Lecture 5- Road Congestion and City Structure

Question 1

How many homogenous zones should we think of a city as?

Answer 1

3 islands

Question 2

On the bridges separating the islands, is there congestion?

Answer 2

There is congestion on the bridges separating the islands.

Question 3

What is urban sprawl?

Answer 3

Urban sprawl, also called sprawl or suburban sprawl, is the rapid expansion of the geographic extent of cities and towns, often characterized by low-density residential housing, single-use zoning, and increased reliance on the private automobile for transportation.

Question 4

What is urban sprawl in part caused by?

Answer 4

The need to accommodate a rising urban population; in many metropolitan areas, it results from a desire for increased living space and other residential amenities.

Question 5

What has urban sprawl also been linked to?

Answer 5

Urban sprawl has been linked to increased energy use, pollution, destruction of green areas, traffic congestion and a decline in community distinctiveness and cohesiveness

Question 6

What is the general link between urban development densities and petroleum consumption?

Answer 6

More dense areas use less petroleum on average than less dense areas.

Question 7

What do we allow to happen when we consider a city in a 3 island model?

Answer 7

This simplified setting allows to include congestion in the monocentric model in a tractable way.

Question 8

What are the 3 islands called?

Answer 8

• Central
• Midtown
• Suburban

Question 9

Where is the vacant land?

Answer 9

On the right of the suburban island

Question 10

What are the 2 bridges called?

Answer 10

• Midtown bridge
• Suburban bridge

Question 11

Where is the CBD?

Answer 11

On the left of the central island.

Question 12

What is the exogenous population size?

Answer 12

𝑁 = 𝑛𝑐 + 𝑛𝑚 + 𝑛s

Question 13

How does everyone commute to the CBD?

Answer 13

By using cars

Question 14

What is true about the sizes of the Central and Midtown islands?

Answer 14

They are both limited in size, which is normalised to one

Question 15

What is true about the size of land in the suburban region? What is true about its elasticity of supply of land?

Answer 15

In the Suburban island, there is always free space, and the supply of land is perfectly elastic (ie there is always land available at a given price.) We assume that the price is set to zero.

Question 16

What is the travel cost on the suburban bridge?

Answer 16

𝑡𝑠(𝑛𝑠)

Question 17

What is the travel cost on the midtown bridge?

Answer 17

𝑡𝑚(𝑛𝑚 + 𝑛𝑠)

Question 18

What is the utility function?

Answer 18

• 𝑣 (𝑞𝑙) + 𝑒𝑙 where 𝑙 = 𝑐, 𝑚, s
• 𝑣 . increasing and concave
• 𝑞𝑙 is the size of individual dwelling on island 𝑙

Question 19

What happens to the proceeds of the tolls?

Answer 19

They are reinjected back into society

Question 20

What is the total cost from travelling from midtown to central/

Answer 20

𝜏𝑚 + 𝑡 (𝑛𝑠 + 𝑛𝑚)

Question 21

Does everyone rent or own their house?

Answer 21

They all rent, at either 𝑟𝑐 , 𝑟𝑚 and 𝑟s = 0 (exogneous in S)

Question 22

What is the individual budget constraint for someone residing in S?

Answer 22

𝑦 + 𝐿 = 𝑒𝑠 + 𝜏𝑚 + 𝑡𝑚(𝑛𝑠 + 𝑛𝑚) + 𝜏𝑠 + 𝑡(𝑛s)

Question 23

What is the individual budget constraint for someone residing in M?

Answer 23

𝑦 + 𝐿 = 𝑒𝑚 + 𝑟𝑚.𝑞𝑚 + 𝜏𝑚 + 𝑡(𝑛𝑠 + 𝑛𝑚)

Question 24

What is the individual budget constraint for someone residing in C?

Answer 24

𝑦 + 𝐿 = 𝑒𝑐 + 𝑟𝑐.𝑞𝑐

Question 25

What does a traffic cost represent?

Answer 25

Congestion

Question 26

What is the first-best toll policy?

Answer 26

𝜏𝑚 >0, 𝜏𝑠 >0
ie all links in the network are tolled

Question 27

What is the Cordon toll (second best policy?)

Answer 27

𝜏𝑚 > 𝜏𝑠 = 0
ie only the link between Midtown and Central is tolled

Question 28

List the 3 equilibrium conditions:

Answer 28

• Utility of an individual invariant with location (as in the standard monocentric model)
  𝑈𝑐 = 𝑈𝑚 = 𝑈𝑠
• Space on C and M has to be fully utilized
  𝑛c𝑞c = 1 𝑛𝑚𝑞𝑚 = 1
• Utility maximization (housing/land size)
  𝑣′(𝑞𝑖) = 𝑟𝑖 𝑖 = 𝑐,𝑚,𝑠

Question 29

What does 𝑣′(𝑞𝑖) = 𝑟𝑖 infer?

Answer 29

Marginal utility = rental price

Question 30

Is v(.) increasing and convex?

Answer 30

It is increasing, but concave not convex

Question 31

What is the surplus gained by an individual of living in the middle area rather than central equal to?

