∞j
Vt =Et∑∏(1+rt+k)-1{(1-t )[Yt+
t t t+ c t+ j
j=0 k=0
-wte+jLt+j]-It+j}
(24)
subject to the technology, the adjustment cost and the capital accumulation constraint. Define with
λk the multiplier associated with the constraint on capital respectively. Differentiating the
objective function with respect to Kt+j, Jt+j, Lt+j (j=0,1....), gives the following system of
stochastic Euler equations (subject to the transversality condition)
(1 - tj(1 - α ) Yt
Kt+j
( J ï2
(rt+.. + δ)λL. - ° ' - E [λk+j.+1 - λkk+] ]
∖ t + j t t + j ɔ t + j L t + j+1 t + j J
21 Kt+j )
Jt+j
Kt+j
Yt+j
Lt+j
=w
e
t+j
(25)
(26)
(27)
Equation (25) is the equation of motion of the marginal shadow value of capital λk. Equation (26)
is the first order condition for total investment and it implies that the cost of a marginal unit of
capital, including both its purchase and adjustment costs, must equal the shadow value of capital
λk. It has been shown by Hayashi (1980) that marginal and average value of Tobin’s Q coincides
under the technology and market structure assumed here, i. e. λk = q . The cost of capital includes
both the pure rental price and adjustment costs. Equations (27) defines labour demand
Unlike in the goods market we assume imperfect competition in the labour market in both
countries. Instead of deriving a labour supply equation from the household optimisation problem
we assume a standard wage rule which can be derived from various labour market models. As
discussed by Pissarides (1998) for example, the following generic wage rule
could be derived from alternatively from search model, a union bargaining or an efficiency wage
model of the labour market. According to this rule, wage costs are a weighted average of the
reservation wage (wu) labour productivity plus an additional mark-up term that depends positively
on labour market tightness.
(Y
wt = (1- χ)wt + χ αj-+
V Lt
vt
LFt / Lt -1
(28)
The government provides three types of transfers to households: it pays unemployment benefits
(LFt - Lt)wtu , subsidises pension transfers at the amount ( TRPENt) and provides lump sum
transfers (TRt). In addition the government purchases goods and services ( Gt ). Expenditures are
financed by labour income and company taxes plus taxes on consumption. The tax rates on wages
(tl ) and corporate income (tc ) as well as the consumption tax rate (tv ) are assumed to be constant.
Alternatively, the government can issue debt. Thus the government budget constraint is given by
Bt+1 =(1+rt)Bt+(LFt - Lt)wtu +Gt +TRt+TRPENt-tlwtLt -t (Yt -wteLt)-t Ct (29)
t+ t t t t t t t t t t c t t t v t
77
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