Technical Appendix to "Macroeconomic effects of public sector unions"

Munich Personal RePEc Archive

Technical Appendix to "Macroeconomic effects of public sector unions"

Vasilev, Aleksandar

AUBG

June 2013

Online at https://mpra.ub.uni-muenchen.de/68235/

MPRA Paper No. 68235, posted 09 Dec 2015 03:02 UTC

Technical Appendix to ”Macroeconomic effects of public sector unions

Aleksandar Vasilev

December 6, 2015

1 Technical Appendix

1.1 Optimality conditions

1.1.1 Firm’s problem

The profit function is maximized when the derivatives of that function are set to zero.

Therefore, the optimal amount of capital - holding the level of technologyAt and labor input N_t^p constant - is determined by setting the derivative of the profit function with respect to K_t^p equal to zero. This derivative is

(1−θ)A_t(K_t^p)^−θ(N_t^p)^θ(K_t^g)^ν−r_t = 0 (1) where (1−θ)At(K_t^p)^−θ(N_t^p)^θ(K_t^g)^ν is the marginal product of capital because it expresses how much output will increase if capital increases by one unit. The economic interpretation of this First-Order Condition (FOC) is that in equilibrium, firms will rent capital up to the point where the benefit of renting an additional unit of capital, which is the marginal product of capital, equals the rental cost, i.e the interest rate.

r_t = (1−θ)A_t(K_t^p)^−θ(N_t^p)^θ(K_t^g)^ν (2)

∗Asst. Professor, American University in Bulgaria, Department of Economics, Blagoevgrad 2700, Bul- garia. E-mail for correspondence: alvasilev@yahoo.com.

Now, multiply byK_t^p and rearrange terms. This gives the following relationship:

K_t^p(1−θ)At(K_t^p)^−θ(N_t^p)^θ(K_t^g)^ν =rtK_t^p or (1−θ)Yt =rtK_t^p (3) because

K_t^p(1−θ)At(K_t^p)^−θ(N_t^p)^θ(K_t^g)^ν =At(K_t^p)^1−θ(N_t^p)^θ(K_t^g)^ν = (1−θ)Yt

To derive firms’ optimal labor demand, set the derivative of the profit function with respect to the labor input equal to zero, holding technology and capital constant:

θAt(K_t^p)^1−θ(N_t^p)^θ−1(K_t^g)^ν −w^p_t = 0 or w^p_t =θAt(K_t^p)^1−θ(N_t^p)^θ−1(K_t^g)^ν (4) In equilibrium, firms will hire labor up to the point where the benefit of hiring an additional hour of labor services, which is the marginal product of labor, equals the cost, i.e the hourly wage rate.

Now multiply both sides of the equation by N_t^p and rearrange terms to yield

N_t^pθAt(K_t^p)^1−θ(N_t^p)^θ−1(K_t^g)^ν =w_t^pN_t^p or θYt =w^p_tN_t^p (5) Next, it will be shown that in equilibrium, economic profits are zero. Using the results above one can obtain

Πt = Yt−rtK_t^p−w^p_tN_t^p =Yt−(1−θ)Yt−θYt= 0 (6) Indeed, in equilibrium, economic profits are zero.

1.1.2 Consumer problem

Set up the Lagrangian

L(Ct, K_t+1^p , N_t^p; Λt) =E0

∞

X

t=0

(

(Ct+ωG^c_t)^ψ(1−Nt)^(1−ψ) 1−α

−1

1−α + (7)

+Λt

"

(1−τ^l)(w_t^pN_t^p+w^g_tN_t^g) + (1−τ^k)rtK_t^p + +τ^kδ^pK_t^p−(1 +τ^c)Ct−K_t+1^p + (1−δ)K_t^p

#)

This is a concave programming problem, so the FOCs, together with the additional, bound- ary (”transversality”) conditions for private physical capital and government bonds are both necessary and sufficient for an optimum.

To derive the FOCs, first take the derivative of the Lagrangian w.r.t Ct (holding all other variables unchanged) and set it to 0,i.e. L_C

t = 0. That will result in the following expression β^t

(1−α 1−α

(C_t+ωG^c_t)^ψ(1−N_t^h)^{(1−ψ) −}α

×

ψ(Ct+ωG^c_t)^ψ−1(1−N_t^h)^(1−ψ)−Λt(1 +τ^c) )

= 0 (8)

Cancel the β^t and the 1−α terms to obtain

(C_t+ωG^c_t)^ψ(1−N_t)^{(1−ψ) −}α

ψ(C_t+ωG^c_t)^ψ−1(1−N_t)^(1−ψ)−Λ_t(1 +τ^c) = 0 (9) Move Λt to the right so that

(Ct+ωG^c_t)^ψ(1−Nt)^{(1−ψ) −}α

ψ(Ct+ωG^c_t)^ψ−1(1−Nt)^(1−ψ)= Λt(1 +τ^c) (10) This optimality condition equates marginal utility of consumption to the marginal utility of wealth.

