The estimation of present bias and time preferences using land-lease contracts∗ Pieter A. Gautier†
Aico van Vuuren
‡
First version 2011, this version 2014
Abstract When agents have present bias, they discount more between now and the next period than between period t (> 1) and t+1. We show how the shortand long-run discount rates can be identified with contracts that specify payments that take place at various points in time in the future and which are traded and priced in a competitive market. We use a unique data set containing information about sales of houses in Amsterdam with landlease contracts that have the above properties. We estimate the long-run discount rate to be 1.91% and the short-run discount rate to equal 20.4%.
∗ We would like to thank Carla Flemmincks for valuable help with the land-lease data and the NVM for sharing their housing data with us. We thank various conference and seminar participants, in particular Bj¨orn Br¨ ugemann for useful comments. † VU University Amsterdam, Tinbergen Institute ‡ VU University Amsterdam and Tinbergen Institute .
1
Introduction
In Amsterdam, many homeowners pay land-lease rents to the municipality. The amounts and frequency of those payments are specified in a land-lease contract. The land-lease costs are paid either annually or in advance for up to 50 years. The pre-payments are non-refundable and are transferred to the new owner when the house is sold. In this paper, we show how and under which conditions this source of exogenous variation in land-lease contracts can be used to test for present bias and to estimate the long- and short-term discount rates. Specifically, suppose that we observe two identical houses (I and II) that are sold at a particular moment in time.1 Suppose further that house I is located on privately owned land, while house II must pay 2000 Euros land-lease rent per year, starting 10 years from now. In a competitive housing market, the selling price of house I will obviously be higher.2 What do we have to assume to use this price difference (and that of other identical houses with different land-lease contracts) to identify the short- and long-run discount rates? Obviously, we need some form of credit constraints because else, buyers can neutralize any difference in the timing of the payments between the two houses by either borrowing or saving against the market interest rate.3 As such, the difference between the selling prices is only affected by the interest rate and not by the time preferences of the buyers. As in many other countries, almost all houses are financed using 1
The data contain many observable characteristics that jointly explain 95% of the variance in house prices. So in the empirical part of the paper, we can control for almost all differences in house characteristics. 2 In the reduced form analysis of Gautier and van Vuuren (2014), it is shown, as expected, that a house on private land is more expensive than a house with land lease. In addition, the number of years that no land-lease rent has to be paid (because the previous owner paid this in advance) has a significant and positive effect on the house price. 3 A similar point has been made by Frederick et al. (2002).
2
a mortgage in the Netherlands. It is however, almost impossible to accumulate large amounts of debts when one does not own a house (which can be used as a collateral). Therefore, it is reasonable to assume that agents face credit constraints on consumption, i.e. that agents can save at the market interest rate but that they can only borrow to finance a house.4 Other assumptions that we need to make in order to estimate the discount rates are that: (i) the housing market functions well so that there are no arbitrage opportunities in equilibrium and (ii) the utility function is additively separable in housing and consumption. In terms of the above example, the agent who buys the house with land lease faces lower mortgage payments today but she must pay more after 10 years because a new land-lease term starts then. Therefore, she has more income available for consumption in the first 10 years and less after 10 years than the buyer of the more expensive house on private land. Her willingness to pay for each of the two houses then depends on her short- and long-run discount rate and on the shape of the utility function. Without making restrictions on the utility function, it is also impossible to identify present bias. This is because both present bias and the shape of the utility function affect the data only through the way in which consumption is smoothed after a shock in income available for consumption. Andersen, et al. (2008) have also shown with experimental data that not allowing for risk aversion, strongly overestimates the discount rate.
5
Therefore, we allow for risk aversion
by using a CARA specification in our estimation procedure with linear utility 4
This assumption is almost identical to assuming that it is possible to borrow without collateral but against a much higher rate than the savings and mortgage interest rate. Moreover, credit card debt is low in the Netherlands. About half of the population above 18 owns a credit card and the average credit is only about 200 Euros per card. 5 See also Andreoni and Sprenger (2012, a,b) and Harrison, et al. (2012) on this issue.
3
as a special case. In general, an agent with present bias realizes that his desire to smooth consumption over time will be frustrated by his future self who will consume a large share of his wealth tomorrow and therefore, he puts less weight on smoothing. Identification comes from the fact that this effect becomes stronger over time. So for an exponential discounter, consumption falls linearly over time while for an agent with present bias, consumption falls at an increasing rate. Moreover, the effect of risk aversion on consumption is also constant over time. Different contracts imply different consumption paths and from the willingness to pay for those contracts, we can estimate the short- and long-run discount rate. Furthermore, we show that the short-run discount rate can be seen as a lower bound in the sense that if agents have decreasing absolute risk aversion (like in the CRRA utility function), the estimated short run discount rate would be higher. Our theoretical consumption model with quasi-hyperbolic discounting is closely related to Harris and Laibson (2001) except that we consider a finite horizon problem and they assume a CRRA utility function. We find that the short-run discount factor equals 0.85 and the long-run discount factor equals 0.98. This corresponds to an 20.4% short-run and a 1.9% long-run discount rate. We reject the null of no present bias. As a robustness check, we also estimate our model under the (unrealistic) assumption that agents can also not borrow to finance their house, i.e. only wealth can be used. In that case, we find the short-term discount factor to be equal to 0.37 and the long-term discount factor to be equal to 0.98 (i.e. a short-run discount rate of 172.9% and a long-run discount rate of 2.5%). This suggests that a misspecified model that does not allow agents to borrow for housing, blows up the estimated short-run 4
discount rate. Our estimation results are surprisingly close to what Laibson et al. (2007) find even though their data, model and identification strategy differs from ours. They consider a CRRA utility model with present bias and identify the shortand long-run discount rates by matching moments on wealth accumulation, credit card borrowing, and excess sensitivity of consumption to predictable movements in income. When they estimate the coefficient of relative risk aversion, they find a short-term discount factor equal to 0.90 and a long-term discount factor equal to 0.96, implying a short-term discount rate of 15.8% percent and a long-term discount rate of 3.8%. Most of the evidence for present bias comes from lab experiments, see Frederick et al. (2002) for an overview. Although some of the evidence they discuss is convincing, we believe that evidence from the field can give important additional information. As Laibson et al. (2007) note, hundreds of studies have estimated general discount functions with lab data but only few papers use field data to estimate the degree of present bias.6 Our data are interesting because the stakes are a lot higher than what is typically offered in the lab. Housing is the largest investment spending for most people and land lease is a substantial part of that. When the stakes are high, the incentives to invest in information and to calculate (or let an expert calculate) the payoffs over time are large as well. In addition, our population differs from the typical undergraduate population. Finally, experiments can be vulnerable to framing. We do of course face the general drawbacks of using field data in terms of lack of control of the environment, and we must 6
Examples include, Ahumada and Garegnani (2007), Attanasio and Weber (1995), Attanasio, et al. (1999), Paserman (2008), Fang and Silverman (2008) and Shui and Ausubel (2004).
