Rent Control and Misallocation

Abstract

Rent control is still an important type of government regulation of housing markets in many countries and numerous researchers have studied its implications for allocation, welfare and investments in housing. The present paper aims to improve our understanding of the effect of second-generation rent control when it is applied only to one sector of the rental market. It is diagrammatically shown that the welfare effects are very different between a universal and a limited application of rent control. Studying the Danish case of second-generation rent control, lower rents are found in controlled sectors and a minor increase of the rent in the uncontrolled sector. Using the area of living space in the dwelling as a measure for housing consumption, evidence is also produced of both overallocation and underallocation of housing in the rent-controlled sectors; as envisaged by economic theory.

1. Introduction

In the middle of the 1990s, Arnott (1995) and Olsen (1998) pointed out the scarcity of research on the detriments and benefits of rent control. They argued that, even though there is broad agreement about the detrimental effects of first-generation rent control, second-generation rent control might not be that harmful. Furthermore, given the variety of rent control programmes, the best way to judge second-generation rent control is through empirical studies (Arnott, 1995). Since Arnott’s (1995) paper, several papers on rent control have appeared in economic journals.¹ For a critical review of eight recent theoretical papers, see Lind (2007), who argues that these models only provide modest additions to our understanding of rental housing. There are relatively few empirical papers on the subject and most are on the New York City case of rent control. Early (2000) estimated the average level of benefits based on a sample from New York City and concluded that tenants in the uncontrolled sector would actually be better off if rent control was never introduced. Glaeser and Luttmer (1997 and 2003) point out that first-generation rent control in a perfectly competitive housing market incurs not only the well-known welfare loss from reduced supply, but also a welfare loss from misallocation among households of the controlled dwellings. Most rent controls applied today are, however, of the second-generation type. Thus, the Los Angeles Rent Control Act of 1978 exempted newly constructed units (see Fallis and Smith, 1984) and rent control in Vancouver, British Columbia, exempted units built after 1974 (see Marks, 1984). In the New York case studied by Glaeser and Luttmer (2003), rent control regulations excluded apartments in buildings with fewer than six apartments. Sims (2007) examines the effects of rent control on the supply and maintenance of rental dwellings by use of the sudden abolishment of rent control in Massachusetts in 1995. His results suggest that rent control has a negative impact on the supply and maintenance of rental units. Different types of second-generation controls will have different impacts on the market. Institutional detail and mechanism design are important. The empirical results presented here for the Danish case should not be extended uncritically to other countries.

The reason for the scarcity of empirical studies can usually be attributed to a lack of suitable data. Denmark has the advantage of having register-based data, which can be merged to give detailed information on households and housing units. Thus, a couple of empirical studies of Danish rent control have already been published. Munch and Svarer (2002) examine the effect of rent control on household mobility and show that rent control has a significantly negative effect on mobility. Svarer et al. (2005) find that the probability of finding a local job increases with rent control, whereas the probability of finding a job outside the local labour market decreases with rent control. Yet another paper based on Danish data by the Economic Councils (2001) demonstrates that abolishing the Danish rent control will lead to heavy increases in rents in the Copenhagen area. The results furthermore show that persons with high income and long educational attainment among those living in controlled dwellings obtained the highest benefit from control.

The aim of the present paper is twofold. First, it adds to the existing literature by providing a diagrammatic presentation, which shows that the welfare loss from second-generation partial rent control may be very different from the welfare loss from first-generation rent control. Secondly, it presents new empirical evidence that shows the impact of Danish rent control on the various sub-sectors of the rental market. It presents figures of the difference between the estimated demand for square metres under controlled and free market rent versus the actual allocated square metres in the controlled sectors. The data are from a 20 per cent random sample of the Danish rental market.

Among the results, we find that the rent is on average close to 20 per cent lower in controlled dwellings compared with the free market rent, and this price wedge causes overallocation of square metres in the controlled sectors. However, disentangling allocation by household type reveals overallocations for some households, who consume more than they would under free market conditions and underallocations for other households who crowd themselves in smaller than optimal units.

The paper is structured as follows. Section 2 gives a brief overview of the Danish rental housing market. Section 3 illustrates welfare effects of rent control when applied in a sector of a rental housing market. Section 4 presents the data and a hedonic regression on rents. Section 5 discusses and presents the estimations of household demand for square metres of rental housing. Based on the estimated housing demand, and using the hedonic equation to predict free market rents, section 6 gives various indications of misallocation in the rent-controlled market sectors. Finally, section 7 presents the conclusion.

2. Rent Control in Denmark

The Danish housing market has, with some modifications, kept rent control on the rental housing market since the beginning of the Second World War. However, in 1991, the Danish parliament decided that new dwellings could be let without control. It was believed to be a decision with no real effect because new dwellings with free market rent would be unable to compete with rent-controlled dwellings. However, new dwellings have since then been built and let at free market rent, and today letting without rent control accounts for close to 10 per cent of the rental market, which covers nearly half of the total housing in Denmark.

