http://www.econ.vt.edu/tideman/mbr.pdf

T. Nicolaus Tideman
February 1991

This paper describes systems for assigning rental value to land through markets for the use of land that are created and managed by government officials. The central idea of the paper is that the full rent of land can be collected while achieving an efficient allocation of land if the rent for improved sites is revised annually, based on offers for the use of similar unimproved sites for the current year. The efficient allocation of land requires neither the sale of land nor leases of long duration at fixed rents.

Two systems for assigning rental value to land are presented. One system employs a market in which land is actually turned over to bidders for their use, while the other employs a market in options to use land. While these systems are applicable in many settings, they are particularly applicable to current conditions in the Soviet Union, where land is being transferred from public to private management. The new manager of each site will receive the profit or bear the loss from production on that site. The main question under consideration concerns the process to be used to determine how much must be paid for the use of each site.

A standard response of economists to such questions is, "Sell everything to the highest bidder." In the context of land in the Soviet Union, this would mean selling, for a lump-sum amount, the right to use each site into the indefinite future. However, for a number of reasons, such an approach is less attractive than selling the right to use sites for rent to be paid annually.

• First, the person who can make best use of the site may be unable to match a purchase bid from someone else, because of unequal access to lending markets (due to inequalities in wealth, for example). Selling land-use rights for annual rent makes sites more available to users with relatively little in assets.
• Second, uncertainty about future political conditions would tend to depress purchase offers. Selling land-use rights for annual rent requires the state to accept the risk of future unfavorable public policies, and thereby tends to promote favorable future conditions.
• Third, the high degree of general uncertainty about future economic conditions can lead to offers of less than the expected value of opportunities, simply because people have no idea how to guess what the opportunities are worth. Selling land-use rights on an annual basis permits many of the future uncertainties to be resolved before bids must be made.
• Finally, the returns from future use of a site can more justly be claimed by future generations than by the present generation.

Selling land-use rights for annual rent allows each year's population to claim that year's rent. While the proceeds of sales can be invested for the benefit of future generations, not collecting the money in advance helps preserve the heritage of the future against political predations. For all of these reasons, I assume that what is desired is not a purchase price but a rental price to be assigned to each site for each year.

II. Renting Land One Year at a Time

The idea of renting a site to the highest bidder is conceptually more complex than selling it to the highest bidder, because different users can be expected to desire to use a site for different spans of time. Suppose that one person bids 1,000 rubles per year for 20 years and another bids 1,100 rubles per year for 30 years. Which bid is higher? To compare these bids, one must know what someone else would pay for the use of the site twenty years hence, and one must also know the appropriate discount rate for the next thirty years. These difficulties of comparing bids for different terms can be avoided by renting sites just one year at a time. This annual rent-setting procedure also guarantees future generations that they will not be disadvantaged by being bound by the terms of agreements in which they did not participate, and which disposed of their heritage for a small fraction of its value.

The idea of renting sites one year at a time may sound unpromising at first. The efficient use of land generally requires improvements that last for many years. How can entrepreneurs be expected to improve land if they are only permitted to rent it for one year at a time? The answer is that entrepreneurs can be expected to make durable improvements to land even under these circumstances, provided that they have the opportunity to continue to use the land they have improved, for a price that is determined in a fair and impartial manner.

The risks associated with uncertainties are customarily borne, for a price, by entrepreneurs. Uncertainty in the future price of using land, like uncertainty in the future price of any other input or product, can be accommodated by entrepreneurs.

The kind of uncertainty that will significantly deter potential investors is uncertainty associated with a climate that invites tax increases or other rule changes that extract additional money from investors, taking advantage of the fact that investment in durable improvements has occurred. To attract investment, governments must do what they can to commit themselves not to take confiscatory action once investment has occurred. This commitment is accomplished by settled rules, by public respect for property in a democratic system, and by a system of public finance that provides adequate revenue for necessary expenditures and curbs the tendency of politicians to spend excessively. The future rent of land can accordingly be uncertain without discouraging investment, provided that the process used to determine rent is settled and does not confiscate capital.

The question at issue can thus be restated as, "What process can be used to determine the rental value of land, one year at a time, in such a way that the full rent of land is collected, while at the same time insuring that entrepreneurs are not required to pay extra amounts by virtue of having made durable improvements to land?"

