Notes on Leontief exchange economy

Analysis of equilibrium in Leontief and CES exchange economies

November 28, 2025

Introduction

Recall in the consumer theory, we say a consumer has Leontief preferences if he has a UMP

$$\begin{align*} \max_{x_i\in\mathbb{R}_+^n} ~& u(x_i) \\ \mathrm{s.t.}~ ~& p^\top x_i \le w_i, \end{align*}$$

where $u$ is a Leontief utility function:

$$u(x_i) = \min_{j\in[n], c_{ij} > 0} \left\{ \frac{x_{ij}}{c_{ij}} \right\}.$$

At least one piece will be active. So for $i \in [m]$, an exchange economy is encoded by a so call Leontief utility matrix $C \in \mathbb{R}_+^{m \times n}$:

$$C = [c_1; c_2; \cdots; c_m]^\top.$$

In the Eisenberg-Gale program, we could use $u$ instead of $x$,

$$\begin{align*} \max~ & \sum_{i\in[m]} \log(u_i) \\ \mathrm{s.t.}~~ & C^\top u \le \mathbf{1}, \\ &u \ge 0. \end{align*}$$

In the Arrow-Debreu case, we know,

$$u_i \sum_{j\in[n]} c_{ij} p_j = w_i, \quad \forall i\in[m].$$

This is to say, in the $n\times n$ case, the initial endowment is $I$: player $i$ brings 1 unit of good $i$ into the market. The *necessary* condition for an Arrow-Debreu equilibrium is

$$\begin{align*} U C p = w = p, u \ge 0 \\ 0 \le p \perp \mathbf{1} - C^\top u \ge 0 \end{align*}$$

In ^[1], it is shown that the system above always has a solution, but may not be a Arrow-Debreu equilibrium. We could have a partition on the goods,

$$B = \{j: p_j > 0\}, ~N = \{j: p_j = 0\}, ~ C = \begin{bmatrix} C_{BB} & C_{BN} \\ C_{NB} & C_{NN} \end{bmatrix}$$

and because $p_N = 0$, we know $u_N = 0$, so we have

$$C^\top u = \begin{bmatrix} C^\top_{BB} & C^\top_{NB} \\ C^\top_{BN} & C^\top_{NN} \end{bmatrix} \begin{bmatrix} u_B \\ u_N \end{bmatrix} = \begin{bmatrix} C^\top_{BB} u_B \\ C^\top_{BN} u_B \end{bmatrix}$$

Then, one can write,

$$\begin{align} U_B C_{BB} p_B = p_B, \\ C_{BB}^\top u_B = \mathbf{1}_B,\\ C_{BN}^\top u_B \le \mathbf{1}_N.\end{align}$$

One can say, cf. Theorem 8.3.4 in ^[2]

• $(U_B C_{BB})$ satisfies

  $$(U_B C_{BB})^\top \mathbf{1} = C_{BB}^\top u_B = \mathbf{1}_B$$

  i.e., a column stochastic matrix, and $\mathbf{1}$ is the left eigenvector of $U_B C_{BB}$ corresponding to the eigenvalue $1$. It is known that, because the positivity, the spectral radius (Perron root) of $U_B C_{BB}$ must be $1$

• Besides, we know $p_B$ is the (right) eigenvector of $U_B C_{BB}$ corresponding to the eigenvalue $1$.

• $C_{BB}$ is irreducible because $U_B C_{BB}$ irreducible and $U_B \succ 0$, and $U_B C_{BB}$ is irreducible because $p_B > 0$ and with the partition it is simple. We have the nice theorem.

The takeaway is,

• You can find a rational AD equilibrium from a rational $C$, but maybe a rational Fisher equilibrium does not exist.

• If one trader likes his or her own good more than any other good, that is, $c_{ii} \geq c_{ij}$ for all $j$, then $u_i=1 / c_{i i}, p_i=1$, and $u_j=p_j=0$ for all $j \neq i$, that is, $B=\{i\}$, makes a Leontief economy equilibrium.