5
make additional identifying assumptions (all agents have identical discount functions and the housing market is competitive). However, when lab participants choose between donations over time, also strong additional assumptions must be made, i.e. that those donations are used on consumption and that they cannot borrow. We therefore view our results as being complementary to the lab evidence. Our paper is organized as follows. The next Section introduces the concepts of present bias and hyperbolic discounting. Section 3 describes the land-lease system in Amsterdam while 4 discusses our data sources. Section 5 discusses our main estimation method. Then, in Section 6 we carry out a number of robustness checks (different nominal income growth and interest rates). Finally, Section 7 concludes.
2
Present bias
Standard (exponential) discounting implies that if an agent is indifferent between receiving 100$ now or 105$ next year, she is also indifferent between receiving 100$ in ten years or 105$ in eleven years. When agents have present bias, they discount more between now and the next period than between t (>1) and t + 1. Strotz (1956) was the first to suggest that people are more impatient in the short than in the long run. One implication of present bias is that it can be rational to voluntarily impose constraints on oneself. The (quasi-)hyperbolic discounting literature replaces the constant discount utility function with a particular alternative form, see Laibson (1997). Following the more recent literature, i.e. Harris and Laibson (2001), we focus on the discrete 6
time quasi-hyperbolic specification 1, βδ, βδ 2, ...βδ T where if β = 1, discounting
is constant over time and if β < 1, discounting is quasi hyperbolic.
3
Land lease
In our context, land lease is defined as the right to hold and to use the land of the city of Amsterdam. For this right the leaseholder must pay the city an annual fee. This is called the land-lease rent. Land lease is different from tenancy because it can be traded without the intervention of the owner. The city of Amsterdam uses land-lease contracts since 1896. Before that period all land was sold, while after that period the city of Amsterdam always remains the owner of the land. Some houses were property of the city before 1896 and were sold afterwards and the city also frequently buys land belonging to pre-1896 houses. Therefore, there are also land-lease contracts in the older neighborhoods. Houses built after 1896 are always under a land-lease contract unless these houses were built on land that was already sold before 1896. This implies a non-perfect concentration of houses on privately owned land in the older (and usually more popular) neighborhoods of the city. We take this into account in our empirical analysis by focusing on the neighborhoods that were built around the year 1896 and therefore have a mixture of different house types. The duration of a land-lease contract is typically 50 years which we call the period of lease. At the beginning of the lease period the terms are specified in a contract called the “general conditions” (GC). At present, new contracts are based on the General conditions for perpetual land leases in Amsterdam 2000 (GC2000). However, land-lease contracts with an earlier starting date belong to 7
different GC’s. The newest conditions are valid at the date of termination. An important difference between the recent GC’s and older ones concerns the landlease rent which was always a fixed amount before 1966 and is sometimes fixed after 1966 and always variable under GC2000.7 Another difference between the period before 1966 and the period afterwards is that in the contracts from the earlier periods the land-lease period was 75 years while it is only 50 years for the more recent contracts. At the end of the period of lease, the municipality offers a new contract.8 As Veen (2004) documents, this is based on the consultation of independent experts who are typically real-estate agents. Their procedure is as follows: (1) estimate the total value of the property and multiply this by the land ratio, which is a parameter that depends on the neighborhood of the house, and (2) multiply the result with 0.6 in order to take into account that land with property has less value than land without property.9 A final aspect of the price determination of the experts is that they try to follow the long-term market trends by determining the total value of the property (in step (1)) instead of the short-term fluctuations of the market. After determining the land price, the (yearly) land-lease rent is simply calculated as a percentage of the land price.10 7
In the period between 1966 and 2000 different systems were effective from variable to fixed over the period of lease. 8 In very special cases, the municipality is able to terminate further land lease after the end of the period of lease. In that case the municipality pays the value of the houses. However, these cases happen very rarely and hence we can safely assume in our empirical analysis that leaseholders expect their right to lease the land to last indefinitely. 9 In practice, the land ratio differs between 0.20 and 0.25, see Veen (2004) for more details. 10 The exact percentage depends on the specific contract and on the district within the city.
8
4
Data
We have data from three different sources. The first source comes from the city of Amsterdam. It contains all the information about land-lease contracts that were effective between 2007 and 2011 with a distinction between those houses for which the land-lease rent is paid in advance and the houses for which the landlease rent must be paid immediately. In total, we have 158,380 houses for which the land-lease rent is paid in advance. For these houses we have the identifier of the house from the Dutch register, the beginning and the end date of the contract and the beginning and the end date for which the land-lease rent has been paid in advance. Finally, we have information on the general conditions of the land-lease contract, we have the special conditions of the payment period and the exact amount that has to be paid annually during the years of observation. In total there are 56,242 houses for which the land-lease rent was not paid in advance and for which immediate payments must be made. Our second data set is from the Dutch association of real-estate agents (NVM). It contains more than 70 percent of the houses that were sold in Amsterdam within our observation window. The start of the data set is the 1st of January 1985 and the end of the observation window is the 31st of June of 2011. The total number of sales available is 117,176. A large set of characteristics is available for every house, i.e. the address, zip code, the selling price, size (both in square and cubic feet) as well as a large set of other features that may have an impact on the price of the house, see Appendix D. We also have an indicator that specifies whether the house is under a land-lease contract or not. Our third data set is an official list from the municipality of all houses reg-
9
istered in Amsterdam in 2010. This data set contains the identifier from the Dutch register as well as the address and zip code. We use this data set to match the other two data sets since it contains both the address, zip-code and the registration code which is used for the land-lease contracts of our first data set. We merge the second and third data set in order to obtain the identifiers from the registers of all the houses in the second data set. Here, we use street address (with house number and addendum) and zip code. We were able to match 99,093 out of the 117,176 house sales that appeared in the data from the real-estate association. The main reason we were not able to match all the houses is that the house identifier used by the real-estate agent is not always equal to the house identifier used by the municipality. Especially, the addendum used by the real-estate agent often differs from the addendum used by the municipality. Even though we put a lot of effort in matching those houses with different addendums we have been quite conservative in order to minimize the number of wrong matches. Another reason for not being able to match all houses is that some houses sold between 1985 and 2009 do not exist in 2010 and the newly built houses during 2010 did not yet have an official address. We also delete the houses that are not a single unit for the Dutch land register. The problem with these houses is that they do not have a private contract for land lease with the municipality, but instead have a collective contract together with the other houses. Hence, we cannot identify the exact amount of land-lease rent that must be paid by the owner of such a house. This resulted in a deletion of 28,464 houses. Next, we merge the resulting data set with the first data set of all the landlease contracts. Houses located on privately owned land are identified as houses 10
without a corresponding land-lease contract. There are however a few cases where the real-estate agent indicates that the house is not located on private land despite the fact that there is no land-lease contract. If this occurs, we delete the observation from our data set (about 2 percent). In addition, we merge houses based on the contract that was in place at the moment of the sale. Since we only have information on contracts that were effective over the period 2007 to 2010, some house sales are lost since we do not have any information about the land-lease contract of the house at the moment the house was sold. In total this leaves us with 60,998 house sales and after deletion of the houses that do not have all information necessary for estimation (such as size), we keep 53,397 observations. Table 1 below gives some descriptive statistics of our final data set. We make a distinction between private land and land lease. This last category is further subdivided into paid in advance and variable and fixed land lease. Table 1 shows that less than half of the houses are on private land and that the prices of these houses are higher than the prices in any other category. More surprising is the fact that houses with a variable land-lease rent are more expensive than those paid in advance. The most likely reason for this is the fact that those houses are typically sold after 2000. There are not many differences in size between the different categories. However, there are differences between the neighborhoods. For example, private land is overrepresented in the more expensive areas in the city center and the western and southern parts close to that city center. Also for variable land rent, the old western and southern parts of the city are overrepresented, while paid-in-advance and fixed land-lease rent are overrepresented in less popular areas. 11
For the empirical analysis we divide Amsterdam into 90 different neighborhoods. Those neighborhood definitions are based on Statistics Amsterdam and within a neighborhood houses and economic status of the owners are approximately homogeneous. Our empirical implementation is based on comparisons within neighborhoods. If some neighborhoods only contain houses on private land or only houses with land lease, then we cannot separately identify a neighborhood and a land-lease effect. Therefore, we deleted all neighborhoods that do not have at least 10 observations from either houses on private land, land-lease rent paid in advance or land-lease rent not paid in advance. This results in a deletion of 53 neighborhoods. Descriptive statistics of the final data set can be found in the lower panel of Table 1.
