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Appendix 1: Proofs
Proof of the Equilibrium characterization theorem
First, we note that since the supply of private real estate is perfectly elastic, the discount rate
for private real estate is always r ∗ and in turn, the price of private real estate is always:
V P ∞
(r, L) = E e
0
−r∗
t
Ltdt|L0 = l, r0 = r = L/(r ∗ − µL). (23)
We can now turn to the REITs price V R (r, L). By Bellman’s optimality principle, for any
(small) θ > 0 we have:
⎡
V R (r, L) = sup E ⎣
θ
Now, by Ito’s lemma,
0
e − t
0 Rs(Is)ds Ltdt −
τn≤θ
e − τn
0 Rs(Is)ds ckn−1, kn Lτn + e − θ
19
⎤
0 Rs(Is)ds V Iθ (rθ, Lθ) ⎦ .
(24)
V Iθ (rθ, Lθ) = V Iθ
θ
+ LV
0
Iθ
θ
dt + ▽V
0
IθΣdWt, (25)
where Σ is the volatility process for (rt, Lt) jointly, W = (W r , W L ) ′ is the Brownian motion driving
(rt, Lt) and the differential operator L is:
Lf(r, L) = (1/2)σ 2 rfrr + ρσrσLfrL + (1/2)σ 2 LfLL + µrrf + µLfL.
Since the expectation of the Brownian integral is zero, the Bellman’s optimality principle becomes:
inf E V R (r, L) −e− θ
0 RsdsV Iθ(r, L) − e− θ
0 Rsds θ
0 LV Iθdt
− θ
0 e− t
0 RsdsLtdt +
τn≤θ e− τn
0 Rs(Is)dsckn−1, knLτn
= 0.
We now distinguish three possibilities. In the first, it is optimal to switch from REITs to private
real estate at the current time. If this is the case, then as we push θ → 0,
(26)
V R (r, L) = V P (r, L) − cRP L. (27)
In the second, we just switched from the private state and thus: