Bargaining Power and Equilibrium Consumption

Proposition 1 Suppose that the household’s budget constraint is never binding. If
0 < α < α* < 1, then one of the following two assertions holds:

(i) Uι(xh(α)) = Uι(xh(α*)), U2(xh(α_t)) = U2(xh(α*)).

(ii) Uι(xh('ɪ)) < U√x√α*)), U2(x√α_t)) > U2(x√α*)).

The proof of Proposition 1 is given in the appendix. We next examine the case
when the budget constraint is binding.

3.3 Binding Budget Constraint

If the budget constraint is binding for household h, then (9) still holds true whereas

∂F        ∂F

∂U1 ∙ DXi^U1 ⁺ ∂U2 ∙ DXi^U2 = ʌ (^α )p (^α ) ^,

with positive Lagrange multiplier λ(α). Moreover, with binding budget constraints,

p(α) ∙ [x 1(α) + x2(α) - ω_h,] = 0,

hence

p(α) [x⁰₁(α) + x⁰₂(α)] = -p⁰(α) [x1(α) + x2(α) - ωh] .               (12)

Substituting (11) and (12) into (9) yields

Without further qualification, it is impossible to sign Φ⁰(α). Under additional assump-
tions, however, one can gain some detailed insights. To this end, let us decompose the
effects of a change of consumer 1’s relative bargaining power from α to α + e into two
parts:

1. a pure bargaining effect when α is changed to α + e whereas the price system
stays at p(α);

2. a price effect when relative bargaining power remains constant at α + e while
the price system adjusts from p(α) to p(α + e).3

³ Of course, the price effect could be further decomposed into a substitution and an income effect.
But that is immaterial to our analysis.

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