Federal coordination of local housing demolition in the presence of filtering and migration

Abstract

Vacant housing and even housing demolition have recently become an issue in a number of countries. Given this renewed interest in demolition, this paper contributes to the literature on (i) housing demolition and (ii) policy coordination. The paper extends Sweeney’s (Econometrica 42:147–167, 1974a) analysis of demolition and filtering, by letting households also choose their location. Then when demolishing part of its housing stock, a city effectively evicts some of its residents not just out of the housing quality it demolishes but out of every other of its qualities, too. The paper shows that demolition’s coordination strengthens local governments’ incentive to demolish part of their stock, by shutting down inter city migration within qualities.—A case study on Germany’s East illustrates the effects of coordinated, simultaneous every-city demolition.

Appendix

Lemma 1

(Indirect utility properties)

• (i) V exhibits strictly increasing differences with respect to θ and (s,q).

• (ii) V _q =v _q =−h(q _i ,s _i )<0, by the envelope theorem.

• (iii) V _θs >0, from the definition of V in (2).

• (iv) V _θq =0, from the definition of V in (2).

• (v) V _s =v _s =r _s (h(q _i ,s _i ),s _i )>0, by the envelope theorem.

Proof of Lemma 1

(i) Consider comparing the maximum utilities that result for two distinct combinations of quality and rent (s′,q′) and (s″,q″), where s′,s″∈{s ₁,…,s _I } and $q',q'' \in\mathbb{R}_{+}$. Suppose s″>s′ and q″>q′. Now surely for all θ″≷θ′ we have θ″s″−θ′s″≷θ″s′−θ′s′. But then

$$\bigl(\theta''s'' + v \bigl(q'',s''\bigr)\bigr) - \bigl(\theta's'' +v\bigl(q'',s'' \bigr) \bigr) \gtrless \bigl(\theta''s' + v\bigl(q',s'\bigr)\bigr)- \bigl(\theta's' +v\bigl(q',s'\bigr)\bigr) $$

also. This proves that

Put differently, indirect utility V exhibits strictly increasing differences with respect to θ and (s,q). Proofs of (ii) to (v) are indicated in Lemma 1. □

Proof of Proposition 1

Given Lemma 1, V(θ _i,i+1,s _i+1,q _i+1)−V(θ,s _i+1,q _i+1)≷V(θ _i,i+1,s _i ,q _i )−V(θ,s _i ,q _i ) for all θ≶θ _i,i+1. Inserting indifference condition (3), and also spelling out the resulting pair of inequalities for segment i−1, translates into

(21)

(22)

Hence residents with a taste index θ in the interval [θ _i−1,i ,θ _i,i+1] prefer segment i to either of i’s neighbors i−1 and i+1.

In fact, residents in [θ _i−1,i ,θ _i,i+1] prefer i to every other quality. Consider, for example, a much better segment s _k , with k>i+1. Specifying the pair of inequalities in (22) for i=k gives V(θ,s _k ,q _k )≷V(θ,s _k−1,q _k−1) for all θ≷θ _k−1,k . Since residents in [θ _i−1,i ,θ _i,i+1] exhibit θ<θ _k−1,k , they are more strongly attracted to segment k−1 than to k. Applying this argument to successively lower segments until segment i+1 is reached, shows that V(θ,s _k ,q _k )<V(θ,s _k−1,q _k−1)<⋯<V(θ,s _i ,q _i ) for all residents with θ in [θ _i−1,i ,θ _i,i+1]. A similar argument applies to the case where k<i−1. □

Proof of Proposition 2

We divide the proof into three steps. First we derive the right derivative of $n_{i}^{j}$ with respect to $q_{i}^{j}$ at $\bar{q}_{i}^{j}$, then we derive its left derivative, then we compare the two. (i) Consider a rent $\hat{q}_{i}^{j}$ which strictly exceeds $\bar{q}_{i}^{j}$. Resulting variable values are denoted $\hat{\theta}_{i,i+1}^{j}= \theta(\hat{q}_{i}^{j}, \bar{q}_{i+1}^{j})$, etc. Further, define

$$\hat{m}_i =\bar{V}_i^{j+1} - \hat{V}_i $$

as the pull that segment (i,j+1) exerts on those initially in segment (i,j). Surely only those with taste parameter θ in $[\hat{\theta}_{i-1,i}^{j}, \hat{\theta}_{i,i+1}^{j}]$ and mobility cost m beyond $\hat{m}_{i}$ will remain in segment (i,j). In this case, and given independence between m and θ, $\hat{n}_{i}^{j}$ simply equals $\bar{n}^{j}(F(\hat{\theta}_{i,i+1}^{j})-F(\hat{\theta}_{i-1,i}^{j})) (1-F(\hat{m}_{i}))$. In contrast, $\bar{n}_{i}^{j}$ equals $\bar{n}^{j}(F(\bar{\theta}_{i,i+1}^{j})-F(\bar{\theta}_{i-1,i}^{j}))$.

