The Effects of a Targeted Financial Constraint on the Housing Market

Abstract

We study how financial constraints affect the housing market by exploiting a regulatory change that increases the down payment requirement for homes selling for

$\$$

1M or more. Using Toronto data, we find that the policy causes excess bunching of homes listed at

$\$$

1M and heightened bidding intensity for these homes, but only a muted response in sales. While difficult to reconcile in a frictionless market, these findings are consistent with the implications derived from an equilibrium search model with auctions and financial constraints. Our analysis points to the importance of designing macroprudential policies that recognize the strategic responses of market participants.

This paper examines how financial constraints targeting a specific housing market segment affect house price formation. A growing class of “targeted” policies aim to cool a red-hot housing segment rather than the overall market. In Toronto and Vancouver, a higher down payment is required to qualify for government-guaranteed mortgage insurance for homes purchased for over CAN

1M (since 2012). In New York and London, so-called “mansion taxes” have been imposed on purchases of all homes valued over US

$\$$

1M (since 1989) and over £1.5M (since 2014), respectively.¹ While varied in their design, these policies impose additional financial constraints on prospective homebuyers in a particular segment relative to those in other segments, which in turn can affect a seller’s decision to list a house, their choice of asking price, and the process of final price determination. The central role of financial constraints makes them an appealing macroprudential vehicle for policy makers to intervene in housing markets, often for the purpose of “[ensuring] that shocks from the housing sector do not spill over and threaten economic and financial stability” (IMF, 2014).² While financial constraints represent a recurring theme in the finance literature, there remains no micro analyses of the links between financial constraints and search behavior among housing market participants. Moreover, policies targeting a particular housing segment have just begun to attract serious attention from economists (e.g., Kopczuk and Munroe 2015). This paper fills the gap in the literature by examining how financial constraints affect price formation in the targeted segment of a frictional housing market. Our empirical methodology exploits a natural experiment arising from a mortgage insurance policy change implemented in Canada in 2012. The interpretation of our results is motivated by a search-theoretic model of sellers competing for financially constrained buyers.

Canada experienced one of the world’s largest modern house price booms, with house prices more than doubling between 2000 and 2012. In an effort to cool this unprecedentedly long boom, the government implemented the so-called “million dollar” policy that restricts access to mortgage insurance when the purchase price of a home exceeds one million Canadian dollars (⁠

1M). Note that lenders are required to insure mortgages with loan-to-value (LTV) ratios over |$80$||$\%$|⁠. As such, the minimum down payment jumps from |$5$||$\%$| to |$20$||$\%$| of the entire transaction price at a threshold of

$\$$

1M, increasing the minimum down payment by

$\$$

150,000 for million dollar homes. The existence or absence of bunching around the threshold should provide compelling and transparent evidence about how home buyers and sellers respond to a targeted financial constraint.

Understanding the mechanisms that generate bunching requires an equilibrium analysis of a two-sided market. To this end, we preface the empirical work with a search-theoretic model that features financial constraints on the buyer side.³ Sellers pay a cost to list their house and post an asking price, and buyers allocate themselves across sellers subject to search frictions governed by a many-to-one meeting technology. Prices are determined by an auction mechanism: a house is sold at the asking price when a single buyer arrives; but to the highest bidder when multiple buyers submit offers to purchase the same house. In that sense, our model draws from the competing auctions literature (McAfee 1993, Peters and Severinov 1997, Julien, Kennes, and King 2000, Albrecht, Gautier, and Vroman 2014, Lester, Visschers, and Wolthoff 2015). The distinguishing feature of the model is that the million dollar policy tightens the financial constraints faced by a subset of buyers and limits how much they can bid on a house.⁴

We characterize the pre- and post-policy equilibria and derive a set of empirical predictions. The post-policy equilibrium features a mass of sellers with asking prices at the

1M threshold. These price adjustments can come from either side of

$\$$

1M. In some circumstances, sellers lower the asking price from above

$\$$

1M to attract both constrained and unconstrained buyers to compete for their homes. In other circumstances, sellers increase the asking price from below

$\$$

1M to extract a higher payment in bilateral situations. In both cases, the policy generates an excess mass of homes listed at

$\$$

1M. As the bunching response passes through to the sales price distribution, however, the effect on sales prices is mitigated by search frictions and bidding wars. For example, even though some sellers lower the asking price to

$\$$

1M, the induced competition among constrained and unconstrained buyers creates a heated market just under

$\$$

1M that both pushes the sales price above

$\$$

1M and leads to shorter time-on-the market.

Ultimately, the magnitude of the impact of the policy on prices is an empirical question. We test the model’s predictions using the 2010–2013 housing market transaction data in the Greater Toronto Area, Canada’s largest housing market. This market provides an ideal setting for this study for two reasons. First, home sellers in Toronto typically initiate the search process by listing the property and specifying a date on which offers will be considered (often 5–7 days after listing). This institutional practice matches well with our model of competing auctions. Second, the million dollar policy caused two discrete changes in the market: one at the time the policy was implemented, and another at the

1M threshold. The market thus provides a natural experimental opportunity for examining the price response to targeted financial constraints.

Figure 1 presents the distribution of listings (left column) and sales (right column) in the segments around the

1M threshold. Panels A and B display frequency counts of asking prices in each

$\$$

5,000 dollar bin during the pre- and post-policy periods, respectively.⁵ In both periods, there is a substantial mass of listings right below

$\$$

1M, possibly because of a psychological bias associated with the million dollar threshold. Panels C and F net out the time-invariant threshold effects by presenting the difference in the frequency of listings and sales between the post- and pre-policy periods, along with confidence interval bars.⁶ The results are striking. First, there is a substantial and statistically significant positive jump in the number of listings in the

$\$$

995,000–

$\$$

999,999 bin, suggesting that the policy induces excess bunching of listings at

$\$$

1M. Second, this excess bunching appears to come from both sides of the threshold, as reflected by the reduction in the number of homes listed in bins just to the left and right of the million dollar bin. Finally, the mass of sales in the million dollar segment is much less pronounced, and the difference is statistically insignificant. The evidence here, in its most descriptive form, lends support to the key implications of our model and forms the basis for our empirical estimation design.

Despite the appealing first-cut evidence presented in Figure 1, identifying the million dollar policy’s impact on asking and sales prices is difficult for several reasons. First, housing composition may have shifted around the time the policy was implemented. As a result, changes to the distributions of prices between pre- and post-policy periods may simply reflect the changing characteristics of houses listed/sold rather than buyers’ and sellers’ responses to the policy. Second, the implementation of the policy coincided with a number of accompanying government interventions,⁷ complicating the challenge of attributing any changes in the price distributions to the million dollar policy.

Our solution relies on a two-stage estimation procedure that examines changes in the price distribution. First, leveraging the richness of our data on house characteristics and using the well-known reweighting approach introduced by DiNardo, Fortin, and Lemieux (1996), we decompose the observed before-after-policy change in the distribution of house prices into: (1) a component inspired by changes in house characteristics and (2) a component inspired by changes in the price structure. The latter represents the quality-adjusted changes in the distribution of house prices that would have prevailed between the pre- and post-policy periods if the characteristics of houses remained the same as in the pre-policy period. Next, we measure the effects of the

1M policy by comparing the actual changes in the quality-adjusted price distributions to counterfactual changes in the quality-adjusted price distributions that would have prevailed in the absence of the policy, separately for asking and sales prices. We employ a recently developed bunching estimation approach (Chetty et al. 2011, Kleven and Waseem 2013).⁸ The key idea is to use price segments that are not subject to the policy’s threshold effects to form a counterfactual near the

$\$$

1M threshold. By comparing the counterfactual changes in distributions with the actual changes in distributions around

$\$$

1M, bunching estimation allows us to difference out impacts of contemporaneous factors on house prices, such as other mortgage rule changes and market trends. Further, working with changes in price distributions over time allows us to net out any time-invariant threshold price effects unrelated to the policy, such as psychological bias.

Our main findings are as follows. In the single-family housing market, the asking price distribution features large and sharp excess bunching right at the

1M threshold with corresponding holes both above and below

$\$$

1M. In particular, the policy adds |$86$| homes to listings in the million dollar bin (from

$\$$

995,000 to

$\$$

999,999) in the post-policy year for the city of Toronto, which represents about a |$38$||$\%$| increase relative to the number of homes that would have been listed in this

$\$$

5,000 bin in the absence of the policy. Among these, half would have otherwise been listed below

$\$$

995,000; the other half above

$\$$

1M. In contrast, the policy adds only about |$11$| homes to sales in the million dollar bin, which is economically small and statistically insignificant. These findings are robust to an extensive set of specification checks, including a counterfactual constructed using only data below

$\$$

1M, alternative functional forms, estimation windows and excluded regions, different definitions of pre- and post-policy periods, and allowing for possible spillovers near the

$\$$

1M segment. We also find similar patterns in the condominium and townhouse markets.

