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*hi = |
2n(α + ε) |
for |
h= |
1, . . |
ʌ |
2nα + 2hε | ||||||
* |
* |
2n(1 — α — ε) |
for |
h= |
1, . . |
ʌ . , h; |
yh |
= yh2 = |
n (1 — α ) — h ε | ||||
* |
* = 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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