SOLUTIONS University of Victoria

Economics 312

Urban Land Economics Martin Farnham Problem Set #2 Note: Answer each question as clearly and concisely as possible. Use of diagrams, where appropriate, is strongly encouraged. Problem sets are ungraded. However, developing a careful understanding of the problems will, on average, dramatically improve your exam performance. True/False/Uncertain For each question, state clearly whether you find the statement to be true, false, or uncertain. Then provide a clear explanation. Answers without explanation will be given zero points. 1) Improvements in local schools will benefit renters. False. To the extent that the improvements in local school are attractive to renters, this will increase the willingness to pay of renters for locating near the local schools. Therefore, landlords will benefit from the improvements in the schools, because they will be able to increase their rents. Note that if renters sign long-term contracts, they can benefit from the improvements without paying higher rent (until the contract runs out). 2) The leftover principle says that firms’ overall profits will rise when their non-rent costs of production fall. False. When non-rent costs of production fall, firm’s willingness to pay for rental space rises. Since the landlord will rent to the highest bidder, firms will pay their entire willingness to pay to the landlord. The “leftover principle” says that whatever revenues are leftover after non-rent costs are paid for will go to the landlord. Now there are some exceptions here. If you are a firm with a technology that no one else has (say you’re the lowest cost producer), you may be able to win the bidding war for space without bidding your entire willingness to pay. In this case you get to keep some of the leftovers too. 3) If monthly commuting costs are $100 per roundtrip km and a household pays monthly rent of $2 per square foot and lives in a 1000 sq. ft. apartment, the housing price gradient is 10% The slope of the housing price curve is given by

ΔPΔx

= −t

h(x)

where x denotes distance from jobs, t denotes commuting costs per km per month, P denotes the price of housing (per sq ft), and h denotes the size of the house (in sq feet). The slope is given in dollars per km. The slope in this case is -$0.10. This means that the cost of housing per square foot falls by 10 cents per km further away from the employment district. This compensates the household for the greater commuting costs. Another measure of how housing prices change with commuting distance is the housing price gradient (the percentage change in house price per km). To get the housing price gradient, we just want to take the slope of the housing price curve and divide by the base price level, P. This is just the housing price where you start. So the house price gradient is given by ΔP / ΔxP

= −t

P *h(x)*100%

So the housing price gradient is -100/2(1000)*100%=[-$0.10/$2]*100%=-5% This tells us that if commuting costs are $100 per roundtrip km, the price per square foot of a 1000 square foot house must fall by 5% per km that we move away from the employment district, in order to compensate for the increased commuting costs. 4) Firms face a basic tradeoff between freight costs and labour costs in the monocentric model. True. Firms could pay their workers a lower wage if they located nearer the workers. The workers would accept the lower wage in exchange for lower commuting costs. But freight costs to transport output to the central city rail station would be too high if firms moved too close to workers. 5) Population density gradients have increased over time in cities around the world. False. Population density gradient have fallen, as cities have become more decentralized. This is largely due to the introduction of the car. Short Answers 1) Consider  a  region  with  2  cities.    Everyone  lives  in  City  A  or  in  City  B  and  people  can  migrate  between  the  two  cities.    Utility  per  worker  curves  are  given  for  each  city  below.    In  the  diagram  below  we  see  the  region  in  initial  equilibrium.    

Consider  a  change  in  policy  in  City  A  that  reduces  social  welfare  for  residents  of  City  A  instantaneously  (at  all  levels  of  population).    Assume  that  the  utility  per  worker  curve  of  City  A  remains  above  that  of  City  B  after  the  policy  change.  Illustrate  the  new  equilibrium  after  the  wasteful  policy  is  implemented  in  City  A.        The  new  equilibrium  has  City  A  at  point  e  on  its  new  curve.    It  has  City  B  at  point  d  on  its  original  curve.    Which  city  grows  and  which  shrinks?    City  B  grows  and  City  A  shrinks.    What  happens  to  utility  in  city  B  as  a  result  of  the  poor  policy  change  in  City  A?   Utility per worker falls in City B. This is due to people moving from City A to City B and making City B more crowded and expensive. Interestingly, poor policy choices in City A hurts City B through equilibrium migration.

