| Sorry, I have sent to you a backup instead of the emergency file :-(
--
Prof. Murat Yildizoglu |
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\begin_body
\begin_layout Title
Structure and components of the model
\end_layout
\begin_layout Section
Households
\end_layout
\begin_layout Standard
Consume the final good and supply labour, which can be skilled or unskilled.
\end_layout
\begin_layout Section
Capital goods production and market
\begin_inset CommandInset label
LatexCommand label
name "sec:Capital-goods-production"
\end_inset
\end_layout
\begin_layout Standard
At the end of the period, the firm computes its cash-flow
\begin_inset Formula
\[
CF_{t}=\Pi_{t}+\left(1+i^{D}\right)D_{t-1}-\left(1+i^{C}\right)C_{t-1}
\]
\end_inset
\end_layout
\begin_layout Standard
and computes the budget necessary for financing the labour and invest for
the next period, and the resulting credit demand
\begin_inset Formula
\[
C_{t}^{d}=max\left\{
w_{t+1}N_{t+1}^{d}+p_{K,t+1}^{e}I_{t+1}^{d}-CF_{t},0\right\}
\]
\end_inset
\end_layout
\begin_layout Section
Final good production and market
\end_layout
\begin_layout Standard
Production using intermediate goods and unskilled labour.
\end_layout
\begin_layout Standard
No innovation in this sector, but different vintage of intermediate goods
with increasing productivity.
\end_layout
\begin_layout Section
Banking
\end_layout
\begin_layout Section
R&D and innovations
\end_layout
\begin_layout Standard
Only K-firms do R&D and innovations
\begin_inset Formula
\[
R_{l}=n_{R,l}N_{l}^{S}
\]
\end_inset
\end_layout
\begin_layout Standard
Sharing between radical and incremental innovation/imitation orientated
R&D using the proportion
\begin_inset Formula $v$
\end_inset
for the radical R&D:
\begin_inset Formula
\[
R^{r}=vR_{l}
\]
\end_inset
\end_layout
\begin_layout Standard
and incremental R&D is
\begin_inset Formula
\[
R^{i}=(1-v)R_{l}
\]
\end_inset
\end_layout
\begin_layout Subsection
Innovations
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
noindent
\end_layout
\end_inset
\series bold
Incremental innovations
\end_layout
\begin_layout Standard
Number of incremental innovative draws are given by a Poisson law given
by the parameters
\begin_inset Formula $\psi^{inc}R^{inc}$
\end_inset
, where
\begin_inset Formula $\psi^{inc}$
\end_inset
is the productivity of the incremental R&D expenditure:
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
n_{\tau}^{inc}\rightsquigarrow max\left\{
0,\mathcal{P}\left(\psi^{r}R^{r}\right)-\Omega_{\tau}^{inc}\right\}
\]
\end_inset
where
\begin_inset Formula $d\Omega_{\tau}^{inc}/d\tau$
\end_inset
>0.
\begin_inset Formula
\[
\Omega_{\tau}^{inc}=\psi^{inc}v(0)R^{inc}(0)-1+\varpi^{inc}\tau
\]
\end_inset
In the case of an incremental innovation, the new production technology
of the firm is drawn from a multinomial normal distribution:
\begin_inset Formula
\[
P_{K}^{K}\rightsquigarrow
N\left(P_{K,\tau},\sigma_{K,\tau}\right),P_{\tau}^{K}\rightsquigarrow
N\left(P_{L,\tau},\sigma_{K,\tau}\right)
\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\tau$
\end_inset
: a trajectory, firms can change trajectory by doing enough radical R&D.
Only the technology yielding the highest expected average profit of of
the potential customers is adopted (unit profit computed using average
values of
\begin_inset Formula $p,w,P_{L}^{K},P_{K}^{K}$
\end_inset
and its price for the capital good):
\begin_inset Formula
\begin{equation}
\pi_{l}=\left(\bar{p}-\frac{\bar{w}}{P_{L,l}^{K}}\right)P_{K,l}^{K}-p_{l}^{K}/\theta\label{eq:profit-rate-technique}
\end{equation}
\end_inset
\end_layout
\begin_layout Standard
with only the first term being relevant when comparing different machines:
\begin_inset Formula $\left(p_{l}^{K}/m\right)$
\end_inset
is the same for all technologies.