The lack of excess bunching in the sales price, together with the sharp bunching in the asking price, suggests that the intended cooling impact of the policy is mitigated by sellers’ listing decisions and buyers’ bidding behavior. Consistent with this interpretation, we find that housing segments right below the

1M threshold experience a shorter time-on-the-market and a higher incidence of bidding wars, confirming the notion that the million dollar policy created a “red-hot” market for homes listed just below

$\$$

1M (Marr, 2013).

Together, our findings contribute to a better understanding of policies that use targeted financial constraints to temper a heated market segment. We find that the million dollar policy did not achieve the specific goal of cooling the housing boom in the million dollar segment. This is not because market participants did not respond to the policy. In fact, quite the opposite appears to be true: it is precisely the strategic responses of home sellers and buyers that interact to undermine the intended impact of the policy on sales prices. Our analysis thus points to the importance of designing policies that recognize the endogenous responses of buyers and sellers in terms of listing strategies, search decisions and bidding behavior.

While our main focus is on segments around

1M, we also go beyond the bunching estimation and examine the policy effects in segments further above

$\$$

1M. An extensive margin response would imply that some transactions above

$\$$

1M did not occur due to the additional financial constraint. An intensive margin response would imply depressed prices in at least some segments of the market above

$\$$

1M. Either of these responses should manifest as a systematic discrepancy between the counterfactual post-policy price distribution and the observed post-policy price distribution for price bins above the

$\$$

1M threshold. Using a distribution decomposition method, we do not find such discrepancies, which suggests that transactions above

$\$$

1M are not markedly affected by the policy.

Despite failing to curb house price appreciation, the policy may have nonetheless succeeded in improving the creditworthiness of homebuyers. A key implication of the model is that, when facing the

1M policy, less constrained buyers have an advantage over constrained ones in multiple offer situations and hence have incentive to participate in the segment below

$\$$

1M. Consistent with this, we observe a large share of homebuyers with LTV ratios below 80|$\%$| even in the segment just below

$\$$

1M after the implementation of the policy. Thus, the policy improves borrower creditworthiness in segments above

$\$$

1M without compromising borrower creditworthiness in the segment slightly below

$\$$

1M via its influence on buyers’ and sellers’ search and listing behaviors. More broadly, by reallocating million dollar homes from more constrained to less constrained homebuyers, the policy effectively prevents lenders from making riskier loans. As such, our analysis is also related to the recent literature on macroprudential interventions in mortgage markets. Significant contributions have been made toward our understanding of how these policies affect mortgage market outcomes (Allen et al., 2016, DeFusco, Johnson, and Mondragon, 2020), as well as financial stability and mortgage market efficiency (Elenev, Landvoigt, and Van Nieuwerburgh, 2016, Van Bekkum, Gabarro, and Irani, 2017, Elenev, Landvoigt, and Van Nieuwerburgh, 2018, Acharya, Berger, and Roman, 2018). From that perspective, the effects of the

$\$$

1M policy may help reduce the likelihood of mortgage crises and safeguard the stability of the financial system, which would be welfare enhancing in the long run. Studying such benefits by quantifying the effects of the policy on mortgage market outcomes may prove to be an important area for future research.⁹

1. Background

1.1 Mortgage insurance

Mortgage insurance is an instrument used to transfer mortgage default risk from the lender to the insurer and represents a key component of housing finance in many countries including the United States, the United Kingdom, the Netherlands, Hong Kong, France, and Australia. These countries share two common features with Canada: (a) the need to insure high LTV mortgages and (b) the central role of the government in providing such insurance. The combination of these two requirements gives the government the ability to influence the financial constraints faced by homebuyers.

In Canada, all financial institutions regulated by the Office of the Superintendent of Financial Institutions (OSFI) are required to purchase mortgage insurance for any mortgage loan with an LTV above |$80$||$\%$|⁠. The mortgage insurance market is comprised of the government-owned Canada Mortgage and Housing Corporation (CMHC) as well as two private insurers, Genworth Financial Mortgage Insurance Company Canada and Canada Guaranty. All three institutions benefit from guarantees provided by the Canadian government and therefore are subject to federal regulations through the OFSI.

In practice, although buyers can obtain uninsured residential mortgages with a loan-to-value ratio greater than 80|$\%$| from unregulated lenders, we find that private lending accounted for only 4|$\%$| of all loans in the Greater Toronto Area in 2013, and this sector did not experience any noticeable growth around the million dollar policy period. The reason is that, compared to traditional mortgages from regulated lenders, private mortgages on average have one-fifth duration, over three times higher interest rates, and loan amounts that are one-third of the size. Hence, they operate in a small disparate niche corner of the Canadian mortgage market.¹⁰ In addition, anecdotal evidence suggests that it is generally difficult for a borrower to obtain a second mortgage at the time of origination to reduce the down payment of the primary loan below 20|$\%$| in Canada, making this strategic circumvention of macroprudential regulation less of a concern.¹¹ The pervasiveness of government-backed mortgage insurance within the housing finance system makes it an appealing macroprudential policy tool for influencing housing finance and housing market outcomes.

1.2 The million dollar policy

Figure 2 plots the national house price indexes for Canada and the United States reflecting Robert Shiller’s observation in 2012 that “what is happening in Canada is kind of a slow-motion version of what happened in the U.S.” (Macdonald, 2012). As home prices in Canada continued to escalate post-financial crisis, the Canadian government became increasingly concerned that rapid price appreciation would eventually lead to a severe housing market correction.¹² To counter the potential risks associated with this house price boom, the Canadian government implemented several rounds of housing market macroprudential regulation, all through changes to the mortgage insurance rules.¹³ This paper examines the impact of the so-called “million dollar” policy that prevents regulated lenders from offering mortgage loans with LTV ratios above 80|$\%$| when the purchase price is

1M or more. The objective of the regulation was to curb house price appreciation in high price segments of the market and at the same time improve borrower creditworthiness. The law was announced on June 21, 2012, and enacted July 9, 2012. Anecdotal evidence suggests that the announcement of the policy was largely unexpected by market participants (Nelson and Perkins, 2013).

2. Theory

To understand how the million dollar policy affects strategies and outcomes in the housing market, we present a two-sided search model that incorporates auction mechanisms and financially constrained buyers. We characterize pre- and post-policy directed search equilibria and derive a set of empirical implications. The purpose of the model is to guide the empirical analyses that follow. As such, we present a simple model of directed search with auctions and bidding limits that features heterogeneity only along the financial constraints dimension. The clean and stylized nature of the model allows for a quick understanding of the intuition underlying plausible strategic reactions among buyers and sellers to the implementation of the policy.

2.1 Environment

2.1.1 Agents

There is a fixed measure |$\mathcal{B}$| of buyers, and a measure of sellers determined by free entry. Buyers and sellers are risk neutral. Each seller owns one indivisible house, their value of which is normalized to zero. Buyer preferences are identical; a buyer assigns value |$v>0$| to owning the home. Buyers cannot pay more than some fixed |$u\leq v$|⁠, which can be viewed as a common income constraint or debt-service constraint.¹⁴

2.1.2 Million dollar policy

The introduction of the million dollar policy causes some buyers to become more severely financially constrained. Post-policy, a fraction |$\Lambda$| of buyers are unable to pay more than |$c$|⁠, where |$c<u$|⁠. Parameter restrictions |$c<u\leq v$| can be interpreted as follows: all buyers may be limited by their budget sets, but some are further financially constrained by a binding wealth constraint, such as a minimum down payment requirement, following the implementation of the policy.¹⁵ Buyers with financial constraint |$c$| are hereafter referred to as constrained buyers, whereas buyers willing and able to pay up to |$u$| are termed unconstrained.

2.1.3 Search and matching

2.1.4 Price determination

The price is determined in a sealed-bid second-price auction. The seller’s asking price, |$p\in\mathbb{R}_+$|⁠, is interpreted as the binding reserve price. If a single bidder submits an offer at or above |$p$|⁠, they pay only |$p$|⁠. In multiple offer situations, the bidder submitting the highest bid at or above |$p$| wins the house but pays either the second highest bid or the asking price, whichever is higher. When selecting among bidders with identical offers, suppose the seller picks one of the winning bidders at random with equal probability.

2.1.5 Free entry

The measure of sellers is determined by free entry so that overall market tightness is endogenous. Supply side participation in the market requires payment of a fixed cost |$x$|⁠, where |$0<x<c$|⁠. It is worthwhile to enter the market as a seller if and only if the expected revenue exceeds the listing cost.

2.2 Equilibrium

2.2.1 The auction

When a seller meets |$k$| buyers, the auction mechanism described above determines a game of incomplete information because bids are sealed and bidding limits are private. In a symmetric Bayesian-Nash equilibrium, it is a dominant strategy for buyers to bid their maximum amount, |$c$| or |$u$|⁠. When |$p>c$| (⁠|$p>u$|⁠), bidding limits preclude constrained (and unconstrained) buyers from submitting sensible offers.

