Welfare Measures to Assess Urban Quality of Life

Welfare measures to assess urban quality of life

Francesco Andreolia,b,∗ and Alessandra Michelangelic

aLISER, 3 avenue de la Fonte, L4364 Esch-sur-Alzette, Luxembourg bUniversity of Verona, DSE, Vicolo Campoﬁore 2, 37129 Verona, Italy. cUniversity of Milano-Bicocca, DEMS, Piazza Ateneo Nuovo 1, Milan I-20126, Italy.

March 20, 2015

Abstract The standard index of urban quality of life is an approximated measure of the liveability of a city. In fact, this index associates a bundle of amenities with the implicit marginal prices of these amenities, and not with the prices of infra-marginal units. In this paper, we adjust the standard measure to value bundles that can diﬀer from the reference bundle of amenities for which marginal prices are estimated. Our methodology relies on a welfare measure representing individual willingness to give up (accept) to ensure (forego) that a change in the current distribution of amenities across areas will take place, keeping the level of utility unchanged. We obtain a new measure, the value-adjusted quality of life index, which can be identiﬁed from parametric models of consumer preferences. We use this index to measure the quality of life in the city of Milan.

Keywords: Hedonic Models, Quality of Life, Structural models

JEL Codes: D6, H4, R1, R2.

∗Corresponding author: [email protected].

1 1 Introduction

The most common framework used by urban economists to assess urban quality of life is the hedonic approach first developed by Rosen (1979) and Roback (1982). The quality of life index defined by this approach measures the estimated value of a set of amenities characterizing a given area. To determine this value, the implicit prices of the amenities are estimated using housing and wage hedonic regressions. Since the pioneering work of Rosen (1979) and Roback (1982), additional re- finements to this approach have been developed in different directions, both on theoretical and empirical grounds, to address the problems of the cost-of-living, tax-adjusted earnings and incomes from sources other than labor.1 These issues have been treated separately, although recent work incorporates some of these issues into a single framework. For example, Albouy (2008) and Albouy, Leibovici and Warman (2013) incorporate cost-of-living, tax-adjusted earnings and income from sources other than labor, in order to obtain much more sensitive estimates of quality of life. In empirical applications, implicit prices are usually computed at the average quantities of the amenities in the sample areas. They are then used as weights in the sum of the amenity quantities of each area to obtain the Roback (1982) quality of life index by area. The index calculated in this way provides an exact evaluation solely for the bundle with amenity average quantities computed on the overall sample areas. For all other bundles specific to each sample area, the Roback (1982) index gives an approximated value of the quality of life, as the amenity quantities specific to each area are associated with the implicit marginal prices computed on the overall average quantities and not with the prices of infra- marginal units. In other words, the value of the index has no direct interpretation in terms of willingness to pay for bundles different from the overall average bundle, as marginal prices do not correspond to a proper evaluation of these units. As Roback (1982) acknowledges, the vector of marginal prices is used as weights

1See Andreoli and Michelangeli (2015) for a recent survey of the measurement of urban quality of life following the hedonic approach.

2 for amenities and “merely shows the order of magnitude of expenditure in the average budget”(Roback 1982, p. 1274). In this paper, we develop a new measure, the value-adjusted quality of life index, providing the proper evaluation of infra- marginal units of amenities. Our methodology, developed focussing on a single city2 relies on the compensating benefit, a welfare measure introduced by Palmquist (2006) in the hedonic property value models to evaluate environmental values. In our framework, the compensating benefit gives the amount of money that a representative consumer living in a given neighborhood of the city would like to give up (accept) to insure (forego) a change in the current distribution of amenities across neighborhoods, keeping the level of utility unchanged. The identification of the compensating benefit is based on the bid function, developed by Rosen (1974), for which we assume an Inverse Almost Ideal Demand System (IAIDS) specification. The IAIDS, developed by Eales and Unnevehr (1994), is a model for consumer preferences, which explicitly into account a set of restrictions on consumer tastes required by welfare analysis. The IAIDS allows the identification of the inverse demand for housing attributes as a function of consumer utility parameters. The compensating benefit for a neighborhood is the difference between the bid function for amenities in that neighborhood and the bid function for the overall average quantities of the amenities in the city. The value-adjusted quality of life index for a neighborhood is obtained by augmenting the Roback (1982) index computed at the citywide average bundle of amenities by the compensating benefit for that neighborhood. The latter measures the willingness to pay for the average bundle; the former provides an evaluation of the units of amenities, which differentiate the neighborhood bundle from the city-average bundle. We illustrate our methodology through an empirical application to the city of Milan, using a dataset on housing transactions that occurred between 2004

2A large number of studies have focused on a single city to study issues especially related to the environment (air or water pollution, proximity to noxious sites, etc.). See for, example, Kohlhase (1991), and Michaels and Smith (1990).

3 and 2010. We consider a set of local amenities collected at the neighborhood level, such as green areas, educational and health services, recreational activities, public transport, security services, facilities and socio-demographic composition. We calculate the value-adjusted quality of life index for Milan at the neighborhood level and compare the ranking of neighborhoods based on this index with the ranking obtained with the standard hedonic index. The two indices turn out to be strongly positively correlated, although only the value-adjusted quality of life index is able to give a measure of the impact of each amenity on quality of life, explicitly taking into account the particular composition of the amenity bundle characterizing the neighborhood where the change occurs. The rest of the paper is organized as follows. In Section 2 we present the theoretical framework. Section 3 discusses the identiﬁcation of the value-adjusted quality of life index, the data and the empirical strategy. Section 4 shows empirical results. Finally, Section 5 presents the conclusions.

2 Theoretical Framework

We consider a city partitioned into N neighborhoods, and a representative agent who has to choose the neighborhood to live in to maximize utility. The focus on a single city allows to develop our arguments on a simpler framework, thereby avoiding inter-city differences in non-housing costs, intergovernmental transfers and local taxes. Moreover, labor market heterogeneity across neighborhoods is assumed to be negligible. The next section briefly reviews the housing purchase decision in the hedonic model. In Section 2.1 we introduce the compensating benefit that is used in Section 2.3 to define the new index.

2.1 The consumption decision in the hedonic model

For a given housing unit, we denote with q = (zA, zH ) RA+H the vector of the ∈ unit’s characteristics, where zA is a vector of local amenities and zH a vector of intrinsic characteristics of the unit. Henceforth, we use j = 1,...,A to indicate

4 the set of amenities while j = A+1,...,A+H indicates the set of housing-specific characteristics. All attributes in q are considered as normal goods. The consumer decides how to allocate the disposable income m between attributes q and a composite good x R+. The consumer preferences over amenities, housing-specific ∈ attributes and the composite good are represented, as usual, by an increasing and strictly concave utility function U(q, x), characterized by a decreasing marginal rate of substitution between goods along an indifference surface. Let P (q) be the observed equilibrium price schedule associated with the housing unit with attributes q, while the composite good is chosen to be the numéraire. The optimal bundle (q0, x0) maximizes the utility of the consumer subject to the budget constraint and corresponds to the solution of the following problem:

0 0 (q , x ) := arg(q, x)∈ A+H+1 max U(q, x): m P (q) + x . (1) R+ { ≥ }

First order conditions for internal solutions imply the following set of equations:

0 0 0 ∂P (q ) U(q , x )qj = 0 0 , j = 1,...,A + H (2) ∂qj U(q , x )x ∀ P (q0) = m x0 −

where U( )q is the marginal utility associated with the generic attribute qj be- · j longing to q. At the optimum, the marginal rate of substitution between the jth attribute and the composite good must equalize the marginal price of the attribute j implicitly determined by the housing market. The marginal price gives the consumer’s marginal willingness to pay for an additional amount of the jth attribute at the consumer’s optimal choice.

