Bargaining Power and Equilibrium Consumption



where 0 < α < 1.

The aggregate demand function of household h (xh = xh1 +xh2, yh = yh1 +yh2) is given
by

xh = α(2 + p1)/p1,

yh = (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

* __   *

xh = xh1


=1,


**

yh = yh2 = 2,

p*1 = (2 α ) / (1 — α ).

The utilities of the members of each household are Uh1 = 1, Uh2 = 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*hi =

2n(α + ε)

for

h=

1, . .

ʌ
.,h;

2nα + 2hε

*

*

2n(1 — α — ε)

for

h=

1, . .

ʌ

. , h;

yh

= 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.

yh

= yh2 =

n (1 — α ) — h ε

24



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