2.2.2 Expected payoffs

Expected payoffs are computed taking into account the matching probabilities in (1) and (2). These payoffs, however, are markedly different depending on whether the asking price, |$p$|⁠, is above or below a buyer’s ability to pay. Each case is considered separately in Internet Appendix B.1. In the submarket associated with asking price |$p$| and characterized by market tightness |$\theta$| and buyer composition |$\lambda$|⁠, let |$V^s(p,\lambda,\theta)$| denote the sellers’ expected net payoff. Similarly, let |$V^c(p,\lambda,\theta)$| and |$V^u(p,\lambda,\theta)$| denote the expected payoffs for constrained and unconstrained buyers.

The second term reflects the surplus from a transaction if they meet only one buyer. The third and fourth terms reflect the surplus when matched with two or more buyers, where the last term is specifically the additional surplus when two or more bidders are unconstrained.

The first term is the expected surplus when competing for the house with other unconstrained bidders, and the second term is the additional surplus when competing with constrained bidders only. In that scenario, the unconstrained bidder wins the auction by outbidding the other constrained buyers, but pays only |$c$| in the second-price auction. The third term represents the additional payoff for a monopsonist. Closed-form solutions for the other cases are derived in Internet Appendix B.1.

2.2.3 Directed search

Agents perceive that both market tightness and the composition of buyers depend on the asking price. To capture this, suppose agents expect each asking price |$p$| to be associated with a particular ratio of buyers to sellers |$\theta(p)$| and fraction of constrained buyers |$\lambda(p)$|⁠. We will refer to the triple |$(p,\lambda(p),\theta(p))$| as submarket|$p$|⁠. When contemplating a change to their asking price, a seller anticipates a corresponding change in the matching probabilities and bidding war intensity via changes in tightness and buyer composition. This is the sense in which search is directed. It is convenient to define |$V^i(p)=V^i(p,\lambda(p),\theta(p))$| for |$i\in\{s,u,c\}$|⁠.

 

The definition of a DSE is such that for every |$p\in\mathbb{R}_+$|⁠, there is a |$\theta(p)$| and a |$\lambda(p)$|⁠. Part 1(a) states that |$\theta$| is derived from the free entry of sellers for active submarkets (i.e., for all |$p\in\mathbb{P}$|⁠). Similarly, parts 1(b) and 1(c) require that, for active submarkets, |$\lambda$| is derived from the composition of buyers that find it optimal to search in that submarket. For inactive submarkets, parts 1(b) and 1(c) further establish that |$\theta$| and |$\lambda$| are determined by the optimal sorting of buyers so that off-equilibrium beliefs are pinned down by the following requirement: if a small measure of sellers deviate by posting asking price |$p\not\in\mathbb{P}$|⁠, and buyers optimally sort among submarkets |$p\cup\mathbb{P}$|⁠, then those buyers willing to accept the highest buyer-seller ratio at price |$p$| determine both the composition of buyers |$\lambda(p)$| and the buyer-seller ratio |$\theta(p)$|⁠. If neither type of buyer finds asking price |$p$| acceptable for any positive buyer-seller ratio, then |$\theta(p)=0$|⁠, which is interpreted as no positive measure of buyers willing to search in submarket |$p$|⁠. The requirement in part 1(a) that |$V^s(p)\leq 0$| for |$p\not\in\mathbb{P}$| guarantees that no deviation to an off-equilibrium asking price is worthwhile from a seller’s perspective. Finally, part 2 of the definition makes certain that all buyers search.

2.2.4 Pre-policy directed search equilibrium

The solution to problem |$\text{P}_0$| can therefore be summarized as |$p_0=\min\{p^*_u,u\}$| and |$\theta_0$| satisfying |$V^s(p_0,0,\theta_0)=0$|⁠.

The following proposition provides a partial characterization of the pre-policy DSE constructed using this solution, per the algorithm in Internet Appendix B.2.

 

As buyers’ ability to pay approaches their willingness to pay (i.e., as |$u\rightarrow v$|⁠), the equilibrium asking price tends to zero (i.e., |$p_0=p^*_u\rightarrow0$|⁠), which is the seller’s reservation value. This aligns with standard results in the competing auctions literature in the absence of bidding limits (McAfee, 1993, Peters and Severinov, 1997, Albrecht, Gautier, and Vroman, 2014, Lester, Visschers, and Wolthoff, 2015). When buyers’ bidding strategies are somewhat limited (i.e., |$p_0=p^*_u\leq u<v$|⁠), sellers set a higher asking price to capture more of the surplus in a bilateral match. The equilibrium asking price is such that the additional bilateral sales revenue exactly compensates for the unseized portion of the match surplus when two or more buyers submit offers but are unable to bid up to their full valuation. This is the economic interpretation of Equation (4). When buyers’ bidding strategies are too severely restricted (i.e., |$p_0=u<p^*_u$|⁠), the seller’s choice of asking price is constrained by the limited financial means of prospective buyers. Asking prices in equilibrium are then set to the maximum amount, namely, |$u$|⁠. In this case, a seller’s expected share of the match surplus is diminished, and consequently fewer sellers choose to participate in the market (i.e., |$\theta_u>\theta^*_u$|⁠).

If |$p_0=p^*_u\leq u$|⁠, the equilibrium expected payoff |$\bar{V}^u$| is independent of |$u$| (in particular, |$\bar{V}^u=\theta^*_u e^{-\theta^*_u} v$|⁠). As long as the constraint remains relatively mild, a change to buyers’ ability to pay, |$u$|⁠, will cause the equilibrium asking price to adjust in such a way that market tightness and the expected sales price remain unchanged. This reflects the fact that the financial constraint does not affect the incentive to search. When |$p_0=u<p^*_u$|⁠, the constraint is sufficiently severe that it affects the ability to search in that it shuts down the submarket that would otherwise achieve the mutually desirable trade-off between market tightness and expected price. This feature highlights the distinction between the roles of financial constraints and reservation values, since a change to buyers’ willingness to pay, |$v$|⁠, would affect the incentive to search, the equilibrium expected payoff, and the equilibrium trade-off between market tightness and expected sales price.

2.2.5 Post-policy directed search equilibrium

Let |$\{p_1,\lambda_1,\theta_1\}$| denote the solution to problem |$\text{P}_1$| when |$\bar{V}^u$| is set equal to the maximized objective of problem |$\text{P}_0$|⁠. The bidding limit once again limits the set of worthwhile submarkets. In particular, the optimal submarket for constrained buyers must feature an asking price less than or equal to |$c$|⁠. If the solution is interior, it satisfies the two constraints with equality and a first-order condition derived in Internet Appendix B.3. This interior solution is denoted by |$\{p^*_c,\lambda^*_c,\theta^*_c\}$|⁠. The corner solution is denoted by |$\{c,\lambda_c,\theta_c\}$|⁠, where |$\lambda_c$| and |$\theta_c$| satisfy the free entry condition |$V^s(c,\lambda_c,\theta_c)=0$| and an indifference condition for unconstrained buyers |$V^u(c,\lambda_c,\theta_c)=\bar{V}^u$|⁠. In summary, the solution to problem |$\text{P}_1$| is |$p_1=\min\{p^*_c,c\}$| with |$\lambda_1$| and |$\theta_1$| satisfying |$V^s(p_1,\lambda_1,\theta_1)=0$| and |$V^u(p_1,\lambda_1,\theta_1)=\bar{V}^u$|⁠.

As long as the aggregate share of constrained buyers, |$\Lambda$|⁠, does not exceed |$\lambda_1$|⁠, we can construct an equilibrium with two active submarkets associated with the asking prices obtained by solving problems |$\text{P}_0$| and |$\text{P}_1$| in the manner described above.

 

Intuitively, constrained buyers would prefer to avoid competition from unconstrained buyers because they can outbid them. For the same reason, some unconstrained buyers prefer to search alongside constrained buyers. The equilibrium search decisions of constrained buyers take into account the unavoidable competition from unconstrained buyers to achieve the optimal balance between price, market tightness, and the bidding limits of potential auction participants.