Now consider the ratio $(\hat{n}_{i}^{j}-\bar{n}_{i}^{j})/(\hat{q}_{i}^{j}-\bar{q}_{i}^{j})$. This ratio can be expanded as follows:

(23)

where $\bar{m}_{i}=0$.

By the implicit function theorem, θ _i,i+1 is a differentiable, and hence continuous, function of $\hat{q}_{i}^{j}$ for rents $\hat{q}_{i}^{j}$ sufficiently close to $\bar{q}_{i}^{j}$. Then as we let $\hat{q}_{i}^{j}$ approach $\bar{q}_{i}^{j}$, $\hat{\theta}_{i,i+1}$ approaches $\bar{\theta}_{i,i+1}$. Since F(θ) is differentiable, then the first ratio on the first line of (23) converges to $f(\bar{\theta}_{i,i+1}^{j})$.

Taking limits of all other terms in (23) and applying similar arguments eventually gives the right derivative of $n_{i}^{j}$ with respect to $q_{i}^{j}$ at $\bar{q}_{i}^{j}$:

(24)

(ii) Consider a rent $\check{q}_{i}^{j}$ strictly below $\bar{q}_{i}^{j}$. Resulting variable values are $\check{\theta}_{i,i+1}^{j}= \theta_{i,i+1} (\check{q}_{i}^{j}, \bar{q}_{i+1}^{j})$, etc. Given $\check{q}_{i}^{j} < \bar{q}_{i}$, segment (i,j) clearly is more attractive than segments of comparable quality elsewhere. We decompose $\check{n}_{i}^{j}$ into natives and immigrants. The ratio of the change in natives to the change in rent can be written as

$$\bar{n}^j\frac{ (F(\bar{\theta}_{i, i+1}^j)-F(\bar{\theta}_{i-1,i}^j) )- ( F(\check{\theta}_{i, i+1}^j)-F(\check{\theta}_{i-1,i}^j) )}{\bar{q}_i ^j - \check{q}_i^j} $$

Letting $\check{q}_{i}^{j}$ approach $\bar{q}_{i}^{j}$ gives the derivative of natives’ numbers with respect to $q_{i}^{j}$ at $\bar{q}_{i}^{j}$, i.e.,

(25)

Next we turn to immigrants to (i,j). Figure 1 (in the text) points to which natives from city j−1 may be attracted to (i,j). The figure shows maximum utility in the three segments (i−1,j−1), (i,j−1) and (i+1,j−1), as well as maximum utility in competing segment (i,j), as functions of the taste index θ. Define

$$\check{m}_i =\check{V}_i^{j} - \bar{V}_i^{j-1} $$

as the pull that segment (i,j) exerts on those initially in segment (i,j−1). On the one hand, the unknown number of those wanting to emigrate to (i,j), dE, certainly is smaller than $\bar{n}^{j-1} (F(\check{\theta}_{i,i+1}^{j-1}) - F(\check{\theta}_{i-1,i}^{j-1}) ) F(\check{m}_{i})$. On the other hand, this number certainly is greater than $\bar{n}^{j-1} (F(\bar{\theta}_{i,i+1}^{j-1}) - F(\bar{\theta}_{i-1,i}^{j-1}) ) F(\check{m}_{i})$. I.e.,

The first and the last term in this series of inequalities converge to the same expression as $\check{q}_{i}^{j}$ approaches $\bar{q}_{i}^{j}$. Hence so does the middle term, by the “squeezing rule”. The derivative of the number of migrants from j−1 to j thus is the limit of either the first or the last term in the preceding series of inequalities, and hence can be derived as:

$$ \bar{n}^{j-1} \bigl(F\bigl(\bar{\theta}_{i,i+1}^{j-1}\bigr)- F\bigl(\bar{\theta}_{i-1,i}^{j-1} \bigr) \bigr) f(\bar{m}_i) \frac{\partial V(\bar{q}_i^j)}{\partial q_i^j} $$

(26)

Adding (25) and (26) gives the left derivative of $n_{i}^{j}$ with respect to $q_{i}^{j}$ at $\bar{q}_{i}^{j}$:

(27)

(iii) Comparing (27) with (24) shows that the derivative at $\bar{q}_{i}^{j}$ exists, if the initial allocation is symmetric. □

Proof of Proposition 3

The proof is similar to that of Proposition 2. □

Proof of Proposition 4

(Existence) We first document an existence property not explicitly mentioned in the Proposition. Generally, the coefficients of $\partial q_{i}^{j}/\partial\alpha^{j}$, i=2,…,I, are stacked into the (i−1)th column of A ^j:

$$ \begin{array}{l} \\ \\ \\ \\ \\ \mbox{row }i-2 \\ \\ \\ \mbox{row }i-1 \\ \\ \\ \mbox{row }i \\ \\ \\\\ \\ \\ \end{array} \left ( \begin{array}{c} 0 \\ \vdots \\ 0 \\ \\ n^j f( \theta_{i-1,i}^j) \frac{\partial \theta_{i-1,i}^j}{\partial q_i^j} \\\\ n^j f( \theta_{i,i+1}^j ) \frac{\partial \theta_{i,i+1}^j}{\partial q_i^j} - (\sigma_i^j + \varepsilon_i^j + \eta_i^j ) - n^j f( \theta_{i-1,i}^j)\frac{\partial\theta_{i-1,i}^j}{\partial q_{i}^j} \\ \\ - n^j f( \theta_{i,i+1}^j) \frac{\partial \theta_{i,i+1}^j}{\partial q_i^j}\\\\ 0\\ \vdots \\ 0 \end{array} \right ) $$

(28)

where, to be sure, the first and the last column of A ^j look slightly different.

The effects of a one Euro change in $q_{i}^{j}$ on segments i−1, i and i+1 are found in column (i−1)’s three consecutive rows i−2, i−1, and i, respectively, whereas the column’s remaining entries are zero. Now, the element in column i−1 and row i−1, also shown in (28), is a diagonal element of A ^j. In absolute value this (negative) element exceeds the sum of the two (positive) off-diagonal elements found in the column’s rows i−2 and i. This property applies to any of A ^j’s columns. Thus A ^j is diagonally dominant.

But then A ^j also is non-singular (Graybill 1983, Theorem 8.11.2). Hence (A ^j)⁻¹ exists, and with it a solution to (10). This solution is unique.

(i) (Raising Low Quality Rent) Note that (A ^j)⁻¹ has non-positive entries everywhere (Takayama 1985, Theorem 4.D.3, parts (I″) and (III″), or Simon and Blume 1994, Theorem 8.14). Thus every component of the solution vector (A ^j)⁻¹ b ^j is simply the product of a non-positive number, taken from the first column of (A ^j)⁻¹, and the negative shock (11). Thus every element of the solution y ^j is non-negative.

Next we show that the elements of y ^j in fact are strictly positive. Consider $\partial q_{2}^{j}/\partial q_{1}^{j} >0$ first. Assume, to the contrary, that $\partial q_{2}^{j}/ \partial q_{1}^{j} =0$. Then if also setting i=2, choosing α=q ₁, and dropping the city index, Eq. (10) becomes

(29)

where ∂t _2,3/∂q ₁ is given in (13) when setting α=q ₁ and i=2.

On the one hand, the l.h.s. of (29) must be non-negative. After all, ∂q ₃/∂q ₁≥0, as established above, while ∂q ₂/∂q ₁=0, by assumption. Hence ∂t _2,3/∂q ₁≥0.

On the other hand, the r.h.s. of (29) is strictly negative (see (5)). We conclude that ∂q ₂/∂q ₁>0. Repeating this argument for successively higher quality segments reveals that ∂q _i /∂q ₁>0 for all i=2,…,I.

(iii) (Demolishing low quality stock) The proof for showing that ∂q _i /∂z ₂>0 for all i=2,…,I is similar to (ii). □

Proof of Proposition 5

(i) We focus on the case where α=q ₁ first. The Ith segment version of (14) reads

$$ nf(t_{I-1,I})\frac{\partial t_{I-1,I}}{\partial q_1} = - ( \sigma_I + \varepsilon_I + \eta_I) \frac{\partial q_I}{\partial q_1} $$

(30)

Following Proposition 4, Part (i), the r.h.s. of (30) is strictly negative. Then so must be its l.h.s. Thus ∂t _I−1,I /∂q ₁<0.

Proceeding in this fashion towards successively lower qualities will show that derivatives of all boundaries (except for t _0,1) with respect to q ₁ are strictly negative.

(ii) Now focus on the case where α=z ₂. Analysis of derivatives of boundaries with respect to z ₂ is identical to the analysis in the previous paragraph for i=3,…,I. These derivatives are all strictly negative, too.

Treating the case of ∂t _1,2/∂z ₂ is even simpler. The i=2 version of (13) reduces to

$$\frac{\partial t_{1,2}^j}{\partial z_2^j} = \frac{\partial \theta_{1,2}^j}{\partial q_{2}^j} \frac{\partial q_{2}^j}{\partial z_2^j} $$

which is strictly positive. □

Proof of Proposition 6

(i) (Migration) (We sketch the proof only because the details of the full proof would require introducing additional notation.) To solve for rents in every segment i=2,…,I in every city j=1,…,J we need to set up a system of (I−1)J equations. Differentiating this system with respect to α ^j=α gives a linear system of (I−1)J equations in the (I−1)J unknowns contained in y=(y ¹,…,y ^J).