2.2 Welfare measures for changes in amenities

Consider now the problem of evaluating a variation in the quantities of amenities associated with the housing unit chosen at the optimum by the representative consumer. This variation shifts the preferred bundle from (q0, x0) towards any

0 A+H+1 other bundle (q, x ) R+ . We are interested in assessing the representative ∈

5 consumer’s evaluation of (q, x0) in monetary terms. For this purpose, we adopt the compensating benefit (CB), a welfare measure also used by Palmquist (2006) to evaluate environmental goods in a hedonic framework. The CB measures the maximum quantity of the composite good that the consumer is willing to give up (accept) to insure (forego) the shift in amenities. Formally, the CB is defined in terms of the benefit function, originally formulated by Luenberger (1992), which maps changes in the vector of attributes q into changes of the composite good quantity at the optimum, x0, such that utility is

0 0 0 A+H+1 held ﬁxed at u . The beneﬁt function B(q; x ;u ): R+ R R is related to × → the distance function3 and corresponds to the solution of the following program:

0 0 0 0 B(q; x ;u ) := max β R : U(q; x β) u . (3) { ∈ − ≥ }

The representation of the benefit function in problem (3) is due to the particular choice of the reference bundle, set to be (0, 1), so that it is possible to compensate for changes in the housing attributes and amenities in the same (monetary) units of consumption good.4 Therefore, the benefit function is the monetary equivalent of the change in utility due to the change in amenities. The compensating benefit corresponds to the difference of the benefit function computed in any q and the benefit function computed in q0, keeping the level of utility constant at u0. Formally:

CB(q; q0, x0, u0) = B(q; x0, u0) B(q0; x0, u0). −

In the next section we use the compensating beneﬁt to assess quality of life.

3The distance function is a measure, denoted δ = D(q, u), of the distance of a vector of attributes q from a proportionally scaled bundle q/δ providing utility u that can be attained with a unitary expenditure (after normalization). When the budget is linear, this function becomes: D(q, u) := maxδ δ U(q/δ) u = minp p q E(p, u) = 1 . Therefore, the distance function expresses a direct{ measure| of≥ utility} by mean{ · of| budget distance.} 4We assume throughout the paper that compensation for the initial quantity x0 is always possible.

6 2.3 Quality of life measurement

The Roback’s (1982) quality of life index, denoted QoL, is the sum of a set of amenities, weighted by the implicit marginal prices of the amenities. Usually, in empirical applications the implicit prices are calculated for the overall average quantities of the amenities in the sample areas, to minimize the predicted error. The quality of life associated with a given area (neighborhood in our case) endowed with a bundle of amenities zA is the following:

A X ∂P (q) QoL := zA . (4) j ∂q j=1 · j

We stress that the implicit prices ∂P (q) , j = 1, ..., A, correspond to the market ∂qj

value of the overall average quantity of amenities zj, j = 1, ..., A, but they are used to assess the quantities specific to each area. The quantities of amenities can vary considerably across areas while their value is held constant at ∂P (q) and are ∂qj used as a metric for quality of life comparisons. As mentioned in the Introduction, the quality of life literature recognizes that the vector of prices acts as a weighting scheme for amenities, which is consistent with the market evaluation but “merely shows the order of magnitude of expenditure in the average budget” (Roback 1982, p. 1274). As a consequence, the standard quality of life index has no direct interpretation in terms of the representative consumer’s willingness to pay for bundles of amenities that are far from the average bundle, as marginal prices do not correspond to proper evaluations of these units. To cope with this problem, we propose an adjustment to the Roback index that uses a term corresponding to the compensating benefit. As in Roback (1982), we suggest to evaluate this change by taking the perspective of the representative consumer with income m = m and consuming the bundle (q0, x0) = (q, x), which grants utility u0 = u. The new index, named the value-adjusted quality of life index (va-QoL) is the sum of two components. The first component is the Roback index computed at the

7 overall average amenity bundle, zA, which also represents the level of the quality of life in the city as a whole, i.e. QoL = PA zA ∂P (q) . The second component is the j=1 j ∂qj compensating beneﬁt calculated for each neighborhood with amenity quantities that might diﬀer from zA. The value-adjusted quality of life index for a given neighborhood characterized by amenities zA is the following:5

va-QoL := QoL + CBzA, zH ; q, x, u. (5)

Suppose that the bundle zA has a lower quantity of amenities than the average bundle. The compensating benefit will be negative and will correspond to the amount of money that the representative consumer would like to accept as compensation for the less-endowed neighborhood in which he lives. Suppose now the opposite, i.e., a neighborhood is more endowed with amenities than the average bundle. The compensating benefit will be positive and will represent the amount of money that the representative consumer is willing to give up to enjoy the richer bundle of amenities located in the neighborhood. What about the case in which a neighborhood is more endowed with certain amenities and less with others? The compensating benefit will be a monetary compensation if the neighborhood has a lower quantity of amenities that are more important for the consumer than the amenities whose quantity is higher; otherwise, the compensating benefit will represent the willingness to pay if the neighborhood has a higher quantity of amenities that are more important for the consumer. It is worth noting that the compensating benefit is the amount of money that the consumer is willing to give up or accept for not moving from the neighborhood where he lives, as the utility he would receive from moving is equal in all neighborhoods and coincides with the level of utility attained with the overall average bundle of amenities.

5Notice that the compensating beneﬁt evaluates a shift from q0 to q. Throughout the analysis we exclusively consider shifts in amenities. Therefore the component zH of q is always held ﬁxed at zH0.

8 3 Identiﬁcation, Data and Empirical Strategy

The va-QoL index can be identified from the estimation of inverse demands for local amenities and housing-specific attributes. In this section, we derive these inverse demands and discuss the relation between them and the compensating benefit.