Table 1 shows the distribution of rental housing in Denmark based on a 20 per cent random sample from register data on Danish dwellings from 2004 combined with data on the rents paid from 1999. The rents were collected by the municipalities together with rent information from the register of the tax authorities, which is used for real estate appraisals. The information on rents has been used in a study by the Danish Ministry of Housing and Urban Affairs (2000).

Table 1.

Distribution of rental housing in Denmark based on a 20 per cent random sample

Market	Rental housing type	Observations in sample (percent)		Observations with rent (percent of sample)
		Number	Percentage	Number	Percentage
Non-private sector		125 419	67
	Social housing	92 947	48	85 400	92
	Co-operative ownership	32 472	16	—	—
Private sector		65 878	33	38 394	58
	Controlled rent	29 809	16	25 518	86
	Adjusted rent	22 084	12	8 314	38
	Market rent	13 985	8	4 562	33
Total		191 297	100	123 794	65

Notes: In the table, private housing let at market rent consists of housing units either built after 1991 or situated in municipalities with no rent control. Private housing let at adjusted rent consists of housing units in small buildings (six or less apartments per building) in municipalities with rent control. Unlike the rent information from the tax authorities, the study by the Danish Ministry of Housing and Urban Affairs (2000) only covers buildings with three and more let apartments, which explains the comparatively low coverage for adjusted rent, mostly found in smaller buildings that are typically privately let. Moreover, the time lag between 1999 and 2004 has the consequence that apartments with market rent built between these years are excluded.

Source: A 20 per cent random sample from register data on Danish dwellings from January 2004.

Rents in the social housing sector² provided by housing associations that receive public support are fixed according to special cost-based rules. Anyone of age 18 and above can apply for a dwelling in the social sector. However, allocation rules do not allow single-person households to get more than one bedroom and one living room. Three rooms may in some cases be offered to married or cohabitating couples, but apart from this, three and more bedrooms are reserved for households with children. Once a household has moved in, it has a preference position vis-à-vis outsiders for choosing another apartment and is allowed to stay in an apartment if the household size decreases. A third form of tenancy in Denmark is tenancy under co-operative ownership. Because co-operative ownership differs significantly from ordinary rental housing, it is disregarded in the following analysis.³ For rent-controlled private rentals, the rent covers running, maintenance and capital costs, plus a ‘fair’ interest on invested capital.

Out of 275 municipalities, 130 have no rent control for the private rental sector, and private rental housing built after 1991 has uncontrolled or free market rent. This is also the case for apartments in buildings used earlier for commercial purposes and newly established roof apartments in older buildings. We are unable to identify these dwellings with the consequence that we classify some as controlled, which are let at market rent.⁴ In the municipalities that apply rent control, private owners of buildings with less than seven apartments and older, thoroughly renovated apartments are exempted from cost-based rent control, but tenants may complain about high rents to the local rent control board. We term rent in this sector ‘adjusted rent’.

3. Welfare Effects from Misallocation of Rent Controlled Housing Units

A graphical treatment of the welfare loss from misallocation under a universal maximum price on a competitive market seems to have been first presented by Ng (1979) and later by Glaeser (1996) and Glaeser and Luttmer (1997 and 2003). They do not, however, consider rent control in one single sector of a homogeneous rental housing market. Figure 1 supplements their analysis with this case. Housing units are measured on the horizontal axis H with rent r on the vertical axis. Declining marginal valuations of housing units explain the negative slope of households’ demand curve D. With no income effects assumed, Marshall’s consumer surplus is used to demonstrate welfare effects of the introduction of rent control.

Figure 1.

Rent control in one sector of a rental housing market.

Under universal rent control at rent $r_{c}$ , supply is reduced from $H_{0}$ to $H_{c}$ and this gives a welfare loss equal to the triangle bEc if the controlled housing units are allocated to those households who value the units most highly. A rationing mechanism—for example, queuing, lottery, nepotism—will allocate housing in the controlled sector in such a way that households who value housing the most cannot be sure of becoming tenants in these dwellings. If the rent-controlled dwellings are allocated randomly⁵ among all the households who in total demand $H_{cd}$ units, only the area under the dashed line adc can be included in the consumer surplus. The total welfare loss from rent control is then the area aEc of which the area abc, due to misallocation, may be the bigger part depending on the elasticities of the demand and supply curves.

Imagine now that an initial equilibrium between demand and supply is reached in the point c after which rent control is introduced at the level $r_{c}$ , but with newly built dwellings exempted from control.⁶ After the introduction of control, demand is assumed to increase until the demand curve D is reached. The new supply follows the supply curve S. If the controlled (old) housing units are allocated to the households who value them most highly, the demand for new units will follow the demand curve D and the new equilibrium will be E. This is identical to the equilibrium without rent control and there is no overall welfare loss from this version of control. However, the occupants of the controlled housing units are typically those households who lived there when control was introduced, mixed with newcomers selected by queuing, nepotism and the like. Random allocation is probably the best description of this allocation mechanism, which means that the consumer surplus obtainable in the controlled sector of the market is measured below the dashed line ac. The random allocation leaves a residual demand for new housing units, which appears in the uncontrolled or free market sector to the right of $H_{c}$ as the dashed curve going through $E'$ . The demand for new housing units is thus lifted because of the random allocation of the controlled units and this gives an increase of new housing supply in excess of $H_{0}$ . The free market rent is lifted from $r_{0}$ to $r_{f}$ as a consequence of the random allocation of controlled housing.

