Housing Markets with Foreign Buyers

Abstract

There are cities in the world which have experienced substantial numbers of foreign buyers in the local housing markets, thereby pushing up the real estate prices to the levels beyond the affordability of local residents. To suppress foreign influences in the forming of housing bubbles, governments have resorted to short-term measures of stamp duty or raising the duty rate for non-local buyers, increasing down payments and restricting or even forbidding non-local purchases. These new measures may help contain the demand for housing, but short of being the first-best optimal housing policy for an open economy with significant non-local and foreign buyers. We argue that the first-best policy is to tax non-local and foreign buyers and then use the tax revenue generated to subsidize domestic low- and middle-income buyers. The optimal tax rate under this compensated scheme is smaller than the tax rate under the lump-sum transfer of tax revenue to all residents.

Appendix

• 1. Stability

  Following Dei (1985), the adjustment for the housing price is:

  $$ \overset{\cdot }{p}=\rho Z(p), $$

  where the dot denotes a time derivative, ρ is the speed of adjustments and Z is the excess demand for housing, i.e., Z = E ₂(1,p,u) + C ^∗₂ (p,α) − R ₂(1,p,V). From (1) and (2), u is a function of p and so is Z for given α and V. A necessary and sufficient condition for stability is: dZ/dp < 0. Solving (1) and (2), we obtain: dp/dZ = − 1/Δ, where Δ = R ₂₂ − E ₂₂ − ∂C ^∗₂ /∂p − (m/p)C ^∗₂ . By the stability condition, it requires that Δ < 0.

  Letting η[ = (∂R ₂/∂p)(p/R ₂)] denote the price elasticity of the housing supply and also letting ε [= − (∂C ₂/∂p)(p/C ₂)] and ε ^∗[= − (∂C ^∗₂ /∂p)(p/C ^∗₂ )] be respectively the price elasticities of housing demand by domestic and foreign buyers, we can rewrite Δ as Δ = − (R ₂/p)(m + ε − ε ^*)[ρ ^* − (η + e)/(m + ε − ε ^*)], where ρ ^∗( = C ^∗₂ /R ₂) is the ratio of housing bought by foreigners in the economy. Hence, stability requires that ε ^* < m + ε and ρ ^* < (η + ε)/(m + ε − ε ^*).

• 2. Derivation of (7)

  Totally differentiating (5) and (6), we have:

  $$ \begin{array}{l}{E}_u du+\left\{{E}_2-{R}_2-{\tau}^{\ast }{C}_2^{\ast }-{\tau}^{\ast}\left(1+{\tau}^{\ast}\right)p\left[\partial {C}_2^{\ast }/\partial \left(1+{\tau}^{\ast}\right)p\right]\right\} dp=p\left\{{C}_2^{\ast }+{\tau}^{\ast }p\left[\partial {C}_2^{\ast }/\partial \left(1+{\tau}^{\ast}\right)p\right]\right\}d{\tau}^{\ast },\hfill \\ {}\hfill {E}_{2u} du+\left\{{E}_{22}-{R}_{22}+\left(1+{\tau}^{\ast}\right)\left[\partial {C}_2^{\ast }/\partial \left(1+{\tau}^{\ast}\right)p\right]\right\} dp=-p\left[\partial {C}_2^{\ast }/\partial \left(1+{\tau}^{\ast}\right)p\right]d{\tau}^{\ast }.\hfill \end{array} $$

  Solving these two equations by using Cramer’s rule and then combining the terms, we obtain du/dτ ^* given in (7).

• 3. Derivation of (9)

  Totally differentiating (1) and (2), we have:

  $$ \begin{array}{l}\hfill {E}_u du-{C}_2^{\ast } dp={R}_V dV,\hfill \\ {}\hfill {E}_{2u} du+\left({E}_{22}-{R}_{22}+\partial {C}_2^{\ast }/\partial \mathrm{p}\right)\mathrm{dp}={R}_{2V}\mathrm{d}V.\hfill \end{array} $$

  Solving these equations yields the expression of dp/dV in (9).

• 4. Derivation of (13)

  Totally differentiating (10) and (11), we have:

  $$ \begin{array}{l}\left({E}_u- sp{E}_{2u}\right) du+\left\{\left(1+{s}^{\ast}\right){C}_2^{\ast }+s\left(1+s\right)p{E}_{22}+s\left(1+{s}^{\ast}\right)p\left[\partial {C}_2^{\ast }/\partial \left(1+{s}^{\ast}\right)p\right]\right\} dp=\hfill \\ {}\begin{array}{cc}\hfill \hfill & \hfill \hfill \end{array}p\left\{{C}_2^{\ast }+s{p}^{\ast}\left[\partial {C}_2^{\ast }/\partial \left(1+{s}^{\ast}\right)p\right]\right\}d{s}^{\ast },\hfill \\ {}{E}_{2u} du+\left\{{E}_{22}\left(1+s\right)-{R}_{22}+\left(1+{s}^{\ast}\right)\left[\partial {C}_2^{\ast }/\partial \left(1+{s}^{\ast}\right)p\right]\right\} dp=-p\left[\partial {C}_2^{\ast }/\partial \left(1+{s}^{\ast}\right)\mathrm{p}\right]d{s}^{\ast },\hfill \end{array} $$

  By using Cramer’s rule to solve these two equations, we obtain du/ds ^* in (13).