5
Testing for present bias with land-lease data
In this section we first present a model of consumption and housing in section 5.1. Then, we discuss identification in 5.2 and we estimate our model in section 5.3.
5.1
A model of consumption and housing
Agents can spend their income on housing or on a composite consumption good. Let H be a vector of housing characteristics and cs be the number of units of the composite consumption good bought in period s. Assume that utility is additively separable in housing and the composite consumption good. Let U0 (H, ω0 ) be lifetime utility of a buyer who buys a house with a land-lease contract, where ω0 is initial wealth. Let p(H) be the price of a house with characteristics H in
12
Private land
Paid in advance
Not paid in advance Fixed Variable
Including neighborhoods with few observations Number of observations Price Size in square feet Number of neighborhoods Neighborhoods City center West East North New-West South-East South
31973 254427 930 90
14305 5764 249374 176092 1088 955 90 90
30.6 26.5 10.4 1.6 0.9 0.2 30.0
6.2 9.1 23.2 12.7 30.3 11.2 7.3
1355 280275 1036 90
0.6 9.2 1.5 12.5 25.7 9.8 40.8
3.4 16.4 9.7 12.7 2.1 15.4 40.4
2821 783 240010 123169 931 813 27 27
474 268875 952 27
Excluding neighborhoods with few observations Number of observations Price Size in square feet Number of neighborhoods Neighborhoods City center West East North New-West South-East South
16975 257948 953 27 32.3 14.8 11.9 0.8 0.9 0.0 39.3
23.1 16.7 24.6 19.9 4.0 0.0 11.7
4.3 66.1 11.0 0.6 14.6 0.0 3.3
Table 1: Descriptive statistics of the data set.
13
6.6 42.7 26.6 3.1 1.5 0.0 19.5
case this house is sold with a land-lease contract and pb(H) the counterfactual
price in case that house would be situated on private land. Moreover, let uH (·) be the felicity function for housing and uC (·) be the felicity function for the composite consumption good. For the moment, we abstract from income growth and inflation but we incorporate them in the empirical analysis.11 Let r be the rate at which agents can borrow and lend. The agent’s lifetime utility from consumption is then given by U0 (H, ω0 ) =
max
Cs ;s=0,...,T
(
uH (H) + uC (c0 ) + β
= uH (H) + β
T X
[uH (H) + uC (cs )] δ s
s=1
T X
)
(1)
uH (H) δ s + W0 (H, ω0 )
s=1
where H contains all relevant characteristics that make the house attractive to a potential buyer, and W0 (H, ω0 ) is the life-time utility out of consumption. The payments are specified in the land-lease contract and they affect (foregone) consumption of the composite consumption good at different periods in time. We assume that conditional on the timing and size of those payments, the buyer receives no additional utility from a house on private land (i.e. the land-lease contract is not a part of H). The amount of rent that must be paid for land lease in period s is denoted by Ls . Finally, we assume that individual income equals y. Consider a neighborhood with a large number of identical houses that are sold either including the land or with a land-lease contract. In a competitive equilibrium where houses with and without land lease are simultaneously traded, 11
See Section 5.3 and our webappendix for more details.
14
a buyer-indifference (or no-arbitrage) condition must hold, i.e. c0 (H, ω0 ), W0 (H, ω0 ) = W
(2)
c0 is lifetime utility out of consumption associated with a house on priwhere W
vately owned land.
c0 only depend on H through their selling prices. Since Note that W0 and W
we focus on identical houses for the remainder of this subsection, we drop the H
for ease of notation and write W0 (H, ω0 ) = W0 (ω0 ). We deal with the case of heterogeneous houses in Subsection 5.3. In the absence of borrowing constraints and when borrowing and lending rates are equal, δ and β drop out of this condition. Therefore, we need some form of credit constraints in order to identify these parameters. For our baseline analysis, we assume that there are only credit constraints for the composite consumption good but that the agents can borrow for housing services (this can be motivated by the fact that houses can be used as collateral). We discuss identification under these assumptions in Section 5.2.12 In Section 6 we show that this assumption is important: we find an extremely high short-term discount rate and a fit that is substantially worse in comparison to our baseline model under the unrealistic assumption that buyers cannot borrow for housing either. Equation (1) is a dynamic programming model with liquidity constraints and present bias. We draw heavily on Harris and Laibson (2001) to solve it. Compared to their model, we make two assumptions that make our model more tractable. 12
Alternatively one could assume that buyers face different lending rates for housing and consumption. Such a model would be equivalent to our case whenever the short-run discount rate is smaller than the lending rate for consumption. The short-run discount rate that we find in our empirical application is lower than the typical interest rate that is paid on credit cards.
15
First, we assume a finite time horizon and second we assume deterministic income of the buyers. The first assumption allows us to use backward induction which results in a unique optimal consumption path. We use deterministic income paths since we are not able to observe income in our empirical implementation. Consider a buyer of a house with a land-lease contract. The law of motion for wealth is, ωs = (ωs−1 + y − cs−1 − rp − Ls−1 )(1 + r). This period’s wealth depends on previous period’s wealth plus income minus the sum of consumption, mortgage payments and land-lease payments. Let Vs be the continuation value of the buyer (i.e. the future value evaluated today). It equals Vs (ωs ) = u (cs (ωs )) + δVs+1(ωs+1 ),
(3)
where cs (ωs ) is the optimal consumption for a buyer with wealth level ωs in period s. In line with equation (1), let Ws be the current value for a buyer with a land-lease contract, Ws (ωs ) = u (cs (ωs )) + βδVs+1(ωs+1 ).
(4)
Moreover, c(ωs ) = arg max {u (c) + βδVs+1(ωs+1 )} . c∈[0,ωs ]
From this equation, we obtain the first-order condition of the consumer’s problem, ′ u′ (cs (ωs )) ≥ βδ(1 + r)Vs+1 (ωs+1 ),
with equality whenever cs (ωs ) < ωs + y − Ls , i.e. when the liquidity constraints 16
are not binding. For the moment we make this assumption and look at the case in which the borrowing constraints are binding afterwards. Differentiating (4) with respect to ωs and using the first-order condition and the envelope theorem gives ′ Ws′(ωs ) = βδ(1 + r)Vs+1 (ωs+1 ) = u′ (cs ) .