The lost welfare because of misallocation of the controlled housing units consists of two triangles. Households who value housing above $r_{f}$ can buy housing in the uncontrolled sector, but the allocation of housing to households who value housing between $r_{c}$ and $r_{f}$ leads to a loss from misallocation. The consumer surplus due to this part of the random allocation is measured below the dashed line dc. Under perfect allocation, it would be measured under the line de, which is a section of the demand curve D, moved to the left in the figure. This gives a welfare loss from misallocation equal to the triangle dec. Furthermore, there is a welfare loss from misallocation equal to the triangle $EE' f$ because of the increased supply of new uncontrolled housing from $H_{0}$ to $H_{f}$ .

To sum up, there are important differences between first-generation rent control and the second-generation case where rent control is applied only in a single sector of the market. Assuming that the best description of allocation under control is random allocation, three important differences can be listed. First, compared with a situation with no rent control, a first-generation rent control can be expected to reduce supply, whereas rent control confined to one market sector can be expected to increase supply. A corollary to this is a higher uncontrolled rent when control is confined to one sector of the market. The increased free rent corresponds to empirical observations by Fallis and Smith (1984) and Marks (1984).⁷ With no reduction of supply, the welfare loss from reduced supply (the triangle bEc) disappears. Secondly, because all households who value housing highly find housing in either the controlled or the uncontrolled sector, the lost consumer surplus because of misallocation is considerably reduced (from the triangle abc to the triangle dec). Thirdly, under a sector rent control and random allocation the ‘over supply’ of uncontrolled housing units gives an additional welfare loss where the marginal costs of supply are higher than marginal benefits under perfect allocation (the triangle EE’f ).

Existing rent-controlled housing units stay on the market after the introduction of control in Figure 1 and this—with random allocation—leads to an increase of the total supply. However, for the controlled units to stay in the market, it is necessary that the controlled rent $r_{c}$ keeps its real value and is designed to allow landlords to obtain an interest on the invested capital, which can match the interest on alternative investments with equal risk. In the long run, housing capital lasts only as long as total production costs are covered by rents and, as documented by Gyourko and Linneman (1990), if profit is negative under control, housing units can be expected to gradually deteriorate until they are ready for demolition.⁸ In Denmark, a large share of private controlled rental housing has disappeared from the market, but an increased supply of social housing has filled the gap. This illustrates that institutional details are important for the market effects of second-generation rent control.

4. Determination of Rents

Figure 1 shows that rent control in a single sector of the rental market can be expected to give a spread of rents with controlled rents below and uncontrolled rents above the rent in a market with no control. In this section we use a hedonic regression to find the determinants of the controlled and uncontrolled rents in the Danish rental market.

The available data make it possible to calculate the actual paid rent per square metre of living space for the rented dwellings, with some limitations. One is that the reported rent is from the year 1999. This implies that all dwellings built after this year are excluded from the sample, but also that one must expect some differences between dwelling characteristics in the year 2004 and the actual characteristics in 1999. We do not consider this to be a serious problem, but obviously some dwellings have been better equipped over the years, implying that the reported rents may be too low for these dwellings and this may give a downward bias for coefficients on some variables. Finally, the households occupying the dwelling in January 2004 may be different from those who occupied the dwellings in 1999. The assumption behind our analysis is that the distribution of rents remains unchanged over the four-year time-span so that, for example, a high-rent dwelling in 1999 is also a high-rent dwelling in 2004.

The chosen form for the hedonic function is log-linear, which gives a straightforward interpretation of coefficients as elasticities. Similarly to Hoffmann and Kurz (2002) we estimate the following hedonic model

\begin{matrix} \ln r e n t_{i} = β_{0} + β_{1} {a d j u s t e d}_{i} + β_{2} {c o n t r o l l e d}_{i} + β_{3} {s o c i a l}_{i} \\ + \sum_{k = 1}^{K} β_{k + 3} X_{i} + \sum_{k = 1}^{K} γ_{k} {a d j u s t e d}_{i} * X_{i} \\ + \sum_{k = 1}^{K} δ_{k} {c o n t r o l l e d}_{i} * X_{i} + \sum_{k = 1}^{K} θ_{k} {s o c i a l}_{i} * X_{i} + ε_{i} \end{matrix}

where, $\ln ren t_{i}$ refers to the natural logarithm of the actual rent paid per year; the variables adjusted, controlled and social are dummy indicators for the three rent-regulated sectors; and $X_{i}$ is a set of control variables.

