Strategic monetary policy in a monetary union with non-atomistic wage setters

and

.—.

where Ph =

^u ]β=3u

(n — 1)σ — 1
n — a

(57)

г ,     =       (^k + ^pF $f ) [¹ + ⁽ⁿH ^{— 1}M

^ⁿ β=β∏   PfΘf (пнΘf — 1 + Θn) + к(пн — 1 + Θn)

(58)

k(k+β_p θF )
(1-θ_w )θ_f (fc+β_p θ_f )

i о      к ггл    ∙      i    r._ii ∙-jji

and Pu = _α(ι_-α) ∙ There is no value of пн belonging to the

relevant domain in which the elasticity (57) and (58) cross each other⁵⁰. The expression

(57) evaluated at пн = 1 yields

∙^' ^,βu=^βu '" ¹     1 _{— a}

(59)

Note that expression (58) is an increasing function of foreign CB hence we evaluate it
when Pf → ∞ as follows:

^f

[^ⁿ^h =β∏Λn=1ΛβF→∞ ^- 1 _{— a}'                     ((

It is apparent that expression (59) is always larger than (60).which proves Proposition 7.
■

Welfare and macroeconomic institutions. It is straightforward to compute that
welfare level as follows:

-= I τ l(1 — ɪ )(∣ — (1 — ɪ ))] = ∣ ⅞ (1 — ⅛ )∙       -)

Now consider the problem of maximizing the individual welfare on the constraint set as
follows:

max Uj_i

Fi ,β

s.to n_H ≥ 1 Λ P ≥ 0.

(62)

The solution of the Kuhn-Tucker conditions yields

if σ  >

if σ  <

к + P_f θF       _a, „

——      ---r^{, p} = ^{0 λ n}H > ¹

kθ n + P f ^f (1 — a)

к + P_f θf       _a, „

——      ----^{, p} > ^{0 λ n}H = ¹

kθ n + P f ^f (1 — a)

⁵⁰Such a value is in fact n = —ɪ.
σ

34