Answer 31

The total transportation costs of commuting to the central region

Question 32

List the order of islands in terms of rent prices.

Answer 32

𝑟𝑠 < 𝑟𝑚 < 𝑟c

Question 33

How do we know that 𝑞𝑐 < 𝑞𝑚 < 𝑞s

Answer 33

As individuals living further away must gain extra utility from their housing, as although they have to pay transport costs, the condition 𝑈𝑐 = 𝑈𝑚 = 𝑈𝑠 must be satisfied.

Question 34

Is the 3-island model consistent with the monocentric model in its basic predictions? Why?

Answer 34

Yes it is, and this is irrespective of tolls:
- Dwelling size increases with distance from centre
- Unit price of land (housing) decreases with distance from centre
- Density of population decreases with distance from centre

Question 35

What happens to the central island when the toll on the midtown bridge rises?

Answer 35

Price of land in the central region will rise, as the region is more desirable to avoid the rise in travel costs.

Question 36

What happens to the midtown island when the toll on the midtown bridge rises?

Answer 36

M does npt becmoe more attractive to anyone in C or S (who would have to pay that toll anyway), so the population of M falls, as does the price of its land.

Question 37

What happens to the suburban island’s population when the toll on the midtown bridge rises?

Answer 37

The effect is generally ambiguous, however under reasonable assumptions on functions and parameters, the effect is negative (reduce suburban population and city size).

Question 38

What happens to the Midtown and Central islands when the suburban bridge toll is implemented?

Answer 38

• The suburban toll raises land prices and population in both central and midtown
  zones (so, it also makes those areas more densely populated)
• So this toll reduces population in suburban zone, 𝑛𝑠 = 𝑁 − 𝑛𝑐 − 𝑛𝑚, and the
  geographical size of the city (sprawl)

Question 39

What is the formula for maximising welfare with respect to tolls? How else can this be written?

Answer 39

• 𝑀𝑎𝑥 𝜏𝑚,𝜏𝑠𝑊 = 𝑛𝑐𝑈𝑐 + 𝑛𝑚𝑈𝑚 + 𝑛𝑠𝑈𝑠
  Which using the equal utilities equilibrium condition can be written as
  𝑀𝑎𝑥𝜏𝑚,𝜏𝑠 𝑁(𝑣(𝑞𝑐) − 𝑟𝑐𝑞𝑐 + L)

Question 40

What is the optimal first-best toll on the Midtown bridge?

Answer 40

𝜏𝑚𝐹𝐵 = 𝑡′𝑚 ∙ (𝑛𝑠 + 𝑛𝑚)

Question 41

What is the optimal first-best toll on the suburban bridge?

Answer 41

𝜏s𝐹𝐵 = 𝑡′𝑚 ∙ 𝑛𝑠

Question 42

Under first-best pricing. What do the tolls on each bridge equall?

Answer 42

The tolls on each bridge = the marginal external cost of congestion

Question 43

What happens without tolls?

Answer 43

We have excessive congestion on both bridges, as external costs of congestion are ignored by travellers on midtown and suburban bridge.

Question 44

What happens to urban sprawl with no tolls?

Answer 44

The suburban island gets too large as there is a lack of incentive to live in C or M, so urban sprawl becomes excessive.

Question 45

How do tolls help deal with congestion and urban sprawl?

Answer 45

Tolls make people internalise the external cost of congestion, making more individuals live closer to the centre, reducing urban sprawl.

Question 46

Why is second-best pricing usually applied as a cordon around a city centre?

Answer 46

It is more practical this way.

Question 47

What is the difference between the second-best pricing compared to in lecture 2?

Answer 47

Here the roads are in a series, whereas in lecture 2 they were parallel.

Question 48

What is the expression for the optimal cordon toll?

Answer 48

𝜏𝑆𝐵 = 𝑡′𝑚 (𝑛𝑚 + 𝑛𝑠) + 𝑡′𝑠𝑛𝑠 . 𝑞𝑚/𝑞𝑚 − 𝑡′𝑠𝑍𝑚

Question 49

What does 𝑞𝑚/𝑞𝑚 − 𝑡′𝑠𝑍𝑚 > 0 mean?

Answer 49

This toll is higher than the MEC of congestion on the midtown road

Question 50

Why is the cordon toll adjusted?

Answer 50

To make suburban commuters internalise the cost of their congestion, however, this isn’t fully adjusted as 𝑞𝑚/𝑞𝑚 − 𝑡′𝑠𝑍𝑚 < 1 to reflect the fact that midtown commuters do not cause suburban congestion.

Question 51

Are the 2 road links complements?

Answer 51

Yes, for suburban commuters they are, as they cannot use the midtown road without the suburban road.

Question 52

Why is the 2 roads being complements relevant?

Answer 52

Unlike if they were substitutes, raising the toll on 1 road helps reduce the DWL on the other road.

Question 53

What are the effects of the second best cordon toll on city size, density and
housing market?

Answer 53

• Make centre more dense and expensive
• Make midtown less dense and expensive
• Ambiguous effect on S and city size
• Though typically the effect is to reduce city size overall, compared to no
  toll at all