 + N + N 3 100 = 22 + 2( N + N 3 ) 100 = N + N 3 = 800 Since 2 N + N 3 = 1000 we have so T 1 = T 2 = T 3 = 38. N = 200 = N 3 = 600 Economics 336 (Toronto) Lecture 19: Common property resources 18 / 19
So another perverse result: Adding linkages to the road network can cause travel time to increase for everyone. This is the Braess paradox. The problem is externalities at the congested nodes. Adding the linkage increases the number of commuters through each node, and they ignore the impact of their decision on those taking route 1 or route 2. The faster the connection between C and D, the more people take it, and the slower is their trip. This idea is behind the recent closure of Broadway in Manhattan. Because it runs diagonally across the grid, it creates many complex intersections, slowing down traffic on other streets. Planners expected closure to increase speeds on Seventh Avenue by 17 per cent. (But it didn t.) Economics 336 (Toronto) Lecture 19: Common property resources 19 / 19