Now take the derivative of the Lagrangian w.r.tK_t+1^p (holding all other variables unchanged) and set it to 0, i.e. L_Kt+1^p = 0. That will result in the following expression

β^t (

−Λt+EtΛt+1

(1−τ^k)rt+1+τ^kδ^p + (1−δ^p) )

= 0 (11)

Cancel the β^t term to obtain

−Λt+βEtΛt+1

(1−τ^k)rt+1+τ^kδ^p+ (1−δ^p)

= 0 (12)

Move Λt to the right so that βEtΛt+1

(1−τ^k)rt+1+τ^kδ^p+ (1−δ^p)

= Λt (13)

Using the expression for the real interest rate shifted one period forward one can obtain rt+1 = (1−θ)Yt+1

K_t+1^p βEtΛt+1

(1−τ^k)(1−θ)Yt+1

K_t+1^p +τ^kδ^p+ (1−δ^p)

= Λt (14)

This is the Euler equation, which determines how consumption is allocated across periods.

Take now the derivative of the Lagrangian w.r.t N_t^p (holding all other variables unchanged) and set it to 0, i.e. L_Np

t = 0. That will result in the following expression β^t

(1−α 1−α

(Ct+ωG^c_t)^ψ(1−Nt)^{(1−ψ) −}α

×

(1−ψ)(C_t+ωG^c_t)^ψ(1−N_t)^−ψ(−1) + Λ_t(1−τ^l)w_t^p )

= 0 (15)

Cancel the β^t and the 1−α terms to obtain

(Ct+ωG^c_t)^ψ(1−Nt)^{(1−ψ) −}α

(1−ψ)

C_t+ωG^c_t 1−Nt

ψ

(−1) + Λt(1−τ^l)w^p_t = 0 (16) Rearranging, one can obtain

(Ct+ωG^c_t)^ψ(1−Nt)^{(1−ψ) −}α

(1−ψ)(Ct+ωG^c_t)^ψ(1−Nt)^−ψ = Λt(1−τ^l)w_t^p (17) Plug in the expression for w^h_t, that is,

w^p_t =θ Yt

N_t^p (18)

into the equation above. Rearranging, one can obtain

(Ct+ωG^c_t)^ψ(1−Nt)^{(1−ψ) −}α

(1−ψ)(Ct+ωG^c_t)^ψ(1−Nt)^−ψ = Λt(1−τ^l)θ Y_t

N_t^p (19) Transversality conditions need to be imposed to prevent Ponzi schemes, i.e borrowing bigger and bigger amounts every subsequent period and never paying it off.

tlim→∞β^tΛ_tK_t+1^p = 0 (20)

1.1.3 The Objective Function of a Public Sector Union: Derivation

This subsection shows that the objective function in the government sector is a generalized version of Stone-Geary monopoly union utility function used in Dertouzos and Pencavel (1981) and Brown and Ashenfelter (1986). The utility function is

V(w^g, N^g) = (w^g −w¯^g)^φ(N^g−N¯^g)^(1−φ), (21) where φ and 1−φ are the weights attached to public wage and hours, respectively, and ¯w^g and ¯N^g denote subsistence wage rate and hours. Since there is no minimum wage in the model, ¯w^g = 0. Additionally, as public hours are assumed to be unproductive, it follows that ¯N^g = 0 as well. Therefore, the utility function simplifies to

V(w^g, N^g) = (w^g)^φ(N^g)^(1−φ). (22) Doiron (1992) uses a generalized representation, which encompasses (2) as a special case when ρ→0.