2) Illustrate the following situations using a labour supply and demand diagram. You may find it useful to state the assumptions on which you base your diagram. a) A city initiates a reform that improves the quality of education, at no extra cost. This should make the city a more attractive place to live for families. This may cause people to move in from the suburbs or from other cities. It’s unlikely to have much effect on the labour demand side (in the very long run, it could increase the productivity of workers, but this would take many years). An outward shift (right) of the LS curve should result. Employment will rise and wages will fall. b) A city increases the property tax on homeowners. Then it takes the money and throws it in the ocean. Ah, profligate waste! Here the property tax does no one any good by being spent on something like schools. It just raises the cost of living in the city vs. living some other place. The tax should cause people to move out of the city, in search of lower tax environments. This can be represented by a shift back (left) in the labour supply curve. Equilibrium wage will rise, while employment will fall.

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c) A city uses a grant from the Federal government to improve its road system which means people and goods can move more easily in and out of the city. If the grant comes from the Federal government, it costs the residents essentially nothing. But it improves the commuting time for city residents, which makes the city a more desirable place to live. So it shifts the LS curve out (right). It also improves the infrastructure available to firms, which should lower their cost of doing business in the city. This shifts out the labour demand curve (both by attracting new business in, and by making existing businesses want to hire more people). The two shifts cause employment to increase unambiguously. It’s uncertain whether wage will rise or fall (depends on relative magnitudes of the shifts in LD and LS. d) A city uses a Federal grant to institute a job retraining program that raises the productivity of workers by developing skills that are useful to employers. By increasing worker productivity, this should increase labour demand. It’s possible the job retraining program would encourage some workers to move into the city. The overall effect will depend on the relative magnitude of the shifts of the curves.

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e) A city reduces its restrictions on polluters. This leads to increased smog. This policy change should have two effects. It lowers the cost of doing business which should increase labour demand. It also makes the city a less pleasant place to live. This should decrease labour supply. The overall effect should be to cause wages to rise unambiguously. The overall effect on employment will be ambiguous. It will be determined by the relative magnitude of the shifts. 3) For the following questions, use a supply and demand diagram to illustrate the effects on employment and wage of an increase in exports. a) First, assume that labour is perfectly mobile, so that any increase in the wage in your city will immediately be met with an inflow of labour from other cities. Decompose the demand shift into a direct effect of the export demand increase plus a multiplier effect.

LS

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The first labour demand shift shows the direct effect of the increase in export demand. The second shows the multiplier effect. Due to the perfectly elastic LS, no wage increase results from from the demand increase. But large employment effects result. b) Now assume the reverse case of (a). A negative demand shock reduces labour demand. Show the direct effect and the multiplier effect. Here the decline in export demand gets multiplied through the urban economy by the reduction in income that results in less demand for local goods. This leads employment in the local goods sector to fall further. Due to perfectly elastic LS, there is no resulting decline in the wage. People migrate out of the city immediately in response to any downward movement in the wage, so the wage remains fixed. All effects are employment effects (negative in this case). c) Now assume that there are also agglomeration externalities, so that productivity falls as the city gets smaller. What does this do to the LD curve in (b)? Illustrate this.

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agglom

This would further multiply the negative demand shock. As people move out of the city, positive agglomeration externalities are reduced. Remember that firms may benefit from locating near other firms for a variety of reasons. As the city shrinks (or at least as its firms shrink) these agglomeration externalities will be reduced, which will further decrease demand for labour. Reduction of agglomeration externalities increases firms costs, and so shifts in labour demand. d) Now go back to the case in (a). What happens to employment and wage if you assume the same export demand shock as in (a) but assume that LS is upward sloping? How does the result differ? Employment increases are smaller now, and outward LD shifts now result in wage increases. The more inelastic LS is, the smaller the employment increase will be for a given increase in LD. 4) Assume there is no inflation and that annual rents are constant over time into the infinite future. Find the market value of an acre of land under the following assumptions: a) annual rent is $5000 and the interest rate is 5%. V=5000/0.05=100,000 b) annual rent is $3000 and the interest rate is 10%. V=3000/0.1=30,000 c) annual rent is $1000 and the interest rate is 2%. V=1000/0.02=50,000 5) Explain why the manufacturer’s bid-rent function might be convex. What is the intuition behind the shape of the bid-rent function? Why does it slope down? Why is it curved the way it is?

