Is there a fair and stable outcome?
K . Eriksson et al. Mathematical Social Sciences 39 2000 109 –118
115
Sattinger 1979, 1993 considers an assignment model, called the Ricardian differen- tial rents model, where there is a continuous distribution of workers’ skills and a
continuous distribution of machine sizes. Given this model and the above conditions on the productivity function f, Sattinger argues that the equilibrium assignment of workers
to jobs will be strictly top–down, i.e. the nth worker, in order of decreasing skill, will be assigned to the nth machine, in order of decreasing size. From the continuous
distributions of skills and machine sizes it then follows that the relationship between them can be described by a monotonic function k g determining which machine size
will be assigned to a worker of skill g.
Then Sattinger proceeds by determining the wage function in equilibrium, i.e. the amount w g of the productivity that the employer will pay for a worker of skill g. He
shows that the wage function must satisfy ≠f g,k
]] w9 g 5
5
F G
≠g
k 5k g
This equation determines the wage function up to a constant term. The constant term, describing the absolute level of wages, will be determined by the conditions for the last
match in order of decreasing skill and machine size, basically by choosing a minimum wage.
Taking assortative matrices in the limit, Eq. 1 says that f g, k is increasing in both
2
g and k, while Eq. 2 says that the mixed partial derivative ≠ f ≠g ≠k is positive. Hence, we have obtained the differential rents model as the limit case of assortative
markets. Let us now derive the differential equation Eq. 5 for the wage function w g given
by w g 5 u . In the worker-optimal outcome we had u 2 u 5
a 2 a while in
i i
i i 21
i,i i,i 21
the employer-optimal outcome we had u 2 u 5
a 2
a . The function f g,k
i 11 i
i 11,i i,i
being continuously differentiable means that in the limit the left-hand derivative equals the right-hand derivative:
a 2 a 5
a 2
a
i,i i,i-1
i 11,i ii
Hence we have that both the employer-optimal and worker-optimal wage functions satisfy the differential equation Eq. 5, so in fact they coincide except for a constant
term and thus we have a unique wage function in equilibrium, up to the absolute level of wages.
3. Is there a fair and stable outcome?
The two most famous guiding principles for choosing outcomes to a game like the assignment game are the utilitarian principle, which tells you to maximize the overall
productivity, and the egalitarian principle which tells you to be as fair as possible cf. Moulin, 1988, e.g. by minimizing envy or maximizing the outcome of the most
unfortunate. These principles are often in conflict with each other. Our approach, as most standard methods in game theory, is utilitarian since stable outcomes maximize
116 K
. Eriksson et al. Mathematical Social Sciences 39 2000 109 –118
productivity. However, we will now try to analyze how far such outcomes can satisfy the demands of egalitarians. We will take the approach of minimizing envy.
3.1. Minimizing envy Envy arises when one values another’s basket of goods more than one’s own, cf.
Feldman and Kirman 1974 and Varian 1974. Here, we are only looking at payoffs, so envy refers to inequalities of payoffs. We will make the natural assumption that all envy
is between pairs of P-agents or pairs of Q-agents, so that a P-agent is never envious of a Q-agent or vice versa. If u . u then agent p ’s envy of p is u 2 u . The total envy E in
i j
j i
i j
a market is the sum of all these differences, i.e.
1 1
E 5
O
[u 2 u ] 1 [v 2 v ] ,
i j
i j
i, j 1
where [a 2 b] 5 a 2 b if a 2 b . 0, and 0 otherwise. It turns out that assortative markets are particularly well-behaved with respect to envy:
every stable outcome will yield the same total envy
Theorem 3.1. In an assortative market
a , the total envy in any stable outcome is
ij n
E 5
O
2k 2 n 2 1 a
kk k 51
Proof. In any stable outcome of an assortative market we have u 1 v 5 a , so
i i
ii 1
1
u 2 u 1 v 2 v 5 a 2 a . The envy terms [u 2 u ]
and [v 2 v ] are non-zero only
i j
i j
ii jj
i j
i j
if i . j, by Proposition 2.3. Hence,
n
E 5
O
a 2 a 5
O
2k 2 n 2 1 a
ii jj
kk i .j
k 51
by simple verification. h
This invariance implies that the smaller the envy among P-agents, the greater the envy among Q-agents. If we actually want to minimize the envy among the P-agents, we shall
minimize the sum o
u 2 u . But from Section 2.3.2 we already know a way to
i .j i
j
minimize this sum: just choose the P-worst outcome By symmetry, we also have the analogous result for Q-agents.
Corollary 3.2. The P-worst outcome minimizes the envy among P-agents. The Q-worst outcome minimizes the envy among Q-agents.
K . Eriksson et al. Mathematical Social Sciences 39 2000 109 –118
117
This is an illustration of the conflict between the utilitarian and the egalitarian principles.