(5)
Subtracting (4) from β times (3) yields, Ws (ωs ) − (1 − β)u (cs (ωs )) = βVs (ωs ).
(6)
Taking first-order derivatives of (6) with respect to ωs and substituting (5) in yields βVs′ (ωs ) = u′ (cs (ωs )) (1 − (1 − β) c′s (ωs )) .
(7)
Next, use the first-order condition for s − 1, and substitute it into (7) yields, 1 ′ ′ log [u(cs (ωs ))] = log [u(cs−1 (ωs−1))] + log ((1 + r)βδ) + log cs (ωs ) + (1 − cs (ωs )) . β (8) Repeated backward substitution yields, log [u(cs (ωs ))] = log [u(c0(ω0 ))] + s log ((1 + r)βδ) + s−1 X 1 ′ ′ log cs−k (ωs−k ) + 1 − cs−k (ωs−k ) . β k=0
(9)
Using (8), we see that the consumption function is decreasing over time as long as13 δ<
1 , 1+r
13
This can be done by rewriting the last two terms of the right-hand side of (8) as log ((1 + r)δ) + log (1 − (1 − β)c′s (ωs )). The final term is non-positive.
17
while it is increasing if βδ >
1 . 1+r
since the last term in equation (8) is non-negative. When δ lies in between those values, it can be either increasing or decreasing depending on the time that is still left until the end of the time horizon. For the remainder of this section, we assume for convenience that δ < 1/(1 + r). How do things change in the case of a binding credit constraint for consumption? For the moment, only consider those buyers who bought a house that is pre-paid for σ years. Moreover, assume that buyers do not have any initial wealth (i.e. ω0 = 0). Note that the latter assumption is only relevant if a > 0. In that case, buyers with a constant real income stream for consumption consume their income minus their mortgage payments minus the payments for land lease. However, since they face a downward jump in their income available for consumption in period σ, they prefer to smooth their consumption by consuming less than their income available for consumption before σ in order to use their wealth for consumption in the periods after σ. This implies the following consumption pattern y − rp s ≤ s∗ c(s) = C(s) s∗ < s ≤ s∗∗ , y − Ls (b p) − rp s > s∗∗
18
(10)
where log [u (C(s))] = log[u(cs∗ (ωs∗ ))] + (s − s∗ ) log ((1 + r)βδ) ∗ s−s X−1 1 ′ ′ 1 − cs−k (ωs−k ) , + log cs−k (ωs−k ) + β k=0
(11)
and where s∗ and s∗∗ are the boundary points between which the buyer is smoothing her consumption (before and after, all available income is consumed). The boundary points follow from, cs∗ +1 (ωs∗ +1 ) < y − rp < cs∗ (ωs∗ ), and cs∗∗ +1 (ωs∗∗ +1 ) < y − rp − Ls (b p) < cs∗∗ (ωs∗∗ ), while cs∗ is chosen such that ω(s∗∗ ) = 0. Note that Ls is allowed to depend on pb,
which is in line with the procedure that we discussed in Section 3.
Figure 1 illustrates the boundary points s∗ and s∗∗ . The solid line is income available for consumption. There is a drop in income at σ due to the fact that the buyer has to pay for land lease from that period onwards. The dotted line represents consumption. Up to s∗ the buyer uses her available income for consumption. Then at s∗ , she starts to save for future consumption and hence accumulates wealth. From σ onwards, she is consuming those assets up to s∗∗ when all assets are depleted. Then, from s∗∗ till the end of the time horizon, she consumes again all her available income. The way in which consumption is smoothed between s∗ and s∗∗ depends on the shape of the utility function. Define A(c) = −u′′ (c)/u′ (c) as the absolute risk aversion and define the utility function as increasing absolute risk aversion 19
Figure 1: Illustration of the boundary points
Income/consumption
Available income Consumption
σ
s*
s**
Years after purchase
(IARA) when A(c) > A(c − ∆) for every ∆ > 0 and c > 0. Under CARA, A(c) is constant and under decreasing absolute risk aversion, A(c) < A(c − ∆) for every positive c and ∆. Note that our definitions are global and we rule out utility functions in which these properties can change with the level of consumption. We have the following lemma. Lemma 1 Define ∆(s) = C(s + 1) − C(s). Then 1. When β = 1, IARA implies that ∆(s) > ∆(s − 1) while CARA implies equality and DARA replaces the >-sign with a <-sign. 2. When β < 1, IARA and CARA imply that ∆(s) > ∆(s − 1) while the inequality is undetermined for DARA. 20
Proof: See Appendix A.
Lemma 1 states that if β = 1, then IARA gives a concave relationship of the consumption function over time whereas DARA exhibits a convex relationship. Finally, CARA gives a linear pattern of consumption smoothing. When there is present bias, both CARA and IARA give a concave consumption smoothing pattern. Utility functions with the DARA property can give a convex and concave relationship depending on the magnitude of present bias. Figure 2 illustrates the impact of present bias on consumption in the case of CARA. Again, the solid line represents available income. The dashed line represents consumption under present bias, while the dotted line represents the case without present bias where the short and long-run discount rates are the same. The latter case always results in a linear evolution of consumption over time (see Lemma 1-1). In contrast, under present bias, consumption decreases rapidly just before s∗∗ . In addition, individuals with present bias smooth their income for a shorter period of time. Lifetime utility out of consumption for a buyer with present bias equals W0 (0) = u(c0 (0)) + β
T X
u(cs (ωs ))δ s
(12)
s=1
A buyer with a house on private land receives an income available for consumption equal to y −rb p. If the long-term discount rate is large enough (i.e. δ < 1/(1 + r)), the buyer is consuming exactly this amount every period. Hence, lifetime utility
21
0.82
Available income Consumption for β = 0.95 Consumption for β = 1
0.815
Income/consumption
0.81 0.805 0.8 0.795 0.79 0.785 0.78 0.775 0
20
40 60 Years after purchase
80
100
Figure 2: Relationship between consumption and income available for consumption, r = 0.045, δ = 0.05, a = 10, y = 1, pb = 4.1, p = 4, σ = 40
out of consumption for this buyer equals
c0 (0) = u(y − rb W p) + β
T X
u (y − rb p) δ s .
(13)
s=1
The model can be closed by using the no-arbitrage condition (2). Using this condition, it follows that pb > p because else income available for consumption of
the buyer without a land-lease contract is always higher and no houses with land lease would be sold. In addition, the left-hand side of (2) is always larger than the discounted stream of income available for consumption, i.e. u(y − rp) + β
σ X
u (y − rp) δ s +
s=1
T X
u (y − rp − Ls (b p)) δ s .
s=σ
The left-hand side of (2) decreases at a slower rate than the right-hand side when p) is small in comparison to r. In our data set we p is increasing as long as L′s (b find that the land-lease rent that must be paid increases very slowly with the
22
value of the house (less than half a percent). This implies that a unique solution for pb using equation (2) is guaranteed.
