Abstract
This paper is concerned with the computation of equilibrium for an exchange economy with constant returns production technologies. We convert such an economy into a pure exchange economy by allocating the production to each consumer’s endowment evenly. In this way, the market clearing condition of the original economy is reformulated as that of a pure exchange economy, together with an additional complementarity condition to ensure the feasibility of production plans. A homotopy method is proposed to solve these two problems simultaneously. With this approach, the economic equilibrium model with constant returns production can be handled in a similar way to the pure exchange economy. A path-following algorithm is then developed for computing equilibria in these economies.
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Notes
Simplicial methods are also theoretical sound. Any problem that can be solved by homotopy methods is also solvable with simplicial method. One main advantage of homotopy method is that the differentiability of the problem is considered, which makes it numerically more efficient in computation.
Utilities are usually desired to be (at least) three times differentiable. If \(u^i\) is non-smooth, we need to use some regularization techniques in mathematical optimization. As we will see, utilities need not be strictly quasi-concave and strictly increasing. With our approach, the algorithm works if they are quasi-concave and non-decreasing.
Since the matrix A(p) is defined according to the function \(a^k\), for each firm k, A(p) can be different if different \(a^k\) is chosen. For ease of reference, we use A(p) in the definition of equilibrium. This means that the equilibrium is defined according to the matrix function A(p), rather than the production functions \(\{f^k\}\). However, in many cases, it is non-trivial to determine A(p) with given \(\{f^k\}\). This issue will be addressed in the next section.
In the literature, it is usually assumed that firms have owners. Consumer i’s profit share in firm k is given by \(\theta ^i_k\ge 0\), and \(\sum _i\theta ^i_k=1\) for each firm k. However, when firms have constant returns production technologies, they make zero profit from producing commodities at equilibria, which makes equilibria independent from consumers’ profit shares. If production has [for example, in Garg et al. (2018)] decreasing marginal returns, different values of \(\theta ^i\) would result in different equilibria.
The cost function \(c^k:{\mathbb {R}}^m_+\rightarrow {\mathbb {R}}\) indicates the minimal cost to produce one unit of output of commodity \(\pi (k)\), given the prices for the inputs \(p^{-\pi (k)}\). From production theory, cost functions are continuous, see Varian (1992), but not necessarily differentiable.
The introduction of variable q is not necessary. But as we will see in the next section, it makes easier the determination of a starting point of the path-following algorithm.
According to this lemma, under the assumption of no production without input, there must exist a solution \(\xi \) to the system \(-A(q)^\top \xi \in {\mathbb {R}}^{m+1}_{++}\) with \(\xi \in {\mathbb {R}}^l_{++}\).
If \(g:{\mathbb {R}}^m\rightarrow {\mathbb {R}}\) is concave and \(h:{\mathbb {R}}\rightarrow {\mathbb {R}}\) is convex and non-increasing, then the composition h(g(x)) is convex.
(9) must have an optimal solution when \(t\in [0,1)\). The derivatives of the objective function in (9) decrease to \(-\infty \) as \(\alpha ^{-\pi (k)}\) nears the boundary, and converge to \(p>0\) as \(\alpha ^{-\pi (k)}\) becomes extremely large. From the continuity of the derivatives in the feasible set, there must exists \(\alpha ^{-\pi (k)}\) such that the derivatives equal zero.
The logarithmic barrier terms approximate the 0-infinity indicator function on set \(C^k(p,t)\) as \(t\rightarrow 1\), and thus converge to zero if \(\alpha ^{-\pi (k)}\in C^k(p,t)\). For the use of logarithmic barrier function in constrained optimization, we refer interested readers to Chapter 11.2 in Boyd et al. (2004).
The no production without input assumption (Assumption 1) is made on the original production activity matrix A(p), and hence on each \(f^k\). It is not hard to verify that this property also holds for the perturbed production matrix A(p, t). From (9), we have \(f^k(\alpha ^{-\pi (k)})>1\) and/or \(\sum _{j\ne \pi (k)}q^0_j \alpha _j>1\). In the former case, the production activity with A(p, t) is more inefficient. In the latter case, we have \(\sum _{j\ne \pi (k)}\alpha _j>1/\max _{j'}\{q^0_{j'}\}>1\). Then, \((A(p,t)y)_j=\sum _{k:\pi (k)=j}y_k-\sum _{k:\pi (k)\ne j} y_k \alpha ^k_j\) and we have \(\sum _j(A(p,t)y)_j = \sum _k y_k- \sum _k y_k\sum _{j\ne \pi (k)}\alpha ^k_j = \sum _k y_k(1- \sum _{j\ne \pi (k)}\alpha ^k_j)\le 0\) if \(y\ge 0\). Therefore, \(A(p,t)y\ge 0\) and \(y\ge 0\) implies \(y=0\).
To compute an equilibrium, the utility functions need not to be strictly increasing or even differentiable. The logarithmic term in (5) makes the agents face strictly increasing and concave utilities in the artificial economies along the homotopy path. The nondifferentiablity can be smoothed with the regularization technique in Zhan and Dang (2018).
The activity matrix A violates our definition in Sect. 2, as there are two commodities as outputs and the quantities of outputs are not normalized to one. However, it can be handled with our approach.
The values of endowments and coefficients of utilities are picked uniformly at random from [1, 2]. For the activity matrix A, to satisfy Assumption 1, half of the commodities are set as inputs while others are outputs. Absolute value of each element in the matrix is drawn uniformly at random from [1, 2].
The algorithm in Zhan and Dang (2018) is applicable to all utility functions that can be expressed as \(u(x)=\min \{ f^1(x),f^2(x),\dots ,f^k(x)\}\) with \(f^\ell (x)\) being a smooth concave function for all \(\ell \). The efficiency of this algorithm is shown with a numerical comparison to the complementarity pivot algorithm in Garg et al. (2015).
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Appendix
Appendix
Theorem 4
(Transversality Theorem) Let \(L:S\times {\mathbb {R}}^l\rightarrow {\mathbb {R}}^s\) be \(C^r\), where \(S\subset {\mathbb {R}}^n\) is an open set and \(r\ge 1+\max \{0,n-s\}\). If zero is a regular value of L, then zero is a regular value of \(L(\cdot ,{\hat{\gamma }}):S\rightarrow {\mathbb {R}}^s\) for almost all fixed \({\hat{\gamma }}\in {\mathbb {R}}^l\).
The proof for Lemma 3.
Proof
According to (8), the Jacobian matrix of \(H(\theta ,t)\) is given by
where \(I_{m+1}\) and \(I_l\) are identity matrices of \((m+1)\times (m+1)\) and \(l\times l\), respectively, U is the derivative of \(\sum _ix^i(p,q,y,t)\) with respect to p, \(W_q\) and \(W_y\) are derivatives of \(\sum _i{\overline{\omega }}^i(q,y,t)\) with respect to q and y, respectively, and Q is the derivative of A(q, t) with respect to q. At \(t=0\) and \(H(\theta ,0)=0\), \(U=-diag((1/p^0_j)_{j\in M})\), \(W_q=\mathbf{0} \) and \(W_y=\mathbf{0} \). With the fact that \(p=p^0\in {\mathbb {R}}^{m+1}_{++}\) and \(v=v^0\in {\mathbb {R}}^l_{++}\), the Jacobian matrix is clearly of full row rank. \(\square \)
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Zhan, Y., Dang, C. Computing equilibria for markets with constant returns production technologies. Ann Oper Res 301, 269–284 (2021). https://doi.org/10.1007/s10479-020-03733-2
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DOI: https://doi.org/10.1007/s10479-020-03733-2