φ(N^g)^−ρ+ (1−φ)(w^g−w)¯ ^{−ρ −}1/ρ

, (23)

when ¯w= 0, the function simplifies to

φ(N^g)^−ρ+ (1−φ)(w^g)^{−ρ −}1/ρ

, (24)

Union objective function used in the paper is very similar to Doiron’s (1992) simplified version:

(N^g)^ρ+η(w^g)^ρ 1/ρ

, (25)

can be transformed to

(N^g)^ρ+ φ

(1−φ)(w^g)^ρ 1/ρ

, (26)

Collecting terms under common denominator (1−φ)

(1−φ)(N^g)^ρ+ φ

(1−φ)(w^g)^ρ 1/ρ

, (27)

Factoring out the common term 1

1−φ 1/ρ

(1−φ)(N^g)^ρ+φ(w^g)^ρ 1/ρ

, (28)

Note that the constant term

1 1−φ

1/ρ

>0 can be ignored, as utility functions are invariant to positive affine transformations. After rearranging terms, the equivalent function

V˜ =

φ(w^g)^ρ+ (1−φ)(N^g)^ρ 1/ρ

. (29)

Take natural logarithms from both sides to obtain ln ˜V = 1

ρln

φ(w^g)^ρ+ (1−φ)(N^g)^ρ

. (30)

Take the limit ρ→0

limρ→0ln ˜V = lim

ρ→0

ln

φ(w^g)^ρ+ (1−φ)(N^g)^ρ

ρ (31)

Apply L’Hopital’s Rule on the R.H.S. to obtain

limρ→0ln ˜V = lim

ρ→0

∂

∂ρln

φ(w^g)^ρ+ (1−φ)(N^g)^ρ

∂ρ

∂ρ

(32) Thus

ln ˜V = lim

ρ→0

φ(w^g_t)^ρlnw^g+ (1−φ)(N^g)^ρlnN^g

/

φ(w^g)^ρ+ (1−φ)(N^g)^ρ

1 (33)

Simplify to obtain

ln ˜V =

limρ→0

φ(w^g_t)^ρlnw^g+ (1−φ)(N^g)^ρlnN^g

limρ→0

φ(w^g)^ρ+ (1−φ)(N^g)^ρ

= φlnw^g + (1−φ) lnN^g

φ+ (1−φ) (34)

Therefore,

ln ˜V =φlnw^g+ (1−φ) lnN^g. (35)

Exponentiate both sides of the equation to obtain

e^{ln ˜V} =e^{φlnwg+(1−φ) lnNg}. (36)

Thus

V˜ =e^{ln(wg)φ+ln(Ng)(1−φ)}. (37)

or

V˜ =e^{ln(wg)φ(Ng)(1−φ)}. (38) Finally,

V˜ = (w^g)^φ(N^g)^(1−φ) (39) Furthermore, government period budget constraint serves the role of a labor demand func- tion. Additionally, the public sector demand curve will be subject to shock, resulting from innovations to the fiscal shares. The balanced budget assumption is thus important in the model setup. Since wage bill is a residual, if wage rate is increased, then hours need to be decreased. Additionally, government period budget constraint can be expressed in the form N^g =N^g(w^g) as

N^g = τ^lw^pN^p +τ^k(r−δ^p)K^p+τ^cC−G^c−Gⁱ−G^t

(1−τ^l)w^g (40)

Therefore, the problem in the government sector is reshaped in the standard formulation in the union literature:

wmax^g,N^gV(w^g, N^g) s.t. N^g =N^g(w^g) (41)

Since union optimizes over both the public wage and hours, the outcome is efficient. The solution pair is on the contract curve (obtained from FOCs), at the intersection point with the labor demand curve (government budget constraint).

1.1.4 Public sector union optimization problem

The union solves the following problem:

wmax_t^g,N_t^g

(N_t^g)^ρ+η(w^g_t)^ρ 1/ρ

(42) s.t

G^c_t+G^t_t+Gⁱ_t+w^g_tN_t^g =τ^cCt+τ^krtK_t^p−τ^kδ^pKt+τ^l[w^p_tN_t^p +w^g_tN_t^g] (43) Setup the Lagrangian

V(w^g_t, N_t^g;νt) = max

w^g_t,N_t^g

(N_t^g)^ρ+η(w^g_t)^ρ 1/ρ

(44)

−νt

G^c_t +G^t_t+Gⁱ_t+w^g_tN_t^g−τ^cCt−τ^krtK_t^p+τ^kδ^pKt−τ^l[w_t^pN_t^p+w_t^gN_t^g]

Optimal public employment is obtained, when the derivative of the government Lagrangian is et to zero, i.e V

N_t^g = 0 (1/ρ)

(N_t^g)^ρ+η(w_t^g)^ρ

(1/ρ)−1

ρ(N_t^g)^ρ−1−(1−τ^l)ν_tw^g_t = 0 (45) or, when ρ is canceled out and (1−τ^l)νtw^g_t put to the right