The incentive to search alongside constrained buyers in a submarket distorted by a binding financial constraint is increasing in the share of buyers constrained by the policy. If the fraction of constrained buyers is not too high (i.e., |$\Lambda<\lambda_1$|⁠), the DSE features partial pooling. That is, only some unconstrained buyers search for homes priced at |$p_1$| while the rest search in submarket |$p_0$|⁠.¹⁸ As |$\Lambda\rightarrow\lambda_1$|⁠, one can show that |$\sigma(p_0)\rightarrow 0$| and the DSE converges to one of full pooling, with all buyers and sellers participating in submarket |$p_1$|⁠. Finally, if |$\Lambda>\lambda_1$|⁠, market clearing (part 2 of Definition 1) is incompatible with unconstrained buyer indifference between these two submarkets, which begets the possibility of full pooling with unconstrained buyers strictly preferring to pool with constrained buyers. We restrict attention to settings with |$\Lambda\leq \lambda_1$| for the analytical characterization of equilibrium and rely on numerical results for settings with |$\Lambda>\lambda_1$|⁠.¹⁹

2.3 Empirical predictions

This section summarizes the housing market implications of the million dollar policy by comparing the pre- and post-policy directed search equilibria. Since financial constraint |$c$| is intended to represent the maximum ability to pay among buyers affected by the million dollar policy, parameter |$c$| corresponds to the

1M threshold and |$\Lambda$| reflects the share of potential buyers with insufficient wealth from which to draw a |$20$||$\%$| down payment.²⁰

Four cases are possible depending on whether financial constraints |$u$| and |$c$| lead to corner solutions to problems |$\text{P}_0$| and |$\text{P}_1$|⁠. In this section, we focus on the most empirically relevant case in which the financial constraint is slack in problem |$\text{P}_0$| but binds in problem |$\text{P}_1$|⁠. In other words, we consider the possibility that preexisting financial constraints are mild , but that the additional financial constraint imposed by the policy is sufficiently severe . Under this assumption, the equilibrium asking prices are |$p_0=p^*_u$| and |$p_1=c$|⁠. Two subcases are still possible, namely, (a) |$p^*_u\leq c$| and (b) |$p^*_u>c$|⁠, which we use to derive several testable predictions that we bring to the data in Section 4.

 

Per Propositions 1 and 2, the set of asking prices changes from just |$\mathbb{P}=\{p_0\}$| pre-policy to |$\mathbb{P}=\{p_0,p_1\}$| post-policy. Following the introduction of the policy, some or all sellers find it optimal to target buyers of both types by asking price |$p_1=c$| . The million dollar policy can thus induce a strategic response among sellers in market segments near the newly imposed financial constraint. If |$p_0< p_1$|⁠, some sellers who would have otherwise listed below |$c$| respond to the policy by increasing their asking price to the threshold. The intuition for bunching from below is the following: as buyers become more constrained, the distribution of possible sales prices features fewer extreme prices at the high end. Sellers respond by raising their asking price to effectively truncate the distribution of prices from below. The higher price in a bilateral situation can offset (in expectation) the unseized sales revenue in multiple offer situations arising from the additional financial constraint. Constrained buyers tolerate the higher asking price because they face less severe competition from unconstrained bidders in submarket |$p_1$|⁠. If instead |$p_0>p_1$|⁠, the policy induces some sellers who would have otherwise listed above |$c$| to drop their asking price to exactly equal the threshold. In the case of bunching from above, the reduction in asking prices is designed to attract constrained buyers. Because there is pooling of both buyer types in submarket |$p_1$|⁠, these sellers may still match with unconstrained buyers and sell for a price above |$c$|⁠.

 

The frictional matching process between buyers and sellers results in some homes failing to sell. With probability |$e^{-\theta_1}$|⁠, a seller listing a home post-policy at price |$p_1=c$| does not meet even a single buyer. The auction mechanism further reduces the mass of sales relative to listings at price |$c$|⁠. With probability |$1-e^{-(1-\lambda_1)\theta_1}-(1-\lambda_1)\theta_1e^{-(1-\lambda_1)\theta_1}$|⁠, competition among unconstrained bidders in submarket |$p_1$| escalates the sales price up to |$u$|⁠.

This bidding war effect intensifies (diminishes) in response to the million dollar policy if |$p_0>p_1$| (⁠|$p_0\leq p_1$|⁠). This is related to the ratio of unconstrained buyers to sellers and relies on the indifference condition for unconstrained buyers between submarkets |$p_0$| and |$p_1$|⁠. If |$p_1<p_0$|⁠, the ratio is higher in submarket |$p_1$| (i.e., |$\theta_0<(1-\lambda_1)\theta_1$|⁠), which shifts the Poisson distribution that governs the random number of unconstrained buyers meeting each particular seller in the sense of first-order stochastic dominance. The policy therefore increases the probability of multiple offers from unconstrained buyers and the overall share of listed homes selling for |$u$|⁠. The intuition for this is that unconstrained buyers enter the pooling submarket until the lower sales price when not competing against other unconstrained bidders (that is, |$p_1$| instead of |$p_0$|⁠) is exactly offset by the higher incidence of price escalation, resulting in indifference between the two submarkets. If instead |$p_0<p_1$|⁠, the indifference condition for unconstrained buyers implies the opposite, namely, |$\theta_0\geq (1-\lambda_1)\theta_1$|⁠. In that case, the policy raises asking prices but lowers the probability of multiple offers from unconstrained buyers.

In both cases, the effect of the policy on sales prices via sellers’ revised listing strategies (Prediction 1) is partly neutralized by the endogenous change in bidding intensity. We should therefore expect a more dramatic impact of the million dollar policy on asking prices than sales prices.

 

At asking price |$p_1=c$|⁠, the presence of constrained buyers does not alter the payoff to an unconstrained buyer. This is because, in a second price auction with reserve price exactly equal to constrained buyers’ ability to pay, offers from constrained buyers affect neither the probability of winning the auction nor the final sales price when an unconstrained buyer bids |$u$|⁠. For submarkets priced above |$c$|⁠, these constrained buyers cannot afford to participate. Given that |$\bar{V}^u$| is unchanged by the policy, it follows that the ratio of unconstrained buyers to sellers is also unaffected by the policy in any submarket asking |$c$| or more. The policy, however, induces the participation of constrained buyers in submarket |$c$| and a range of inactive submarkets below |$c$|⁠. Submarkets that attract both constrained and unconstrained buyers post-policy feature higher market tightness because the presence of constrained buyers does not deter unconstrained buyers. On the contrary, unconstrained buyers are drawn to these submarkets because they have an advantage when competing bidders face tighter financial constraints. The resulting discontinuous drop in market tightness at asking price |$c$| can be understood as discontinuous reductions in both the probability of selling and the probability of receiving multiple offers and hence selling above asking. The inverse of the probability of selling in the static model proxies for expected time-on-the-market in a dynamic setting. Prediction 3 therefore summarizes the implications for time-on-the-market. Specifically, the million dollar policy causes homes listed just below

1M to sell faster, as well as induces a discontinuous increase in average selling time at the threshold.

Appendix B.5 illustrates Predictions 1, 2, and 3 by simulating a parameterized version of the model that has been extended to incorporate a form of seller heterogeneity. Specifically, sellers with different reservation values implement different asking price strategies, which permits the characterization of equilibria featuring bunching from both above and below simultaneously. Figures B.1 and B.2 plot the asking and sales price distributions. These simulated distribution functions reveal an excess mass of listings at

1M from both above and below the threshold (Prediction 1), and a much less pronounced excess mass of sales at

$\$$

1M (Prediction 2). Figures B.3 and B.4 present visualizations of Prediction 3 by plotting expected time-to-sell and the probability of selling-above-asking as functions of the asking price.

Our analysis so far has exclusively focused on housing market outcomes. On the normative side, the model implies that the policy reduces the social surplus derived from housing market activity, as it affects the entry decision of sellers and hence market tightness, distorting the total number of housing market transactions.²¹ It is worth noting that the million dollar policy was introduced not only to cool housing markets but also to improve financial stability and mortgage market efficiency. The latter is a central theme in the recent macrofinance literature on macroprudential policies.²² Although the model is not designed to assess the policy’s impact on borrowers’ creditworthiness, it nevertheless offers an important insight in this regard. In particular, less constrained buyers have an advantage over constrained ones in multiple offer situations, and as such we would expect post-policy homebuyers to be wealthier and hence more “creditworthy.”²³

 

By reallocating million dollar homes from financially constrained buyers to less financially constrained buyers, the policy effectively improves borrower creditworthiness and prevents lenders form making more risky loans. A normative argument in favor or against the million dollar policy would weigh these credit market benefits against the distortions introduced in the housing market.

2.4 Caveats about modeling assumptions

Further discussion of some features of the model is in order. First, the asking price is assumed to represent a firm commitment to a minimum price, which results in a sales price either above or at the asking price. In practice, sales prices can be above, at, and below asking prices. Embellishing the price determination mechanism may allow for transaction prices below asking prices without compromising the asking price-related implications of the theory.²⁴ The theory of asking prices advanced in Khezr and Menezes (2018), for example, considers the situation wherein sellers learn their reservation value after setting an asking price and observing buyers’ interest. As in our setting, transactions at the asking price arise in bilateral meetings; but, unlike our model, multilateral meetings can, in some circumstances, result in transactions below the asking price. Alternative price determination mechanisms would add considerably to the analytical complexity of the model. Such extensions, however, would not affect our theoretical results substantively as long as (a) the asking price remains meaningful (in expectation) for price determination in a bilateral match, and (b) competition among bidders in a multilateral match tends to drive up the sales price. The former is to ensure the directing role of the asking price, which is key for establishing Prediction 1. The latter is to allow for price escalation in multilateral matches so that sellers can list at

1M in response to the policy but may still end up selling for more. This is important for Prediction 2.