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Bid-rent functions slope down for users who see an advantage to being close to some location (a highway for manufacturers; the central business district for office firms). To be more precise, the bid-rent function slopes down as one moves away from this key location. This is because firms are willing to pay more for land that is most profitable (in a pre-rent profits sense). If the manufacturer’s bid-rent is convex, it is due to factor substitutability. If a producer can substitute away from land and toward non-land inputs in its production process, it can utilize smaller pieces of land and thus save on rent. A producer who can substitute away from land can then outbid (on a per acre basis) a producer who can’t. As demand for land gets higher near highways, factor substitution leads to manufacturers using smaller plots of land and thus offering more per acre. This causes the bid-rent to be convex. The basic intuition to the shape of the bid-rent is that firms are willing to pay more for land rent as they get closer to a key location (a highway, in the case of manufacturers) and that the willingness to pay increases at an increasing rate if firms have the ability to cram onto smaller and smaller plots through factor substitution. Note that there’s more opportunity for office firms and residential developers to engage in factor substitution than there is for manufacturers to do so. This is because manufacturing technology (especially heavy manufacturing) typically calls for single-story production facilities. This limits the degree of factor substitution that manufacturers can engage in. 6) Explain why the residential bid-rent function might be convex. What is the intuition behind the shape of the bid-rent function? Why does it slope down? Why is it curved the way it is? There are two reasons why the residential bid-rent function can be convex: factor substitutability and consumer substitutability. If developers can build up on a piece of land in order to save on rent costs, their willingness to pay (per acre) for land will rise. They can do more with an acre of land, after all, so its value to them goes up. This leads to convexity of the bid-rent function as described with firms above. There’s another element of the residential bid-rent function. The residential bid-rent is partly determined by consumers’ willingness to pay for housing. This can be represented by the housing price function. (see Fig 8-6 in text) The housing price function is linear if consumers don’t substitute to smaller houses as housing prices rise. If consumers substitute to smaller units as housing prices (per square foot) rise, this leads to convexity in the housing price function. The residential bid rent is derived from the housing price function (see Fig 8-7 in text). Therefore the convexity of the housing price function is reflected in the residential bid rent. It is further compounded by developer factor substitution. 7) Explain why office firms locate at the very center of the city in the monocentric model.

Office firms locate at the very center because they face the highest travel costs. Manufacturers face travel costs in the form of getting freight to the train station. But it’s even more costly to transport highly educated workers than it is to transport freight. This is because highly educated workers (office workers going off during the work day to meet with clients) are paid a high wage. So their time spent getting to meetings is very expensive for their employers at the office firms. This makes these firms particularly willing to locate near the center (where the meetings of those workers take place). Suppose an office firm did all its consulting for a manufacturer on the west side of the Central Business District. Then it probably would want to locate next to that manufacturer. If, on the other hand, it consulted for manufacturers all over the CBD, it might make sense to stay in middle (this would be the median location). 8) Discuss the divergence between the predictions of the monocentric model and what we observe of modern cities today. Why are modern cities so different from the monocentric representation of 19th century cities? The monocentric model predicts a highly centralized city with all employment occurring at the center, and segregation of residential areas from business areas. Workers live outside the core city in the suburbs and commute into the center each day. In the modern city, employment is distributed throughout the metro area, not just located in the center. Instead of having one center, modern cities tend to have a main center and several suburban subcenters. Employees often commute from one suburb to another rather than commuting from suburbs into the main center. Why are modern cities so different? For one, the transportation system is different. The replacement of the central rail station with trucks for intra- and inter-city transit made it less essential for firms to locate near the train station. Now firms locate along highways which lowers commuting costs for their workers (and hence the wages firms must pay), and lowers procurement and distribution costs. Also, the car allows people to live further away from work. This leads to people spreading out more, thus lowering the density gradient. Finally, technology associated with workplaces has changed in important ways. Manufacturing facilities using heavy equipment and large assembly lines that can’t operate well above ground floor. This has lead manufacturers to shift toward ground floor plants, which are better suited to location outside of the center city where land is cheaper. Also, telecommunications have reduced the need for all office firm operations to be located near the center city. Telecommunications reduces the need for some types of face to face contact and therefore for location near the city center. These are some of the major causes of decentralization of cities over the past century.

