A smooth homotopy method for incomplete markets

Abstract

In the general equilibrium with incomplete asset markets (GEI) model, the excess demand functions are typically not continuous at the prices for which the assets have redundant returns. The reason is that, at these prices, the return matrix drops rank and households’ budget sets collapse suddenly. This discontinuity results in a serious problem for the existence and computation of general equilibrium. In this paper, we show that this problem can be resolved with a new return matrix, which has constant rank. As a function of the price vector, the continuity of this new return matrix is ensured on a subset of the price space. This enables us to handle incomplete markets using a standard homotopy path-following argument by restricting the price vector to such a subset. The proposed approach naturally provides a constructive proof for the generic existence of general equilibrium. A homotopy method can then be applied to compute equilibria in the GEI model. Numerical experiments are presented to illustrate its efficiency.

Appendix

1.1 A. Linear algebra

For a real matrix $A\in {\mathbb {R}}^{m\times n}$,

• $AA^\top \in {\mathbb {R}}^{m\times m}$ is positive semi-definite, since for any $x\in {\mathbb {R}}^m$, $x^\top AA^\top x = \left\Vert A^\top x\right\Vert \ge 0$;

• with any matrix $B\in {\mathbb {R}}^{m\times t}$, rank $A\le$ rank $[A,\;B]$, and with any matrix $C\in {\mathbb {R}}^{n\times t}$, rank $A\ge$ rank AC;

• rank $A=$ rank $A^\top =$ rank $AA^\top =$ rank $A^\top A$; and

• span $A=$ span $AA^\top$, i.e. $\left\{ A\theta \;|\;\theta \in {\mathbb {R}}^n\right\} =\left\{ AA^{\top }\phi \;|\;\phi \in {\mathbb {R}}^m\right\}$.²⁰

1.2 B. Some useful theorems

For completeness, below we state several useful theorems.

Theorem 3

(Regular Level Set, [16]) Every regular level set of a smooth map is a closed embedded submanifold whose codimension is equal to the dimension of the range.

Theorem 4

(Transversality, [18]) Suppose U and W are $C^r$ manifolds and $f:U\times W\rightarrow {\mathbb {R}}^k$ is a $C^r$ function with $r>\text {dim }U-k$. If y is a regular value of f, then for almost every $w\in W$, $f(\cdot ,w)$ has y as a regular value.

Theorem 5

(Generalized implicit function theorem, Theorem I in [3]) Let $F:U\rightarrow {\mathbb {R}}^m$ be a smooth function on the manifold $\mu \in U$. Let $\Pi :U\rightarrow {\mathbb {R}}^{t\times m}$ and $G:U\rightarrow {\mathbb {R}}^{t\times n}$ be smooth functions such that, for all $\mu \in U$, $\Pi (\mu )$ and $G(\mu )$ have constant rank t and n respectively. If:

• 1. $F=0$ implies rank DF is at least $k=m+n-t$, and,

• 2. $\Pi (\mu )F(\mu )\in$ span $G(\mu )$ for all $\mu$,

then $F^{-1}(0)$ is a smooth submanifold of U with codimension k. In addition, rank $DF=k$ on this set.

In the proof of Lemma 6, we need to apply the generalized implicit function theorem with $m=M-1$, $n=N$, $t=S$ and $k=M+N-S-1$.

1.3 C. Proofs

We will show that generically the path will not run into a doubly bad price.²¹

Proof

Suppose there exists a sequence in the path that converges to a point $(p^*,t^*)$ where $p^*$ is a doubly bad price. The excess demand $\tilde{Z}^h(p,T(p),t)$ will converge to $\tilde{Z}^h(p^*,G,t^*)$, where G is the limit of T(p) along the path. Since T(p) always has rank N, its span must converge to some N-dimensional subspace of ${\mathbb {R}}^S$. That is,

$$\begin{aligned} \text {span }T(p)\rightarrow \text { span }\Phi \left[ \begin{array}{c} I\\ V \end{array}\right] , \end{aligned}$$

where $\Phi$ is a permutation matrix and $V\in {\mathbb {R}}^{(S-N)\times N}$. Therefore, we have

$$\begin{aligned} G=\Phi \left[ \begin{array}{c} I\\ V \end{array}\right] \Gamma \;[I\quad V^\top ]\Phi ^\top , \end{aligned}$$

(11)

where $\Gamma \in {\mathbb {R}}^{N\times N}$ is a diagonal matrix. At a doubly bad price, the eigenvalues of G will be uniquely determined by $R(p^*)R(p^*)^\top$ and $t^*$. Then, $\Gamma$ is uniquely determined by $p^*$, V and $t^*$. Note that $\Gamma$ is a smooth function of $(p^*,V,t^*)$, since $\Phi [I\quad V^\top ]^\top$ always has full rank N. Thus, G can be regarded as a smooth function of $(p^*,V,t^*)$. From the continuity of $\tilde{Z}^h$ we have

$$\begin{aligned} Z^u(p^*)+\sum _{h\ne u}\tilde{Z}^h(p^*,G(p^*,V,t^*),t^*)=0. \end{aligned}$$

(12)