The analysis changes slightly for those houses that have a land-lease contract
that is not pre paid since land-lease rent must also be paid before the end of the present period σ. Still, there is a downward jump in income available for consumption at σ and hence similar techniques as above can be used for those houses.
5.2
Identification
Identification of our model parameters β and δ comes from the way in which prices are affected by those different parameters taking the moment of the drop in income available for consumption (i.e. σ) into account. A necessary condition is that there should not be two (or more) sets of β and δ that results in the same consumption pattern over time. Lemma 1 is very informative about the potential problems facing identification since it shows that a concave relationship of consumption smoothing can be explained by for example a combination of DARA and a high level of present bias but it can also be explained using no present bias and IARA. This implies that β and δ cannot be identified as long as we do not make additional restrictions with respect to the utility function. For the remainder of this paper, we assume that our utility function exhibits CARA. i.e. u(c) =
1 − exp(−ac) . a
The CARA felicity function reduces in the limit to a linear utility function when a ↓ 0. We also estimate the model for this limiting case.
23
Our non-identification result is very general. It implies that β and δ can only be identified for a particular utility function. This still holds even if we would have had income and consumption data for every period. Note that there is almost no independent empirical evidence supporting IARA. Therefore, any observed convex consumption pattern would directly reject present bias in the case of CARA even though a DARA utility function could explain such a convex consumption pattern in combination with present bias. This implies that our estimates of β < 1 using CARA can be seen as an upper bound of the real size of β. We obtain the following result. Theorem 2 Define
∗
δ (β) =
1 a
Ps∗∗ −s∗ −1 t=0
with
Ψ(β) = −
1 (1+r)t
P ∗∗
Ps∗∗ −s∗ −1 t=0
Ps∗∗ −s∗ Ps∗∗ −s−s∗ −1 s=0
k=0
Ps∗∗ −s∗ s=0
Qt−1 1 ∗∗ ∗ −s −s +t k=0 1−(1−β)c′s−k Q t−1 1 1 Ψ(β) × k=0 1−(1−β)c′ (1+r)t
s −s∗ s(1+r)s Ps=1 s∗∗ −s∗ (1+r)s s=1
s−k
c′s∗ −s−k 1−(1−β)c′s∗ −s−k
(1 + r)s
+
∗ s−s X−1
k=0
c′s−k (ωs−k ) 1 − (1 − β)c′s−k (ωs−k )
In case of a CARA utility function with known absolute risk aversion, a sufficient condition for the identification of β and δ is δ < δ ∗ (β). Proof: See Appendix B. The condition of Theorem 2 always holds if β is not extremely small or s∗∗ − s∗ is not too small (larger than three periods). The intuition behind the identification under the CARA assumption can be explained using Figure 2. As stated by Lemma 1, CARA utility without present bias implies that consumption smoothing is linear around the period in which there is the drop in income whereas it 24
is concave in the presence of present bias. The way in which consumption is smoothed around the date of the drop in income affects the differences in the house prices. Therefore, δ can only affect the slope of the consumption pattern while β not only affects the slope but also the curvature of the consumption smoothing over time.
5.3
Empirical implementation
In our empirical application, Ls is not constant, not all houses are identical and we do not observe all relevant characteristics. Another practical issue is that we do not know the exact amounts to be paid in future land-lease contracts so we have to form expectations about them (which will be based on the institutional details that we discussed in Section 3). Figure 3 gives a schematic overview of our object of investigation. Time is indicated by s and the year that a particular house is bought is denoted by t. Note that this is a slight generalization in comparison to the previous subsection since there we assumed for reasons of exposition that t equals zero. Suppose that the number of years since the beginning of the period of lease equals τ and that the original contract was for J years, so t − τ is the year in which the last period of lease started. Let σ = J − τ be the number of years until the present period of lease expires. This implies that the current period of lease expires in period t + σ. We use i for an observation in the data set and assume that the observations are ordered as follows; first the houses on private land, next the houses for which the land-lease rent is paid in advance and finally the houses for which the landlease rent is not paid in advance. In addition, the number of observations for 25
t−τ
t
t+σ =t−τ +J
s ✲
Figure 3: Time line for our empirical implementation these three different groups are assumed to be respectively n1 , n2 and n3 . Since the different contract types may come with different levels of uncertainty about the future payments, it is important to control for them. Next, we must make additional identifying assumptions to deal with the fact that houses are not identical. Let xi be the observed individual characteristics of a house (including a full set of regional dummies). Assume that the price of a house on private land is given by, log pbi = xi α + υi ,
(14)
where υi is interpreted as a term that represents the unobserved characteristics of the house and assume that υi = ξi for any house that is built on privately owned land, while υi = η1 + ξi ,
υi = η2 + ξi ,
υi = η3 + ξi
(15)
for respectively houses with the land lease paid in advance, fixed-land-lease and variable-land-lease contracts. Here, υi is interpreted as a term that represents the unobserved characteristics of the house and η1 , η2 and η3 capture the mean unobserved heterogeneity levels of houses with different types of contracts. This specification implies that the full set of characteristics of a house (i.e H) can be 26
decomposed into x and υ. We assume that E(υi |xi ) does not depend on contract type up to a constant η1 , η2 or η3 . So conditional on observed characteristics, the unobserved characteristics are the same between contract types. The ξi ’s are i.i.d. with mean zero. We make the additional assumption that the price of a house on private land changes from our base year, 1985, with a growth rate ψt . Hence, the (log) price P of a house t periods from now is: log pbi,t = log pbi + ts=1 log(1 + ψs ).
The left-hand side of (14) is only observed for the houses that are built on
private land. Still, we can calculate pb for the houses that are not on private
land conditional on β, δ, the observed levels of the selling prices (i.e. pi ), the termination date (i.e. σi ) and the future land-lease payments of this house. We
use the no-arbitrage condition (2) for this calculation together with (12) and (13) and use the notation pbi (β, δ) wherever appropriate. Therefore, the biggest hurdle
we must take is to determine future values of Li,s . As we discussed in Section
3, the land-lease rents in the periods between t − τ and t + σ are in practice calculated as a fixed percentage, γi , of the value of the house on private land in those periods. As was also discussed in that section, γi may be different for different neighborhoods. Hence, γi = γJ(i) . Based on these assumptions, we derive the following regression equation for houses for which the land-lease rents are not paid in advance (i.e. observations i = n1 + n2 + 1, . . . n1 + n2 + n3 ) in Appendix C, log Li −
t−1 X
ϕt′ di − xi α = log γJ(i) + υi ,
(16)
t′ =τ
where di is a dummy variable that equals one for variable land lease. This makes
27
it possible to estimate log γJ(i) and ψt (for t < 1985) using OLS.14 We use the overall price deflator published by Statistics Netherlands, which is also used by the municipality.15 The final hurdle that we have to take is to include inflation into our empirical analysis. We assume that the inflation rate equals nominal income growth (ϕ) which we assume to be constant for all future periods.16 The theoretical model changes somewhat by including inflation. We set the inflation rate equal to 2 percent per year and finally we set the interest rate equal to 4.5 percent per year. We look at the robustness of our results with respect to these assumptions in Section 6. Our estimation procedure is based on the simultaneous minimization of the sum of squared residuals (i.e. the estimates of ξi ) in (14) and (16) to estimate α, β, γj , δ, η1 , η2 , and η3 .