3.1 Identiﬁcation

The uncompensated inverse demand for a generic attribute qj, either an amenity or a housing-speciﬁc characteristic, can be derived by linearizing the budget constraint around the optimal choice q0. As in Palmquist (2006), we introduce an

income measure, mlin and a vector of constant marginal prices p RA+H where ∈ 0 ∂P (q0) p := ∂q , such that:

mlin := m + (p0 q P (q)). · −

The linearized income has a ﬁxed component m and a variable part that depends on the distance between the actual price schedule (the non-linear part of the budget constraint) and its linear approximation. By using the linearized budget constraint at the optimum and substituting price vectors in the budget constraint with equation (2) we obtain the following:

1 mlin x0 0 0 := H+A − . U(q , x )x P 0 0 j=1 U(q , x )qj qj

Substituting this result into the optimal solution (1) gives the uncompensated inverse demand system for housing attributes: p0(q, mlin). The uncompensated demand system depends on the quantities of attributes that are consumed and the linearized income parameter. Substituting the direct demand functions into the utility function, we reach the indirect utility function V (p, mlin), which is decreasing in prices and increasing in income. For any point lying on the contour curve of V (p, mlin) = u0 generated by (q0, x0), an increase in prices makes the

9 initial bundle unaffordable and reduces the level of utility. Conversely, an increase in income makes available the bundles that are preferred to (q0, x0) and increases the level of utility. Because prices are always normalized by the price of the composite good, the indirect utility function is quasi-convex in amenity prices and linearized income. Moreover, the slope of its indifference surface must be positive.6 The inverse compensated demand system p(q, u0) gives the willingness to pay for a bundle of attributes q keeping utility as fixed at u0 (Cornes 1992, p. 79). Making use of linearized budgets, the system can be regarded to as the solution of a problem of minimization of the expenditure in the composite good, taking u0 as a constraint.

0 lin 0 lin (p(q, u ), m ) := arg lin min x : U(q, x) u , x m p q . (6) p,m { ≥ ≤ − · }

At the optimal quantity q0, compensated and uncompensated demand coincide and x0 is the level of composite good that solves problem (6). To understand the logic underlying the compensating benefit, we make use of a simple example with just one amenity, denoted by q. The left-hand side of Figure 1 represents the optimal choice of the consumer and, in particular, it shows the effect of a change from the preferred bundle A = (q0, x0) to another bundle B = (q, x0), where q > q0. This shift induces an increase in utility from u0 to u, while the marginal price of the amenity q decreases from p(q0, m) to p(q, m), as the utility is assumed to be concave. The question we address is the extent to which the consumer should be compensated to keep his utility at u0 when the amenity shifts from q0 to q. To answer this question, we decompose the shift from A to B into two movements. The first movement is from A to C, where the utility of the individual is kept fixed at u0. Since the representative consumer likes amenities, the evaluation of the amenity

6To prove this conjecture, see the results on the indirect utility function in Cornes (1992). In lin lin this case, note that for any vector (p0, m0 ) and (p1, m1 ),

max V (p , mlin); V (p , mlin) V (λ(p , mlin) + (1 λ)(p , mlin)), 0 λ 1. { 0 0 1 1 } ≥ 0 0 − 1 1 ∀ ≤ ≤

10 x, mlin

q 0 U(q, x) = u u B u 0 m q

q0 A0 u0 = V (p, mlin) C0 B A x0 m P (q0) −

C p(q, m) mlin p(q, u0)q m θ(q, m, u0) − −

p(q, u0) 0 q p(q , m) p q q0 p(q, u0) p(q, m) p(q0, m)

Figure 1: Compensated non-marginal changes in housing attributes and amenities and the relation with the beneﬁt function

quantity q must be positive, so that to hold u0 fixed given q q0 > 0, the consumer − has to reduce the overall consumption of composite good to x = mlin p(q, u0) q. − · The price differential between p(q0, m) = p(q0, u0) and p(q, u0), along with the required change in income, reflects how much the consumer is willing to forego in units of the composite good to receive more units of the amenity, keeping the level of utility fixed at u0. The second movement is from C to B. The change in marginal bids associated with this shift is positive, reflecting the fact that, given q, the marginal utility of a unit of composite good is larger than its price, which is defined in income units. Hence, the consumer has to increase her marginal bid from p(q, u0) to p(q, m) to make her budget m x0 binding, given that utility has − already reached the level u.7 Of the two movements, only the variation between A and C is relevant for assessing a welfare-consistent measure of the change in amenities. As illustrated in the figure, both marginal prices and linearized income respond to changes in quantities. Instead of looking at variations in the expenditure function (which is meaningful when the budget constraint is linear), we focus on the compensated

7In general, the inverse demands p(q, u0) and p(q, m) coincide only under a strong separability assumption on the utility function. If preferences are quasilinear in the composite good (U(q,x) = f(q) + x with f increasing and strictly concave) the exchange rate between utility units and composite good units is unitary, therefore this bid diﬀerential disappears.

11 change in the share of income that the consumer does not use for buying the housing unit. This amount is (m p(q0, u0) q0) (mlin p(q, u0) q) and has a − · − − · natural counterpart in the dual setting. It corresponds to the Rosen’s (1974) bid function θ(q; m, u0), identifying the maximal (minimal) amount of money that the consumer is willing to give up (to accept) to buy q, given a level of income m and attained utility u0. The bid function is implicitly deﬁned by the following:

U(q, m θ(q; m, u0)) = u0. (7) −

The right hand-side panel of Figure 1 illustrates the effect of a change in the amenity from q0 to q in the dual space, where marginal bids and composite goods are reported in the horizontal and vertical axes, respectively. The difference θ(q; m, u0) P (q0) measures the minimal drop in composite good consumption that − leaves the consumer indifferent between the amenity quantity q0 (point A0) and q (point C0). As shown by the graph, this quantity coincides with the expenditure variation associated with the shift from A to C. Overall, the figure clarifies that the compensation due to the change in amenities from q0 to q can be studied within the dual setting with traditional instru- ments. The following proposition formalizes the relationship between compensating benefit, bid function and compensated inverse demand at the optimal choice of the consumer.

Proposition 1 Assume the utility function U is quasi-concave, monotonic and continuous and consider the reference bundle (0, 1), then the following must hold:

CB(q; q0, x0, u0) := B(q; x0, u0) = θ(q; m, u0) P (q0) (8) − 0 0 0 0 0 0 qCB(q; q , x , u ) 0 = p(q , u ) = qθ(q; m, u ) 0 . (9) ∇ |q=q ∇ |q=q

Proof. The relation in (8) is a direct result of the deﬁnitions of the beneﬁt and bid functions, provided that (q0, x0) is the reference bundle,

U(q, m θ(q; m, u0)) = u0 = U(q, x B(q; x0, u0)), − −

12 and by using the fact that B(q0; x0, u0) = 0 and m x0 = P (q0) = θ(q0; m, u0) − from the dual of problem (6). The ﬁrst equality in (9) is a result of Proposition 2 in Luenberger (1996) and Lemma 1 in Chambers (2001) at the optimum with linearized constraints. The second equality follows by implicitly diﬀerentiating

0 U(q; m θ(q, m, u )) with respect to qj for every j = 1,...,A + H, which gives − the following:

0 0 0 ∂ θ(q; m, u ) U(q , x )j ∂P (q) 0 0 0 0 q=q = 0 0 = q=q = pj(q , u ). ∂ qj | U(q , x )x ∂qj |

The sequence of equalities can be established only at the optimum q = q0, which concludes the proof.

The implementation of these results requires the formulation of assumptions about preferences. The next section discusses the identiﬁcation of preferences using a parametric model.

