Varian's theorems

In welfare economics, Varian's theorems are several theorems related to fair allocation of homogeneous divisible resources. They describe conditions under which there exists a Pareto efficient (PE) envy-free (EF) allocation. They were published by Hal Varian in the 1970s.^[1][2]

Examples

All examples involve an economy with two goods, x and y, and two agents, Alice and Bob.

A. Many PEEF allocations: Alice and Bob have linear utilities, representing substitute goods:

$u_A(x,y)=2x+y$,

$u_B(x,y)=x+2y$.

The total endowment is (4,4). If Alice receives at least 3 units of x, then her utility is 6 and she does not envy Bob. Similarly, if Bob receives at least 3 units of y, he does not envy Alice. So the allocation [(3,0);(1,4)] is PEEF with utilities (6,9). Similarly, the allocations [(4,0);(0,4)] and [(4,0.5);(0,3.5)] are PEEF. On the other hand, the allocation [(0,0);(4,4)] is PE but not EF (Alice envies Bob); the allocation [(2,2);(2,2)] is EF but not PE (the utilities are (6,6) but they can be improved e.g. to (8,8)).

B. Essentially-single PEEF allocation: Alice and Bob have Leontief utilities, representing complementary goods:

$u_A(x,y)=u_B(x,y)=\min(x,y)$.

The total endowment is (4,2). The equal allocation [(2,1);(2,1)] is PEEF with utility vector (1,1). EF is obvious (every equal allocation is EF). Regarding PE, note that both agents now want only y, so the only way to increase the utility of an agent is to take some y from the other agent, but this decreases the utility of the other agent. While there are other PEEF allocations, e.g. [(1.5,1);(2.5,1)], all have the same utility vector of (1,1), since it is not possible to give both agents more than 1. ^[3]

C. No PEEF allocations: Alice and Bob have concave utilities:

$u_A(x,y)=u_B(x,y)=\max(x,y)$.

The total endowment is (4,2). The equal allocation [(2,1);(2,1)] is EF with utility vector (2,2). Moreover, every EF allocation must give both agents equal utility (since they have the same utility function) and this utility can be at most 2. However, no such allocation is PE, since it is Pareto-dominated by the allocation [(4,0);(0,2)] whose utility vector is (4,2).

Existence of PEEF allocations with monotone convex preferences

Varian's theorem says that:^[1]:80

If the preferences of all agents are monotone and convex, then PEEF allocations exist.

In the #Examples, the preferences are always monotone. However, only in examples A and B the preferences are convex.

The proof relies on the existence of a competitive equilibrium with equal incomes. Assume that all resources in an economy are divided equally between the agents. I.e, if the total endowment of the economy is $E$, then each agent $i\in 1,\dots,n:$ receives an initial endowment $E_i = E/n$.

Since the preferences are convex, the Arrow–Debreu model implies that a competitive equilibrium exists. I.e, there is a price vector $P$ and a partition $X$ such that:

• (CE) All agents maximize their utilities given their budget. I.e, if $P\cdot Y \leq P\cdot X_i$ then $Y \preceq_i X_i$.
• (EI) All agents have the same income in the equilibrium prices: for all $i,j: P\cdot X_i = P\cdot X_j$.

Such an allocation is always EF. Proof: by the (EI) condition, for every $i,j: P\cdot X_j \leq P\cdot X_i$. Hence, by the (CE) condition, $X_j \preceq_i X_i$.

Since the preferences are monotonic, any such allocation is also PE, since monotonicity implies local nonsatiation. See fundamental theorems of welfare economics.

References

1. Varian, Hal R (1974). "Equity, envy, and efficiency". Journal of Economic Theory 9: 63. doi:10.1016/0022-0531(74)90075-1.
2. Varian, Hal R. (1976). "Two problems in the theory of fairness". Journal of Public Economics 5 (3–4): 249. doi:10.1016/0047-2727(76)90018-9.
3. Note that a similar economy appears in the 1974 paper^:70 as an example that a PEEF allocation does not exist. This is probably a typo - the "min" should be "max", as in example C below. See this economics stack-exchange thread.