5.4
Results
Table 2 presents the main results of our structural model. Since the models are nested, the sum of squared errors should go down by going from the left to the right of the table. Hence, the magnitude of the decrease gives information about the importance of the parameters. We find that the model with a CARA utility function performs significantly better (in terms of residual sum of squares, RSS) than the model with linear utility. The model with present bias and a 14
We use the true house price for the period after 1985 and before 1985 we use the P increases Pt−1 1985 ′ trend, so pi,τ follows from, t′ =1985 ψt 1(t ≥ τ ) pi,τ =E(pi,t ) + υi t′ =τ ψ + 15 We test this assumption for the years that we had sequential data of the land lease (20012009) and find a perfect fit. 16 We abstract here from real income growth and growth in the price for land because this simplifies the CARA utility case a lot. This is less restrictive if both land price and income grow at the same rate.
28
CARA utility function performs slightly better than the model without present bias. Still, we reject the hypothesis that there is no present bias (β = 1). This conclusion is based on the standard deviation of β. An F-test based on the residual sum of squares gives a value equal to 11.2 and hence we can also reject the exponential model based on this statistic. Note that both tests have the caveat that they are on the boundary of a parameter space although β’s larger than 1 are statistically possible. The estimated long-run discount rate equals 39.43% for the linear utility model and it is respectively only 2.33% and 1.91% for the CARA utility models without and with present bias. In the first two columns of Table 2, the short-run discount rate is restricted to be equal to the long-run discount rate. When we estimate β in column 3, we find a short-run discount rate of 20.4%. Laibson et al. (2007) report very similar long- (3.8%) and short-run (15.8%) discount rates for the US when they jointly estimate the risk aversion parameter (like we do here).17 Our estimate of the coefficient of absolute risk aversion is similar to what Gertner (1991) finds and in between what Cohen and Einav (2007) find for the mean and median individual. The unobserved heterogeneity coefficients become smaller once we allow for risk aversion and present bias. Table 2 also lists the R2 values for different parts of the data set. Overall, we do extremely well with respect to the prediction of our model although it should be stated that most of the predictive power comes from the hedonics of our model. Nevertheless, the high level of predictive power of the hedonics is 17
Their identification comes from matching moments on wealth accumulation and credit card borrowing.
29
δ β a(×1000) η1 η2 η3 RSS
Linear CARA 0.7172 0.9772 0.9813 (0.0006) (0.0042) (0.0017) 1 1 0.8463 (0.0307) 0 0.2907 0.3405 (0.0003) (0.0001) -0.0402 -0.0109 -0.0176 (0.0004) (0.0001) (0.0004) -0.0087 0.0059 0.0067 (0.0002) (0.0109) (0.0003) 0.0610 0.0770 0.0727 (0.0002) (0.0064) (0.0002) 715.57 714.12 713.74
R2 -values of the different parts of the data set R2
0.9284
0.9285
0.9286
Private land Paid by previous owner Fixed land rent Variable land rent
0.9668 0.9223 0.9723 0.9163
0.9668 0.9256 0.9726 0.9166
0.9668 0.9264 0.9726 0.9189
0.8152 0.8210 0.5109
0.8157 0.8218 0.5114
0.8161 0.8228 0.5111
Land rent Overall Fixed land rent Variable land rent Number of observations
21053
Table 2: Results of the structural form model for present bias assuming no credit constraints on housing. The price of a house on private land depends on size, neighborhoods, squared footage times neighborhood, nine categories of the period of construction, period of construction × neighborhood, year, year × neighborhood, year2 × construction, nine categories for type of house, nine categories for maintenance inside, nine categories for maintenance outside, dummy for monumenst, dummy for (partly) let, five categories for location, dummy for new house, dummy for investment project, three categories for heating system, five categories for isolation. Details can be found in Appendix D and in the web appendix. Estimates can be found in our web appendix. 30
encouraging since it allows us to price the various land-lease contracts (given the observed characteristics of the house). For all versions of our model, we have the same pattern with respect to how well we predict the values of pb: the houses with
fixed land rent are predicted best, next houses with private land, then the houses with land rent paid by the previous owner and the houses with a variable land rent are predicted worst. The differences are small. The predictive power of the land rent (i.e. equation (16)) is somewhat lower, especially for the houses with
variable land rent.
6
Robustness checks
In this section, we present robustness checks on some of the assumptions we made in the previous section. First, in Section 6.1, we use different values for the expected inflation rate and the house price trend. Second, in Section 6.2 we assume that agents are not only credit constrained on consumption but also on housing.
6.1
Estimation with different calibrated parameters
Table 3 lists the results when we change the calibrated model parameters, ϕ (inflation) and the real interest rate r − ϕ. Keeping the parameter of absolute risk aversion constant, a lower interest rate makes it harder to explain the observed price differences between houses with and without land lease and therefore the estimates of the (short- and long-run) discount rates go up. As Table 3 shows, small changes in the interest and/or inflation rate mainly affect the estimates of the risk-aversion parameter.18 From this sec18
If we would raise the nominal interest rate above 5.5%, consumption can become non-
31
Inflation/growth rate
0.01
0.02
0.03
0.7887 0.8635 0.3161 715.79
0.9731 0.8284 0.5268 714.47
0.9494 0.8457 0.3011 715.16
0.9810 0.8465 0.3084 716.55
0.9813 0.8463 0.3405 713.74
0.9813 0.8463 0.2997 714.87
Real interest rate = 0.01 δ β a RSS Real interest rate = 0.025 δ β a RSS
Table 3: Results of the estimated discount rate when we change the calibrated model parameters tion, we conclude that our estimates are robust to small changes in inflation and the interest rate.
6.2
Credit constraints on housing
So far, we have estimated our models using the assumption of no credit constraints on housing. In this section, we continue to assume that agents can save and face credit constraints on consumption but now we make the (strong) assumption that they face full credit constraints on housing as well. So mortgages are no longer allowed. We believe this case is less relevant but it is interesting to see how it affects the estimated discount rates. In the original model with no credit constraints on housing, we assumed that buyers do not have any wealth at the moment of buying a house. Now we must assume that the buyers of a particular monotonic over time and this would complicate our estimation procedure substantially. See the discussion below equation (9).