Consequently, the machine with the highest value of the first term
\begin_inset Formula
\[
P_{K,l}^{K}\left(1-\frac{\bar{w}}{\bar{p}}\frac{1}{P_{L,l}^{K}}\right)
\]
\end_inset
\end_layout
\begin_layout Standard
will be adopted as the new product (machine) to sell.
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
noindent
\end_layout
\end_inset
\series bold
Radical innovations
\end_layout
\begin_layout Standard
In the case of the radical innovation, the firm can change the trajectory
\begin_inset Formula $\tau=\left(P_{\tau}^{K},P_{\tau}^{L}\right)$
\end_inset
.
\begin_inset Formula $\tau$
\end_inset
is increasing with the distance of the technology to the origin.
\end_layout
\begin_layout Standard
Initially, we draw
\begin_inset Formula $\varrho$
\end_inset
different trajectories using
\begin_inset Formula
\[
P_{\tau}^{K}\rightsquigarrow
U\left(P_{min}^{K},P_{max}^{K}\right),\thinspace\thinspace\thinspace
P_{\tau}^{L}\rightsquigarrow U\left(P_{min}^{L},P_{max}^{L}\right)
\]
\end_inset
\end_layout
\begin_layout Standard
and
\begin_inset Formula
\[
\sigma_{K,\tau}=25P_{\tau}^{K},\sigma_{L,\tau}=25P_{\tau}^{L}
\]
\end_inset
\end_layout
\begin_layout Standard
in order to exhaust any technology
\begin_inset Formula $\tau$
\end_inset
in approximately 25 years in both dimensions.
\end_layout
\begin_layout Standard
Given the current technology
\begin_inset Formula $\tau_{t}$
\end_inset
of the firm, the firm draws a number of radical innovation draws that follows
the truncated Poisson law:
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
n_{\tau}^{rad}\rightsquigarrow max\left\{
0,\mathcal{P}\left(\psi^{rad}R^{rad}\right)-\Omega_{\tau}^{rad}\right\}
\]
\end_inset
where
\begin_inset Formula $d\Omega_{\tau}^{rad}/d\tau$
\end_inset
>0.
\begin_inset Formula
\[
\Omega_{\tau}^{rad}=\psi^{rad}v(0)R^{rad}(0)-1+\varpi^{rad}\tau
\]
\end_inset
\end_layout
\begin_layout Standard
Each of the
\begin_inset Formula $n_{\tau}^{rad}$
\end_inset
technologies discovered by the firm is drawn using a roulette-wheel where
the proportion of each technology is decreasing with its distance to the
current technology.
\end_layout
\begin_layout Standard
The firms must now discover a technique from this new technology
\begin_inset Formula $\tau'$
\end_inset
:
\begin_inset Formula $a^{î}$
\end_inset
\end_layout
\begin_layout Standard
It draws a new technique
\begin_inset Formula $\left(P^{K},P^{L}\right)$
\end_inset
between its current technique and the technology
\begin_inset Formula $\tau'$
\end_inset
following a uniform distribution and draws a new technique around this
point, using a normal distribution with means
\begin_inset Formula $\left(P^{K},P^{L}\right)$
\end_inset
and with standard deviations
\begin_inset Formula $\sigma_{K,\tau'}$
\end_inset
and
\begin_inset Formula $\sigma_{L,\tau'}$
\end_inset
.
\end_layout
\begin_layout Standard
At the end the firm compares the
\begin_inset Formula $m_{\tau}$
\end_inset
new techniques discovered and chooses the technology that yields the highest
profit using current economic data (average prices and wages, see
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:profit-rate-technique"
\end_inset
).