As a doubly bad price, there exist at least two columns being redundant, say the ith and the jth, of the matrix $[R(p^*)\; Y^u(p^*)]$. (the case that $Y^u(p)\in R(p)$ at a bad price can be treated in a similar way, so omitted here.) Then, we have

$$\begin{aligned}{}[R_{-i}(p^*)\quad Y^u(p^*)]\alpha _1= & {} 0, \end{aligned}$$

(13)

$$\begin{aligned} \alpha _2= & {} 0, \end{aligned}$$

(14)

for some vector $\alpha _1$ and $\alpha _2$ in a unit sphere of ${\mathbb {R}}^N$. For $G(p^*,V,t^*)$, each column of $[R(p^*)\; Y^u(p^*)]$ must be in the span of $G(p^*,V,t^*)$. Thus, we have

$$\begin{aligned}{}[-V\quad I]\Phi ^\top R_{-ij}(p^*)=0. \end{aligned}$$

(15)

The four equations above are a system of $M-1+2S+(S-N)(N-2)$ equations with unknowns $(p^*,t^*,V,\alpha _1,\alpha _2)$. These unknowns belong to a manifold of dimension $M+(S-N)N+2(N-1)$. Note that from Lemma 3 the left side of (12) is smooth and we do not need to restrict the price to be in E. Taking derivative of (12) with respect to $\omega ^0$ shows it has rank $M-1$; taking the derivative of (13) and (14) with respect to $A_j$ and $A_i$, respectively, shows them each to have rank S; taking derivative of (15) with respect to $A_{-ij}$ shows it has rank $(S-N)(N-2)$. Then, generically the set of solutions is a manifold of

$$\begin{aligned}{}[M+(S-N)N+2(N-1)]-[M-1+2S+(S-N)(N-2)]=-1, \end{aligned}$$

which means that generically these equations have no solution and the path will not converge to a doubly bad price. $\square$

The following proves that the path will generically converge to a good price at $t=1$.

Proof

Suppose by contradiction that ${p}^*\in \triangle _b$. From the continuity of the homotopy, we have

$$\begin{aligned} Z^u(p^*)+\sum _{h\ne u}\tilde{Z}^h(p^*,T(p^*),1)=0. \end{aligned}$$

Since $T(p^*)$ is only smooth on E, we shall replace it with another positive semi-definite matrix G. Similar to the approach used in the previous proof, G can be defined as (11), where it is a function of $p^*$ and V. That is, we have $T(p^*,1)=G(p^*,V)$ for some $V\in {\mathbb {R}}^{(S-N)\times N}$. Then, from the equation above, we have

$$\begin{aligned} Z^u(p^*)+\sum _{h\ne u}\tilde{Z}^h(p^*,G(p^*,V),1)=0. \end{aligned}$$

Since the left side is smooth for any $p\in \triangle ^M_{++}$, parameters $\chi _0$ can be used in applying the transversality theorem. At a bad price $p^*$, suppose the ith column of R(p) is redundant, we have

$$\begin{aligned} R(p^*)\alpha =0, \end{aligned}$$

where $\alpha$ is a vector in the unit sphere of ${\mathbb {R}}^N$. According to the definition of T(p), each column of R(p) and $Y^u(p)$ must be in the span of $G(p^*,V)$, we have

$$\begin{aligned}{}[-V\quad I]\Phi ^\top [R_{-i}(p^*)\quad Y^u(p)]=0. \end{aligned}$$

For the three equations above, the unknowns $(p^*,V,\alpha )$ belong to a manifold of dimension $M-1+(S-N)N+N-1$. Taking derivatives with respect to $\omega ^h$ for some constrained household, to $A_{i}$, and to $A_{-i}$ and $\omega ^0$, respectively, shows that they have rank $M+N-S-1$, S, and $(S-N)N$. Therefore, the set of solutions of these three equations is generically a manifold of dimension

$$\begin{aligned}{}[M-1+(S-N)N+(N-1)]-[M+N-S-1+S+(S-N)N]=-1. \end{aligned}$$

This completes the proof. $\square$

1.4 D. An explanation on Example 3

It was shown in [17] that the model of a stochastic finance economy can essentially be reduced to a two-period model with many goods (the GEI model studied in this paper). Only slightly changes in our algorithm are necessary to compute equilibria for multi-period models. For the particular problem in Example 3, the parameters (endowments and payoffs) are listed in Table 4, where $\pi ^j_{t,s}$ indicates the price of asset j at (t, s). Under the assumption of no-arbitrage, the asset prices can be determined by the dividends and state prices $p\in {\mathbb {R}}^M_{++}$, that is

$$\begin{aligned} \pi ^j_{t,s}=\frac{1}{p_{t,s}}\sum _{(t',s')\in \mathscr {D}_{t,s}} p_{t',s'}d^j_{t',s'}, \end{aligned}$$

where $\mathscr {D}_{t,s}$ is the set of states that can occur after (t, s). The five columns ($r^{stock}$) indicate the payoffs (in the commodity) of the stock in the $M=21$ states. Together with that for asset 1 (the bond), we can construct the payoff matrix $W(\pi )\in {\mathbb {R}}^{M\times N}$, where $N=10$ is the total number of assets traded. $r^{bond}_{1,1}$, $r^{bond}_{1,2}$ and $r^{bond}_{1,4}$ are not listed in the table. The nominal return matrix $R(p)\in {\mathbb {R}}^{(M-1)\times N}$ in this model is given by $R(p)=P_1 W(\pi )$, where $P_1\in {\mathbb {R}}^{(M-1)\times M}$ is constructed by deleting the first row of diag(p). R(p) is clearly a linear function of price vector p. This stochastic finance economy can be analyzed in the same framework of the standard GEI model, that is, households are facing problem (2), where the return matrix R(p) is formulated in a different way.

Table 4 Parameters in Example 3