3.2 Implementation

As mentioned in Section 2.3, the benefit function is related to the distance function between bundles q0 and q. The metric of the distance is defined in the space of the composite good. We can express this distance as the ratio between the actual consumption of the composite good and the level of consumption that the consumer with preferences U would attain if the bundle q is offered while utility is fixed at u0: m P (q0)/m θ(q; m, u0). If q is preferred to (least preferred − − than) q0, then this distance is greater (smaller) than one. We assume a very flexible parametric specification for this distance, which is consistent with the Inverse Almost Ideal Demand System (IAIDS) developed by Eales and Unnevehr (1994). Solving the distance for θ(q; m, u0), we obtain the IAIDS consistent specification for the bid function associated with any bundle q

13 as follows:

h 0 0 i−1 θ(q; m, u0) = m a(q)1−u b(q)u m P (q0), (10) − · · − where, for i and j going from 1 to A + H:

X 1 X X ln a(q) = α + α ln q + γ∗ ln q ln q (11) 0 j j 2 ji j i j j i

Y −βj ln b(q) = β0 qj + ln a(q) (12) j

The formulation of the bid function in (10) is consistent with the properties outlined by Rosen (1974) and Palmquist (2006): the function must be linear- homogeneous, concave, non-decreasing in quantities, decreasing in utility and linear in income. Most of the properties make intuitive sense, as already mentioned in Palmquist (2006). Linear homogeneity and concavity result from the IAIDS representation of preferences. The third property holds whenever the amenities are normal goods. An increase in consumption of local amenities holding income and utility level constant must increase the total bid for the housing unit when local amenities are substitutes of consumption. This relationship corresponds to a change in the marginal rate of substitution along an indifference curve. The fourth property suggests that an increase in utility is possible, leaving local amenity consumption and income as fixed and increasing only the composite good consumption. By the substitution effect, the marginal bid for other attributes must decrease. The last property is obviously satisfied. All of these properties induce testable restrictions on the model. Under integrability conditions, the utility level at the optimum (q0) is obtained by solving (10) for u0, which gives the following:

ln a(q0) u0 := . (13) [ln a(q0) ln b(q0)] − The compensating beneﬁt CB(q; q0, x0, u0) can be formulated using result (8)

14 in the proposition and substituting u0 for (13), as follows:

h 0 0 i−1 CB(q; q0, x0, u0) = 1 a(q)1−u b(q)u m P (q0) . (14) − · · −

Identification of the compensating benefit above is parametric and depends on the IAIDS structural form. The underlying assumption here is that every consumer’s demand admits the same IAIDS representation. The parameters of the bid function in (10) can be estimated from the inverse compensated demand functions, exploiting the result (9) in Proposition 1. To obtain these demands, consider the elasticity %j of the demand for the composite good with respect to the demand for the attribute j, when the utility is held fixed at u0. Making use of a log-transformation of (10), we are able to show that this elasticity can be equivalently expressed as the attribute-to-consumption value ratio, for any attribute j = 1,...,A + H. As attributes are treated as substitutable normal goods, this elasticity must be negative, as provided in the following:

0 0 ∂ ln(m θ(q; m, u )) qj ∂θ(q; m, u ) %j := − = 0 . (15) − ∂ ln qj m θ(q; m, u ) · ∂qj − At the consumer’s optimum, it holds that q = q0 and m θ(q0; m, u0) = m P (q0). − − Using the results in Proposition 1, %j in (15) is identiﬁed as follows:

0 0 0 0 0 ∂P (q ) q pj(q , u ) qj % = j · = · ∂qj . (16) j m P (q0) m P (q0) − −

Each element %j is a measure of the expenditure for attribute j, obtained by a ﬁrst order approximation of the price schedule around q0, expressed in units of the composite good. Under the IAIDS identiﬁcation assumption and using (13), it can be shown by solving the derivative in (15) that the elasticity %j is a linear

15 function of the parameters of the IAIDS, as follows:

X %j = αj + γji ln qi + βj ln Q, j = 1,...,A + H (17) i ∀ X 1 X X ln Q = α + α ln q + γ ln q ln q , (18) 0 j j 2 ji j i j j i

∗ ∗ γji+γij where γji = 2 for all i and j. This system of equations allows us to identify the parameters of the bid function, which enters in the formulation of the compensating beneﬁt (14) and the value-adjusted quality of life index (5).

3.3 Identiﬁcation of the va-QoL index

The result in Proposition 1 holds for any consumer with utility U, and therefore can be reformulated in terms of the optimal choice of the representative consumer holding income m, who chooses the market average bundle of amenities and housing attributes, i.e., when (q0, x0) = (q, x). The va-QoL index expresses the representative consumer’s welfare evaluation of a change in amenities from his bundle q towards any other bundle (zA, zH ) oﬀered in one of the neighborhoods of the city. Using the result in (8), it is possible to identify the va-QoL indicator by the bid function when the reference utility is set to u, as follows:

va-QoL = QoL + θzA; zH , m, u P (q) . (19) −

Under the IAIDS assumption, parametric identiﬁcation of the representative consumer’s preferences is achieved. Using (13), the reference level of utility of the representative consumer is u = ln a(q)/ ln a(q) ln b(q). Using (14), the IAIDS − representation of the va-QoL index associated with any bundle of amenities zA becomes the following:

h i−1 va-QoL = QoL + 1 azA, zH 1−u bzA, zH u (m P (q)) . (20) − · · −

16 The value of the index associated with any neighborhood of the city (oﬀering amenities zA) is identiﬁed if structural parameters and information on the average bundle of amenities, housing quality and income in the city are available. The empirical exercise shows how to obtain this information from housing market data.

3.4 Data

The empirical analysis relies on a data set of individual housing transactions provided by the real estate observatory Osservatorio del Mercato Immobiliare (OMI), which is managed by the public agency Agenzia del Territorio within the Italian Ministry of Economy. We focus on housing transactions that occurred during the period from 2004 to 2010 in the city of Milan. In addition to housing market values, the data set provides a detailed description of structural attributes of the surveyed housing units, including total floor space, age of the building in the year of sale, number of bathrooms, whether the housing unit needs to be renovated, whether the housing unit has independent heating, the floor above street level at which it is located, the presence of an elevator or a garage, and building quality. Transactions occur in the 55 neighborhoods of the city, which are identified by the OMI. We have information about income at the neighborhood level expressed in constant 2010 Euros. Neighborhood-level data on amenities and socioeconomic conditions were taken from public authority records (see Table 4 in the Appendix for details). We consider 8 neighborhood-level amenities. Environmental conditions are proxied by the green areas relative to the area of the neighborhood (Green). Ed- ucation is proxied by the number of secondary schools per 10, 000 inhabitants (Schools). Public transport is represented by the number of metro and railways stations per 10, 000 inhabitants (Transport). Security is measured through the number of police stations per 10, 000 inhabitants (Security). Health is proxied by the number of health centers per 10, 000 inhabitants (Health). The presence of facilities in the neighborhood is proxied by the number of pharmacies and post offices per 10, 000 inhabitants (Facilities). The recreational dimension is proxied

