- n [kαiug (Pi - Po) + kαpo {υgg - up) + λτ (1 - α) k' + ηk] = 0. (A.13)
When the recruitment constraint is not strictly binding, we have η = 0. Using
the government budget constraint τk = (ι-α=) W to eliminate τk, the reader may
verify that (A.11) through (A.13) then lead to (3.1) and (3.2) in section 3.1.19
Consider now the case of autarky and suppose that W>wso that η =0.
Setting n = 1 and noting from the government budget constraint that W = τk∕α
under autarky, we may then write (3.1) and (3.2) in the form
αu0
w (r (α, τ) + τ) + r (α, τ) k) - (α +
δ) u0 (—
α
+ r (α, τ
) k)
=0,
(A.14)
α2g0 (α)(1+δ)+α2
μ τk
α
+ r (α, τ
) k)
- u (w (r (α, τ) + τ) + r (α, τ) k)
- (α + δ) τku0
μ τk
α
+ r (α, τ
) k)
=0,
(A.15)
where the derivatives of the function r (α, τ) are given by (2.13) and (2.14). Taking
total differentials of (A.14) and (A.15), evaluating the derivatives in an initial
equilibrium where δ =0(so that W = w, ug = up and u0g = u0p initially), and
defining b ≡ -τk0∕k, we get (using (2.13) and (2.14) with n = 1 plus the facts
that τk = αw and g0 = u0w in the initial undistorted equilibrium):
wu" [1 + * ( T=S Γ
.«/ - wu0 ( 1⅛)2 + w⅛0 £1 + = (ɪ)]
-ku00 dα
- wu00k (1 - α) dτ
u0 ∙ dδ
wu0 (1 - α) ∙ dδ
Applying Cramer’s rule to this system, we find that
∂α
Ы =0,
∂δ
(A.16)
∂τ = - ( )2 - αu0 .. + w2u00) > 0
∂δ u00k £a (g00 + w2u00) - g0]
(A.17)
Thus the introduction of a small political distortion will drive up the tax rate but
leave public sector employment unchanged, as reported in section 3.1.
19A detailed derivation of (3.1) and (3.2) from (A.11) through (A.13) is provided in a supple-
mentary appendix available from the authors.
34