# Urban Economics, Questionnaire Example

**Exercise 4.1**

Suppose that landowners have the power to restrict *x*, the distance tothe edge of the city, in order to increase the land rent they earn. Supposethat, with no restriction, the urban land rent function is given by *r *=100 – *x*, where *x *is distance in blocks to the CBD. Suppose that agriculturalland rent *r *A is equal to 20.

Solution, r = 100 – x, but r = rA + b

With no restriction, then r ≥r A, hence r = 20. Therefore, r = 100 – x but r = r A = 20, hence, x = 100 – 20 = 80 blocks,

Given the restriction of 65 block, then r = 105 – 65 = 40, at 65 blocks r = 40

Thus, with the restriction, the urban sprawl is controlled.

In the figure, X0 = 80, without restriction, X1 = 65 with restriction and r = 20 without restriction and r = 40 with restriction.

The above diagram shows the area corresponding to land rent loss resulting from the restriction. This is indicated by area shown by V.

Area is equal to 65 * 40 = 2600 with restriction and 20 * 80 = 1600 without restriction.

Therefore, net loss from imposing restriction is 2600 – 1600 = 1000

Given r = 110 – x and x = 50, then land rent rises to 110 –50 = 60

Then area 50 * 60 = 3000.

The further restriction is beneficial to the landlords since the total land rent rises thus preventing further urban sprawl. This means that the restriction will be imposed.

The answer is different since further restriction leads to an increase in total land rent. This aims at restricting further urban sprawl.

**Exercise 5.1**

Suppose there are three potential users of a freeway: Mr. 1, Mr. 2, andMr. 3. The cost of the best alternative route for each commuter is asfollows:

Commuter Alternate cost

Mr.1 $7

Mr. 2 $5

Mr. 3 $3

The average cost AC of using the freeway (i.e., the cost per car) as a function of traffic volume *T *is as follows:

*T *AC

1 $2

2 $5

3 $9

Using this information, answer the following questions:

Find the equilibrium allocation of traffic between the freeway andalternate routes.

The equilibrium will be 2 clusters, that is, Mr. 1 and Mr. 2 will use the freeway since the alternate costs are equal or greater than the AC. Only Mr. 3 will use the alternate route since the freeway has 2 clusters at an average cost of $5 greater than $3.

On freeway On alternate routes total commuting cost

No one Mr. 1, Mr. 2, Mr. 3 = 7 + 5+ 3 = $15

Mr. 1 Mr. 2, Mr. 3 = 2 + 5 +3 =$ 10

Mr. 1, Mr. 2 Mr. 3 = 5+ 3 = $8

Mr. 1, Mr. 2, Mr. 3 No one = $9

The socially optimal allocation is one in which total commuting cost cannot be reduced by switching any commuter between routes.Mr. 1, Mr. 2 on the freeway, and Mr. 3 on the alternate route give the socially optimal allocation of traffic. This equal to 5+ 3 = $8. This allocation is socially optimal. This is similar to the equilibrium allocation of traffic.

**Exercise 5.2**

The relationship betweenbenefit per person (measured in utils) and the number of people in thehot tub (denoted by *T *) is

AB = 2 + 8 *T *– *T*^{2}, where AB denotes “averagebenefit.”

Average Benefits AB Total Benefits( T*AB) MB

When T = 1 AB = 2 + 8(1) – 1 = 9 1*9 = 9 9

T = 2, AB = 2 + 8(2) – 4 = 14 2*14 = 28 19

T = 3, AB = 2+ 8(3) – 9 = 17 3* 17 = 51 23

T = 4, AB = 2 + 8(4) – 16 = 18 4* 18 = 72 21

T = 5, AB = 2 + 8(5) – 25 = 17 5 *17 = 85 13

T = 6, AB = 2 + 8(6) – 36 = 14 6* 14 = 84 – 1

T = 7, AB = 2 + 8(7) – 49 = 9 7* 9 = 63 – 21

T = 8, AB = 2+ 8(8) – 64 = 2 8* 2 = 16 – 47

T = 9, AB = 2 + 8(9) – 81= – 7 9 * – 7 = -63 – 79

T = 10, AB = 2 + 8(10) – 100 = – 18 10* – 18 = – 180 – 117

Total benefit is given by AB*T

MB at *T *= *T = *– 117, at the entry of the last person

The equilibrium people using the hot tub equal to the highest average benefit. This is equal to 5 people with AB = 17utils.

The optimal size is 3 people using the hot tub. This differs because the MB at 3 users is high for all individuals from the equilibrium position.

The average benefit AB = 17, Therefore, the equilibrium group of hot tub users is 8 which areCheryl, David, Emily, Frank, Gail, Henry, Ivan, and Juliet since their benefits from the video game are greater than the benefits from the hot tub.

The total benefits for the bot tub users will be Adam and Brigit = 28 from (a)

For the video users, it is = 18 + 20 + 27+ 30+ 31+ 34+36+37 = 233

The equilibrium is at 5 with AB = 17 and MB = 13. Hence, the toll is AB – MB = 17 – 13 = 4. Thus the subsidy paid is 4 for the park.

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