III. Defining "The Rent of Land"

To proceed, there must be agreement upon a definition of "the rent of land." From a conceptual or theoretical perspective, the rent of land is a residual. It is the difference between total revenue and costs other than land, when land is managed in such a way as to yield the greatest present value of net returns.

There are two practical difficulties with this definition. It yields an answer in terms of the present value of rent rather than an annual amount, and it does not provide an algorithm for public officials to follow. The two difficulties can both be overcome by using an operational concept of rent, defined in terms of what people are willing to pay. But before proceeding, it is useful to define some symbols.

Define V(0) as the present discounted value of the net returns from the use of a site from the present time onward, assuming that the site is now unimproved. Define V(1) as the present value, discounted to today, of the net returns from the use of the site from one year from now onward, assuming that the site will be unimproved one year from now. The rental value of the site for the year beginning today, R(0), may then be defined as V(0) - V(1). The logic of this definition is that the rent for a site for the current year is the most that anyone is willing to pay for use of the site for the current year, if use of the site after the current year is contingent upon payment of amounts that have a present value of V(1). Alternatively stated, rent for this year is what must be added to the value, as of today, of using a site from one year hence into the indefinite future, to produce a total equal to the value of using the site from today into the indefinite future.

This definition corresponds to the following operational conditions. Suppose that there are a number of identical sites, and suppose that people have the opportunity to bid for the use of one of these sites, under a rule that the use of the site will go to the highest bidder, who will be obliged to make an immediate payment equal to the amount offered by the second highest bidder.1 This payment will entitle the highest bidder to use the site for one year. The highest bidder will also have the option of using the site into the indefinite future, upon payments of annual amounts R(1) for use in the year beginning at time 1, R(2) for use in the year beginning at time 2, and so on, where the amounts R(1), R(2), ... will be determined by bids, under corresponding conditions, for identical sites in future years.

A bidder will reasonably expect that the sum of present values of the amounts R(1), R(2), ... will be V(1). If it is any less, an opportunity for profit has been left unexploited. And if it is any more, at least one bidder is heading for a loss. In these circumstances, a rational bidder who is able to put the land to its best use will offer V(0) - V(1) for use of the site for the year beginning at time 0. And this is the definition of the rent for the site for the year beginning at time 0.

The identification of one year as the period for which the rent applies in this analysis is arbitrary. A similar analysis can be undertaken using a month or a decade as the time period for the first rental payment. In practice, the appropriate period is chosen by balancing the higher administrative costs of frequent rent changes against the costs of inaccuracies in rent from less frequent rent changes.

IV. Consequences of Foreseeable Increases in Rental Value

If the value of a site is expected to rise in the future, bidders for the current use of the site will take that into account in their current offers. A potential user who wishes to make long-lasting improvements to the site, of a sort that cannot take full advantage of the future higher value of the site, will offer less for the site this year than if the site did not have future prospects that will raise future offers from competing uses. On the other hand, a potential user who wishes to use a site as something like a parking lot, requiring very little in durable improvements, will be virtually unconcerned with the prospect of higher future rents. Thus, to the extent that bidders can foresee future bids, sites where optimal use requires a change in use in the near future to achieve greatest efficiency will tend to be acquired by users who can relinquish use without suffering loss.

Of course, the future cannot be seen perfectly. People will sometimes experience losses as a result of unanticipated rises in the rent of the land under their durable improvements. To compensate for this risk, people who wish to use land for durable improvements will reduce, to some extent, their bids for land. They may also seek to purchase insurance against rent increases, just as they now purchase fire insurance.2 The practice of setting each year's rent at the amount that someone is willing to pay in that year for the use of a similar site ensures that some profit-oriented person (either the investor or the seller of insurance) has an incentive to estimate carefully the opportunity cost of a site in future years before the site is committed for a long span of time to a particular durable improvement.

V. Relaxing the Assumption of Identical Sites

The system described so far presumes that there are many identical sites, so that the rents offered for ones that are relinquished can reveal the value of the ones in continued use. In fact, every site has unique characteristics, especially in terms of the distances from other locations that are economically relevant, and often in other ways as well. Fortunately, the initial assumption of identical sites, which was made to explore the system, is not necessary. It is possible to make good estimates of the rental value of all land from information about the rental value of a small percentage of sites, because the rental value of land tends to be a smooth function of location. Rents are highest in the centers of cities, diminishing gradually with distance from the center. Rents are also higher in places with characteristics such as proximity to transit facilities or especially good views, and tend to be lower in places that are close to the source of a noxious smell or noise.

