Appendix 1: Proof of Investors’ Binding Borrowing Constraint

PROOF:

Use the Kuhn-Tucker condition to check whether the collateral constraint is binding. We have

${\mu }_{t}\left[{m}_{t}{p}_{t}^{h}\left({h}_{t}^{I}+{h}_{t}^{RI}\right)-{b}_{t}^{I}\right]=0$

If (11) is not binding, then µt = 0. We can write the investor’s FOC Equation (18) as:

$\begin{array}{cc}{U}_{t,{c}^{I}}^{I}\left({c}_{t}^{I},{h}_{t}^{I},{n}_{t}^{I}\right)={\beta }_{I}{𝔼}_{t}\left[\left(1+{i}_{t}\right){U}_{t+1,{c}^{I}}^{I}\left({c}_{t+1\right)}^{I}{h}_{t+1\right)}^{I}{n}_{t+1}^{I}\right)\right]& \left(42\right)\end{array}$

${\beta }_{I}\left(1+i\right)=1$

However from (6), we know βR(1 + i) = 1 at steady state. With parameter restrictions that βR > βI, therefore βI (1 + i) < 1, contradiction. Therefore we cannot have µt = 0. Therefore, µt > 0, and thus we have ${b}_{t}^{I}={m}_{t}{p}_{t}^{h}\left({h}_{t}^{I}+{h}_{t}^{RI}\right)$. Q.E.D.

Appendix 2: System of Steady-State Conditions

This appendix lays out the system of equilibrium conditions in steady state.

$\begin{array}{cc}{c}^{R}+{p}^{r}{h}^{R}=\frac{Y}{N}+i{d}^{R}& \left(43\right)\end{array}$
$\begin{array}{cc}{U}_{{h}^{R}}^{R}\left({c}^{R},{h}^{R}\right)={p}_{t}^{r}{U}_{{c}^{R}}^{R}\left({c}^{R},{h}^{R}\right)& \left(44\right)\end{array}$
$\begin{array}{cc}{U}_{{n}^{R}}^{R}\left({c}^{R},{h}^{R}\right)=-W{U}_{{c}^{R}}^{R}\left({c}^{R},{h}^{R}\right)& \left(45\right)\end{array}$
$\begin{array}{cc}1={\beta }_{R}\left(1+i\right)& \left(46\right)\end{array}$
$\begin{array}{cc}{c}^{I}+{p}^{h}\delta \left({h}^{I}+{h}^{RI}\right)+i{b}_{t}^{I}=\frac{Y}{N}+I+{p}^{r}{h}^{RI}& \left(47\right)\end{array}$
$\begin{array}{cc}\left[1-{\beta }_{I}\left(1-\delta \right)\right]{U}_{{c}^{I}}^{I}\left({c}^{I},{h}^{I}\right){p}^{h}={U}_{{h}^{I}}^{I}\left({c}^{I},{h}^{I}\right)+\mu m{p}^{h}& \left(48\right)\end{array}$
$\begin{array}{cc}\left[1-{\beta }_{I}\left(1-\delta \right)\right]{U}_{{c}^{I}}^{I}\left({c}^{I},{h}^{I}\right){p}^{h}={U}_{{c}^{I}}^{I}\left({c}^{I},{h}^{I}\right){p}^{r}+\mu m{p}^{h}& \left(49\right)\end{array}$
$\begin{array}{cc}\left[1-{\beta }_{I}\left(1+i\right)\right]{U}_{{c}^{I}}^{I}\left({c}^{I},{h}^{I}\right)=\mu & \left(50\right)\end{array}$
$\begin{array}{cc}{b}^{I}=m{p}^{h}{h}^{I}& \left(51\right)\end{array}$
$\begin{array}{cc}{b}^{Dev}=W{n}^{Dev}+{p}^{l}l& \left(52\right)\end{array}$
$\begin{array}{cc}{Y}^{h}=A{\left({n}^{Dev}\right)}^{\gamma }{\left(l\right)}^{1-\gamma }& \left(53\right)\end{array}$
$\begin{array}{cc}W=\frac{\gamma A{p}^{h}{\left({n}^{Dev}\right)}^{\gamma -1}{\left(l\right)}^{1-\gamma }}{1+i}& \left(54\right)\end{array}$
$\begin{array}{cc}{p}^{l}=\frac{\left(1-\gamma \right)A{p}^{h}{\left({n}^{Dev}\right)}^{\gamma }{\left(l\right)}^{-\gamma }}{1+i}& \left(55\right)\end{array}$
$\begin{array}{cc}Y=A{\left({N}^{G}\right)}^{\alpha }& \left(56\right)\end{array}$
$W=Aα(NG)α−1(57)$
$δ(hI+hRI)=yh(58)$
$(1−ϕ)hRI=ϕhR(59)$
$(1−ϕ)bDev+(1−ϕ)bI=ϕdR(60)$
$NG+(1−ϕ)NnDev=N(61)$
$(1−ϕ)Nl=L¯(62)$

Appendix 3: Parameter Calibration

The discount factor of the renters (βR,ss) is calibrated by matching the steady-state interest rate to the average deposit rate in China. To proxy for the steady-state level, we use the average one-year benchmark deposit rate in China from 2000 to 2018 (from People’s Bank of China), which equals 2.42 percent.