Second, entry on the supply side of the market is a common approach to endogenizing housing market tightness in directed search models with auctions (e.g., Albrecht, Gautier, and Vroman, 2016; Arefeva, 2016). This assumption equates the seller’s expected surplus with the listing cost. Keeping instead the measure of sellers constant pre- and post-policy would further reduce the seller’s expected payoff and hence sales prices. A third alternative is to allow entry on the demand side, as in Stacey (2016). Buyer entry would be less straightforward in our context given that the demand side of the market is homogeneous pre-policy but heterogeneous thereafter. With post-policy entry decisions on the demand side, buyers would self-select into the market in such a way that the effects of the policy would be mitigated or even nonexistent. Suppose for a moment that both types of buyers face entry decisions subject to an entry fee or search cost. Provided there are sufficiently many unconstrained potential market participants, unconstrained buyers would enter the market until they reach indifference about market participation: their expected payoff would equal the participation cost. Because constrained buyers are outbid by unconstrained buyers, the expected payoff for a constrained buyer would be strictly less than the cost of market participation. It follows that constrained buyers would optimally choose not to participate in this segment of the housing market and consequently the post-policy equilibrium would be indistinguishable from the pre-policy equilibrium with identically unconstrained buyers. In contrast, we have shown in the preceding analysis that the policy does affect equilibrium strategies and outcomes when entry decisions are imposed on the supply side of the market.

Finally, the scope of the model shrinks to a narrow segment of the market around

1M if the parameter values for |$v$|⁠, |$u$| and |$c$| are close to the seller’s reservation value, which is normalized to zero. The 20|$\%$| down payment constraint also reduces the maximum affordable price for buyers in segments well above

$\$$

1M. The model’s implications for these segments are the same as the

$\$$

1M segment, albeit with a reinterpretation of parameter |$c$|⁠. More specifically, the perceived reduced ability to pay among prospective buyers generates incentives for the sellers to adjust their asking prices, and heightened competition among less constrained buyers can further drive sales prices above asking. In these segments well above

$\$$

1M, however, the down payment requirement is continuous in the sales price, which makes it empirically challenging to identify the policy effects on buyers’ and sellers’ strategies and price formation. In contrast, the policy creates a discrete change in the down payment requirement at the

$\$$

1M threshold. The degree of excess bunching at

$\$$

1M in the data, which we turn to next, provides evidence on the extent to which buyers and sellers respond to this targeted financial constraint.

3. Data and Methodology

3.1 Data

Our data set includes transactions of residential homes in the Greater Toronto Area from January 1st, 2010, to December 31st, 2013. For each transaction, we observe asking price, sales price, days on the market, transaction date, location, as well as detailed housing characteristics. In particular, we define a number of variables to control for house quality. We create indicator variables for whether the house is detached, semidetached, condominium or townhouse. Houses in our data are coded in 16 different styles. We condense this information into three housing styles (two-story (⁠|$\approx65$||$\%$|⁠), bungalow (⁠|$\approx25$||$\%$|⁠), other (⁠|$\approx10$||$\%$|⁠)), where the style “other” includes 1-1/2 story, split-level, backsplit, and multilevel. We observe the depth and width of the lot in meters, which we convert to the total size of the lot by taking their product. We create a categorical variable for the number of rooms in a house that has seven categories, from a minimum of 5 to |$\ge\,$|11, and another for the number of bedrooms that has five categories from 1 to |$\ge\,$|5. We create an indicator for the geographic district of the house listing. For our main sample of the city of Toronto, this district variable identifies 43 districts corresponding to the MLS district code.

We observe the final asking price posted in each listing, but not the changes in the asking price. From a local brokerage office’s confidential database, we learned that about |$12$||$\%$| of overall listings experienced revisions to the asking price. This number reduces to |$2$||$\%$| when it comes to the estimation sample of properties around

1M. For our analysis, we split our data into two mutually exclusive time periods. We define a post-policy period from July 15, 2012, to June 15, 2013. Our pre-policy period is similarly defined as July 15, 2011, to June 15, 2012. That is, we choose one year around the policy implementation date, but we omit a month covering the pre-implementation announcement of the policy. For the purposes of assigning a home to the pre- or post-policy period, we use the date the house was listed.²⁵ We do so because a seller’s listing decision depends on the perceived ability to pay among potential buyers, which in turn depends on whether the policy is implemented. We assess the sensitivity of our results to different time windows of the pre- and post-policy periods in a later robustness section and in Internet Appendix E.4.

For the main analysis, we focus on single-family homes in the city of Toronto.²⁶Table 1 contains summary statistics. Panel A, containing information on all districts, includes |$22{,}244$| observations in the pre-policy period and |$19{,}061$| observations in the post-policy period. The mean sales price in Toronto was

723,396.82 in the pre-policy period and

$\$$

760,598.15 in the post-policy period, reflecting continued rapid price growth for single family houses (all figures in CAD). Our focus is on homes near the

$\$$

1M threshold, which corresponds to approximately the |$86$|th percentile of the pre-policy price distribution. There were |$1{,}448$| homes sold within

$\$$

100,000 of

$\$$

1M in the pre-policy period and |$1{,}423$| in the post-policy period. Panel B of Table 1 shows summary statistics for the central district only. The central district of Toronto is more expensive than suburban markets in general; in the post-policy period, a

$\$$

1M home is at the |$56$|th percentile of the sales price distribution in the central district. The central district contains nearly 40|$\%$| of the homes sold within the

$\$$

0.9M–1.1M price range. In the empirical analysis below, we will examine the policy impact for the city of Toronto and for central Toronto separately.

3.2 Empirical methodology

To measure price responses, we use a bunching approach recently developed in the public finance literature (e.g., Saez 2010; Chetty et al. 2011; Kleven and Waseem 2013). Our theoretical model established that the down payment discontinuity can create incentives for bunching at the

1M threshold in terms of listings (Prediction 1), but less so in terms of sales (Prediction 2). To test these predictions, we use the price segments which are not subject to the policy’s threshold effects to form a valid counterfactual near the

$\$$

1M threshold. The two underlying assumptions are that (1) the policy-induced incentives for bunching occur locally in segments near the

$\$$

1M threshold, leaving other parts of the price distributions unaffected by threshold consequences; and (2) the counterfactual is smooth and can be estimated using these other parts of the price distributions. In forming the counterfactual, we use a two-step approach: first constructing counterfactual price distributions that would have prevailed if there were no changes in the composition of the housing stock using a common reweighting method; then applying the bunching approach to the difference between each composition-constant post-policy price distribution and the observed pre-policy distribution.

3.2.1 First step: Controlling for housing composition

If houses listed or sold in the million dollar segment in the post-policy year differ in terms of quality from those in the previous year, then the difference between price distributions in the two periods could simply reflect the changes in the composition of housing rather than the effect of the policy. We alleviate this concern by leveraging the richness of our data to flexibly control for a set of observed house characteristics to back out a counterfactual distribution of house prices that would have prevailed if the characteristics of houses in the post-policy period were the same as in the pre-policy period.

3.2.2 Second step: Bunching estimation

With the estimated |${\hat{\Delta}_{S}(y_j)} = \hat F_{Y\langle 1|0\rangle}(y_j) - \hat F_{Y\langle 0|0\rangle}(y_j)$| in hand, we are now ready to estimate the policy effects on asking and sales price using a bunching estimation procedure. This procedure requires separation of the observed |${\hat{\Delta}_{S}(y_j)}$| into two parts: the price segments near

1M that are subject to the policy’s threshold effects, and the segments that are not. The affected segments are known as the “excluded region” in the bunching literature. Since knowledge of this region is not known a priori, it also must be estimated, and we will develop a procedure below to do so. Once this region around

$\$$

1M is determined, we use standard methods to estimate the counterfactual difference in distributions by fitting a flexible polynomial to the estimated |${\hat{\Delta}_{S}(y_j)}$| outside the excluded region. We use the estimated polynomial to predict or “fill in” the excluded region which forms our counterfactual. Our estimates of the policy effects are derived from the difference between the observed |${\hat{\Delta}_{S}(y_j)}$| and the estimated counterfactual.

It is important to note that the interpretation of the total jump at the threshold, as shown in the left-hand-side (LHS) of Equation (8), is not all causal. Since changes in listings and sales between the two periods can be more pronounced in some price segments than others, we should not expect the difference in CDFs to be flat even in the absence of the million dollar policy. In our case, an upward-sloping curve is captured by our polynomial estimates as a counterfactual. Specifically, the first two terms on the right-hand-side of Equation (8) reflect the counterfactual difference at the

1M threshold.

