Capital outflows after financial integration can lead to simultaneous increases in the national savings rate and asset prices of an economy with substantial financing costs. Under autarky, firms invest in risky capital while facing a borrowing constraint that creates a need for precautionary savings. Financial integration provides firms with access to foreign risk-free assets and results in two effects: a substitution effect, whereby firms divert some investments to foreign assets and cause capital outflows; and a wealth effect, whereby they grow richer in equilibrium and thus demand more domestic capital. Savings gluts and asset price booms occur when the wealth effect dominates.
In this Appendix we first detail the system of equations that characterize land price P and firm's portfolio choice α.
Financial Integration: in this equilibrium, firms choose optimal portfolio according to
Firms' net worth follows a geometric Brownian motion:
Define as the Sharp ratio from investing in land. must be larger than 0 in equilibrium, otherwise firms will hold only risk-free bonds. Substituting λ into the dynamics of firm net worth yields
Combining (14) and (15) into firm's value function yields
The solution is , where . The coefficient and payout boundary can be pinned down by matching the refinancing boundary condition and the payout boundary condition .
For the refinancing boundary W, substituting into boundary condition (7) yields
Together, one can solve for
Notice that , which implies the following relationship:
Define as the left hand side (LHS) quadratic function of this equation. therefore determines β given λ, or equivalently, given P. The power of the value function γ is given by .
The geometric Brownian motion with two reflecting barriers and has a stationary distribution. Define . The stationary distribution is characterized by the density function
Finally, PI is the solution to the market clearing condition
Combining Equations (14 –16) and (18) solves for .
Autarky: in this equilibrium, firms invest all their net worth in land. Therefore, when Wt is between the refinancing and payout boundaries. The value function satisfies
Therefore, , where and implies
The dynamics of firm net worth follow another geometric Brownian motion:
There is a stationary distribution for Wt, and its density function is given by (17). The density function implies the aggregate net worth E(W), which pins down PA from the solution to the market clearing condition:
Combining Equations (19–21) solves for PA.
Proof of Lemma 1. Take (16) as a function of β. The function has one negative root and one positive root. The left hand side is smaller than 0 when and larger than 0 when , hence there must be a unique solution β such that , which implies .
For the autarky equilibrium, notice that , since is the price of capital when , i.e. when investment in capital is risk-free. The left hand side of (19) is smaller than 0 when , and therefore (19) has one root larger than 1 and one smaller than 1, that is, there is a unique β such that . □
Proof of Proposition 1. To show the relationship between PI and PA, we start by showing that PA coincides with the solution of PI when . Next, we show that PI is an increasing function of α. By limited enforcement, in equilibrium under financial integration. Therefore it must be that .
Equation (15) implies the geometric Brownian motion of firm net worth Wt satisfies and plus the lower boundary and endogenous upper boundary pinned down by matching the boundary conditions. Notice that is proportional to , which is not surprising given the property of power functions. This reduces the ratio of and into a linear function of . Using the relationship between β and λ derived in (16), standard algebra yields that the market clearing condition can be written as
Notice that from (16), . Substituting this into the simplified market clearing condition yields
Define the left hand side of this equation as , which is now only a function of two variables: firms' effective risk aversion β and their portfolio choice α. The following lemma establishes the link between the autarky equilibrium and the financial integration equilibrium through :
Lemma A1. is the solution to .
Proof: Let , then implies . Plugging this back into (16) implies , which, after rearranging terms, is the same as Equation (19). Substituting (19) into (15) yields and for the geometric Brownian motion of the dynamics of Wt, which is identical to under autarky. Notice that E(W) is defined through η as a function of and only. This implies that the market clearing condition (18) when is identical to the market clearing condition for the autarky equilibrium. Therefore, PA must equal to PI when .
Now comparing PI and PA is equal to comparing the different solutions of in terms of β and α. By the chain rule, . The signs of the two terms can be determined separately in the following two lemmas:
Lemma A2. .
Proof: By the chain rule, . The definition of implies . Rearranging the terms of Equation (16) yields and , sinceand . Therefore .
Lemma A3. under certain parameter conditions.
Proof: from , one can solve for α as a function of β:
Taking logarithms on both sides of the equation gives
Differentiating this equation with respect to β yields
There are three parts to characterizing the derivative of α on β: Part A measures the change of payment boundary , a positive term because lower β implies weaker risk aversion, which lowers the payment boundary; Part B comes from the change of η due to the drift term , which is smaller when α is lower; and Part C is the wealth effect reflected by the change of η due to the diffusion term , which is also smaller when α is lower.
Notice that part A is bounded by
Part A reaches this upper bound, defined as , when . A sufficient condition for is therefore
First, and . Therefore, the inequality of (22) is true as long as
In every step above, the left hand side is made larger and right hand side made smaller and . Finally, since , the last inequality above implies the following sufficient condition under which is satisfied:
Combining Lemmas A2 and A3 implies . By Lemma A1, coincides with the solution where . in the financial integration equilibrium, and therefore it must be that . □
Proposition 1 implies the following corollary.
Corollary 1: .
Proof: From (16), β is an increasing function in λ. λ is a decreasing function in P. Therefore given , this implies . This result is intuitive as precautionary savings with the risk-free asset implies that firms are less risk averse. This Corollary basically states that the wealth effect comes from the reduction of effective risk aversion.
Proof of Proposition 2:
Proof: by the definition of savings rate,
Suppose . Because , it must be that
Dividing both sides by yields
First, , and therefore . That is,
or . Recall that
The left hand side has a minimum when . Therefore, for , a necessary condition is
When condition (23) is violated, cannot be true. i.e., when Condition 2, , is satisfied, it must be that . □
Proof of Proposition 3. The firm's value function solves
subject to the market clearing condition
The market clearing condition implies as an explicit function of α. The first order condition of HJB Equation (24) with respect to α is
The first two terms are the first order derivative of V with respect to α taking P as a constant. Setting them equal to zero gives , the solution of α to the decentralized economy where firms take the equilibrium price P as given. The third and fourth terms come from the additional market clearing constraint for solving , that firms take into account the effect of choosing α on the equilibrium price P. Using the same procedure of refinement as in Section III, we look for the solution where for some β. Substituting out and simplifying the first order condition implies
Let . Equation (25) defines as a function of α:
Notice that, since is the solution of setting . Suppose that . The proof of Proposition 1 shows that under Condition 1, implying that . However, from Equation (19), algebra shows that . According to Lemma A3, is increasing in α, and therefore , a contradiction. Then, it must be that . □
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