Bargaining Power and Equilibrium Consumption

where 0 < α < 1.

The aggregate demand function of household h (x_h = x_h₁ +x_h₂, y_h = y_h₁ +y_h₂) is given
by

xh = α(2 + p1)/p1,

y_h = (1 - α)(2 + p₁).

where good 2 has been used as the num´eraire. If α is the same value across households,
market equilibrium does exhibit zero net trades since excess demands are identical for
all households. Thus, market equilibrium is given by

* __ *

x_h = x_h₁

=1,

y_h = y_h₂ = 2,

p*₁ = (2 α ) / (1 — α ).

The utilities of the members of each household are U_h₁ = 1, U_h₂ = 2.

Next consider 0 < α < α+ε < 1 and 1 ≤ h ≤ n. Suppose that in the first h households,
bargaining power shifts by ε from consumer 2 to consumer 1.

Market equilibrium for the first commodity obtains if

( n — h )( α (2 + p i)) + h ( α + ε )(2 + p ɪ ) = np ɪ, (19)

^ 2 nα + 2hε

Pi⁽^{ε, h}) = —-----:---j-∙ ⁽²⁰⁾

n(1 — α) — h ε

The equilibrium allocation is given by

* x^*_h	* = x^*_h_i =	2n(α + ε)	for	h=	1, . .	ʌ .,h;
		2nα + 2hε
^*	^*	2n(1 — α — ε)	for	h=	1, . .	ʌ . , h;
y_h	⁼^yh2 ⁼	n (1 — α ) — h ε
* _x*_h	* ⁼^x^*hi ⁼	2nα	for	h=	ʌ h +	1, . . . , n;
		2nα + 2he
*	*	2 n (1 — α )	for	h=	ʌ h +	1, . . . , n.
y_h	= y_h₂ =	n (1 — α ) — h ε

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