We make use of two different sets of controls in $X_{i}$ : dwelling-specific controls and local market specific controls. Dwelling specific controls include traditional characteristics such as the size of dwelling, type of housing and age of building. A full set of interaction terms between sector indicators and dwelling characteristics is used to make the hedonic model more flexible. Even though we account for a rich variety of controls, there may still be unobserved variables that influence rent differences apart from control. One limitation is that the location of the household within a specific geographical area—a major determinant of housing quality—cannot be identified in the data used. The register data allow insight into precise data of, for example, building characteristics of dwellings, but due to anonymity requirements precise geographical identifiers are omitted, which hinders identification of important locational characteristics such as school quality, distance to a central business district and various local amenities.⁹ Only the degree of urbanisation is included. Another potential problem in identifying the effect of rent control is the spillover effects to other sectors of the housing market. The introduction of control in one part of the rental market could be expected to lead to higher demand in the uncontrolled parts of the market, which would increase the rent level in the uncontrolled sector (see Fallis and Smith, 1984; and Early, 2000). The magnitude of the spillover effects will, however, depend on the allocation mechanism in the controlled sectors, so nothing can be said for sure. To account for spillover effects in the hedonic regression, we have a variable for rent control versus no rent control (in the municipality) and a variable for the share of social housing in the municipality. Besides locational variables, tenure length is included in the regression because long-time tenants may receive discounts on their rent compared with new tenants (see Hubert, 1995; Hoffmann and Kurz, 2002; and Larsen and Sommervoll, 2009, on this).

The regression (see Table 2) explains the rent reasonably well with an R ² value reaching 61 per cent. The most interesting coefficients are those giving the difference between rents in the uncontrolled sector and the three rent-controlled sectors. The baseline or reference dwelling is let at market rent in the capital area with low lagged vacancy rate. The property is a multifamily building no more than 10 years old and without reinforced concrete walls. The dwelling is equipped with toilet and bathroom. The current tenant has been living there for less than 6 years. Given an average dwelling size of 80 square metres, we can calculate the annual rent for this dwelling in the market sector to be 77 000 DKK¹⁰ and the annual rent for the baseline dwelling in the social sector to be 57 300 DKK or 25 per cent lower compared with the estimated market rent. The rent is slightly higher in the private rent-controlled sector, and the adjusted rent is even higher, but all are below the free market rent.

Table 2.

Determinants of the annual rent per dwelling (N = 123 545)

Dependent: ln of annual rent	Coefficient (S.E.)	Explanatory variables	Coefficient (S.E.)
Constant	8.81 *** (0.050)
Type of rent		Landlord–tenant relation
Market rent	reference	Tenure length 1 (1–5 years)	reference
Adjusted rent	−0.39*** (0.031)	Tenure length 2 (6–10 years)	0.03** (0.013)
Controlled rent	−0.98*** (0.055)	Tenure length 3 (11–15 years)	−0.03** (0.016)
Social housing	−1.49*** (0.051)	Tenure length 4 (16–max)	−0.17*** (0.020)
		Adjusted rent * Tenure length 2	−0.06*** (0.018)
Spillover effects		Adjusted rent * Tenure length 3	−0.08*** (0.023)
No rent control	−0.02*** (0.002)	Adjusted rent * Tenure length 4	−0.14*** (0.025)
High share of social housing	−0.04*** (0.002)	Controlled rent * Tenure length 2	−0.05***(0.014)
		Controlled rent * Tenure length 3	−0.04** (0.017)
Dwelling features		Controlled rent * Tenure length 4	(0.020)
ln living area (square metres)	0.55*** (0.010)	Social housing * Tenure length 2	−0.02 (0.013)
Controlled rent * ln living area	0.14*** (0.012)	Social housing * Tenure length 3	0.03** (0.016)
Social housing * ln living area	0.27*** (0.010)	Social housing * Tenure length 4	0.14***(0.020)
No toilet or bath	−0.12*** (0.005)
		Locationalcharacteristics
Social housing * No toilet/bath	0.05*** (0.008)	Low vacancy rate	reference
		Medium vacancy rate	−0.18*** (0.010)
Property features		High vacancy rate	−0.20*** (0.020)
Multifamily	reference	Adjusted rent * Medium vacancy	0.16*** (0.018)
Semi-detached housing	0.01*** (0.002)	Adjusted rent * High vacancy rate	0.11*** (0.030)
Single-family	−0.06*** (0.008)	Controlled rent * Medium vacancy	0.14*** (0.015)
Farmhouse	−0.30*** (0.110)	Controlled rent * High vacancy	0.05 (0.036)
Reinforced concrete walls	−0.01*** (0.001)	Social housing * Medium vacancy	0.12*** (0.013)
Semi-detached housing * Non-reinforced concrete	0.09*** (0.005)	Social housing * High vacancy	0.11*** (0.022)
Single-family * Reinforced concrete	0.12** (0.052)
		Urbanisation level
Vintage		Town size 1 (Capital area)	reference
Vintage 1 (1994 or later)	reference	Town size 2	−0.31*** (0.046)
Vintage 2 (1984–93)	−0.08***(0.015)	Town size 3	−0.22*** (0.022)
Vintage 3 (1974–83)	−0.25*** (0.019)	Town size 4	−0.32*** (0.021)
Vintage 4 (1964–73)	−0.32*** 0.019)	Adjusted rent * Town size 2	0.26*** (0.050)
Vintage 5 (1944–63)	−0.31*** (0.016)	Adjusted rent * Town size 3	0.17*** (0.028)
Vintage 6 (1904–43)	−0.28*** (0.016)	Adjusted rent * Town size 4	0.25*** (0.029)
Vintage 7 (1854–03)	−0.25*** (0.021)	Controlled rent * Town size 2	0.39*** (0.047)
Controlled rent * Vintage 3	−0.13*** (0.018)	Controlled rent * Town size 3	0.26*** (0.022)
Controlled rent * Vintage 4	−0.16*** (0.018)	Controlled rent * Town size 4	0.37*** (0.024)
Controlled rent * Vintage 5	−0.16*** (0.014)	Social housing * Town size 2	0.10** (0.046)
Controlled rent * Vintage 6	−0.10*** (0.014)	Social housing * Town size 3	0.09*** (0.022)
Controlled rent * Vintage 7	−0.07*** (0.023)	Social housing * Town size 4	0.16*** (0.022)
Social housing * Vintage 2	0.00 (0.015)
Social housing * Vintage 3	−0.00 (0.019)
Social housing * Vintage 4	−0.09*** (0.019)
Social housing * Vintage 5	−0.04*** (0.020)
Social housing * Vintage 6	0.16*** (0.016)
Social housing * Vintage 7	0.15*** (0.019)