32
Linear δ β a(×1000) η1 η2 η3
RSS
0.6866 (0.0551) 1 0
CARA 0.6210 (0.0898) 0.5849 (0.1116) 0
-0.0390 -0.0391 (0.000001) (0.000001) -0.0194 -0.0200 (0.00001) (0.00001) -0.0025 -0.0025 (0.000001) (0.000001) 715.153
715.152
0.6866 (0.0551) 1 0.0000 (·) -0.0390 (0.000001) -0.0194 (0.00001) -0.0025 (0.000001)
0.6210 (0.0898) 0.5849 (0.1116) 0.0000 (·) -0.0391 (0.000001) -0.0200 (0.00001) -0.0025 (0.000001)
715.153
715.152
Table 4: Results of the structural form model for present bias assuming full credit constraints on housing house have just enough wealth to buy that house if it were located on private land. So buyers of houses with land lease will have more wealth after they buy a house than buyers of houses on private land. Note that the assumption of initial wealth is again not important for the linear felicity function since in that model buyers always consume all wealth in the first period. However, if we assume a CARA felicity function, it implies that buyers of a house with a land-lease contract are able to consume more than the buyers of houses with private land in the first periods after buying a house. With this specification we can now also identify β for the linear case. Table 4 reports the results. The performance of this model is poor in comparison to the model of Section 5. We find that the parameter of absolute risk aversion (i.e. a) goes to zero. This implies that the CARA specification converges to the linear utility case and all other parameter estimates are the same for both 33
specifications. We find that the long-run discount rate in the exponential model is high and equals 45 percent per year. If we allow for present bias, we find a very low estimate of β which implies a short-term discount rate of 178% while the long-run discount rate is equal to 61 percent. This suggests that a misspecified model that assumes no mortgages not only has a higher RSS but it can also strongly overestimate the short-term discount rate.
7
Final remarks
We presented and estimated a CARA utility model with present bias. For our estimation, we use land-lease contract data from Amsterdam. Those contracts specify payments at different moments in time and are traded and priced in a competitive market. For a given utility function, the price of those contracts depends on both the long- and short-run discount rates and this is what allows us to estimate both. The specification where agents can borrow for housing performs best. For that case we find a long-term discount rate of 1.9% and a short-term rate of 20.4%. Although our model is stylized, we believe that structural estimation of the discount rates with data from the field is a welcome complement to the many estimates from the lab.
References
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convex budgets”, American Economic Review, 102, 3333-3356. Andreoni, J. and C. Sprenger (2012b), “Risk preferences are not time preferences”, American Economic Review, 102, 3357-3376. Ahumada, H.A. and M.L. Garegnani (2007), “Testing hyperbolic discounting in consumer decisions: evidence for Argentina”, Economics Letters, 95, 146–150. Attanasio, O.P. and G. Weber (1995), “Is consumption growth consistent with intertemporal optimization? Evidence from the consumer expenditure survey ”, Journal of Political Economy, 103, 1121-1157. Attanasio O.P., J. Banks, C. Meghir and G. Weber (1999), “Humps and bumps in lifetime consumption ”, Journal of Business and Economic Statistics, 17, 22-35. Cohen, A., and L. Einav (2007), “Estimating risk preferences from deductible choice”, American Economic Review, 97, 745-788. Fang, H. and D. Silverman (2009), “Time-inconsistency and welfare program participation, evidence from the NLSY”, International Economic Review 50, 1043-1077. Fang, H. and Y. Wang (2008), “Estimating Dynamic Discrete Choice Models with Hyperbolic Discounting, with an Application to Mammography Decision”, Working paper, Duke University, Durnham. Frederick, S., G. Loewenstein, and T. O’Donoghue (2002), “Time discounting and time preference: a critical review”, Journal of Economic Literature 150, 351–401. 35
Gautier, P.A. and A.P. van Vuuren (2014), “The effect of land lease on house price, Mimeo VU University Amsterdam. Gertner, R. (1991), “Game shows and economic behavior: risk-taking on card sharks”, Quarterly Journal of Economics 108, 507-517. Harris C. and D. Laibson (2001), “Dynamic choices of hyperbolic consumers”, Econometrica 69, 935-957. ¨ m (2012), “Identifying time Harrison, G.W., M.I. Lau, and E.E. Rutstro preferences with experiments: comment”, mimeo Stanford. Laibson, D. (1997), “Golden eggs and hyperbolic discounting”, Quarterly Journal of Economics 112, 443-477. Laibson, D., A. Repetto and J. Tobacman, (2007), “Estimating discount functions with consumption choices over the life cycle”, working paper, Harvard University, Cambridge. Paserman, D. (2008), “Job search and hyperbolic discounting: structural estimation and policy evaluation ”, Economic Journal 118, 1418–1452. Shui, H. and L. Ausubel (2004), “Time inconsistency in the credit card market, working paper ”, University of Maryland, College Park. Strotz, R.H. (1956), “Myopia and inconsistency in dynamic utility maximization”, Review of Economic Studies, 23, 165-180. Veen, I. (2004), “De waarde van een erfpachtrecht in het Amsterdamse stelsel, in het bijzonder van voortdurende rechten bij canonherziening einde tijdvak.”, Working paper, Amsterdam (in Dutch).
36
Appendix A
Proof of Lemma 1
We start with the first part of the lemma. Combining the FOCs for s and s + 1 and using the definition of ∆(s) results in u′ (C(s − 1) − ∆(s) − ∆(s − 1)) u′ (C(s − 1) − ∆(s − 1)) = u′ (C(s − 1)) u′ (C(s − 1) − ∆(s − 1))
(17)
The definition of IARA implies that u′ (c − ε) u′ (c − ε − ∆) > ′ u (c) u′ (c − ∆)
(18)
for every ε > 0 and ∆ > 0.19 Therefore, u′ (C(s − 1) − ∆(s) − ∆(s − 1)) u′ (C(s − 1) − ∆(s − 1)) > u′ (C(s − 1)) u′ (C(s − 1) − ∆(s))
(19)
since both ∆(s) and ∆(s − 1) are positive. Combining (17) and (19) gives u′ (C(s − 1) − ∆(s) − ∆(s − 1)) u′ (C(s − 1) − ∆(s) − ∆(s − 1)) > u′ (C(s − 1) − ∆(s − 1)) u′ (C(s − 1) − ∆(s)) Using the fact that u is convex gives the result. The proofs for CARA and DARA are similar. Now, we turn to the second part of the lemma. If β < 1, then combining the FOC, taking into account that c′s > c′s−1 and using again the definition of ∆(s) we obtain u′ (C(s − 1) − ∆(s − 1)) u′ (C(s − 1) − ∆(s) − ∆(s − 1)) < u′ (C(s − 1)) u′ (C(s − 1) − ∆(s − 1)) For CARA we have inequality (18) but with an equality. Therefore u′ (C(s − 1) − ∆(s − 1)) u′ (C(s − 1) − ∆(s) − ∆(s − 1)) = u′ (C(s − 1)) u′ (C(s − 1) − ∆(s)) But then the result follows using the fact that u is convex. The proof for the IARA-case is trivial.
B
Proof of Theorem 2
We first proof the following lemmas. 19
The proof of this statement is available upon request.
37
Lemma 3 When δ ≤ (≥)δ∗ (β), then ∗∗
s X
′
u (C(s)δ
s=s∗
.
s ∂C(s)
∂δ
∗∗
≤ (≥)
s X
u′ (C(s)δs
s=s∗
∂C(s) ∂β
Lemma 4 Suppose that δ < δ∗ (β). Define h(σ) as ∂ pbσ ∂ pbσ ∂β ∂δ
h(σ) =
Then for any σ1 > σ0 we have h(σ1 ) > h(σ0 ).