To take account of the differences among sites, the officials in charge of renting land can annually examine the results of recent auctions and construct a revised map of land rents. In constructing this map, the rent assigned to each site that is auctioned should be not the highest bid, but rather the second-highest bid, which represents the actual opportunity cost of a site, what it would be worth to have one more vacant site.

Consider an example to highlight this point. Suppose that the highest bid for a site is 1,000 rubles per year, the second-highest bid is 800 rubles per year, and an adjacent site, with virtually identical characteristics, is occupied by someone for whom continued use is worth 900 rubles per year. If charged 1,000 rubles, this person will relinquish the site, and the rent that can then be obtained for it will presumably be the 800 rubles offered for the adjacent site. Thus it is counterproductive to assign rent that is greater than the highest offer that is refused.

VI. Summary of the System Involving Actual Delivery of Land

To operate the system for determining annual rents from bids for sites that are actually delivered for use, the officials in charge of renting land simply auction every site that is relinquished by its previous user. According to the terms of the auction, the bid represents an offer of rent for the first year's use, with use of the site going to the highest bidder at a price of the second highest bid. The highest bidder also receives an option to continue to use the site into the indefinite future, upon payment of rent that will be determined, year by year, by rent maps constructed from bids in auctions that will be conducted in the same way.

The user of any site is permitted to terminate use at any time, provided that the site is restored to a condition of bare land. To protect against the abandonment of sites in a condition in which it is expensive to restore them to an unimproved state, anyone who wishes to build can be required to post a bond against the cost of demolition.

Any user of land is permitted to sell both the improvements on a site and the right to continue renting the site to anyone else, for any price on which they mutually agree. However, it can be expected that the prices that will secure agreement will reflect little if anything more than the value of the improvements, because the right to use sites can be obtained at auction just by offering a small amount more than what others are offering for the first year's rent.

Each year the land-managing bureaucracy will construct a new map of land value, based on the second-highest bids for sites that have been auctioned recently. Rent bills for all land will then be sent out, based on this map.

VII. A System Based on Options to Use Land3

The system described previously presumes that each year enough sites are relinquished to the state for re-leasing to produce a current land value map with adequate accuracy. If such is not the case, it is possible, at the cost of somewhat higher administrative expenses, to employ a system in which a specific market rent is assigned to each site each year. This system involves creating an options market, where the officials in charge of renting land solicit offers that bind potential users to rent specified sites at specified rents, if any of the designated sites become available.

A potential problem with setting rents through options is that entrepreneurs who make durable improvements seem exposed to expropriation by such a practice. A bidder for an option on an improved site can make an artificially high offer, taking advantage of the fact that the person who made the improvements will not want to relinquish the site on which they were made. The way to avoid this problem is to require bidders to bid for many similar sites, including some that have a significant probability of being relinquished.

Consider first how this works if there are a large number of sites that are identical, except for the improvements on them. To motivate bidders to bid as much as the full rent of land, a fee (perhaps 1 per cent of the rent collected) is paid to the highest bidder. But if any site is relinquished, the highest bidder is required either to take over the site and pay the specified annual rent for at least a year, or find someone else to take it over, compensating that person as necessary. As long as there are some sites with a significant probability of being relinquished, a bidder will find it unwise to bid more than the true rent for these sites. And bidding less than the true rent means leaving an opportunity for profit unexploited.

VIII. Mathematical Analysis of a Bidder's Calculations

The calculations of a bidder can be made more formal, in the following way. First, define some terms:

Z = The bidder's expected profit per site.

p = the bidder's subjective probability of being the one who names the highest rent.

A = The rent that the bidder names.

f = The fraction of rent that the highest bidder receives.

g = The bidder's subjective estimate of the average probability that a site will be relinquished by its current users.

R = the bidder's estimate of the true rental value of the site--the rent that the site would yield if its use were offered at auction.

C = the cost to the bidder of developing rent estimates.

Putting all of the above terms together, the bidder's expected profit is

Z = p[Af - g(A - R)] - C. (1)

Both bids that are very high and bids that are very low lead to negative profits. If a bid is very high, then p is nearly 1 and g is large relative to f, so that the - g(A - R) term dominates the expression and is negative, leading to negative profits. On the other hand, if a bid is very low, p is virtually zero and the C term dominates the expression, leading again to negative profits. For intermediate bids, however, positive profits are possible. If A equals R and the bidder is confident of being the only bidder, then the expression for Z reduces to Af - C, which is positive if f is generous enough to cover the cost of submitting a bid.