		R ²	0.61

Notes: Heteroskedasticity robust standard errors are reported. Significance at 1 per cent level: ***; significance at 5 per cent level: **. Some combinations of interaction terms, like between adjusted and vintage, were not included in the final regression as they turned out to be insignificant in the preliminary regressions.

Our two variables for spillover effects both have negative coefficients of small magnitude compared with the coefficients for the rent-controlled market sectors. The interpretation for the variable no rent control is that the free market rent is lower without rent control just as depicted in Figure 1 (equilibrium E compared with equilibrium $E'$ ). Although the coefficient is negligible, it is in line with Fallis and Smith (1984) and Marks (1984) who found an increase in the uncontrolled rent after the introduction of rent control. A high share of social housing, which has controlled rent, implies a high uncontrolled rent according to the model of Figure 1, and a positive coefficient should thus be expected. However, a negative coefficient may appear if a high share of social housing gives negative social stigma to a municipality.

The coefficients on interaction terms indicate that rents are set differently in controlled rental sectors compared with the uncontrolled sector. If we look at the main vintage coefficients, they indicate a U-shaped relationship between the age of the building and the rent in the market sector and that this relationship is deeper in the private controlled sector, while the age of the buildings has a less negative effect on the rent in the social sector. Long-term residents pay lower rents and this relationship is most pronounced in the uncontrolled market sector. This is as expected, as landlords may want to differentiate between good and bad tenants (see Hubert, 1995). It is also evident that the market sector is most sensitive to local market characteristics. The rent level falls sharply with higher vacancy rates and a lower degree of urbanisation in this sector, while these relationships are less pronounced in the other sectors.

We do not know how much of the estimated effect on the rent for a dwelling situated—for example, in the social housing sector—stems from rent control and what may be the effect from unobserved variables. A low rent could be caused by the poor quality of the surrounding area. To detect whether or not such locational effects play a role, a robustness check has been performed with the regression rerun with dummies for parishes¹¹ to account for area-specific heterogeneity. The results revealed only minor changes. Thus, the coefficient for adjusted rent was reduced from -0.39 to -0.36, for controlled rent from -0.98 to -0.85 and for social housing it rose from 1.49 to 1.50. Inclusion of parish fixed effects furthermore watered down the coefficient and thereby the explanatory content and predictive power of other control variables. Because of this, we continue with the estimation reported in Table 2, but the reader should keep in mind that the rent-reducing effect of control may be slightly overstated in the private rental sector.

An interesting question is how much rents in the non-market sectors would increase on average if the dwellings were offered on market terms. To look into this, we calculate a market rent per square metre applying the estimated hedonic rent coefficients in Table 2 for each dwelling in the non-market sectors and compare this predicted free market rent with the actually paid controlled rent in Table 3.¹²

Table 3.

Annual paid rent per square metre compared to the market rent per square metre

Type of rental housing	N	Paid rent ( $r_{c}$ )	Predicted market rent^a ( $r_{f}$ )	S.E.^b	Average rent increase compared with paid rent (percentage)
Social housing	85 284	482.6	594.4	0.45	23.2
Controlled rent	25 431	465.2	586.8	1.21	26.1
Adjusted rent	8 281	469.3	513.1	2.28	9.3

The calculated rent differences depend both on the design of the control system and on the specific characteristics of the dwellings.

Standard error of the mean difference.

Notes: See also Figure 2. Rents are in year 1999 DKK.

As expected, actual rents paid in the controlled sectors are on average lower than the free market rent. The difference is largest for social housing and the controlled private rental sector, whereas housing with adjusted rent seems on average to have rents closer to the market rent. Andersen (2008) interviewed 385 private landlords with properties containing three and more dwellings, who on average predicted a 10 per cent increase in rents if control was abolished. The Economic Councils (2001) used an accounting technique and calculated an average rent increase of 43 per cent in municipalities with controlled rent. Our estimate is within this range.

