Bargaining Power and Equilibrium Consumption

Proposition 4 (Spill-overs) Under separate spheres consumption, there exists a unique
market equilibrium (up to price normalization) for each array (α1 , . . . , αn) of bargain-
ing power parameters. Moreover, for any two households g 6= h:

(i) α_h > α₁ ^ xh 1 > ^ 11.

(ii) α_h = α_g ^ xh 1 = x 11.

(iii) ∂xh 1 /∂α_h > 0, ∂x 11 /∂α_h < 0.

(iv) ∂x²_h2/∂α_h < 0, ∂x22/∂α_h > 0.

The proof of Proposition 4 is given in the appendix. Proposition 4 has clear-cut
implications. Consider the sociological groups “first-members” and “second-members”,
defined by similarities with respect to preferences. If all individuals in the first sociolog-
ical group have the same bargaining power (and as a consequence all “second-members”
as well), all households consume their endowments since we are in an equilibrium with
no active trade. An identical shift of bargaining power across all households has no
effect on utilities of any individual either since we will again arrive at an equilibrium
with no trade.

The situation is completely different when only some members of a sociological
group enjoy higher bargaining power. For instance, a “first-member” suffers when
only other “first-members” gain more bargaining power in their respective households.
Conversely, the “first-member” benefits from higher own bargaining power as long as
other “first-members” do not experience a change of bargaining power. The analogue
holds for the other sociological group. Therefore, the main thrust of Proposition 4
is that each individual of a sociological group tends to benefit if he can increase his
bargaining power but tends to suffer if others in his group are able to do the same.

For separate sphere economies of the type discussed above we obtain as an immedi-
ate consequence a power illusion phenomenon. Consider two separate sphere economies
denoted by E₁ ({α¹_h}₁ⁿ) and E₂({α²_h}₁ⁿ) with households that are homogeneous with re-
spect to individual utility functions and endowments. Equilibrium utilities are denoted
by UJh_l1, Uh₂ and UJh²₁, UJh_ι2, respectively. Then the following holds:

22

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