It can confidently be expected that at least one person will find it profitable to submit a bid, because if there is only one bidder, that person can profit from the cursory effort involved in submitting a bid that is sure to be lower than the actual rent. The number of bidders is in equilibrium if every existing bidder has positive expected profits, but any potential new entrant finds the cost of developing bids not worth the expected return.

When a bidder's expected profit is maximized, the derivative of (1) with respect to A, the variable that the bidder controls, is zero. Thus,

dZ dg dp

-- = p[f - g - --(A - R)] + --[Af - g(A - R)] = 0. (2)

dA dA dA

One use that can be made of this expression is to determine the conditions that lead profit-maximizing bidders to submit bids exactly equal to their estimates of the rent of sites. The excess in a bid is A - R. Solving (2) for A - R yields

dp

p(f - g) + --f

dA

A - R = ------------------. (3)

dg dp

pÄÄ + gÄÄ

dA dA

A bidder will be motivated to report accurate estimates of rent if the numerator of the right-hand side of (3) is zero. Denote the elasticity of p with respect to A by

A dp

s = - --. (4)

p dA

Since dp/dA is non-negative, s is non-negative. The condition for motivating accurate rental estimates (the condition that the numerator of the right side of (3) be zero) can be expressed as

p(f - g + sf) = 0, (5)

or

f = g/(1 + s). (6)

Equation 6 expresses the "commission rate" that must be awarded to the highest bidder to motivate accurate bids. The parameter g can be estimated by the frequency with which sites are redeveloped. The parameter s might possibly be estimated from patterns of bids and assumptions about expectations. If there is only one bidder, because everyone recognizes that person's skill and reliability and thinks it not worthwhile to compete, then s is 0. Since s is non-negative, equation (6) implies that the commission rate that must be paid to motivate accurate assessments is less than or equal to the probability that a site will be relinquished.

IX. Relaxing the Assumption of Uniform Rent in the Options System

The above analysis employs an assumption that there are many identical sites. When this assumption is relaxed, bidders must be permitted to submit different bids for different sites, and the problem of avoiding excessive bids for sites with durable improvements re-emerges. If bids were completely unconstrained, bidders would bid extra amounts for sites with durable improvements, reflecting the low probability that these sites would be relinquished.

The way to avoid this problem is, once again, to take advantage of the fact that land rent varies smoothly over locations. Each bidder can be required to offer bids on enough contiguous sites to ensure that there is more than a minuscule probability of relinquishment for a reasonable number of the sites. The requirement can be stated in a form such as, "A bidder must bid on all sites within a convex polygon of the bidder's choosing, containing at least 200 sites. Each bid takes the form of a land rent function that applies to the whole domain on which the bidder is bidding. And the land rent function is constrained to be a uniform amount per square meter, with additive or multiplicative adjustments for identified factors such as distance from the center of the city, distance from transit stops, and elevation.

X. Operating the System Involving Options to Use Land

To operate the system involving options to use land, the officials in charge of renting the land solicit options from anyone who is interested in bidding. A bidder is required to post a bond amounting to something on the order of 0.25 per cent of the rental value being bid upon. The bidder determines the area on which to offer options, and then provides a bid per square meter as a function of characteristics of the site.

For each site, officials compute the rent implied by each bid that includes the site. The highest such bid is the rent assigned to the site, and a corresponding bill is sent to the user.

One might think that to be consistent with the earlier system, the second-highest rather than the highest bid should be used to determine the rent of the site. However, as long as the site is not relinquished, it can be presumed to be worth more to the existing user than to any other. Thus the highest option bid is in fact the second-highest bid overall, so that using the highest option bid is consistent with the earlier practice. The person to whom the land is worth the most (the existing user) pays what is offered by the person to whom it is worth second-most (the highest option bidder).

As with the previous system, the user of a site is permitted to sell both the improvements and the right to continue renting the site, to any one at any price upon which they mutually agree. The user is also permitted to relinquish the site at any time, upon restoring it to the condition of bare land. And again, anyone who wishes to build on a site can be required to post a bond against the cost of clearing improvements.

