Rationalization, II

Universality of homothetic preferences

Imagine that in an exchange economy we have access to the market excess demand $Z(p)$. Let $\Delta \subset \mathbb{R}_+^n$ be the unit simplex and $X \subset \mathbb{R}_+^n$ be the allocation space. In the exchange economy, the budget of an agent $i$ is given by $w_i = \langle p, b_i \rangle$, where $b_i$ is the endowment vector of agent $i$. The following theorem is due to Rolf Mantel [1].

Suppose that $Z: \Delta \mapsto X$ is a $C^2$ function satisfying the Walras's law $p^\top Z(p) = 0$, and let $K \subset \mathrm{int}\,\Delta$ be a compact set. Then there exists an exchange economy with $n$ consumers, in which each consumer $i$'s preferences $\succeq_i$ is convex, homothetic, and monotonic, such that the sum of the consumers' excess demand functions coincides with $Z$ on $K$.

The above theorem is a stronger version of the Sonnenschein–Mantel–Debreu theorem, which says $Z$ can be any continuous function satisfying the Walras's law and homogeneity of degree zero. We show the above theorem; the presentation here is adapted from Chambers and Echenique [2].

We construct an exchange economy with $n$ agents, each with $m$ units of one good, i.e., the endowment vector is $b_i = m\mathbf{1}_i$. Let $A$ be a matrix with unit column sums, $A^\top \mathbf{1} = \mathbf{1}$. Define

$$g_i(p) = \tfrac{1}{m}Z_i\!\left(\tfrac{p}{\sum_j p_j}\right) - \langle A_i, \log(Ap) \rangle, \tag{1}$$

where $A_i$ is the $i$-th column of $A$. For any $p \in K$, $\nabla^2 Z_i(p)$ is bounded, so there exists a sufficiently large $m:=m(K)$ such that $g_i$ is convex for all $p \in K$. Let $u_i$ be the utility function of consumer $i$,

$$u_i(x) = \inf_{p} \, g_i(p), \quad \text{s.t.}~ \langle p, x \rangle \le 1. \tag{2}$$

Hence, for any $\lambda > 0$,

$$\begin{align*} u_i(\lambda x) &= \inf_{p} \, g_i(p), \quad \text{s.t.}~ \langle p, \lambda x \rangle \le 1 \\ &= \inf_{q} \, g_i(q/\lambda), \quad \text{s.t.}~ \langle q, x \rangle \le 1 \\ &= \inf_{q} \, \tfrac{1}{m}Z_i\!\left(\tfrac{q}{\sum_j q_j}\right) - \langle A_i, \log(Aq/\lambda) \rangle \\ &= \inf_{q} \, \tfrac{1}{m}Z_i\!\left(\tfrac{q}{\sum_j q_j}\right) - \langle A_i, \log(Aq) - \log(\lambda)\mathbf{1} \rangle \\ &= u_i(x) + \langle A_i, \log(\lambda)\mathbf{1} \rangle = u_i(x) + \log(\lambda), \end{align*}$$

which means $u_i$ is logarithmically homogeneous, $u_i(\lambda x) = u_i(x) + \log(\lambda)$, and thus homothetic. The same computation applied directly to $(1)$ shows that $g_i$ obeys the dual identity $g_i(\lambda q) = g_i(q) - \log(\lambda)$. We now compute the indirect utility,

$$v_i(p, w) := \sup_{x} \, u_i(x), \quad \text{s.t.}~ \langle p, x \rangle \le w, \tag{3}$$

and claim that $v_i(p, w) = g_i(p) + \log(w)$.

Reduction to $w = 1$. Substituting $x = w y$, the budget $\langle p, x \rangle \le w$ becomes $\langle p, y \rangle \le 1$, and log-homogeneity gives $u_i(w y) = u_i(y) + \log(w)$. Hence

$$v_i(p, w) = \sup_{\langle p, y \rangle \le 1} \big(u_i(y) + \log(w)\big) = \log(w) + v_i(p, 1),$$

so it remains to show $v_i(p, 1) = g_i(p)$.

Upper bound. For any $x$ with $\langle p, x \rangle \le 1$, the price $q = p$ is feasible in the definition $(2)$ of $u_i$, so $u_i(x) = \inf_{\langle q, x \rangle \le 1} g_i(q) \le g_i(p)$. Taking the supremum over such $x$ yields $v_i(p, 1) \le g_i(p)$.