(N_t^g)^ρ+η(w^g_t)^ρ

(1/ρ)−1

(N_t^g)^ρ−1 = (1−τ^l)ν_tw_t^g (46) Optimal public wage is obtained, when the derivative of the government Lagrandean is et to zero, i.e V_wg

t = 0 (1/ρ)

(N_t^g)^ρ+η(w_t^g)^ρ

(1/ρ)−1

ηρ(w^g_t)^ρ−1 −(1−τ^l)νtN_t^g = 0 (47) or, when ρ is canceled out and(1−τ^l)νtN_t^g term put to the right

(N_t^g)^ρ+η(w_t^g)^ρ

(1/ρ)−1

η(w_t^g)^ρ−1 = (1−τ^l)νtN_t^g (48) Divide (11.1.46) and (11.1.48) side by side to obtain

(N_t^g)^ρ+η(w^g_t)^ρ

(1/ρ)−1

(N_t^g)^ρ−1

(N_t^g)^ρ+η(w_t^g)^ρ

(1/ρ)−1

η(w_t^g)^ρ−1

= (1−τ^l)ν_tw_t^g

(1−τ^l)νtN_t^g (49)

Cancel out the common terms

(N_t^g)^ρ−1

η(w_t^g)^ρ−1 = w_t^g

N_t^g (50)

Now cross-multiply to obtain

(N_t^g)^ρ

η = (w_t^g)^ρ (51)

Hence

w_t^g = 1

η 1/ρ

N_t^g (52)

The wage bill expression, which is obtained after simple rearrangement of the government budget constraint, is as follows

w_t^gN_t^g = τ^cCt+τ^krtK_t^p−τ^kδ^pKt+τ^lw^p_tN_t^p−G^c_t−G^t_t−Gⁱ_t

1−τ^l (53)

Use the wage bill equation and the relationship between public wage and employment in order to obtain

w_t^g =η^−2ρ1

τ^cCt+τ^krtK_t^p−τ^kδ^pKt+τ^lw_t^pN_t^p−G^c_t−G^t_t−Gⁱ_t 1−τ^l

¹₂

(54) and

N_t^g =η^2ρ1

τ^cCt+τ^krtK_t^p−τ^kδ^pKt+τ^lw^p_tN_t^p−G^c_t −G^t_t−Gⁱ_t 1−τ^l

¹₂

(55)

1.2 Log-linearized model equations

1.2.1 Linearized market clearing

ct+k^p_t+1+g_t^c+g_tⁱ−(1−δ^p)k^p_t = yt (56) Take logs from both sides to obtain

ln[ct+k^p_t+1+g_t^c+g_tⁱ−(1−δ^p)k_t^p] = ln(yt) (57) Totally differentiate with respect to time

dln[ct+k^p_t+1+g_t^c+g_tⁱ−(1−δ^p)k_t^p]

dt = dln(yt) (58)

[ 1

c+g^c +gⁱ+δ^pk^p][dc_t dt

c c +dg^c_t

dt g g + dg_tⁱ

dt gⁱ

gⁱ + dk_t+1^p dt

k^p

k^p −(1−δ^p)dk^p_t dt

k^p

k^p] = dy_t dt

1

y (59) Define ˆz = ^dz_dt^t1_z. Thus passing to log-deviations

1

y[ˆctc+ ˆg_t^cg^c + ˆgⁱ_tgⁱ + ˆk_t+1^p k^p−(1−δ^p)ˆk^p_tk^p] = ˆyt (60) ˆ

ctc+ ˆg_t^cg^c + ˆgⁱ_tgⁱ + ˆk_t+1^p k^p−(1−δ^p)ˆk^p_tk^p = yˆyt (61) k^pˆk^p_t+1 = yˆy_t−cˆc_t−g^cˆg_t^c−gⁱˆg_tⁱ+ (1−δ^p)k^pkˆ^p_t (62) 1.2.2 Linearized production function

y_t = a_t(k^p_t)^1−θ(n^p_t)^θ(k^g_t)^ν (63) Take natural logs from both sides to obtain

lny_t = lna_t+ (1−θ) lnk_t^p+θlnn^p_t +νlnk^g_t (64) Totally differentiate with respect to time to obtain

dlny_t

dt = dlna_t

dt + (1−θ)dlnk^p_t

dt +θdlnn^p_t

dt +νdlnk^g_t

dt (65)

