partial derivative of F(∙) with respect to j = K, E, the Hamiltonian conditions are
|
∂H = |
,, , dm n |
(9) | ||
|
∂V |
O |
•■:> |
-e cυ + μ1 ∂v = o | |
|
∂H |
-∕11 = e~rt[F- - w] - μ1v |
(10) | ||
|
-μι = |
∂E |
•■:> | ||
|
E = |
∂H ⅜ι |
•■:> |
E = m(Û, V) - vE |
(11) |
|
∂H _ |
—e c + μ2 = O |
(12) | ||
|
~∂i = |
O |
•■:> | ||
|
∂H |
-μ2 = e-rt[F- - r] |
(13) | ||
|
-μ2 = |
∂K |
•■:> | ||
|
∂H |
(14) | |||
|
K = |
⅜2 |
•■:> |
K = I | |
|
with the transversality condition | ||||
Iim H(t) = O.
t→∞
The first order condition for capital respectively labor are given by
Fκ(k) = (1 + c1 ) r (15)
ʌ
Fe(k) = w + ʌ ccv r - λ + β (Û - V) + V θf3 (16)
1 — β L ∖ / _l
with F (k) [j = K, E] as marginal products and the right hand sides are marginal
costs of capital respectively labor.
After describing the intertemporal optimization problem of the representative
firm, the model has to be closed by denoting aggregate income and the budget
constraint. Factor income of the households Y is defined as the remuneration of
production factors capital and labor
Y := rK + wE
(17)
with the wage rate w.
The output is used for factor income Y, installation costs cjl and search costs
cvV :
X = Y + c1I + cvV.
(18)
Both of the last terms represent the profit income of firms that is completely used
for installation and search costs (1 — ω)Fε(k) = c√ + cυV.
17