After netting out the counterfactual, we are left with |$\hat \beta_A - \hat \beta_B$|⁠, which is the policy response we aim to measure. A finding of |$\hat \beta_A > 0$| is consistent with bunching from above since it indicates that sellers that would have otherwise located in bins above |$\$1$|M instead locate in the |$\$1$|M bin. A finding of |$\hat \beta_B < 0$|⁠, on the other hand, is consistent with bunching from below since it indicates that sellers that would otherwise locate below the

1M bin now move up to locate in the

$\$$

1M bin. Both are responses to the million dollar policy.

To implement our estimator, we must make several decisions about unknown parameters, as is the case for all bunching approaches. In particular, the number of excluded bins to the left, |$L$|⁠, and right, |$R$|⁠, are unknown, as is the order of the polynomial, |$p$|⁠. In addition, we choose to limit our estimation to a range of price bins around the

1M threshold. We do this because the success of our estimation procedure requires estimation of the counterfactual in the region local to the policy threshold (Kleven, 2016). Using data points that are far away from the excluded region to predict values within the excluded region can be sensitive to polynomial choice and implicitly place very high weights on observations far from the threshold (Gelman and Imbens, 2019,Lee and Lemieux, 2010). Thus, we focus on a more narrow range, or estimation window, |$W$|⁠, of house prices around the policy threshold. Since we are fitting polynomial functions, this can be thought of as a bandwidth choice for local polynomial regression with rectangular weights (Imbens and Lemieux, 2008). Thus, the parameters we require for estimation of the regression coefficients are |$(L,R,W,p)$|⁠.

We use a data-driven approach to select these parameters. The procedure we implement is a five-fold cross-validation procedure, described fully in Internet Appendix C.1. Briefly, we split our individual-level data into five equally sized groups and carry out both steps 1 and 2 of our estimation procedure using four of the groups (i.e., holding out the last group), and then obtain predicted squared residuals from Equation (7) for the hold-out group. We repeat this procedure five times, holding out a different group each time, and average the predicted squared residuals across each repetition. This is the cross-validated mean squared error (MSE) for a particular choice of |$(L,R,W,p)$|⁠. We perform a grid search over several values of each parameter, and choose the specification which minimizes the MSE.²⁹

3.2.3 Caveats about empirical methodology

One legitimate concern is that our bunching estimates pick up threshold effects in pricing that are caused by, for example, marketing convention or psychological bias surrounding

1M, or other macro forces that affected the housing market at the same time as the implementation of the million dollar policy. Our estimation methodology addresses this concern in two ways. First, we examine the post-policy CDF relative to the CDF in the pre-policy period. Time-invariant threshold price effects unrelated to the policy are therefore differenced out in our estimation. Second, we allow for round number fixed effects to capture potential rounding in the price data. Thus, all estimates reported below include a dummy variable for prices in

$\$$

25,000 increments, and another for prices in

$\$$

50,000 increments.

Another potential concern is that the million dollar policy is announced in combination with three other mortgage rule changes, which may complicate the challenge for identification. However, unlike the million dollar policy, these contemporaneous mortgage rule changes apply to the entire housing market.³⁰ Their housing market impacts are accounted for in the counterfactual that would have prevailed in the absence of the million dollar policy. By comparing the actual changes in the quality-adjusted price distributions to the counterfactual changes in the quality-adjusted price distributions, the bunching estimation teases out the effect of the million dollar policy from these confounding factors. To address the possibility that buyer and seller strategies evolve dynamically in ways that are not reflected in the counterfactuals, we implement several placebo experiments that use alternative cutoffs in Section 4.1.2. We also present results based on counterfactuals constructed only from data below

1M.

4. Empirical evidence

The core estimation is presented in Section 4.1 with an aim to test Predictions 1 and 2 by examining asking and sales prices near the

1M threshold. We then explore Prediction 3 in Section 4.2.1 and Prediction 4 in Section 4.2.2. Finally, we extend the analysis to study all segments above

$\$$

1M in Section 4.3.

4.1 Predictions 1 and 2: Asking price and sales price

4.1.1 Main results

The main predictions of the model are that the million dollar policy leads to an excess mass of homes listed at the

1M threshold (Prediction 1), which only partially passes through to the sales price distribution (Prediction 2).

Our main analysis focuses on single-family housing markets. Figures 3 and 4 present graphical results from the first step estimation of asking and sales price distributions based on Equation (6). We first discuss asking prices. Panel A of Figure 3 plots the distribution functions for the asking price between

600,000 and

$\$$

1,400,000 in the pre- and post-policy years. The post-policy CDF lies everywhere below the pre-policy CDF, indicating that all housing market segments experienced a boom. Panel B plots the difference between the two CDFs. If the CDFs were the same pre- and post-policy for a given bin, the difference would show up as a zero in the figure. The displayed difference in CDFs is always below zero, indicating that houses in general are becoming more expensive over time. Following Equation (6), we then decompose the difference in CDFs into two components: (a) price difference due to shifting housing characteristics in each segment (panel C) and (b) price difference due to changes in sellers’ listing strategies (panel D). The latter is the market response that we aim to measure at the

$\$$

1M threshold. As shown in panel C, the price change caused by shifting housing characteristics is small in magnitude and relatively flat. In contrast, panel D shows that the price difference caused by sellers’ updated listing strategies generally changes smoothly with price, with a relatively large jump at

$\$$

1M. Given the minimal composition effect, nearly all of the difference in the observed distribution of asking prices is driven by sellers’ listing behavior.

Turning to the sales prices, we look to the top panels of Figure 4, which plots the distribution of sales price in the pre- and post-policy years and their differences. The bottom panels of Figure 4 show that after accounting for the composition effect, a jump in the sales price at the

1M threshold is hardly apparent, supporting the notion that buyers’ nontrivial search and bidding activities disentangle sales prices from asking prices, potentially mitigating the overall impact of the policy.

The descriptive findings presented in Figures 3 and 4 are consistent with the model. However, this evidence alone does not distinguish the policy effects from the impact of other contemporaneous macro forces. To isolate the million dollar policy’s effects on the price distributions, we now turn to the second step estimation, namely, the bunching estimation. We choose to plot the bunching estimates in both the cumulative distribution functions (CDFs) and the probability density functions (PDFs). While the latter is more standard in the bunching literature, the former allows us to visualize the decomposition of the estimated jump based on Equation (8) in a more transparent way.

Figure 5a presents a graphical test of Prediction 1 based on the estimation of Equation (7). In particular, we plot changes in the CDFs of the asking price, |${\hat{\Delta}_{S}(y_j)} = \hat F_{Y\langle 1|0\rangle}(y_j) - \hat F_{Y\langle 0|0\rangle}(y_j)$|⁠, holding housing characteristics constant. The solid line plots the quality-adjusted observed changes, with each dot representing the difference in the CDFs before and after the policy for each

5,000 price bin indicated on the horizontal axis. The dashed line plots the counterfactual changes in the absence of the policy, while the vertical dashed lines represent the lower and upper limits of the bunching regions (⁠|$\$975{,}000$| and |$\$1{,}025{,}000$|⁠). Note that the width of the estimation widow (⁠

$\$$

100,000 on each side of the threshold), the order of the polynomial (cubic), and the width of the excluded region were chosen based on the cross-validation procedure outlined in Section 3.2.2.

The empirical distribution of asking prices exhibits a sharp discontinuity at the

1M threshold. After the policy, a total of |$0.45$||$\%$| of listings were added to the

$\$$

1M bin. Following Equation (8), Figure 5a decomposes this total jump into three distinct components: |$0.06$||$\%$| of listings reflect the counterfactual change in the absence of the million dollar policy (marked by C), |$0.18$||$\%$| of listings bunched from below (marked by B), and |$0.20$||$\%$| of listings bunched from above (marked by A). Thus, |$87\%$| (i.e., (A+B)/(A+B+C)) of the excess bunching in the asking price at the

$\$$

1M threshold is attributed to the policy.

Figure 5b presents a graphical test of Prediction 1 based on the difference in densities. The spike in homes listed at the |$\$1$|M is accompanied by dips in homes listed to the right and left of

1M. The spike reflects the excess mass of homes listed between |$\$995{,}000$| and

$\$$

1M after the implementation of the policy. The dips reflect missing homes that would have been listed at prices further from the

$\$$

1M in the absence of the policy.

Column 1 of Table 2 reports our baseline bunching estimates underlying the above graphical presentation. The specification used is chosen by the cross-validation procedure outlined above. Standard errors are calculated via bootstrap.³¹ Overall, we find that approximately |$86$| homes that would have otherwise been listed away from

1M were shifted to the

$\$$

1M bin. While seemingly small, |$86$| additional homes represent a |$38.3$||$\%$| increase relative to the number of homes that would have been listed in the million dollar bin in the absence of the policy. Among these additional listings, about half are shifted from below |$\$995{,}000$|⁠; the remaining half come from above

$\$$

1M. Both estimates are significant at the 5|$\%$| level. When viewed through the lens of our search model, price adjustments from both sides of

$\$$

1M are quite sensible. On the one hand, the policy induces some sellers who would have otherwise listed homes below

$\$$

1M to increase their asking prices toward the

$\$$

1M mark. By doing so, these sellers demand a higher price in a bilateral situation to offset any price dampening effect of the policy in multilateral situations. On the other hand, the policy induces other sellers who would have otherwise listed homes above

$\$$

1M to lower their asking price to just below the cutoff, attracting both constrained and unconstrained buyers to compete for their homes.

