Stable cost sharing in production allocation games

Abstract

Suppose that a group of agents have demands for some good. Every agent owns a technology which allows them to produce the good, with these technologies varying in their effectiveness. If all technologies exhibit increasing returns to scale (IRS) then it is always efficient to centralize production of the good, whereas efficiency in the case of decreasing returns to scale (DRS) typically requires to spread production. We search for stable cost allocations while differentiating allocations with homogeneous prices, in which all units produced are traded at the same price, from allocations with heterogeneous prices. For the respective cases of IRS or DRS, it is shown that there always exist stable cost sharing rules with homogeneous prices. Finally, in the general framework (under which there may exist no stable allocation at all) we provide a sufficient condition for the existence of stable allocations with homogeneous prices. This condition is shown to be both necessary and sufficient in problems with unitary demands.

Appendices

Appendixes

A Interpretation of a PAG as a market with indivisible goods

We show how a PAG relates to a market with indivisible goods as modeled by Kaneko (1976). While Kaneko’s model can accommodate multiple goods, we consider here a version with a single, homogeneous good. Kaneko’s model includes a set of buyers ($N^b)$ and a set of sellers ($N^s)$. We have an endowment for the sellers, and each seller $i\in N^s$ has a utility function $u_i$ such that if $k<k'$, $u_i(k)=u_i(k')$ if $k\ge e_i$ and $u_i(k)<u_i(k')$ otherwise. We have a demand vector for buyers, and each buyer $i\in N^b$ has a utility function $v_i$ such that if $k<k'$, $v_i(k)=v_i(k')$ if $k\ge d_i$ and $v_i(k)<v_i(k')$ otherwise. Let u and v be the sets of these utility functions. Agents are also endowed with money and $U_i(k,l)=u_i(k)+l$ and $V_i(k,l)=v_i(k)+l$ are the corresponding quasi-linear utility functions. A market with indivisible goods is characterized by $(N^s, N^b, e,d,u,v)$.

A competitive equilibrium is $(p,\hat{z})$, with such that

$$\begin{aligned}&\displaystyle u_i(\hat{z}_i)+p(e_i-\hat{z}_i) =\max _{0\le z \le e_i} u_i(z)+p(e_i-z)\text { for all }i\in N^s \end{aligned}$$

(15)

$$\begin{aligned}&\displaystyle v_i(\hat{z}_i)-p\hat{z}_i=\max _{0\le z\le d_i} v_i(z)-pz\text { for all }i\in N^b \end{aligned}$$

(16)

$$\begin{aligned}&\displaystyle \sum _{i\in N^s}\hat{z}_i+\sum _{i\in N^d}\hat{z}_i=\sum _{i\in N^s}e_i \end{aligned}$$

(17)

In words, the first two conditions are utility maximizing conditions, while the third ensures that the quantity sold by sellers is equal to the quantity demanded by buyers. If $(p,\hat{z})$ is a competitive equilibrium, $(u^{p,\hat{z}},v^{p,\hat{z}})$ is a competitive imputation, where $u^{p,\hat{z}}_i=u_i(\hat{z}_i)+p(e_i-\hat{z}_i)$ for all $i\in N^s$ and $v^{p,\hat{z}}_i=v_i(\hat{z}_i)-p\hat{z}_i$ for all $i\in N^b$.

For each $S\in N^s\cup N^d$, define W(S) as

$$\begin{aligned} W(S)=\max \left( \sum _{i\in N^s\cap S} u_i(z_i)+\sum _{j\in N^b\cap S} v_j(z_j) \right) \end{aligned}$$

subject to $\sum _{i\in N^s\cap S}\hat{z}_i+\sum _{i\in N^d\cap S}\hat{z}_i=\sum _{i\in N^s\cap S}e_i$. An imputation (U, V) is a core allocation (for the market with indivisible goods) if

$$\begin{aligned} \sum _{i\in N^s\cap S} U_i+\sum _{j\in N^b\cap S} V_j\ge W(S) \end{aligned}$$

for all $S\subseteq N^s\cup N^b$.

Lemma 2

(Kaneko 1976) For any problem $(N^s, N^b, e,d,u,v)$, if $(p,\hat{z})$ is a competitive equilibrium, then $(u^{p,\hat{z}},v^{p,\hat{z}})$ is a core allocation.

We show that any PAG can be written as a market with indivisible goods. Start with a PAG (x, C), with the set of agents N.

We then let

• $N^s=N$

• $N^d=\left\{ i\in N | x_i>0 \right\}$

• $e=X^{N}$

• $d_i=x_i\text { for all }i\in N^d$

• $u_i(k)=uX-C_i(k)\text { for all }i\in N^s$

• $v_i(k)=uk\text { for all }i\in N^d$

where $u=\max _{i\in N}\max _{q=1,\ldots ,X}c_{i}(q).$

Let $P^{MIG}(x,C)$ be the market with indivisible goods built from the PAG (x, C).

For any imputation (U, V), let $y^{U,V}$ be such that

$$\begin{aligned}&y^{U,V}_i=u(X+x_i)-U_i-V_i\text { if } x_i>0 \\&y^{U,V}_i=uX-U_i\text { if } x_i=0. \end{aligned}$$

In particular, for any competitive imputation $(u^{p,\hat{z}},v^{p,\hat{z}})$, let $y^{p,\hat{z}}$ be such that

$$\begin{aligned} y^{p,\hat{z}}_i=C_i(X-\hat{z}_i)+p(X-\hat{z}_i-x_i). \end{aligned}$$

Lemma 3

If $(U,V)\in Core(P^{MIG}(x,C))$, then $y^{U,V}\in Core(x,C)$.

The above lemma is obtained as there are less core constraints in a PAG than in the corresponding market with an indivisible good: we only consider the coalitions that include seller i if and only if it also includes buyer i. In particular, if there exists a competitive equilibrium, the corresponding competitive imputation generates a stable allocation with homogeneous price for the PAG. Typically, the core of the PAG problem will contain more allocations than the core of the market with indivisible goods built from it.

B Counterexample: monopsony with general cost functions

In the case of a single demander (monopsony), Theorems 4 and 7 respectively show that $Core(x,C)={\mathcal {Y}}^{ms}(x,C)$ if we have IRS or DRS. This result is not true with general cost functions: in the following example, we show that ${\mathcal {Y}}^{ms}(x,C)$ is not a subset of the core.

Suppose that $N=\left\{ 1,2,3\right\}$ and $x=(4,0,0)$, with the cost functions given as follows.

One can see that $\bar{q}^{X}=\left( 2,1,1\right)$. The coalitional costs are $\tilde{C} ^{x}(N)=40,$ $\tilde{C}^{x}(N\backslash 2)=55,$ $\tilde{C}^{x}(N\backslash 3)=55$ and $\tilde{C}^{x}(N\backslash \left\{ 2,3\right\} )=55.$ It hence follows that $p^H_2=p^H_3=15$ and $p^L_2=p^L_3=5$.

Therefore, taking $\mathbf p =(15,15)$ in Eq. (3), we get $^{ms}y^\mathbf{p }=(60,-10,-10)\in {\mathcal {Y}}^{ms}(x,C)$. However, it is not difficult to argue that $^{ms}y^\mathbf{p }=(60,-10,-10)\notin Core(x,C)$, since agent 1’s share is higher than her stand-alone cost: $^{ms}y^\mathbf{p }_1=60>55= C_1(4)$. This shows that $Core(x,C)\ne {\mathcal {Y}}^{ms}(x,C)$.