1 y

dyt

dt = 1 a

dat

dt +1−θ k^p

dk^p_t dt + θ

n^p dn^p_t

dt + ν k^g

dk_t^g

dt (66)

Pass to log-deviations to obtain

0 = −yˆ_t+ (1−θ)ˆk^p_t + ˆa_t+θnˆ^p_t +νkˆ_t^g (67)

1.2.3 Linearized FOC consumption

[(ct+ωg_t^c)^ψ(1−nt)^(1−ψ)]^−αψ(ct+ωg_t^c)^ψ−1(1−nt)^(1−ψ) = (1 +τ^c)λt (68) Simplify to obtain

ψ(ct+ωg_t^c)^{ψ−1−αψ}(1−nt)^{(1−α)(1−ψ)} = (1 +τ^c)λt (69) Take natural logs from both sides to obtain

lnψ(c_t+ωg_t^c)^{ψ−1−αψ}(1−n_t)^{(1−α)(1−ψ)} = ln(1 +τ^c) + lnλ_t (70) ln(c_t+ωg_t^c)^{ψ−1−αψ}(1−n_t)^{(1−α)(1−ψ)} = ln(1 +τ^c) + lnλ_t (71) (ψ−1−αψ) ln(c_t+ωg_t^c) + (1−α)(1−ψ) ln(1−n_t) = ln(1 +τ^c) + lnλ_t (72) Totally differentiate with respect to time to obtain

(ψ−1−αψ)dln(ct+ωg_t^c)

dt + (1−α)(1−ψ)dln(1−nt)

dt =

= dln(1 +τ^c)

dt +dlnλt

dt (73)

(ψ−1−αψ) 1

c+ωg^c(dct

dt +ωdg_t^c

dt ) + (1−α)(1−ψ) −1 1−n

dnt

dt = dλt

dt 1

λ (74) (ψ−1−αψ)

c+ωg^c dct

dt c

c +ω(ψ−1−αψ) c+ωg^c

dg_t^c dt

g^c g^c +

−(1−α)(1−ψ) 1 1−n

dn_t dt

n

n = dλ_t dt

1

λ (75)

c(ψ−1−αψ)

c+ωg^c cˆt+ ωg^c(ψ−1−αψ)

c+ωg^c gˆ_t^c−(1−α)(1−ψ) n

1−nnˆ = ˆλt (76) Since

ˆ

n = n^p

n^p+n^gˆn^p+ n^g

n^p+n^gnˆ^g = n^p

n nˆ^p+ n^g

nnˆ^g, (77)

and consumers choose n^p only, pass to log-deviations to obtain c(ψ−1−αψ)

c+ωg^c cˆt+ ωg^c(ψ−1−αψ)

c^c+ωg gˆ_t^c−(1−α)(1−ψ) n 1−n

n^p

n^p +n^gˆn^p = ˆλt (78) Since n=n^p +n^g, it follows that

c(ψ−1−αψ)

c+ωg^c ˆct+ωg^c(ψ−1−αψ)

c+ωg^c ˆg_t^c−(1−α)(1−ψ) n^p

1−nnˆ^p = 0 (79)

1.2.4 Linearized no-arbitrage condition for capital

λt = βEtλt+1[(1−τ^k)rt+1+τ^kδ^p+ (1−δ^p)] (80) Substitute out r_t+1 on the right hand side of the equation to obtain

λt = βEt[λt+1((1−τ^k)(1−θ)yt+1

k_t+1^p +τ^kδ^p+ 1−δ^p)] (81) Take natural logs from both sides of the equation to obtain

lnλt = lnEt[λt+1((1−τ^k)(1−θ)yt+1

k_t+1^p +τ^kδ^p+ 1−δ^p)] (82) Totally differentiate with respect to time to obtain

dlnλt

dt =

dlnEt[λt+1((1−τ^k)(1−θ)^y_k^t+1p

t+1 +τ^kδ^p+ 1−δ^p)]

dt (83)

1 λ

dλ_t dt =E_t

( 1

λ((1−τ^k)(1−θ)_k^yp + 1−δ^p+τ^kδ^p ×

"

((1−τ^k)(1−θ)y

k^p +τ^kδ^p + 1−δ^p)dλt+1

dt λ λ +λ(1−τ^k)(1−θ)

k^p

dyt+1

dt y y −

λ(1−τ^k)(1−θ)y (k^p)²

dk_t+1^p dt

k^p k^p

#)