As noted earlier, we do not observe sellers’ revisions to asking prices in our main data. With a one-time access to a local brokerage office’s confidential database, we find that about |$44$| houses in the estimation sample (2|$\%$| of all houses that sold within

100,000 of

$\$$

1M) were originally listed before the policy, pulled off the market, relisted after the policy and then sold. Restricting attention to these |$44$| houses, Figure 6 shows that |$18$| (41|$\%$|⁠) of them adjusted their asking price to [

$\$$

975,000,

$\$$

1,000,000]. These adjustments come from both sides, complementing our bunching estimation results. Among the |$22$| relisted houses that were originally asking between

$\$$

1M and

$\$$

1.1M before the policy, |$14$| (64|$\%$|⁠) of them reduced their price to just under

$\$$

1M after the policy. These asking price revisions are consistent the intuition that some sellers lower their asking price to invite competition from both constrained and unconstrained buyers.

Turning to Prediction 2, we report the bunching estimates for the sales price in column 2 of Table 2, with a visualization of the estimates shown in Figure 7. Despite sharp excess bunching of asking prices, we do not find evidence of excess bunching of sales prices at the

1M price bin; the estimated total response attributable to the million dollar policy is small and statistically insignificant. This evidence is consistent with Prediction 2, which characterized how the price dampening effect of the policy can be undermined by the strategic search and bidding behavior of market participants in a setting with search frictions and auctions. To test this interpretation, we estimate the policy effect on bidding intensity in Section 4.2.1.

Million dollar homes are concentrated in central Toronto. In columns 3 and 4 of Table 2, we restrict the sample to the central district and repeat the same estimation for asking and sales prices as in columns 1 and 2. Despite the much reduced sample size, the resulting estimates are qualitatively consistent with what we find above for the city of Toronto.

Condominiums and townhouses make up an important sector of the Toronto housing market with |$21{,}768$| transactions in the pre-policy period.³² For this sector, Table 3 shows that the million dollar policy adds |$19$| listings at

1M and |$12$| sales at

$\$$

1M, aligning again with Predictions 1 and 2. Quantitatively, the degree of excess bunching is smaller, because there are much fewer million dollar condominiums than houses. While a

$\$$

1M home corresponds to the |$86$|th percentile in the single-family housing market, it corresponds to the |$99$|th percentile in the condominium and townhouse market. Given this, we focus the analysis hereafter on single-family homes.

4.1.2 Robustness checks

In Internet Appendix E.1, we estimate an extensive set of specifications to assess the robustness of our main results. We briefly review these results here and provide a more extensive discussion in the Internet Appendix. Our first robustness exercise deals with the concern that the bunching estimates could be altered by plausible policy responses above the

1M threshold. Suppose the introduction of the policy hindered potential listings or transactions above

$\$$

1M. In that case, our counterfactuals estimated by excluding an area around the

$\$$

1M threshold would not accurately reflect what would have occurred in the absence of the policy. Note that this is a common issue in the bunching literature (Kopczuk and Munroe, 2015,Best et al., 2020,Best and Kleven, 2018). As suggested by Kleven (2016), we construct the counterfactual using only data below

$\$$

1M under the assumption that the distribution below the threshold is unaffected by the policy, and the results are similar to those presented above.

Our analysis above hinges on the assumption that homes further below

1M are unaffected by the policy. A second concern, then, is unintended policy consequences in market segments below the threshold. This could occur, for example, if a seller of a below-

$\$$

1M home intends to trade-up to an above-

$\$$

1M home. The seller may be constrained in that the proceeds from the sale of their current home must not compromise their ability to make a 20|$\%$| down payment on their next home. To that end, they may alter their listing and selling strategies, which could affect prices within the estimation window but below the excluded region. Given the rate of house price appreciation, however, this trading-up constraint is unlikely to bind except possibly for sellers that bought their home very recently.³³ In Table E4 of Internet Appendix E.2, we report specifications that exclude sellers that bought their current home within the previous 3, 4, or 5 years. The resulting bunching estimates are very similar to those reported in Table 2, alleviating concerns about constrained buyer-sellers in the Toronto market.

To the extent that the policy might have other spillover effects, we rely on our data-driven method for model selection to appropriately determine, among other things, the estimation window and the size of the exclusion region. In Internet Appendix E.1, we perform an extensive set of robustness checks to ensure that our estimates are not overly sensitive to the parameters chosen by our data-driven procedure. In particular, in Tables E1 and E2, we present results based on alternative criteria for parameter selection, with the estimation window widened by

25,000, with a fourth-order polynomial, and with the constraints in Equations (9) and (10) imposed. Reassuringly, the bunching estimates are extremely robust, suggesting that our results are not driven by the selection of the size of the estimation window, order of the polynomial, or the width of the excluded region. Internet Appendix E.1 also contains a graphical depiction of a larger set of robustness checks, providing further support in these regards.

Next, we perform two different types of placebo tests as additional checks of our identification strategy. First, we designate two years prior to the implementation of the million dollar policy as placebo years. No specific changes were made to policies affecting houses around the

1M threshold during this time, so we would not expect to find patterns of excess bunching.³⁴ Second, we designate alternative placebo thresholds at prices well below or above the

$\$$

1M threshold, and again estimate our baseline specification at each of these points. The idea is straightforward: since the million dollar policy generates a notch in the down payment required of buyers at precisely

$\$$

1M, house prices in market segments well below or well above the

$\$$

1M cutoff should not be affected by the policy in a discontinuous manner. Table E3 contains |$50$| placebo estimates: |$24$| during the years overlapping the implementation of the policy for alternative price thresholds (the estimates excluding the

$\$$

1M threshold), and |$26$| during the pre-policy years. Of these, only |$4$| are statistically significant and only |$1$| is economically large. Taken together, these results support the notion that the bunching results presented in Section 4.1 provide an accurate measure of the threshold effects of the million dollar policy on house prices.

Finally, we assess the robustness of our main results to alternative choices of the pre- and post-policy periods. Our baseline specification groups pre- and post-policy periods by listing date and omits the few weeks following the announcement of the policy but before its implementation. In Internet Appendix E.4, we show that our main results are not sensitive to these choices. We also show that our results are qualitatively robust to narrowing the pre- and post-periods from 1 year to 6 or 3 months.

4.2 Predictions 3 and 4: Bidding wars

4.2.1 Sales-above-asking and time-on-the-market

Turning to the policy effects on market liquidity, Prediction 3 states that the million dollar policy reduces expected time-on-the-market and increases the incidence of sales-above-asking for homes listed just under

1M. The policy thus triggers a discontinuous increase in the expected time-on-the-market and a discontinuous decrease in the probability of selling-above-asking right at asking price

$\$$

1M. We bring this prediction to the data by employing a regression discontinuity design. The variables of interest are (1) the probability that a house sold above the asking price conditional on being listed at |$p \geq y^A_j$|⁠; and (2) the probability that a house stayed on the market for more than 2 weeks conditional on being listed at |$p \geq y^A_j$|⁠, where 2 weeks is roughly the median time-on-the-market in the sample. We construct these two variables in three steps.

Using this three-step procedure, we impute the two variables of interest: (1) the change in the probability of being sold above asking, |$\hat S_{Y\langle 1|0\rangle}(y^S \geq y^A | y^A_j) - \hat S_{Y\langle 0|0\rangle}(y^S \geq y^A | y^A_j)$|⁠; and (2) the change in the probability of being on-the-market for more than 2 weeks, |$\hat S_{Y\langle 1|0\rangle}(D \geq 14 | y^A_j) - \hat S_{Y\langle 0|0\rangle}(D \geq 14 | y^A_j)$|⁠. Both are constructed relative to the pre-policy period, conditional on being listed for at least |$y^A_j$| and holding the distribution of housing characteristics constant.