5. Demand for Rental Housing

The demand curves in Figure 1 are drawn on the assumption that households demand their welfare-maximising amount of housing at different prices. However, households may not be on the demand curve. A relocating household has only a limited number of options for new housing and will choose the one that maximises its (present value of) welfare. This may imply that the accepted amount of housing is more or less than the welfare-maximising amount had there been a continuum of options. Households who are offered a limited number of square metres in rent-controlled sectors are more inclined to accept fewer than demanded square metres because of the lower rent.¹³

This result is illustrated in Figure 2 where the two goods in the utility function are housing, h, and other consumption, c. With other consumption having the price 1, y is the value of income measured in units of other consumption. The dashed line from y illustrates the budget constraint under controlled rent and the solid line illustrates the constraint under a free market rent. Demand increases from $h_{1}$ to $h_{3}$ when the rent is reduced from $r_{f}$ to $r_{c}$ . Yet if the household is only offered the amount $h_{c}$ of controlled housing units, this is accepted because welfare is higher in $E_{2}$ than in $E_{1}$ . Supply of controlled housing is often not able to respond fully to the demand increase from $h_{1}$ to $h_{3}$ .

Figure 2.

Rental demand under rent control.

Rationing of this kind can be expected to be present in our sample for the controlled sectors and this tends to give downward-biased coefficients in demand regressions in the rent-controlled sectors. Supply is more able to respond to demand in the uncontrolled sector, which implies that households in this sector should be closer to their demand curve.

We estimate the demand for square metres as a function of rent and characteristics of the households in the free market. This gives a fair approximation of the optimal demand relation for housing—assuming, of course, that housing in this sector is on average allocated optimally. The simplest structural type of demand can be written as

q = f (rent, h)

where, q is the quantity demanded; rent is the price of housing and h is a set of household characteristics.

As approximation for the housing quantity, we use the number of square metres of living space. We use a log-linear specification

\ln q_{i} = β_{0} + β_{1} {rent}_{i} + \sum_{i}^{I} β_{i} h_{i} + ε_{i}

The variable $h_{i}$ is current equivalent income of the household, the number of persons in the household, age, sex, occupation and other characteristics of the breadwinner. We are aware that households’ demand for housing depends more on permanent income than on current income. Yet permanent income depends on expected future earnings and is difficult to estimate, so we rely on the current income as a proxy for permanent income.

Adjustment of rental housing consumption happens by moving, which is costly and, as demonstrated by Muth (1974), households may be forward-looking and therefore consume more or less than they would in a static world. However, Edin and Englund (1991) present empirical evidence which shows that tenants who have recently moved—contrary to owners—are close to their demand function. Consequently, we have chosen to estimate the demand for a sample of recent movers. Given that there is a trade-off between the length of time at the current address and the sample size, we have chosen a two-year limit for the length of time at the current address.

In order to account for the endogeneity of the rent in the demand function, we use lagged vacancy rates at the municipality level and the degree of urbanisation as instruments for the rent. The vacancy rate is measured as the average percentage of vacant rental dwellings over the preceding five years, which is believed to be without influence on the demanded number of square metres. The second instrument is the degree of urbanisation.¹⁴

Lastly, there may be a selection bias because unobservable characteristics of the households may affect the decision to select housing in the uncontrolled sector and are correlated with the observables in the demand relation for this sector. Failing to control for such unobservable characteristics will lead to a sample selection bias. We deal with the problem by modelling the selection process and adjusting the demand estimation for a possible selection effect. The first to control for endogeneity and self-selection bias was Mroz (1987) in the context of female labour supply and later the method was applied, among others, by Renders and Gaeremynck (2006) in the context of corporate governance.

First, we model the selection process by estimating a standard probit model of being in the uncontrolled sector. The inverse Mills ratio, which relates regressors with the probability of being in this sector, is calculated. In order to reduce the collinearity between the inverse Mills ratio and the regressors of the next stages, we choose the age of dwellings as an exclusion restriction when we estimate the probit model. The intuition is that dwellings in the uncontrolled sector are on average newer and that households with a preference for new dwellings will have high probability of choosing this sector. Secondly, we treat the endogeneity of the rent variable by a standard two-stage least squares (2SLS) procedure. In both stages, the inverted Mills ratio is included to account for a self-selection bias.

Table 4 shows the results for three different estimation models: standard ordinary least squares (OLS), 2SLS with control for endogeneity but not for self-selection and, finally, Heckman 2SLS with controls for both. The 2SLS and Heckmans 2SLS regressions show a smaller rent coefficient. These models were subject to Wooldridge’s (1995) robust score test for endogeneity, which showed that homeownership cannot be regarded as exogenous. To test the validity of the instruments, we applied Wooldridge’s score of overidentifying restrictions. The instruments pass the test. The models were also re-estimated using only one instrument (either vacancy rate or degree of urbanisation), which did not result in a significant change of the main results. Moreover, instrumental variables in the first-stage regression have statistically significant effects on the rent variable. All in all, the instruments seem credible. However, the control for self-selection bias is without significant effect on the coefficients and the inverse Mills ratio turned out to be insignificant, so we rely on the 2SLS coefficients in the following.

Table 4.