• 5. Derivation of (17) and (18)

  Totally differentiating (14) yields:

  $$ {E}_u du+{H}_v dv-\left({R}_2-{E}_2\right) dp=-\left({R}_V-{G}_V\right) Vd\beta, $$

  and then using (15), we obtain (17) as:

  $$ {E}_u du+{H}_v dv-{C}_2^{\ast } dp=-\left({R}_V-{G}_V\right) Vd\beta . $$

  Totally differentiating (15) and (16), we have:

  $$ \begin{array}{l}{E}_{2u} du+\left\{\left.{E}_{22}-{R}_{22}+\partial {C}_2^{\ast }/\partial p\right]\right\} dp=-{R}_{2V} Vd\beta, \hfill \\ {}\hfill {H}_{2v} dv={G}_{2V} Vd\beta .\hfill \end{array} $$

  Solving the above three equations, we obtain dp/dβ given in (18).

• 6. Derivation of (22) and (23)

  Totally differentiating (20) and then using (19), we obtain (22) as:

  $$ {E}_u du+{H}_v dv=\left({C}_2^{\ast }+{H}_2\right) dp={C}_2^{\ast } dp+{t}^{\ast}\left[p/\left(p-q\right)\right]{C}_2^{\ast } dp. $$

  Totally differentiating (19), (20) and (21), we have:

  $$ \begin{array}{l}\left(p-q\right){H}_{2v} dv+\left\{{H}_2-{t}^{\ast }{C}_2^{\ast }-{t}^{\ast}\left(1+{t}^{\ast}\right)p\left[\partial {C}_2^{\ast }/\partial \left(1+{t}^{\ast}\right)p\right]\right\} dp=p\left\{{C}_2^{\ast }+{t}^{\ast }{p}^{\ast}\left[\partial {C}_2^{\ast }/\partial \left(1+{t}^{\ast}\right)p\right]\right\}d{t}^{\ast },\hfill \\ {}{E}_u du+{H}_v dv-\left({H}_2+{C}_2^{\ast}\right) dp=0,\hfill \\ {}{E}_{2u} du+{H}_{2v} dv+\left\{{E}_{22}-{R}_{22}+\left(1+{t}^{\ast}\right)\left[\partial {C}_2^{*}/\partial \left(1+{t}^{\ast}\right)p\right]\right\} dp=\left.-p\left[\partial {C}_2^{\ast }/\partial \left(1+{t}^{\ast}\right)p\right]\right\}d{t}^{\ast },\hfill \end{array} $$

  Solving these three equations, we obtain dp/dt ^* expressed in (23).

• 7. Derivation of (24)

  According to (22), the change in social welfare is

  $$ {E}_u\left( du/d{t}^{\ast}\right)+{H}_v\left( dv/d{t}^{\ast}\right)=\left({H}_2+{C}_2^{\ast}\right)\left( dp/d{t}^{\ast}\right). $$

  Substituting dp/dt ^* in (23) into this equation, we have:

  $$ \begin{array}{l}{E}_u\left( du/d{t}^{\ast}\right)+{H}_v\left( dv/d{t}^{\ast}\right)\hfill \\ {}=-p{C}_2^{\ast}\left({C}_2^{\ast }+{H}_2\right)\left(n/q-m/p\right)\left[\left(1+B\right){\varepsilon}^{\ast }-1\right]\left[{t}^{\ast }-1/\left[\left(1+B\right){\varepsilon}^{\ast }-1\right.\right]/\left(1+{t}^{\ast}\right){\varDelta}_{\mathrm{s}}\hfill \\ {}=-p{C}_2^{\ast}\left({C}_2^{\ast }+{H}_2\right)\left(n/q-m/p\right)\left[\left(1+B\right){\varepsilon}^{\ast }-1\right]\left({t}^{\ast }-{t}^{\ast_{\mathrm{o}}}\right)/\left(1+{t}^{\ast}\right){\varDelta}_{\mathrm{s}},\hfill \end{array} $$

  where t ^*o = 1/[(1 + B)ε ^* - 1] from (23). This gives (24).