As a limiting case of CES economy

Consider a CES economy (square) with $n$ goods, and $n$ consumers. Each of the consumer has a CES utility with the same parameter $r$. The utility function is,

$$u_i(x_i) = \left[\sum_{j\in[n]} c_{ij} (x_{ij})^r\right]^{\frac{1}{r}} = \langle c_i, x_i^r \rangle^{\frac{1}{r}}, \quad c_i \in \mathbb{R}_+^n.$$

Let us write the parameters that define the utility as a 'CES matrix' $C \in \mathbb{R}_+^{n \times n}$:

$$C = [c_1; c_2; \cdots; c_n]^\top$$

Row $i$ of $C$ corresponds to a consumer $i$'s utility function. C special $C$ reads,

$$C = \begin{bmatrix} 1 & 1 & \cdots & 1 \\ 0 & 1 & \cdots & 1 \\ \vdots & \vdots & \ddots & \vdots \\ 1 & 0 & \cdots & 1 \end{bmatrix}$$

which is upper triangular except for one entry. In the $n=3$ case, it reads,

$$C^{(3)} = \begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{bmatrix}$$

It is easy to see this matrix is irreducible. $r \to -\infty$ corresponds to Leontief case.

Leontief equilibrium

Its equilibrium price vectors is as following: the goods $(1, n)$ will be the pivot (like basic variables for LP). In the Leontief setting, we can choose $B = {1, n}$, and $N = {2, \cdots, n-1}$, and so it reads,

• Construct the linear feasibility system

  $$\begin{align*} \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} \begin{bmatrix} u_{1} \\ u_{n} \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \end{bmatrix} \\ \begin{bmatrix} 1 & 0 \end{bmatrix} \begin{bmatrix} u_{1} \\ u_{n} \end{bmatrix} \le 1. \end{align*}$$

  this gives you the whole segment $u_1 + u_n = 1, u_1 > 0, u_n > 0$.

• Then for $p_B$, we should have,

  $$\begin{align*} & \begin{bmatrix} u_1 & \\ & u_n \end{bmatrix} \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} \begin{bmatrix} p_{1} \\ p_{n} \end{bmatrix} = \begin{bmatrix} p_{1} \\ p_{n} \end{bmatrix}. \end{align*}$$

  This implies $p_1 = u_1, p_n = u_n$.

• In this case, you set 0 price to the goods $2,..., n-1$, and so the players $2,..., n-1$ can buy nothing.

• For player 1 and $n$, they will not buy infinite, because their income will be limited to $p_1$ and $p_n$. The allocation for player 1 and $n$ are:

  $$x_1 \ge [p_1,..., p_1]^\top, \quad x_n \ge [p_n, 0,...,0, p_n]^\top.$$

  There is always a leftover for good $2, \cdots, n-1$.There is no price that strictly clears the market and allows Pareto-optimal allocations. The only possibility is to allow zero price on some good.

CES: $0 > r > -\infty$.

One can see the equilibrium price is approximately the same as Leontief, except that all $p_2, ..., p_{n-1} \approx 0$.

Numerically, exactly locate $p_1,..., p_n$ is very hard, actually PPAD ^[3]. You also need exponential bit size to encode the equilbrium price vector; typically, this require $p_1$ to be close to the equilbrium within 6-7 digits.

References

1. [1] Ye, Y.: Exchange market equilibria with Leontief's utility: freedom of pricing leads to rationality. Theoretical Computer Science. 378, 134–142 (2007). https://doi.org/10.1016/j.tcs.2007.03.002
2. [2] Horn, R. C., Johnson, C. R.: Matrix Analysis. Cambridge University Press, Cambridge; New York, 2nd edition (2012).
3. [3] Chen, X., Paparas, D., Yannakakis, M.: The complexity of non-monotone markets. J. ACM. 64, 20:1–20:56 (2017). https://doi.org/10.1145/3064810