Determinants of the demand for square metres in the uncontrolled sector (N = 1016)

Dependent: ln of living area	OLS		2 SLS		Heckman 2SLS
	Coefficient	S.E.	Coefficient	S.E.	Coefficient	S.E.
ln rent per square metre	−0.35***	0.036	−0.20***	0.056	−0.20***	0.056
ln income per equivalent person	0.20***	0.037	0.19***	0.035	0.19***	0.036
Single no children	reference
Single with one child	0.23***	0.037	0.26***	0.033	0.25***	0.037
Single with more than one child	0.48***	0.039	0.50***	0.039	0.48***	0.042
Couple no children	0.24***	0.023	0.25***	0.023	0.25***	0.024
Couple with one child	0.42***	0.040	0.42***	0.038	0.41***	0.041
Couple with more than one child	0.50***	0.038	0.52***	0.043	0.51***	0.044
More than two adults	0.61***	0.123	0.67***	0.167	0.65***	0.162
Breadwinner characteristics
Breadwinner is a woman	0.04**	0.018	0.03	0.019	0.04	0.019
Age 1 (18/24)	reference
Age 2 (25/34)	0.10***	0.029	0.11***	0.028	0.11***	0.028
Age 3 (35/45)	0.15***	0.033	0.17***	0.034	0.16***	0.033
Age 4 (46/59)	0.18***	0.038	0.20***	0.038	0.19***	0.039
Age 5 (60/74)	0.29***	0.039	0.30***	0.039	0.30***	0.040
Age 6 (75+)	0.25***	0.042	0.24***	0.041	0.24***	0.039
No education	reference
Short education	0.03	0.020	0.03	0.019	0.04	0.021
Long education	0.06	0.036	0.07**	0.035	0.08**	0.036
Wage-earner low-skilled	reference
Wage-earner medium-skilled	0.07**	0.029	0.05	0.030	0.06	0.030
Wage-earner high-skilled	0.08**	0.040	0.07	0.041	0.07	0.040
Self-employed	0.10	0.090	0.13	0.073	0.13	0.073
Transfer income	0.03	0.028	0.02	0.029	0.02	0.029
Student	0.01	0.062	−0.01	0.063	0.01	0.065
Immigrant	−0.10	0.073	−0.08	0.074	−0.06	0.075
Mills ratio					−0.05	0.036
Constant	3.72***	0.469	6.51***	0.382	3.03***	0.509

R ²	0.49		0.47		0.47
Endogeneity test, chi²(3) and p-value			6.78		6.40
			0.01***		0.01***
Overidentification, chi²(3) and p-value			4.77		4.80
			0.19		0.19

Notes: Significance at 1 per cent level: ***; significance at 5 per cent level: **.

The coefficient of ln rent per square metre gives an elasticity of demand equal to -0.20, indicating that a 10 per cent increase in the rent will reduce by 2 per cent the number of square metres demanded. The variable ln income per equivalent person covers the effect of a change in the household’s disposable income divided by the number of equivalent persons—i.e. the number of adults + 0.6 times the number of children—raised to the power 0.8. A positive income elasticity of demand equal to 0.19 is found for this variable, which tells us that a 10 per cent increase in disposable income per equivalent person will increase by 1.9 per cent the demanded number of square metres.

Naturally, the demand for space increases with the number of persons in the household. Women as household heads demand more space and demand has a reverse-U relationship with age. More education raises the demand for space. Also, demand rises when the household head has higher skills or is self-employed. Being an immigrant, a student and receiving transfer income are without significant effects.

6. Misallocation of Rent-controlled Housing

We use the estimated demand coefficients in Table 4 to calculate the quantity $h_{3}$ —i.e. the optimally demanded quantity under the paid rent—for households in the controlled sectors.

The allocation mechanism does not secure an allocation of square metres $h_{c}$ in accordance with demand. A number of households may be allocated more or less than demanded, given their actual household characteristics.

A comparison based on average figures is presented in Table 5. It shows that neither in social housing nor in controlled private rental housing are households allocated more square metres than demanded under the controlled rent. This indicates that households who are offered a limited number of square metres are inclined to accept less than demanded because of the low rent in these sectors. However, an average overallocation at 6.0 per cent is found in the sector with adjusted rent. This may be because the rent is closer to the market rent, which reduces the inclination to crowd. The presence of relatively many single-family houses in this sector may also explain why on average households tend to occupy more square metres than demanded.

Table 5.

Actual allocation and demanded number of square metres at controlled rent

Type of rental housing	N	Actual allocation ( $h_{c}$ )	Demanded quantity at paid rent^a ( $h_{3}$ )	S.E.^b	Overallocation (percentage)^c
Social	85 114	76.6	78.4	0.05	−2.3
Controlled rent	25 329	74.3	76.1	0.16	−2.5
Adjusted rent	8 241	80.3	75.7	0.35	6.0

Calculated by use of coefficients from Table 4.

Standard errors for the differences between means.

Percentage of quantity demanded.

Notes: The table shows the average number of square metres living area.