(84) Pass to log-deviations to obtain

λˆ_t=E_t (

λˆ_t+1+

(1−τ^k)(1−θ)y ((1−τ^k)(1−θ)_k^yt+1p

t+1 +τ^kδ^p+ 1−δ^p)k^pyˆ_t+1

− (1−τ^k)(1−θ)y ((1−θ)^y_k^t+1p

t+1 +τ^kδ^p+ 1−δ^p)k^pkˆ_t+1^p )

(85) Observe that

(1−τ^k)(1−θ)yt+1

k_t+1^p +τ^kδ^p+ 1−δ^p = 1/β (86) Plug it into the equation to obtain

λˆt = Et

λˆt+1+β(1−τ^k)(1−θ)y

k^p yˆt+1− β(1−τ^k)(1−θ)y k^p ˆk^p_t+1

(87)

λˆt = Etλˆt+1+β(1−τ^k)(1−θ)y

k^p Etyˆt+1− β(1−τ^k)(1−θ)y

k^p Etˆk_t+1^p (88)

1.2.5 Linearized MRS

(1−ψ)(ct+ωg_t^c) = ψ(1−nt)(1−τ^l) (1 +τ^c)θy_t

n^p_t (89)

Take natural logs from both sides of the equation to obtain ln(1−ψ)(ct+ωg_t^c) = lnψ(1−nt)(1−τ^l)

(1 +τ^c)θyt

n^p_t (90)

ln(ct+ωg^c_t) = ln(1−nt) + lnyt−lnn^p_t (91) Totally differentiate with respect to time to obtain

dln(ct+ωg_t^c)

dt = dln(1−nt)

dt +dlnyt

dt −dlnn^p_t

dt (92)

1

c+ωg^c(dc_t

dt +ωdg_t^c

dt ) = − 1 1−n

dn_t dt + 1

y dy_t

dt − 1 n^p

dn^p_t

dt (93)

1 c+ωg^c

dct

dt c

c+ ω c+ωg^c

dg_t^c dt

g^c

g^c = − 1 1−n

dnt

dt n n + 1

y dyt

dt − 1 n^p

dn^p_t

dt (94)

c c+ωg^c

dc_t dt

1

c + ωg^c c+ωg^c

dg^c_t dt

1

g^c = − n 1−n

dn_t dt

1 n + 1

y dy_t

dt − 1 n^p

dn^p_t

dt (95)

Pass to log-deviations to obtain c

c+ωg^ccˆt+ ωg^c

c+ωggˆ^c_t = − n

1−nnˆ+ ˆyt−nˆ^p_t (96) Since

ˆ

n= n^p

n^p+n^gnˆ^p+ n^g

n^p+n^gnˆ^g, (97)

and noting that consumers are only choosing n^p, then c

c+ωg^ccˆt+ ωg^c

c+ωg^cˆg_t^c = − n 1−n

n^p

n^p+n^gˆn^p+ ˆyt−nˆ^p_t (98) c

c+ωg^ccˆt+ ωg^c

c+ωg^cˆg_t^c = − n 1−n

n^p

n^p+n^gˆn^p+ ˆyt−nˆ^p_t (99) c

c+ωg^cˆct+ ωg^c

c+ωg^cgˆ_t^c = −

1 + n 1−n

n^p n^p+n^g

ˆ

n^p+ ˆyt (100) Since n=n^p +n^g, it follows that

c

c+ωg^cˆct+ ωg^c

c+ωg^cgˆ^c_t = −

1 + n^p 1−n

ˆ

n^p+ ˆyt (101) c

c+ωg^cˆct+ ωg^c

c+ωg^cˆg_t^c+

1 + n^p 1−n

ˆ

n^p−yˆt = 0 (102)

1.2.6 Linearized private capital accumulation

k_t+1^p = it+ (1−δ^p)k_t^p (103) Take natural logs from both sides of the equation to obtain

lnk^p_t+1 = ln(it+ (1−δ^p)k_t^p) (104) Totally differentiate with respect to time to obtain

dlnk^p_t+1

dt = 1

i+ (1−δ^p)k^p

d(it+ (1−δ^p)k_t^p)

dt (105)

Observe that since

i=δ^pk^p, it follows that i+ (1−δ^p)k^p =δ^pk^p+ (1−δ^p)k^p =k^p. Then (106) dk_t+1^p

dt 1

k^p = 1 k^p

di_t dt i

i + k^p i+ (1−δ^p)k_t^p

dk^p_t dt

k^p

k^p (107)

