We turn next to the effects of tax coordination with rent creation discussed in
section 4.2. Assume that the equilibrium value of W implied by (2.22), (4.6) and
(4.7) will indeed satisfy the recruitment constraint W ≥ w. From these equations
and the assumption made in section 4.2 that W = w initially, one can then show
that
dα (ε∕τ)(1 — α) {(α + δ)[1 — γσc (1 — α)] — α (1 — α)}
dτ ∆
dW = ){(÷ ) G—a )—(÷-α )—-→'+δ)}
(∆∆^) [a + Yσc (a + δ)] ,
(A.19)
dr = — ') ∣2 — α + α) [ʌ + ' + -¾U
dτ ∆ α 1 — α 1+ δ
+й)
[γσc (α + δ) — δ] ,
(A.20)
∆ = —£k (1 — α)∫2 — α + fα+^ ) ∣-α- + γσc + ɪ]!
∖ α у 1 — α 1 + δ j
— α [α + γσc (α + δ)] < 0,
(A.21)
W u00
Y ≡-----=, σc ≡—C — ,
W + rk u0
σg≡
g0
—α—
g0
where σc is the coefficient of relative risk aversion in private consumption (which
is identical for private and public sector workers in the initial equilibrium) and σg
is the coefficient of relative risk aversion in public consumption. As mentioned,
these results hold provided W ≥ w. If this condition is satisfied initially, it will
continue to be met if dW ≥ dw. Noting from (2.11) that dw∕dτ = —k (1 + ddT)
and using (A.19) and (A.20), we find
—k
∆ J {δ + α [1 + α + γσc (α + δ)]}
dW dw
dτ dτ
— /) ^ + γσc (α + δ) + (α+δ Wτ¾ — 1)1 . (A.22)
α∆ 1 — α α 1+ δ
The proof of Proposition 6 given in Appendix 3 utilises (A.21) and (A.22).
35
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