As a second exercise, we compare $h_{c}$ and $h_{1}$ —i.e. the allocated square metres compared with the quantity demanded, had the dwelling been offered at uncontrolled rent. The increase of the rent from controlled to uncontrolled should by itself reduce the quantity demanded. As Table 6 reveals, rationing of square metres in social housing and the private-rent-controlled sectors have the consequence that allocations do not differ much from the quantity demanded under uncontrolled rent. This corresponds to panel b of Figure 2. In the adjusted rent sector, $h_{c}$ is to the right of $h_{3}$ and so even further away from $h_{1}$ with an average overallocation of 9.1 per cent.¹⁵

Table 6.

Actual allocation and demanded number of square metres at uncontrolled rent

Type of rental housing	N	Actual allocation ( $h_{c}$ )	Quantity demanded at uncontrolled rent^a ( $h_{1}$ )	S.E.^b	Overallocation (percentage)^c
Social	84 055	76.7	75.8	0.05	1.2
Controlled rent	25 001	74.3	72.6	0.15	2.3
Adjusted rent	8 147	80.4	73.7	0.35	9.1

Calculated by use of coefficients from Table 4 with the uncontrolled market rents estimated by use of coefficients from Table 2.

Standard errors for the differences between means.

Percentage of calculated quantity demanded at uncontrolled rent.

Notes: The table shows the average number of square metres.

To give a more nuanced picture of tendencies for misallocation in the controlled sectors, we decompose square metre allocations by selected demographic groups in Table 7. Households are split into income groups and, for each group, four different household types are listed. Misallocation of square metres can go both ways. Households can end up consuming less or more than they would have consumed under free-market conditions because of the lower controlled rent.

Table 7.

Allocations of square metres for household sub-groups by type of rental sector

	Social housing			Private ctr.-rent housing			Private adj.-rent housing
	h_c	h₃	h₁	h_c	h₃	h₁	h_c	h₃	h₁
Low income
Single	80.8	75.2	72.6	70.5	74.6	71.6	80.3	76.3	74.0
Couple with children	94.8	94.8	92.5	87.9	99.0	93.9	104.8	97.5	99.4
Age 25/34	74.2	73.9	73.1	61.9	64.3	61.3	70.3	65.3	64.2
Age 75+	74.1	70.7	68.5	72.2	69.3	65.4	81.4	71.2	66.6
Medium income
Single	84.3	83.4	80.7	80.5	85.7	82.3	86.7	85.4	84.4
Couple with children	95.9	107.4	104.1	104.1	114.6	109.5	114.6	115.8	111.7
Age 25/34	74.4	78.9	75.5	67.1	74.6	71.2	73.7	73.6	72.2
Age 75+	73.3	67.8	66.4	77.3	69.7	67.5	85.7	71.9	69.4
High income
Single	91.5	86.3	81.5	86.6	98.7	94.3	103.0	99.5	96.8
Couple with children	99.1	120.6	117.3	125.9	134.5	126.9	142.8	133.5	129.9
Age 25/34	71.4	78.8	75.1	70.2	79.6	75.7	76.9	79.0	76.7
Age 75+	79.6	77.5	75.9	95.4	81.0	78.2	132.5	87.0	82.9

Notes: Low income covers the lowest income quartile; medium income the second and third quartiles; and high income the forth quartile. The table shows average allocations per household. The number of observations for each group varies greatly between sectors, from about 2000 in the case of social housing to less than 100 in the case of adjusted rent.

Some interesting tendencies in misallocation can be found in Table 7. We define overallocation as incidences where the actual allocation $h_{c}$ exceeds not only demand at uncontrolled rents $h_{1}$ , but also demand $h_{3}$ at controlled rent. In Table 7, these cases are in bold-faced type. Overallocation is found for households with heads of age 75+ in all three sectors. A corollary of this is that moving costs are high and the expected number of years in which to gain a benefit from moving is lower for old tenants. Systematic overallocation is also present for single-person households in social housing and housing with adjusted rent, but not in the private-rent-controlled sector. Underallocations, defined in a similar way as overallocation, are marked in italics. They are most widespread for young households and households with children, but not in the sector with adjusted rent.

7. Conclusion

This paper contributes to the on-going discussion of welfare and allocation effects of second-generation rent control. We have demonstrated diagrammatically that the welfare effects of second-generation rent control may be very different from the welfare loss of first-generation rent control. We also present empirical evidence on the effect of the Danish case of second-generation rent control where control is applied under different schemes in different sectors, leaving one sector uncontrolled. It is difficult to detect exactly what are the effects of rent control because a number of unobserved variables may bias the results. One such bias may stem from locational variations of the area quality. A check for this led us to believe that the estimated negative effect of control was slightly overstated in the private rental market sectors, but not for social housing. With this reservation, we find that control leads to significantly lower rents in the controlled sectors and to a negligible increase in the uncontrolled rent. All in all, our results also support the widespread view that control generates an overallocation of square metres in controlled dwellings. Yet we also show that underallocation is generated under rent control, and this reduces the amount of space to be freed when control is abolished.

Footnotes

Acknowledgements

The authors would like to thank Yew-Kwang Ng, colleagues at their department and the anonymous referees for fruitful comments. Remaining errors and omissions are the authors’ responsibility alone. The paper was written as part of the Centre for Housing and Welfare/Realdania Research Project.

Notes

Funding Statement

Economic support from Realdania is gratefully acknowledged.

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