We collect the values found on the right hand sides of these equations in the (I−1)J×1-vector b=(b ¹,…,b ^J), where b ^j is constant across cities. Note that b exhibits only zeros, except for on the first, and then on every (I−1)th position.

The system’s coefficient matrix, denoted C, essentially is blockdiagonal, with the representative local coefficient matrix A ^j=A repeatedly used as block, for a total of J times.

The only extra novelty to be taken care of is that on the two off diagonals I−1 entries off the main diagonal we now encounter non-zero elements that capture the effects of changes originating in the two neighboring cities’ rents on a given city’s housing market. (These latter effects were not present in (28).)

One can show that the resulting matrix C is diagonally dominant. Hence C is non-singular, too, (again Graybill 1983, Theorem 8.11.2) and its inverse C ⁻¹ exists, and exhibits non-negative entries only.

The solution for y is given by Cb. This product makes use of only the first, Ith, etc. column of C. The coefficients found in these columns repeat themselves every I−1th row. We suggest a solution for y in which rent changes within any given quality segment do not vary across cities.

But when premultiplied by C, this trial solution “works”. Since with C being non-singular no other solution can exist, we conclude that for a given quality segment the rent changes found in y indeed are the same in every city.

(ii) (Rents) The proof is by contradiction. I.e., there must exist a segment i∈{2,…,I} for which both

$$ \biggl(\frac{\partial q_{i}^j}{\partial\alpha^j} \biggr)^c \le \biggl( \frac{\partial q_{i}^j}{\partial \alpha^j} \biggr)^{nc} \quad \mbox{and}\quad \biggl(\frac {\partial q_{i-1}^j}{\partial\alpha^j} \biggr)^c \ge \biggl(\frac{\partial q_{i-1}^j}{\partial\alpha^j} \biggr)^{nc} $$

(31)

hold.

Let us explore i’s upper neighbor i+1. If i+1≤I and if $(\partial q_{i+1}^{j}/\partial\alpha^{j})^{c} \le(\partial q_{i+1}^{j}/\partial\alpha ^{j})^{nc}$ then we continue by investigating i+2. Else we stop. Suppose we stop after l steps, where l≥0. Then our procedure constructs a “connected sequence” M=i,i+1,…,i+l of adjacent quality segments, of length l+1.

Consider M’s lower neighbor i−1 first. By definition of M, this segment either enjoys a stronger rent rise (if i>2), or the same rent rise (if i=2), under coordinated than under isolated action. Hence $(\partial t_{i-1,i}^{j}/\partial\alpha^{j})^{c} \le (\partial t_{i-1,i}^{j}/\partial\alpha^{j})^{nc}$, by inspection of (13).

Likewise, consider M’s upper neighbor i+l+1 next. If i+l+1>I then this upper neighbor does not exist. Then $(\partial t_{i+l,i+l+1}^{j}/\partial\alpha ^{j})^{c} = (\partial t_{i+l, i+l+1}^{j}/\partial\alpha^{j} )^{nc}= 0$. Alternatively, if i+l+1≤I then this upper neighbor exists, and by M’s definition experiences a strictly stronger rent rise under coordination. I.e.,

$$\biggl(\frac{\partial q_{i+l+1}^j}{\partial\alpha^j} \biggr)^c > \biggl( \frac{\partial q_{i+1+l}^j}{\partial\alpha^j} \biggr)^{nc} $$

Hence $(\partial t_{i+l,i+l+1}^{j}/\partial\alpha^{j})^{c} > (\partial t_{i+l, i+l+1}^{j}/\partial\alpha^{j})^{nc}$. To summarize both cases in one statement, $(\partial t_{i+l,i+l+1}^{j}/\partial\alpha^{j})^{c} \ge (\partial t_{i+l, i+l+1}^{j}/\partial \alpha^{j} )^{nc}$ always.

Recursive substitution in (14) l times gives

(32)

in the uncoordinated case, and

(33)

in the coordinated case, with $\gamma_{i}=(\partial z_{2}^{j}/\partial\alpha^{j})^{nc}= (\partial z_{2}/\partial\alpha)^{c}$ if i=2 and zero otherwise.

Now we make use of the rankings of boundary changes under coordinated and under isolated action derived above. Subtracting the l.h.s. of (33) from the l.h.s. of (32) gives a non-negative number. Subtracting the r.h.s. of (33) from the r.h.s. of (32) gives a strictly negative number, given our assumption (31). Either Eq. (32) or Eq. (33) cannot be satisfied. This contradicts the model’s assumptions. □