Lower bound. Differentiating $g_i(\lambda q) = g_i(q) - \log(\lambda)$ at $\lambda = 1$ gives the Euler relation $\langle q, \nabla g_i(q) \rangle = -1$ for all $q$. Set $x^\ast := -\nabla g_i(p)$, which is nonnegative because $g_i$ is decreasing; then $\langle p, x^\ast \rangle = -\langle p, \nabla g_i(p) \rangle = 1$, so $x^\ast$ is feasible. Since $g_i$ is convex, its gradient inequality at $p$ gives, for every $q$,

$$g_i(q) \ge g_i(p) + \langle \nabla g_i(p), q - p \rangle = g_i(p) + 1 - \langle x^\ast, q \rangle.$$

Thus $\langle x^\ast, q \rangle \le 1$ implies $g_i(q) \ge g_i(p)$, so the infimum defining $u_i(x^\ast)$ is attained at $q = p$:

$$u_i(x^\ast) = \inf_{\langle q, x^\ast \rangle \le 1} g_i(q) = g_i(p).$$

As $x^\ast$ is feasible for $(3)$ with $w = 1$, we conclude $v_i(p, 1) \ge u_i(x^\ast) = g_i(p)$. Combining the two bounds, $v_i(p, 1) = g_i(p)$, and with the reduction $v_i(p, w) = g_i(p) + \log(w)$, as claimed. Then, by Roy's identity, the demand function of the $i$-th agent, with $p \in \Delta$ and the budget $w_i = p^\top b_i$, is

$$\begin{align*} x_i(p, w) &= -\frac{\nabla_p v_i(p, w)}{\nabla_w v_i(p, w)} \\ &= -w\nabla_p v_i(p, 1) = -m\langle \mathbf{1}_i, p \rangle \, \nabla_p g_i(p). \end{align*}$$

Note, for $p \in \Delta$,

$$\begin{align*} \nabla_p g_i(p) &= \frac{1}{m} \nabla Z_i(p) - A^\top\mathrm{diag}\!\left(\tfrac{1}{Ap}\right)A_i \\ \Rightarrow \quad x_i(p, w) &= -p_i \nabla Z_i(p) + m p_i A^\top\mathrm{diag}\!\left(\tfrac{1}{Ap}\right)A_i \\ &= -p_i \nabla Z_i(p) + m A^\top \mathrm{diag}\!\left(\tfrac{1}{\sum_i A_i p_i}\right) A_i p_i. \end{align*}$$

So we have,

$$\begin{align*} \sum_{i=1}^n x_i(p, w) - m\mathbf{1} &= -\sum_{i=1}^n p_i \nabla Z_i(p) + m A^\top \mathrm{diag}\!\left(\tfrac{1}{\sum_i A_i p_i}\right)\sum_{i=1}^n A_i p_i - m\mathbf{1} \\ &= -\sum_{i=1}^n p_i \nabla Z_i(p) + m A^\top \mathbf{1} - m\mathbf{1} \\ &= -\sum_{i=1}^n p_i \nabla Z_i(p). \end{align*}$$

Since $Z(p)$ satisfies the Walras's law,

$$p^\top Z(p) = 0 \Rightarrow p^\top \nabla Z(p) + Z(p)^\top = 0.$$

Hence we have the desired result, $\sum_{i=1}^n x_i(p, w) - m\mathbf{1} = Z(p)$.

The same does not hold for a Fisher market, where budgets are fixed money endowments $w_i > 0$ with $\sum_i w_i = 1$ rather than the price-dependent value $\langle p, b_i \rangle$. Because the budgets $w_i$ are now constant, any demand of a homothetic agent is a gradient field, and so is the aggregate,

$$g(p) = -\sum_i w_i \nabla g_i(p) = -\nabla\Phi(p), \qquad \Phi := \sum_i w_i\, g_i.$$

Hence, any non-symmetric operator $g$ is unattainable.
References
  1. [1] Mantel, R. R.: Homothetic preferences and community excess demand functions. Journal of Economic Theory. 12(2), 197–201 (1976).
  2. [2] Chambers, C. P., Echenique, F.: Revealed Preference Theory. Econometric Society Monographs. Cambridge University Press (2016).