We plot each of the two constructed variables above as a function of the asking price, along with third-order polynomials, which are fit separately to each side of

1M. In Figure 8a, changes in the probability of being sold above asking exhibit a discrete downward jump at

$\$$

1M, with an upward sloping curve to the left of

$\$$

1M. In 8b, changes in the probability of staying on the market for more than 2 weeks exhibit a discrete upward jump at

$\$$

1M, with a downward sloping curve to the left of

$\$$

1M. The evident discontinuities at

$\$$

1M for time-on-the-market and sales-above-asking correspond exactly with Prediction 3, suggesting that the policy’s minimal effect on sales prices can be at least partially attributed to heightened competition for homes listed just under

$\$$

1M. Moreover, Figures 8c and 8d display changes in these two variables during two years prior to the policy. It is clear that changes in sales-above-asking can be represented by a smooth function of the asking price before the policy. Changes in time-on-the-market still exhibit a discrete jump at

$\$$

1M even during the pre-policy periods, but the jump is statistically insignificant and much smaller in magnitude than during the policy periods. Together, these patterns are congruent with the finding that the policy caused a discrete jump in bidding intensity at the

$\$$

1M and a heated market right below the

$\$$

1M threshold.

4.2.2 Reallocation of million dollar homes

Prediction 4 implies that the policy encourages an allocation of million dollar homes that favors less constrained over more constrained homebuyers. With one-time access to restricted proprietary mortgage data, we impute the fraction of constrained buyers (defined as having an LTV ratio above |$80\%$|⁠) around the million dollar segment in our sample market during one year before and one year after the policy. For the segment slightly above

1M, the fraction of constrained buyers is reduced to zero, which is as intended. For the segment slightly below

$\$$

1M, the fraction of buyers making down payments of more than |$20\%$| remains high, even after the implementation of the policy. This is consistent with the model’s implication that less constrained buyers have an incentive to participate in the segment below

$\$$

1M because they can outbid constrained buyers in multiple offer situations. Thus, achieving the desired mortgage market outcome above

$\$$

1M did not necessarily compromise the segment just below

$\$$

1M. Lacking suitable micro-level mortgage data, we leave a formal test of these credit market implications for future research.

4.3 Policy responses above

$\$$

1M

The million dollar policy may affect not only homes around the

1M threshold but also homes above

$\$$

1M. Lacking a discrete change in the down payment requirement in the segments far above

$\$$

1M, the bunching approach cannot be used to estimate policy consequences there. In this section, we design an alternative approach to examine the possible price responses above the

$\$$

1M threshold. An extensive margin response would occur if some transactions above the

$\$$

1M threshold did not transpire due to the additional financial constraint. Fewer transactions mean less probability mass in segments above

$\$$

1M, and hence relatively more probability mass in segments below

$\$$

1M. Consequently, the CDF for post-policy prices would diverge above the counterfactual CDF, with the largest discrepancies at and around the

$\$$

1M price threshold. An intensive margin response, on the other hand, would occur if some prices above

$\$$

1M are lower than they would have been otherwise, in which case probability mass shifts to lower prices closer to

$\$$

1M. The post-policy CDF would again diverge above the counterfactual CDF, and the discrepancies would appear above the

$\$$

1M threshold.

Our proposed method for disentangling market trends from other potential policy effects therefore relies on two identification assumptions: (1) that market trends in the absence of the policy can be suitably represented by an intercept and slope shift in the pre-policy quantile function (equivalently, a shifting and rescaling of the pre-policy distribution function); and (2) that these parameters can be estimated using only price segments below |$\tau$|⁠. To assess these assumptions, we apply the same procedure using only pre-policy sample periods. We also apply the same procedure to simulated data that feature no policy response, an extensive margin response, and an intensive margin response to further justify assumptions (1) and (2), and to show that our method can readily detect policy effects above

1M. To further address assumption (2), we assess robustness to a lower |$\tau$| cutoff, namely, |$\$800$|K.

We first summarize the results of the simulation exercises presented in Internet Appendix F.3. In the absence of a policy response, a shape-preserving change in the distribution is well-summarized by a linear transformation applied to the pre-policy quantile function. Moreover, the intercept and slope coefficients are estimated reasonably well using only prices below a cutoff of

900K.³⁸ Reassuringly given the absence of a simulated policy response, the price counterfactual is everywhere close to zero. We further simulate an extensive margin response to the policy by dropping 20|$\%$| of prices above

$\$$

1M in the post-policy sample, and an intensive margin response by lowering prices in excess of

$\$$

1M by 30|$\%$| of this excess amount. Using the proposed distribution decomposition method, both extensive and intensive margin responses are immediately apparent.

We present our empirical results in Figure 9, which plots the observed CDFs and their differences, along with the decomposition. To aid with data visualization, a smoothing algorithm was applied to each curve following Chernozhukov, Fernández-Val, and Melly (2013), and dots corresponding to ventiles of the post-policy price distribution illustrate how many transactions are represented by different segments of each curve. Panel A uses the main sample for the city of Toronto, whereas panel B focuses on the central district of Toronto. Two patterns emerge in both panels. First, the post-policy price distributions lie everywhere below the pre-policy distributions, reflecting an upward price trend in the Toronto market over time. Second, differences attributed to market conditions, as captured by the intercept and slope coefficients estimated using price data below |$\$900$|K and applied to the observed pre-policy distribution, account for nearly all of the observed differences between the pre- and post-policy CDFs in Figure 9. The price counterfactual differences left unexplained by market conditions and house characteristics are thus nearly indistinguishable from zero.³⁹ In particular, there appears to be no visual evidence of positive discrepancies in price segments around or above

1M. Had the policy either inhibited sales or dampened prices in segments above |$\$1$|M, we would expect the price counterfactual to diverge above zero, as discussed above. Given the standard errors, we are unable to reject the hypothesis that the price counterfactual differences in these price segments are zero. This is true for both the city of Toronto (panel A), and the central district (panel B). This analysis suggests that the policy effects in the above |$\$1$|M segments are minimal.⁴⁰ This may not be surprising given that

$\$$

1M was at the |$86$|th percentile of the house price distribution in 2012. Homes priced above

$\$$

1M thus represent very high-end segments. If buyers in these segments tend to be wealthy, many of them may not be financially constrained by a |$20$||$\%$| down payment requirement, in which case high prices could prevail from bidding competition among these less constrained buyers.

5. Conclusion

In this paper, we assess the impact of a financial constraint on price formation in the targeted segment of a frictional housing market. Our empirical methodology exploits a natural experiment arising from a mortgage insurance policy change that effectively imposes a |$20$||$\%$| minimum down payment requirement on homebuyers paying

1M or more. The interpretation of our results is motivated by a search-theoretic model of sellers competing for financially constrained buyers in the

$\$$

1M segment of the housing market. We model the million dollar policy as a targeted financial constraint affecting a subset of prospective buyers. We show that sellers respond strategically by adjusting their asking prices to

$\$$

1M, which attracts both constrained and unconstrained buyers. Because of the interactions of search, bidding and listing strategies of buyers and sellers, asking price effects translate into milder sales price effects.

We exploit the policy’s

1M threshold to isolate the effects of the policy on prices and other housing market outcomes. Specifically, we implement an estimation procedure that combines a decomposition method with bunching estimation. Using housing market transaction-level data from the city of Toronto, we find that the million dollar policy results in excess bunching at asking prices, but not sales prices, of

$\$$

1M. These results, together with evidence that homes listed just below the

$\$$

1M threshold sell faster with a higher incidence selling-above-asking, match the intuition derived from theory. For segments well above

$\$$

1M, we apply a distribution decomposition approach to uncover potential policy effects. We do not find evidence that the million dollar policy affected home sales above the

$\$$

1M threshold.

Overall, we find that the million dollar policy did not achieve the specific goal of cooling the housing boom, but instead heated a narrow segment of the market right below

1M. These findings are difficult to reconcile in a frictionless market, but are fully consistent with an equilibrium model of financial constraints with search frictions and auction mechanisms. Our analysis thus points to the importance of designing macroprudential policies that consider the underlying market microstructure and recognize the strategic responses of market participants.

Authors have furnished an Internet Appendix, which is available on the Oxford University Press Web site next to the link to the final published paper online.

Acknowledgement

We thank the editor, two anonymous referees, Sumit Agarwal, Jason Allen, Thomas Davidoff, John Pasalis, Andrey Pavlov, Tarun Ramadorai, Kathrin Schlafmann, Tsur Somerville, William Strange, and participants at seminars at Arizona State University, Federal Reserve Board, George Washington University, University of Wisconsin-Madison, Stockholm University, McGill University, National University of Singapore, Singapore Management University, University of Alberta, University of Colorado, University of Calgary, University of Toronto, Queen’s University, Lakehead University, Copenhagen Business School, the AREUEA National Conference, Atlanta Fed Conference, Bank of Canada, Carleton Macro-Finance Workshop, Vienna Macro Workshop, pre-WFA conference at Whistler, UBC summer symposium, McMaster’s AWSOME, and the Annual CEA Conference. All errors are our own. This paper was previously circulated under the title “Do Financial Constraints Cool a Housing Boom?” We gratefully acknowledge financial support from the Social Sciences and Humanities Research Council of Canada. The Securities and Exchange Commission disclaims responsibility for any private publication or statement of any SEC employee or Commissioner. This article expresses the authors’ views and does not necessarily reflect those of the Commission, the Commissioners, or other members of the staff.

Footnotes

References