17 by the number of cinemas, theaters, museums, art galleries, academies of music and libraries per 10, 000 inhabitants (Culture). The social composition of neighborhoods Ethnic is measured by the variable used in Brambilla, Michelangeli and Peluso (2013). This variable captures the pattern of ethnic minorities8 across the city’s neighborhoods and can be interpreted in the terms of a segregation index of the exposure class (see Reardon and O’Sullivan 2004). More precisely, the vari- P 2 able for the neighborhood n is constructed as Ethnicn = Itn/( n06=n Immn0 /dnn0 ),

where Itn is the number of Italians living in the neighborhood n. In the denom-

inator, we add to the immigrants living in the neighborhood, Immn, those living in the surrounding neighborhoods. The weight associated with immigrants is the inverse of the squared Euclidean distance between neighborhoods and roughly ap- proximates the probability that Italian residents in neighborhood n interact with foreigners living in the same neighborhood or in the adjacent neighborhoods. Five amenity variables have missing values for a small number of neighborhoods. We replace missing values with the weighted average of values observed in the neighborhoods adjacent to the neighborhood for which the value is missing.

The weights are the inverse of the squared Euclidean distance dnn0 between the centroids of the neighborhoods. Summary statistics for amenities, income and housing prices are provided in Table 1, while a map showing the spatial distribution of income and housing prices across neighborhoods is reported in Figure 5 in Appendix. The empirical application of the theoretical model requires a quite relevant quantity of parameters to be estimated. We derive a more parsimonious specification of the housing price equation and structural model by reducing the number of housing-specific characteristics (which are used as controls in our analysis) using principal component analysis (PCA). Given that several variables referred to housing-specific characteristics are dichotomous, we have first dichotomized continuous and count variables and then applied PCA using tetrachoric correlations.

8Brambilla et al. (2013) consider as foreign residents the ethnic groups most often subject to discrimination, such as African, South American and some Asian communities. Other immigrants - those from, for example, North America or western Europe - have become ’invisible’ in Italy and are included with Italian residents.

18 Table 1: Summary Statistics of Amenities Variable Mean Std. Dev. Min. Max. N Green 14.98 6.73 1 29 55 Education 1.87 2.07 0.22 9.86 55 Culture 1.81 2.49 0.19 14.19 55 Transport 1.09 0.90 0.19 4.49 55 Security 0.77 0.69 0.22 5.09 55 Health 1.44 1.22 0.19 7.67 55 Facilities 3.23 2.24 0.44 11.67 55 Ethnic 9.14 3.75 3.22 21.26 55 Housing rent 14,376 14,253 4,690 250,509 3,949 Income 23,953 12,890 12,087 78,481 55

Source: OMI Milan and our elaborations.

Continuous variables, such as total floor area, were converted to dichotomous variables by splitting the scale at the sample deciles. Then, a continuous variable was associated with 10 dichotomous variables, one for each decile. Count variables, such as floor or number of bathrooms, were dichotomized by splitting the scale at the most frequent values of the variables. We performed PCA on the resulting set of dichotomous variables and used only the first estimated factor, whose load- ing accounted for nearly 18% of the joint correlation structure among more than 50 dummies constructed from the data. The other components are not retained because they turned out to have no significant impact on housing prices.9

3.5 Estimation strategy

Estimation evolves in two-steps. In the first step, implicit prices of housing attributes are estimated by making use of an empirical specification of the housing price function P (q). In the second step, the predicted implicit prices are com- bined with amenity quantities to estimate the parameters of the compensating benefits associated with each neighborhood. Once the compensating benefits are estimated, the va-QoL index will be computed according to (20). Our data are specified at the housing unit level. A housing unit h located in neighborhood n and sold at time t is associated with a transaction price Phnt,

9The results of the PCA analysis can be provided upon request to the authors.

19 H an indicator of the housing characteristics zhnt, as well as a vector of amenities A of the neighborhood n, which is denoted znt. We adopt a semilog functional representation of the housing prices equation, deﬁned as follows:

A H ln Phnt = β0 + z β + β2z + βt + εhnt (21) nt · 1 hnt

2 where βt is a time-fixed effect. We assume that εhnt v N (0, σε ), which implies that the model can be estimated via OLS. To obtain the annual implicit price of each amenity, housing prices were converted into imputed annual rents by applying a discount rate specific to each neighborhood. This rate has been determined by dividing the average imputed rent by the average price of housing in the neighborhood, both expressed in constant 2010 Euros. In the second step, we retrieve the structural parameters of interest by estimating (17) and (18) as a system of simultaneous equations. The dependent variable

10 in (17) is %j, speciﬁed at the housing unit level. For a given amenity j = 1,...,A,

the numerator of %bj is the hedonic value of amenity level qj experienced by housing ˆ unit h in neighborhood n at time t, evaluated at the marginal price ∂P (q) . This ∂qj price is inferred by diﬀerentiating the hedonic price equation (21) with respect to amenity j, as follows:

ˆ ∂P (q) ˆ ˆ = β1j P (q), (22) ∂qj ·

ˆ n ˆ o where P (q) = exp ln Phnt is the predicted market price of the housing unit with characteristics q. To cope with the fact that income is observed exclusively at the neighborhood level, the denominator in expression (16) is computed using the average income and the average transaction price in the neighborhood where the housing unit h is located. We adopt a standard linearization technique, widely exploited in demand analysis (Deaton and Muellbauer 1980, Eales, Durham and Wessells 1997), to estimate

10To avoid cumbersome notation, in what follows we drop the subscripts indicating the housing unit, time of sale, and neighborhood.

20 (17) and (18). Equation (18) is replaced by the Stone quantity index ln Q∗ for the housing-speciﬁc attributes and amenities, deﬁned as:

A+H ∂Pb(q) qj X ∂qj ln Q∗ := · ln q . (23) ˆ j j=1 P (q)

The substitution of ln Q with ln Q∗ gives a system of equations that is linear in parameters and can be estimated by seemingly unrelated regression techniques, P introducing symmetry (λij = λji) and homogeneity ( j λij = 0) constraints on P P (17). Additional restrictions, such as adding up ( j αj = 1, i λij = 0 and P j βj = 0), are also introduced in the regression model to impose rationality restrictions on the behavior of the consumer. We then compute the average income m, the average bundle of amenities zA and housing characteristics zH , and the average predicted housing price P (q) in the city, averaging across all sampled transactions. We use the IAIDS parameters estimated in the first step to calculate the values of ln a(zA, zH ) and ln b(zA, zH ) for each neighborhood separately, using the definitions in (11) and (12), respectively. Then, we calculate the same two coefficients for the average bundle q to obtain the reference utility level u.11 Finally, we compute the compensating benefit for each neighborhood and the value-adjusted quality of life index as in (20). Observable heterogeneity can be accounted for by controlling for the demographic features of the neighborhoods or even the attributes of the buyers and sellers. These controls would allow for a more precise estimation of the structural parameter, but would not add additional heterogeneity in the evaluation. Some precision in the structural parameters estimation is eventually lost as a consequence of our implementation strategy, which involves the following: re- placing of the term ln Q with the Stone quantity index; adopting the average

neighborhood income, not real buyer incomes, in the calculations of %j; estimating the hedonic prices through an arbitrary log-linear speciﬁcation of the hedonic regression. All together, these simpliﬁcations might distort the estimation of the