XI. Using Land Rent Information to Manage Externalities

Once a detailed land rent map has been created, one of the uses to which it can be put is to determine appropriate prices for activities that affect the rental value of surrounding land. For example, if the bidders for land include a negative adjustment for proximity to a factory that pollutes the air, then it is possible to determine, from the bids, how much greater land rents would be if the pollution were terminated. This amount can rightly be charged to the factory. Similarly, if the bidders include a positive adjustment for proximity to a private parking garage in a commercial district, it is possible to compute the amount that the presence of the parking garage adds to total rents. To motivate people to undertake an efficient amount of the activities with such positive effects, they should receive the addition to rent thus generated. Option bids result in detailed maps of land rent that reveal both the locations of activities with positive and negative effects on land rent and the magnitudes of these effects.

XII. Extensions to Other State-Allocated Privileges

The central idea of the systems developed here is that the price to be charged to someone who makes continuing use of a state-supplied privilege can be determined by annual offers for corresponding privileges. The idea can be applied not only to land but to other domains as well. One domain that comes readily to mind is use of the frequency spectrum. As with land, effective use of the frequency spectrum (by radio stations, television stations, cellular telephone networks and so on) requires a social understanding that a particular individual or organization is permitted to have exclusive use of particular parts of what is available. (In the case of the frequency spectrum, this means the exclusive right to broadcast in a given frequency band in a given geographical area.)

People who wish to use frequency allocations generally need to be assured that they will have continuing use of the portions of the frequency spectrum that are assigned to them, so that they will not be left with useless broadcasting and receiving equipment. If so few people ever want to use the frequency spectrum that there is never a shortage of band widths to be allocated, then it is perfectly just and efficient simply to permit anyone who wishes to use the frequency spectrum to state the band width he or she wishes to use and then to require others not to interfere. However, if band-width allocations are scarce, then justice requires that those who receive allocations compensate the rest of society for their privileged status. But how much should they pay?

The systems described in this paper can be applied to determine appropriate payments. If one wishes to use a system involving actual transfer of broadcasting rights, then payments can be determined by occasional (secondprice) auctions of the right to use a given band width for a single year. If one wishes to have the greater information that is available with bids for options, then one asks potential users of the frequency spectrum to make binding offers annually on a large number of contiguous band-widths, rewards the highest bidders and uses the highest bids to determine the payments required of continuing users of the frequency spectrum.

The general characteristics of the domains to which these bidding systems can usefully be applied are the following:

1. The domain contains many similar bundles of rights.

2. A bundle of rights in the domain must be held for a considerable span of time to be used efficiently.

3. The future value of the rights is more uncertain than the current value of the rights.

4. In the cycle of use of the rights in conjunction with other inputs, there are times when use can be terminated efficiently (as when land is cleared for rebuilding).

While land and the frequency spectrum have these characteristics, the first characteristic is generally present neither for mineral deposits nor for enterprises. Thus other systems are be needed for the privatization of these domains.

XIII. Precedents in Existing Institutions

The idea of offering land for rent under terms in which future rent depends on future market conditions and creating a market in options to use land may seem unprecedented. In fact, there are precedents. Many commercial leases contain escalator clauses, which specify that future rent will be determined by the level of an index.4 Indices of construction cost are often used for this purpose. This practice is parallel to the idea of setting future rent according to future bids for the use of land. Options markets have long existed for shares in companies and for commodities. The firm of London Fox recently proposed to set up futures markets based on the sale value of commercial and residential property and on commercial rents (Frampton, 1990). Thus the idea of options in land also has a precedent.

What is new is the suggestion regarding how a society can receive, in all future years, the full rental value of land that it turns over to private management: Just charge for each site, in each year, the amount people offer for the use of similar land for that year when they know that future rents will be determined in the same way. By collecting all rent without burdening those who improve land, this simple rule maintains land as the heritage of all generations.

Endnotes

1. Requiring a payment equal to the second-highest bid motivates each bidder to bid the full value of use to him or her. See Vickrey (1961).

2. See Cord, "Investment Protection Under Land Value Taxation," this volume.

3. This system has been described previously in Tideman (1990). See that paper for some additional characteristics of the system that are not described here.

4. My father has such a lease for the premises he rents for his business.

References

Frampton, Rachel. "News Analysis: London Fox's Proposals to Establish a Futures Market in Commercial Property are Gaining Ground," Estate Times (May 4, 1990), 12.

Tideman, T. Nicolaus. "Integrating Land Value Taxation with the Internalization of Spatial Externalities," Land Economics 66 (August 1990) 341.

Vickrey, William. "Counterspeculation, Auctions and Competitive Sealed Tenders," Journal of Finance 16 (May 1961), 837.