9) Consider the graph of bid-rent functions (for office firms, manufacturing firms, office workers, manufacturing workers, and farmers) shown in the lecture notes (end of Land Rent section) and on page 143 of the text. Redo this diagram using the assumption that the highway runs through the center of the city. Assume that office wages are higher than manufacturing wages. Explain how and why the diagram differs.

The diagram differs because the highway is located at the city center (at the origin in this graph). This means both office firms AND manufacturing firms want to be located near the city centre. This manufacturing workers and office workers also want to be near the city centre. The following assumptions lead to the above land-use pattern: 1) Travel costs for office firms are greater than transport costs for manufacturing firms (because people are more costly to transport than goods). 2) Office workers face higher commuting costs than manufacturing workers (because they are paid higher wages, the opportunity cost of their commuting time is higher). 3) All land users value land near the city centre more highly than farmers value that land. This makes sense because a field is a field, whether it is near the city centre or far away (assuming the farmer doesn’t need to ship his/her crops downtown (imagine they serve the world market rather than the local market). As a result of these assumptions, office firms occupy the city center. The next ring out is occupied by manufacturers. Then office workers, then manufacturing workers, then farmers. This result basically looks the same as the result for the model of the monocentric city described in class. This is because the highway has replaced the railroad station in the center of town. All other aspects of the model are the same. 10) Suppose that the city of Victoria receives a federal grant to offer 1 free whale-watching tour per resident per year. It is expected that this grant will be in place into the infinite future. Each person values a whale-watching trip at $50. Suppose each residence in Victoria houses 2 people (hence each residence essentially comes with 2 free whale-watching trips each year). Assume the housing stock in Victoria is fixed (it can’t expand). a) By how much will annual rents increase, in equilibrium, in response to the introduction of this grant? If each individual values a whale-watching trip at $50, then each person’s willingness to pay for a place in a Victoria residence goes up by $50 per year. This means the collective willingness to pay for each residence in Victoria goes up by $100 per year. By the leftover principle, we should expect equilibrium rents to rise by $100 per year per residence. b) By how much will the value of each residence in the city rise, in response to the introduction of this grant? The market value of a residence should be equal to the present discounted value of the stream of future rents earned on the residence. We don’t need to know the market value of a residence before the grant to know how market value changes. If rents rise by $100 per year, then property values should rise by 100/r.

c) In dollar terms, how much will each renter benefit from the grant? Assume that rental contracts are continuously updated to reflect current market rents. Each renter gets a trip that they value at $50. Each renter pays an extra $50 in rent. As a result, consumer surplus for renters does not change. d) In dollar terms, how much will a residence owner benefit from the grant? If the owner rents out the residence, assume that rental contracts are continuously updated to reflect current market rents. The grant is capitalized into the value of the home, as seen in (b). Owners see the value of their residence rise by $100/r. This is the dollar benefit that owners get from the grant. e) Why does the nature of rental contracts (e.g. whether they are continuously updated or updated once a year) matter for the answers to (c) and (d)? If rental contracts adjust continuously, then rents will adjust upward immediately upon announcement of the policy. In that case, renters capture none of the net benefits of the grant. Suppose rental contracts adjust once every year. Then renters can collect one free whale-watching trip before their rent rises. If rental contracts adjust once every two years, then renters can collect two free trips before their rent rises. The longer it takes for rental contracts to adjust, the more the benefits (or costs in the case of a policy that hurts local residents) of a policy change accrue to renters, and the less they accrue to owners.