11 Note that the parameters α0 and β0 are not identiﬁed. They are substituted with 0 and 1, respectively.

21 Table 2: Hedonic price regressions

Coeﬀ. SE Hedonic prices Housing Characteristics First princip. comp. 0.545*** (0.011) Amenities Green 0.004*** (0.001) 43.94 Schools 0.029*** (0.004) 322.24 Culture 0.014** (0.006) 151.36 Transport 0.115*** (0.014) 1256.91 Security 0.113*** (0.016) 1233.81 Health 0.017** (0.008) 183.15 Facilities 0.028*** (0.005) 303.23 Ethnic 0.013*** (0.002) 147.21 Constant 7.951*** (0.026) Time controls Yes R-squared 0.657 Transactions 3949

Standard errors in parentheses. ∗ p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01 Source: OMI Milan and our elaborations. compensating beneﬁt, biasing the resulting va-QoL index.

4 Results

The hedonic model specified in (21) is estimated by OLS using 3,949 housing transactions recorded in Milan between 2004 and 2010. Robust standard errors are obtained by clustering at neighborhood level to account for within-neighborhood correlation in prices. Estimates are reported in the first two columns of Table 2. The amenity coefficients are both individually and jointly statistically significant (F(8,3933) = 169.88, Prob > F = 0.0000) and have the expected sign. The first principal component is mainly determined by variables positively correlated with housing prices, such as surface area, the number of bathrooms in each apartment or the present of lifts in the buildings, and the presence of at least a parking area or a terrace. This result explains the relatively high and significant coefficient of the first principal component on housing prices.

22 Social variables, such as ethnic composition of neighborhoods may be endoge- nous to the contemporaneous value of housing prices because of reverse causation and omitted variables. We test the endogeneity of the Italian/foreign ratio (corresponding to the variable Ethnic) following the approach developed by Card, Mas and Rothstein (2008) and Saiz and Wachter (2011). This approach consists of using a gravity model that predicts the actual ethnic composition on the basis of the settlement patterns of previous periods. The predicted ratio is used as instrument to test the endogeneity of the Italian/foreign ratio by applying the Hausman test. The results based on instrumental variable estimation fail to reject

2 the null hypothesis of no endogeneity for Italian/foreign ratio (χ(18) = 11.29, Prob > χ2 = 0.7316). We also performed the Moran’s I test to assess the spatial depen- dence of the model, and the result does not allow us to reject absence of spatial correlation in the OLS residuals. This ﬁnding is probably due to the imputation method for missing amenities, described in Section 3.4 and based on the spatial proximity of amenities in neighborhoods for which statistical information is available. The inclusion of data dependent on distance between neighborhoods, could yield patternless residuals over space (Pace, Barry and Sirmans 1998). The estimated implicit prices of urban amenities are reported in the third column of Table 2. They correspond to the partial derivative of the hedonic price function, given by (22), and can be interpreted as the monetary amounts, expressed in Euros at constant 2010 prices, that households would be willing to pay annually for a one-unit change in the corresponding amenity. The results show that the largest hedonic prices are associated with public transportation urban security. The estimated marginal willingness to pay for increasing by one point the number of metro and railways stations per 10,000 inhabitants is e1,256 (or e1,132 per point of standard deviation). Increasing the number of policy stations per 10,000 inhabitants is valued at e1,233 by the owner of the average housing unit of the sample (or e850 per point of standard deviation). The lowest implicit prices are associated with green areas and the Italian/foreign ratio and are equal to e43 and e147, respectively (e2.97 and e5.20 per point of

23 standard deviation, respectively). The estimated implicit prices are used to estimate the attribute-to-consumption value ratio %bj, for each attribute j = 1,...,A + H, given by (16). The linearized IAIDS demands in (17), where ln Q is substituted with the Stone quantity index ln Q∗, is estimated by Seemingly Unrelated Regression (SUR) methods (Eales et al. 1997). The SUR model consists of a system of 9 equations, one for each attribute, estimated on the 3949 housing transactions of our data set. The results are reported in columns (1) to (9) of Table 3. Each equation provides estimates of γbij coefficients associated with the eight urban amenities (i = 2,..., 9), a coefficient for the principal component summarizing housing-specific features (i = 1), and a constant term. Any potential time trend is also controlled for by including year dummies in the specification. The explanatory power of the system is reasonable with almost all covariates statistically significant and most of the 81+18 parameter estimates are more than twice their estimated standard errors. Parameter estimates are used to calculate ln a(q) and ln b(q), given by 11 and 12 respectively, considering the current distribution of amenities. From the logarithms, we derive a(q) and b(q), which enter in the formula of the compensating benefit (14). The two logarithms ln a(q) and ln b(q) are also calculated for the average quantities of housing attributes at the city level and used to quantify the reference level of utility introduced in Section 3.3. We now have all elements needed to calculate the va-QoL index for the 55 neighborhoods of Milan according to (19). The first term of (19) is equal to e6,622 in our sample, and corresponds to the amount that a representative household would be willing to pay to live in a neighborhood with the average quantities of amenities. The value is obtained from 4 calculated at the city-level average quantities of amenities. The second term of (19) is the amount of compensating benefit that ranges from e-9,514 to e7,920. The positive values of the compensating benefit refer to neighborhoods characterized by a richer bundle of amenities than the average bundle, implying that the consumer is willing to pay the amount of the compensating benefit to enjoy a higher endowment of amenities than those offered

24 Table 3: Parameter estimates from the structural model

%1 %2 %3 %4 %5 %6 %7 %8 %9 (1) (2) (3) (4) (5) (6) (7) (8) (9) Amenities and housing-speciﬁc attributes: Principal component 297.564*** -30.818*** -35.137*** -37.932*** -46.590*** -40.881*** -28.796*** -49.018*** -28.390*** [59.05] [-38.50] [-37.70] [-44.48] [-41.91] [-35.97] [-29.41] [-62.77] [-30.62] Green -30.818*** 27.547*** 6.132*** 0.562** -3.688*** -0.638 2.924*** 1.966*** -3.986*** [-38.50] [78.44] [21.00] [2.17] [-9.60] [-1.49] [9.45] [6.23] [-10.03] Schools -35.137*** 6.132*** 37.532*** -1.040*** -2.385*** -14.636*** 5.566*** -1.107*** 5.075*** [-37.70] [21.00] [75.08] [-3.47] [-4.79] [-26.37] [15.83] [-2.94] [10.67] Culture -37.932*** 0.562** -1.040*** 19.617*** 2.817*** 3.638*** 5.161*** 5.857*** 1.319*** [-44.48] [2.17] [-3.47] [56.79] [7.24] [8.45] [17.66] [18.90] [3.34] Transport -46.590*** -3.688*** -2.385*** 2.817*** 74.315*** -10.257*** -5.812*** 7.330*** -15.731*** [-41.91] [-9.60] [-4.79] [7.24] [71.70] [-12.31] [-11.55] [12.56] [-24.84] Security -40.881*** -0.638 -14.636*** 3.638*** -10.257*** 76.175*** -0.965* -2.226*** -10.211*** [-35.97] [-1.49] [-26.37] [8.45] [-12.31] [65.10] [-1.88] [-3.87] [-14.97] Health -28.796*** 2.924*** 5.566*** 5.161*** -5.812*** -0.965* 18.907*** -0.171 3.186***