Pass to log-deviations to obtain

ˆk^p_t+1 = δ^pk^p

k^p ˆit+ (1−δ^p)k^p

k^p kˆ^p_t (108)

kˆ_t+1^p = δ^pˆit+ (1−δ^p)ˆk_t^p (109) 1.2.7 Linearized government capital accumulation

k^g_t+1 = g_tⁱ+ (1−δ^g)k^g_t (110) Take natural logs from both sides to obtain

lnk_t+1^g = ln(g_tⁱ+ (1−δ^g)k^g_t) (111) Totally differentiate with respect to time to obtain

dlnk_t+1^g

dt = 1

gⁱ+ (1−δ^g)k^g

d(gⁱ_t+ (1−δ^g)k_t^g)

dt (112)

Observe that since

gⁱ =δ^gk^g, (113)

it follows that

gⁱ+ (1−δ^g)k^g =δ^gk^g+ (1−δ^g)k^g =k^g. (114) Hence,

dk_t+1^g dt

1 k^g = 1

k^g dgⁱ_t

dt gⁱ

gⁱ + k^g x+ (1−δ^g)

dk^g_t dt

k^g

k^g (115)

Pass to log-deviations to obtain

kˆ_t+1^g = δ^gk^g

k^g ˆg_tⁱ+(1−δ^g)k^g

k^g ˆk_t^g (116)

Cancel out the k^g terms to obtain

ˆk^g_t+1 = δ^gˆg_tⁱ+ (1−δ^g)ˆk_t^g (117) 1.2.8 Public wage rate rule

w^g_t =η^−2ρ1

τ^cct+τ^krtk_t^p−τ^kδ^pk^p_t +τ^lw_t^pn^p_t −g^c_t−g^t_t−g_tⁱ 1−τ^l

¹₂

(118) Take logs from both sides to obtain

lnw^g_t =− 1

2ρlnη− 1

2ln(1−τ^l) + 1

2ln

τ^cct+τ^krtk_t^p−τ^kδ^pk_t^p+τ^lw_t^pn^p_t −g^c_t −g_t^t−g_tⁱ

(119) Totally differentiate with respect to time to obtain

dlnw^g_t dt = 1

2 d dtln

τ^cct+τ^krtk_t^p−τ^kδ^pk_t^p+τ^lw_t^pn^p_t −g^c_t −g_t^t−g_tⁱ

(120) Observe that

τ^krtk^p_t −τ^kδ^pkt+τ^lw_t^pn^p_t =τ^k(1−θ)yt+τ^lθyt−τ^kδ^pk^p_t =

=

τ^k(1−θ) +τ^lθ

yt−τ^kδ^pk_t^p (121) Also

(1−τ^l)w^gn^g =τ^cc+ [τ^k(1−θ) +τ^lθ]y−τ^kδ^pk^p−g^c−gⁱ−g_t^t (122)

Thus dw_t^g

dt 1 w^g = 1

2

1 (1−τ^l)w^gn^g

τ^cdct

dt + [τ^k(1−θ) +τ^lθ]dyt

dt −τ^kδ^pdk^p_t dt − dg_t^c

dt − dg_tⁱ dt − dg_t^t

dt

(123) dw_t^g

dt 1 w^g = 1

2

1

(1−τ^l)w^gn^g ×

τ^cdct

dt c c+

τ^k(1−θ) +τ^lθ dyt

dt y

y −τ^kδ^pdk_t^p dt

k^p k^p − dg_t^c

dt g^c g^c − dg_tⁱ

dt gⁱ gⁱ − dg^t_t

dt g^t g^t

(124)

dw_t^g dt

1

w^g = (1/2)τ^cc (1−τ^l)w^gn^g

dct

dt 1 c +

(1/2)

τ^k(1−θ) +τ^lθ

y (1−τ^l)w^gn^g

dyt

dt 1 y

−(1/2)τ^kδ^pk^p (1−τ^l)w^gn^g

dk_t^p dt

1

k^p − (1/2)g^c (1−τ^l)w^gn^g

dg^c_t dt

1 g^c

− (1/2)gⁱ (1−τ^l)w^gn^g

dgⁱ_t dt

1

gⁱ − (1/2)g^t (1−τ^l)w^gn^g

dg_t^t dt

1

g^t (125)

Pass to log-deviations to obtain

ˆ

w^g_t = (1/2)τ^cc (1−τ^l)w^gn^gcˆt+

(1/2)