The weight on housing services by renters ( j) and weight on housing services by investors (κ) are chosen to match another steady-state object, which is the housing GDP as a share of the total GDP. That is, we set the following expression to the average value-added of housing to GDP ratio in China from 2000 to 2018 in the data (from China Statistical Yearbooks in these years):

$\frac{\left(1-{\varphi }_{ss}\right)N{y}_{ss}^{h}{p}_{ss}^{h}+{\varphi }_{ss}N{h}^{R}{p}^{R}}{\left(1-{\varphi }_{ss}\right)N{y}_{ss}^{h}{p}_{ss}^{h}+{\varphi }_{ss}N{h}^{R}{p}^{R}+{Y}_{ss}}=0.05,$

where φss is the share of renters in households population and equals 0.255, as discussed in the main text; $(1−ϕss)Nysshpssh$ is the housing value-added produced by real estate developers (due to the one-to-one mapping between real estate developers and housing investors, this also equals the market value of houses purchased by housing investors); φssNhRpR is the housing value-added consumed by renters; Yss is the steady-state output of non-housing production. $yssh,pssh,hR,pR,Yss$ are all general equilibrium objects that depend on both j and κ.

The share of labor in housing production (γ) is calibrated to match the average share of labor costin the total cost of real estate developers. There is no official data on this share. Therefore, we manually calculate it for each of the five large real estate developers in China from 2010 to 2019, including China Overseas Land and Investment, China Vanke, China Evergrande Group, Country Garden, and Sunac China Holdings. In each year, we take the weighted average of the labor cost shares (across the five companies) as the share for the whole industry. We then calibrate/estimate 7 as the average share across all available years (2010–2019), which equals 5.74 percent.

Finally, the depreciation rate of houses (δ) takes the standard value from Iacoviello (2005). The discount factor of the investors I) and share of labor in goods production (α) take values from Gete (2020), which also studies the housing market and also assumes that the non-housing sector uses labor as the only input.

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We are grateful for the helpful discussions during the presentation to the China team (with Helge Berger, Fei Han, Kenneth Kang, and other team members); for the comments received during the formal review stage (from Katharina Bergant, Diego Alejandro Cerdeiro, Junghwan Mok, Cian Ruane) and at the surveillance meeting; and for the helpful discussions with Zhuo Chen (Tsinghua), Martin Cihak, Ding Ding, Jack Favilukis (UBC), Wei Guo, Dong He, Zhiguo He (Chicago), Zhongxia Jin, Petya Koeva Brooks, Ivo Krznar, Li Lin, Jianhai Lin, Yang Liu, Wojciech Maliszewski, Paul Mathieu, Alex Murray (Finance Canada), Xiaohui Sun, Rengming Xie, TengTeng Xu, Wentao Xiong (Harvard), Daria Zakharova, Fudong Zhang (Tsinghua), Tianxiao Zheng, and especially Chuqiao Bi and Tao Zha (Atlanta Fed). We also thank Tiana Wang for her research assistance.

The rising trend of the housing price in the US is also not reversed by the COVID-19, as discussed in Zhao (2020).

Incomplete markets are markets where the number of Arrow-Debreu securities is less than that of states of nature (Arrow, 1964). This shortage of securities will likely restrict individuals from transferring the desired level of wealth among states. In the case of financial markets, market incompleteness practically means the available investment channels (for savers) or financing channels (for borrowers) are inadequate for savers/borrowers to optimally allocate resources over time.

We could extend our model to include the stock market; however, for the “rich” households who can afford the down payments for multiple houses, housing is still a preferred investment under reasonable parameterizations. Therefore, for simplicity we did not add this market.

The source can be found here. In addition, China’s overseas investment in 2018 fell 9.6 percent, as the Chinese government’s crackdown on capital flight continued (South China Morning Post, September 13, 2019).

The lively-updated supplementary materials are available here.

Liu and Xiong (2018) review the historical development of China’s real estate market.

Chen and Wen (2017) finds that China’s housing boom can be interpreted as a rational bubble emerging naturally from its economic transition.

Chen, Ren, and Zha (2018) also study the rise of shadow banking activities in China, although their focus is not on the housing market per se.

Based on data from 2001 to 2012, Glaeser and others (2017) document that China’s housing vacancy rate rose sharply after 2009. And as of February 2019, one-fifth of China’s urban housing stock had been bought and left vacant (Bloomberg, February 27, 2019).

For example, the housing price per square meter in Ordos City dropped from RMB 20,000 to RMB 3,000 (QQ News, April 2, 2020).

Separately, Brunnermeier, Sockin, and Xiong (2017) study the macroeconomic implications of China’s financial market liberalization, and find that China’s current, more liberal financial system poses challenges for the government to experiment with a temporary stimulus that could be reversed easily soon after its inception.

The same comment applies to other 2020 data in subsequent figures. The GDP estimate can be found here.

All the numbers presented in Figure 3 and Figure 4 have been cross-checked against numbers published elsewhere.

More specifically, as explained later, the Nt households as a whole (including all renters and all investors) own the non-housing consumption goods producers and equally split the (off-equilibrium) profit from these producers. Hence, for an individual renter (and an individual investor), the total income from these producers equals ${W}_{t}+\frac{{Y}_{t}-{W}_{t}{N}_{t}}{{N}_{t}}=\frac{{Y}_{t}}{{N}_{t}}$, where Wt is the wage.

Below is one analog that may help better understand our proposal: The current financial system in China is like an “arranged marriage” where corporates with dramatically different risk profiles are forced to “marry” the dominant and risk-averse banks; by contrast, what we propose is similar to “free marriage” where “men” (a variety of financial institutions, including banks and nonbanks) and “women” (corporates) can freely match with each other based on their risk profiles/preferences. But to ensure the fairness and efficiency of this “free marriage market”, some strict rules have to be put in place. And the current issue of emerging fake P2P platforms in China is just like disclosing false information on a dating site.

Incomplete Financial Markets and the Booming Housing Sector in China
Author: Mr. Tamim Bayoumi and Mr. Yunhui Zhao