25 [-29.41] [9.45] [15.83] [17.66] [-11.55] [-1.88] [42.06] [-0.43] [7.13] Facilities -49.018*** 1.966*** -1.107*** 5.857*** 7.330*** -2.226*** -0.171 49.186*** -11.817*** [-62.77] [6.23] [-2.94] [18.90] [12.56] [-3.87] [-0.43] [86.94] [-23.83] Ethnic -28.390*** -3.986*** 5.075*** 1.319*** -15.731*** -10.211*** 3.186*** -11.817*** 60.556*** [-30.62] [-10.03] [10.67] [3.34] [-24.84] [-14.97] [7.13] [-23.83] [70.86] Additional regressors:

ln Q∗ 192.252*** -39.887*** -28.104*** -38.167*** 0.662 0.275 -44.768*** -20.868*** -21.395*** [44.62] [-57.90] [-34.76] [-51.64] [0.67] [0.28] [-52.80] [-30.70] [-26.90] Constant 0.599*** 0.007*** 0.020*** 0.040*** 0.134*** 0.124*** 0.035*** 0.056*** -0.015*** [50.68] [2.99] [7.88] [17.55] [41.59] [36.65] [13.29] [24.28] [-4.73] Time controls Yes Yes Yes Yes Yes Yes Yes Yes Yes Joint signiﬁcance (Chi2) 7,961.5 11,666.4 7,570.9 7,424.3 7,993.4 5,239.6 5,483.8 11,827.3 8,098.3

t statistics in parentheses. ∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001 Source: OMI Milan and our elaborations. Notes: The estimated demand system comprises: (1) first principal component, (2) green, (3) number of high schools, (4) cultural services, (5) public transportation, (6) security, (7) health services, (8) other facilities, (9) approximate probability that Italians interact with immigrants at the neighborhood level. Attributes coefficients are multiplied by 103. Estimates are obtained for a sample of 3949 housing transaction observed in a seven years interval. Symmetry, homogeneity and identification constraints are imposed to the model. (a) va-QoL

(b) QoL

Figure 2: Quality of life evaluations across Milan neighborhoods, based on va-QoL estimates and on Roback’s (1982) QoL index (demeaned of the QoL component). Source: OMI data and our elaborations. Note: The legend identiﬁes three groups gathering one-third of the 55 neighborhoods considered in this analysis, ranked by the most to the least preferred according to the quality of life indicator used. Darker gray indicates larger positive deviations of each index from QoL.

at the average. A symmetric argument applies to the interpretation of negative values of the compensating beneﬁt: they represent the monetary compensation for households located in neighborhoods with lower endowments of amenities rather than in the neighborhood with the average bundle of amenities. We also compute the standard hedonic index for the city’s neighborhoods, given by (4). The two indices give exactly the same value for the average quantities of amenities at the city level. The values of the va-QoL index and the standard hedonic index are reported in panels (a) and (b) of Figure 2, respectively. To facili- tate the comparison, the two indices are normalized with respect to the value of the

26 Figure 3: Distribution of the derivatives for the reference amenities, by neighborhood, based on va-QoL estimates. Source: OMI data and our elaborations. Note: For three neighborhoods we register a negative value of the derivatives, never smaller than e-1,000. These values are omitted from the graph. city average bundle, i.e. QoL. Both maps reveal that neighborhoods with higher quality of life are located predominantly in the city-center.12 Households living in these neighborhoods are willing to pay an implicit premium between e7,047 and e7,920 per year to access the amenities of these neighborhoods compared to the city average. The best neighborhoods according the standard hedonic index are the same four neighborhoods, but the ranking is slightly diﬀerent: the best is 4-Diaz Fontana Europa, followed by 2-Brera Duomo Cordusio Torino, 7-Turati Moscova Repubblica, and 8-Venezia Majno Monforte. Bottom-ranked neighborhoods are the same for both rankings: 31-Loreto Turro Padova; 23-Musocco Varesina Certosa; 41-Ronchetto Chiaravalle Ripamonti; and 40-Omero Gabriele Rosa Brenta. Households should receive an annual compensation for living in these more disadvantaged neighborhoods ranging between e6,221 and e9,514 and according the va-QoL index, while the range is between e2,127 and e4,313 for the standard hedonic QoL index.

12The neighborhood with the highest quality of life according the va-QoL index is 2-Brera Duomo Cordusio Torino, followed by adjacent neighborhoods, in particular 4-Diaz Fontana Eu- ropa, 8-Venezia Majno Monforte, and 7-Turati Moscova Repubblica.

27 Overall, the rankings of neighborhoods provided by the two indices are strongly positively correlated, as measured by Spearman’s rank correlation coefficient, which is equal to 0.9284. However, only the va-QoL index is able to give a measure of the impact of each amenity on quality of life that varies with the neighborhood. Suppose a marginal increase in the amenity j occurs in one of the 55 neighborhoods. In the standard hedonic approach, the effect of this change on quality of life is given by the hedonic price, independent of the neighborhood where the increase occurs. Then, for example, an additional school built in a wealthy neighborhood with many schools will have the same impact as a school build in a poor neighborhood without any schools. Our approach is instead able to provide a measure of the contribution of a marginal increase in amenity j on quality of life, taking into account the particular composition of amenity bundle where the increase occurs. Returning to the example of the school, the additional school build in a wealthy neighborhood with many schools is expected to have a lower impact than the school build in the poor neighborhood without schools. In formal terms, the marginal contribution is given by the first derivative of va-QoL index with respect

to qj, and corresponds to the following:

∂va QoL ∂CB(q; q, x, u) − := ∂qj ∂qj ! 1 m P (q) X Y −βi = 1−−u u αj + γij ln qi βju qj (24) qj · [a(q) b(q) ] · − · i i

where the evaluating bundle is meant to be q = (zA, zH ). As the ﬁrst term of the va-QoL index is constant, its derivative coincides with the derivative of the compensating beneﬁt. As a consequence of the identities formulated in Proposition 1, when the evaluation is taken at the average (i.e., when q = q), the marginal impact of amenity j on the quality of life evaluation coincides with the hedonic marginal prices of the amenities: ∂va−QoL = ∂P (q) . ∂qj ∂qj Figure 3 reports the boxplot for each amenity representing the distributions of the derivatives of the va-QoL index calculated for the 55 neighborhoods. The size of these derivatives can be compared with the size of the hedonic prices represented