τ^k(1−θ) +τ^lθ

y (1−τ^l)w^gn^g yˆt

−(1/2)τ^kδ^pk^p

(1−τ^l)w^gn^gˆkt− (1/2)g^c

(1−τ^l)w^gn^ggˆ^c_t − (1/2)gⁱ

(1−τ^l)w^gn^gˆg_tⁱ− (1/2)g^t

(1−τ^l)w^gn^ggˆ_t^t (126) 1.2.9 Public hours/employment rule

n^g_t = η^1ρw^g_t (127)

Take logs from both sides to obtain

lnn^g_t = 1

ρlnη+ lnw^g_t (128)

Totally differentiate both sides to obtain dlnn^g_t

dt = dlnw^g_t

dt (129)

dn^g_t dt

1

n^g = dw^g_t dt

1

w^g (130)

Pass to log-deviations to obtain

ˆ

n^g_t = wˆ_t^g (131)

1.2.10 Total hours/employment

nt = n^g_t +n^p_t (132)

Take logs from both sides to obtain

lnnt = ln(n^g_t +n^p_t) (133)

Totally differentiate to obtain

dlnn_t

dt = dln(n^g_t +n^p_t)

dt (134)

dnt

dt 1 n =

dn^g_t

dt + dn^p_t dt

1

n (135)

dnt

dt 1 n =

dn^g_t dt

n^g

n^g + dn^p_t dt

n^p n^p

1

n (136)

dn_t dt

1

n = dn^g_t dt

1 n^g

n^g

n + dn^p_t dt

1 n^p

n^p

n (137)

Pass to log-deviations to obtain ˆ

nt = n^g

n nˆ^g_t +n^p

n nˆ^p_t (138)

1.2.11 Linearized private wage rate

w^p_t =θyt

n^p_t (139)

Take natural logarithms from both sides to obtain

lnw_t^p = lnθ+ lnyt−lnn^p_t (140) Totally differentiate with respect to time to obtain

dlnw^p_t

dt = dlnθ

dt +dlnyt

dt −dlnn^p_t

dt (141)

Simplify to obtain

dw^p_t dt

1

w^p = dyt

dt 1

y − dn^p_t dt

1

n^p (142)

Pass to log-deviations to obtain

ˆ

w^p_t = ˆy_t−nˆ^p_t (143)

1.2.12 Linearized real interest rate

r_t=θyt

k^p_t (144)

Take natural logarithms from both sides to obtain

lnrt = lnθ+ lnyt−lnk_t^p (145) Totally differentiate with respect to time to obtain

dlnrt

dt = dlnθ

dt +dlnyt

dt − dlnk_t^p

dt (146)

Simplify to obtain

dr dt

1 r = dyt

dt 1 y − dk_t^p

dt 1

k^p (147)

Pass to log-deviations to obtain

ˆ

r_t = ˆy_t−ˆk_t^p (148)

1.2.13 Public/private wage ratio

rwt = w^g_t/w^p_t (149)

Take logs from both sides of the equation

lnrwt = lnw_t^g−lnw^p_t (150)

Totally differentiate to obtain

dlnrwt

dt = dlnw_t^g

dt −dlnw^p_t

dt (151)

drwt

dt 1

rw = dw^g_t dt

1

w^g − dw_t^p dt

1

w^p (152)

Pass to log-deviations to obtain

ˆ

rwt = wˆ^g_t −wˆ^p_t (153)

1.2.14 Public/private hours/employment ratio

rl_t = n^g_t/n^p_t (154)

Take logs from both sides of the equation

lnrlt = lnn^g_t −lnn^p_t (155)

Totally differentiate to obtain

dlnrlt

dt = dlnn^g_t

dt − dlnn^p_t

dt (156)

drlt

dt 1

rl = dn^g_t dt

1

n^g −dn^p_t dt

1

n^p (157)

Pass to log-deviations to obtain

rlˆ_t = ˆn^g_t −nˆ^p_t (158)

1.2.15 Linearized technology shock process

lnat+1 = ρalnat+ǫ^a_t+1 (159) Totally differentiate with respect to time to obtain

dlnat+1

dt = ρa

dlnat

dt +dǫ^a_t+1

dt (160)

dat+1

dt = ρ_adat

dt +ǫ^a_t+1 (161)

where for t = 1 ^dǫ_dt^at+1 ≈ ln(e^ǫat+1/e^ǫa) = ǫ^a_t+1−ǫ^a =ǫ^a_t+1 since ǫ^a = 0. Pass to log-deviations to obtain

ˆ

at+1 = ρaˆat+ǫ^a_t+1 (162)