Proof Lemma 3: Using (11), we obtain for the CARA case that ∂C(s) ∂cs∗ 1 s − s∗ = + ∂δ ∂δ a δ and
(20)
∗
s−s −1 c′s−k (ωs−k ) ∂cs∗ 1 X ∂C(s) = + ∂β ∂δ a 1 − (1 − β)c′s−k (ωs−k )
(21)
k=0
Using the budget constraint s∗∗ −s∗ X
s
cs∗∗ −s (1 + r) =
s∗∗ −s∗ X
s
(y − rp)(1 + r) −
Ls (b p)(1 + r)s
s=0
s=0
s=0
s∗∗ −σ X
we can derive cs∗ and taking the derivatives yields, " # Ps∗∗ −s∗ s s(1 + r) ∂cs∗ 1 s∗∗ − s∗ − Ps=0 =− s∗∗ −s∗ ∂δ aδ (1 + r)s s=0 while
∂cs∗ =− ∂β
1 a
Ps∗∗ −s∗ Ps∗∗ −s−s∗ −1 s=0
k=0
Ps∗∗ −s∗ s=0
c′s∗ −s−k 1−(1−β)c′s∗ −s−k
(1 + r)s
Substitution of (22) and (23) into (20) and (21) results in # " Ps∗∗ −s∗ s s(1 + r) ∂C(s) 1 s − s∗∗ + Ps=0 = s∗∗ −s∗ ∂δ aδ (1 + r)s s=0
and
∂C(s) = Ψ(β) ∂β
38
(22)
(23)
This results in ∗∗ −1 sX
t=s∗
exp(−ac0 ) ∂C = u′ (C(t))δt ∂δ aδ ×
t−1 Y
k=0
s∗∗X −s∗ −1 t=0
1 (1 + r)t !
"P
1 1 − (1 − β)c′s−k
s∗∗ −s∗ s(1 + r)s s=1 Ps∗∗ −s∗ (1 + r)s s=1
− s∗∗ − s∗ + t
#
and ∗∗ −1 sX
t=s∗
∂C u′ (C(t))δt = exp(−ac0 ) ∂δ
s∗∗X −s∗ −1 t=0
1 Ψ(β) × (1 + r)t
t−1 Y
k=0
1 1 − (1 − β)c′s−k
!
and this gives the desired result for δ < δ∗ (β). Proof of lemma 4 Total differentiation of (2) gives the following derivative of β c0 )/∂β ∂(W0 − W ∂ pbσ (β, δ) = c0 )/∂ pb ∂β ∂(W0 − W
Using the definition of (12) and taking derivatives we obtain ∗∗
s X ∂W0 (0) W0 (0) C(ω(s)) = − u(y − rp) + β u′ (∂Cs (ωs ))δs ∂β β ∂β s=s∗
Using equation (2) we obtain that the numerator of ∂p/∂β equals ∗∗
s X c0 (0) ∂W0 (0) ∂C(ω(s)) ∂W u′ (Cs (ωs ))δs − = u(y − rp) − u(y − rb p) − β ∂β ∂β ∂β s=s∗
Similarly, we obtain from (13) that ∗∗
s X c0 (0) ∂W0 (0) ∂W ∂C(ω(s)) u′ (Cs (ωs ))δs − =u(y − rp) − u(y − rb p) − β ∂δ ∂δ ∂δ s=s∗
Hence, we obtain that the division of the derivatives equals P ∗∗ u(y − rp) − u(y − rb p) + β ss=s∗ u′ (C(s)δs ∂C(s) ∂ pbσ /∂β ∂β <1 h(σ) = = Ps∗∗ ′ ∂C(s) s ∂ pbσ /∂δ u(y − rp) − u(y − rb p) + β s=s∗ u (C(s))δ ∂δ
Since δ < δ(β). Finally, since a larger σ does nothing else as shifting s∗ and s∗∗ towards the end of the time horizon, the third terms in the numerator and denominator shrink in absolute terms towards zero when σ becomes larger. This proofs the lemma.
39
Proof of theorem 2 Suppose that we have two different values of σ, say σ0 and σ1 . Moreover let pbσ (β0 , δ0 ) be the prices determined by the real values of β and δ. Then, there cannot be any other combination of β and δ that produces both pbσ0 (β0 , δ0 ) and pbσ1 (β0 , δ0 ) because it must be that, pbσ0 (β0 , δ0 ) = pbσ0 (β, δ)
and
(24)
pbσ1 (β0 , δ0 ) = pbσ1 (β, δ)
(25)
Since pbσ increases in its second argument, (24) implies that we obtain a function δ(β). bσ1 (β, δ(β)) From the implicit function theorem, ∂δ(β) ∂β = −h(σ0 ). If we can prove that p is monotonic, then we have proved that hat (25) can only have one solution. Taking total derivatives, we obtain ∂pσ1 (β, δ(β)) dpσ1 (β, δ(β)) ∂δ(β) = 1 + h(σ1 ) <0 dβ ∂β ∂β where the last inequality follows from Lemma 4.
C
Derivation of equation (16)
Consider two cases: (i) houses for which the land lease has been paid in advance for σ years and (ii) houses for which the land lease has not been paid in advance and for which immediate payments must be made. This last case can be further subdivided into fixed and variable land lease. We start with the first case. For these houses Li,s = 0 as long as s < t + σ. If s ≥ t + σ payments must be made. Let ϕt′ be the expected land rent increase within a period in year t′ , then
Li,s = γJ(i) exp
σ X t′ =1
ψt′ +
s X
t′ =σ
ϕt′
!
pbi,t .
(26)
The payments of houses for which the land lease is not paid in advance is also given by equation (26) as long as s ≥ t + σ . However, it is different when s < t + σ . For
40
the fixed case we obtain,
Li,s = γJ(i) pbi,τ ,
where the house price in period τ follows from pbi,τ variable case it is,
Li,s = γJ(i) pbi,τ exp
t−1 X t′ =τ
(27) Pt−1
t′ =τ
ϕt′
!
.
ψt′ = pbi,t , while for the (28)
Combining these two equations and dropping the subscript results in equation (16).
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D
Description of the variables in the main regressions
Period of Construction (0) Unknown or before 1500 (base), (1) 1500-1905, (2) 1906-1930, (3) 1931-1944, (4) 1945-1959, (5) 1960-1970, (6) 19711980, (7) 1981-1990, (8) 1991-2000, (9) after 2001.
Location/View (0) Not stated (base), (1) Next to large park, (2) Along a canal/river/lake, (3) next to small park, (4) Free view.
Heating system (0) No heating system (base), (2) TradiMaintenance inside tional heating system, (3) Modern cen(0) Bad (base), (1) in between poor and tral heating system, (4) idem with airbad, (2) poor, (3) poor to average, (4) av- conditioning or solar system. erage, (5) average to good or not stated, (6) good, (7) good to excellent, (8) excel- Type of house lent. (0) Simple, (1) single family dwelling, (2) Canal house, (3) mansion, (4) homestead, Maintenance outside (5) bungalow, (6) villa, (7) country house See maintenance inside for the description or cottage, (8) first floor apartment (in of this variable small building) (9) apartment in small building which is not first floor, (10) duYear plex apartment, (11) apartment in every Every year from 1985 (base) to 2011. sized building, (12) apartment in large building, (13) apartment for elderly, (14) Neighborhood apartment in small building, floor not Dummy for all remaining 27 neighbor- stated. hoods of Amsterdam Isolation Dummy variable for the number of isolation techniques used in the house.
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