28 by the dots in the figures. The different evaluations of a marginal increase of qj by neighborhood capture the representative consumer’s preferences for the particular composition of the amenity bundle where the increase occurs. As the figure shows, the greater variability in the derivatives of the va-QoL index tend to be associated with amenities having a higher hedonic price. The correlation coefficient between the hedonic prices of amenities and the coefficient of variation of the va-QoL derivatives is equal to 0.2265. The amenities associated with higher implicit prices are also the least evenly distributed across neighborhoods, as measured by the correlation coefficient equal to 0.6918. Looking at (24), each derivative depends on the whole bundle of amenities available at neighborhood-level. This measure is more informative than the corresponding hedonic price (which is fixed across neighborhoods) in spotting neighborhoods where a marginal increase in the amenity would lead to larger quality of life improvements. The potential improvements are plotted on the Milan maps reported in Figure 4. The maps show that the smallest increases in quality of life are always registered in the central neighborhoods of the city, where quality of life is higher. Conversely, the derivatives reach their maximum values in neighborhoods scoring at the bottom of the quality of life rankings, as they are amenity poor. The va-QoL derivatives are, instead, heterogeneous across peripheral neighborhoods, and the pattern of this heterogeneity changes substantially across amenities. In this case, we speculate that the major driver of heterogeneity is attributable to the composition of amenity bundles in these neighborhoods. The analysis of the variability of the va-QoL derivatives provides information valuable to urban planning policy. An urban policy-maker aiming at increasing quality of life in the city should shape urban intervention to increase amenities availability in those neighborhoods exhibiting larger values of the index derivatives. For instance, if the focus is restricted to the amenities with more heterogeneous impacts on quality of life, or with larger hedonic prices (i.e., transport, culture and security), one speculates from Figure 4 that the largest intervention should take place in the southern and western neighborhoods of Milan. Further

29 (a) Green (b) Schools

(e) Security (f) Health

(g) Facilities (h) Ethnic

Figure 4: Distribution of va-QoL index derivatives across neighborhoods, by amenity. Source: OMI and municipality data, our elaborations.

30 discussion on alternative intervention is bounded by the local policy debate. We can nevertheless conclude that improvements in amenities driven by this logic not only promote quality of life amelioration in the city and in more disadvantaged neighborhoods, but also compress the variability in quality of life across city neighborhoods without necessarily leveling the heterogeneity in amenities composition in each neighborhood.

5 Conclusion

This paper extends the standard hedonic approach to assess urban quality of life using a welfare measure quantifying the gain (loss) for living in an area with a bundle of amenity quantities higher (lower) than the average bundle. The new index adjusts the Roback’s index by properly evaluating infra-marginal units of amenities without demanding more data than those used by the standard hedonic approach to measure quality of life. Indeed, we need to know only household income, in addition to information about housing and amenities, as the willingness to pay for amenities depends explicitly on the share of income that remains from the consumption of all the other goods, represented by the composite good. We use our methodology to assess the quality of life at the neighborhood level for Milan and compare our results with those obtained with the standard hedonic approach. The ranking of neighborhoods produced by the two indices are positevely and significantly correlated. Differences in the rankings are observed for those neighborhoods having a greater quantity of amenities unevenly distributed across neighborhoods. The differences are because only the va-QoL index captures the consumer’s tastes for the particular composition of amenity bundles in the city’s neighborhoods. Some technical caveats are worth mentioning. First, our identification strategy assumes a parametric structure for the preferences of the representative consumer. A desirable extension of this work is to admit heterogeneity in individuals’ preferences by considering structural parameters as a function of observable demograph- ics.

31 Second, the theoretical framework is developed focusing on a single city, although the va-QoL index could be applied to other geographical areas such as state, region or city. In that case, we should reformulate the va-QoL index including in our model the labor market and other important aspects, such as intercity differences in non-housing costs, intergovernmental transfers and local taxes, and housing production.13 Such a general framework constitutes a promising avenue for future research. Another future challenge for the analysis of urban quality of life consists in adapting the tools provided by the literature for policy analysis. The hedonic index is necessarily an ex post evaluation tool to assess quality of life, using observed quantities of amenities and information drawn from the labor and housing markets to infer the implicit prices associated with the amenities. Suppose that the local government of a city implements a policy to increase the quantity of amenities in the less endowed neighborhoods of the city. The standard hedonic approach provides an approximated measure of the change in quality of life, as implicit prices are a proxy for the current distribution of amenities within the city. Our methodology is a promising way to evaluate ex ante the effect on quality of life of policies that modify the distribution of amenities across and within urban areas. Instead of fixing the hedonic prices, which change endogenously with the distribution of amenities, we fix the tastes of the representative consumer and then evaluate the change in welfare from the perspective of this consumer. The analysis of the derivatives of the va-QoL index with respect to each amenity also provides interesting implications from the policy perspective. The derivatives depend on the whole bundle of amenities. Thus, their value changes across the city’s neighborhoods. The policy maker interested in efficiently raising urban quality of life should increase the amenity quantities in those neighborhoods where the derivatives are higher, indicating that the representative consumer has also a larger marginal willingness to pay for those amenities. The application illustrates that, by doing so, the policy maker would also promote a more equitable distribution of quality of life within the city.

13These issues have been considered by Albouy (2008) and Albouy et al. (2013).

32 Acknowledgements

We gratefully acknowledge the Osservatorio del Mercato Immobiliare for data on housing transactions. We thank Eugenio Peluso and Maurizio Pisati for valuable suggestions. Financial support by the Italian Ministry of University and Research is gratefully acknowledged. Francesco Andreoli kindly acknowledges the ﬁnancial support from the FNR Luxembourg (AFR grant 5932132).

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34 Appendix

(a) Income (b) Transaction prices

Figure 5: Average income and transaction prices across neighborhood in Milan. Source: OMI and municipality data, our elaborations.

Table 4: Description and sources of variables

Variable Description Green areas Percentage of urban green over urban area. Source: Milan Mu- nicipality. Schools Number of institutes offering secondary education programs, per 10,000 inhabitants. Source: Authors’ computations. Culture Number of cultural locations per 10,000 inhabitants. Source: Au- thors’ computations. Transport Number of underground and railways stations per 10,000 inhabitants. Source: Milan Transport Agency. Security Number of police departments per 10,000 inhabitants. Source: Authors’ computations. Health Number of hospitals and related services per 10,000 inhabitants. Source: Authors’ computations. Facilities Number of pharmacies and post offices per 10,000 inhabitants. Source: Authors’ computations. Ethnic An exposure index measuring the probability that non-Italian residents have to interact with natives. It is constructed as in Bram- billa et al. (2013) and represents a measure of integration. Source: Milan Municipality. Income Average before taxes household income at the level of the neighborhood. Source: Milan Municipality and University of Milan Bicocca. Housing prices Observed/imputted market prices and rents, offer prices for 3949 transactions in years 2004-2010. Source: OMI. Housing attributes Floor surface, second/third bathroom, renewed unit, gas central heating, second floor or above, low cost/medium/luxory building, presence of a parking slot and elevator, age of the building, distance from the city center. Source: OMI.