\end_layout
\begin_layout Subsection
Imitation
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
noindent
\end_layout
\end_inset
\series bold
Incremental imitation
\end_layout
\begin_layout Standard
Given the current technology
\begin_inset Formula $\tau$
\end_inset
of the firm, the firm draws a number
\begin_inset Formula $m_{\tau}^{inc}$
\end_inset
of imitation trials in the set of firms using the same technology.
\begin_inset Formula $m_{\tau}^{inc}$
\end_inset
follows the truncated Poisson law:
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
m_{\tau}^{inc}\rightsquigarrow max\left\{
0,\mathcal{P}\left(\psi^{inc}R^{inc}\right)-\varGamma_{\tau}^{inc}\right\}
\]
\end_inset
where
\begin_inset Formula $d\Gamma_{\tau}^{inc}/d\tau$
\end_inset
>0.
\begin_inset Formula
\[
\Gamma_{\tau}^{inc}=\psi^{in}v(0)R^{inc}(0)-1+\gamma^{inc}\tau
\]
\end_inset
\end_layout
\begin_layout Standard
Firm
\begin_inset Formula $i$
\end_inset
makes
\begin_inset Formula $m^{inc}$
\end_inset
draws from a roulette-wheel where the probability of being imitated for
competitor
\begin_inset Formula $j$
\end_inset
, using the same technology as
\begin_inset Formula $i$
\end_inset
, is proportional to
\begin_inset Formula $ms_{j}/D_{ij},$
\end_inset
where
\begin_inset Formula $ms$
\end_inset
is the real actual market share and
\begin_inset Formula $D$
\end_inset
is the technological distance between
\begin_inset Formula $i$
\end_inset
and
\begin_inset Formula $j$
\end_inset
.
\end_layout
\begin_layout Standard
It draws a new technique
\begin_inset Formula $\left(P^{K},P^{L}\right)$
\end_inset
between its current technique and the technique of each imitated firm following
a uniform distribution.
\end_layout
\begin_layout Standard
At the end the firm compares the
\begin_inset Formula $m_{\tau}^{inc}$
\end_inset
new techniques discovered through imitation, and chooses the technique
that yields the highest profit using current economic data (average prices
and wages, see
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:profit-rate-technique"
\end_inset
).
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
noindent
\end_layout
\end_inset
\series bold
Radical imitation
\end_layout
\begin_layout Standard
Given the current technology
\begin_inset Formula $\tau$
\end_inset
of the firm, the firm draws a number
\begin_inset Formula $m_{\tau}^{rad}$
\end_inset
of radical imitation draws, following the truncated Poisson law:
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
m_{\tau}^{rad}\rightsquigarrow max\left\{
0,\mathcal{P}\left(\psi^{rad}R^{rad}\right)-\varGamma_{\tau}^{rad}\right\}
\]
\end_inset
where
\begin_inset Formula $d\Gamma_{\tau}^{rad}/d\tau$
\end_inset
>0.
\begin_inset Formula
\[
\Gamma_{\tau}^{rad}=\psi^{rad}v(0)R^{rad}(0)-1+\gamma^{rad}\tau
\]
\end_inset
Firm
\begin_inset Formula $i$
\end_inset
makes
\begin_inset Formula $m^{rad}$
\end_inset
draws from a roulette-wheel where the probability of being imitated for
any competitor
\begin_inset Formula $j$
\end_inset
in the industry is proportional to
\begin_inset Formula $ms_{j}/D_{ij},$
\end_inset
where
\begin_inset Formula $ms$
\end_inset
is the real actual market share and
\begin_inset Formula $D$
\end_inset
is the technological distance between the productivities of
\begin_inset Formula $i$
\end_inset
and
\begin_inset Formula $j$
\end_inset
.
\end_layout
\begin_layout Standard
It draws a new technique
\begin_inset Formula $\left(P^{K},P^{L}\right)$
\end_inset
between its current technique and the technique of each imitated firm following
a uniform distribution and draws a new technique around this point, using
a normal distribution with means
\begin_inset Formula $\left(P^{K},P^{L}\right)$
\end_inset
and with standard deviations
\begin_inset Formula $\sigma_{K,\tau'}$
\end_inset
and
\begin_inset Formula $\sigma_{L,\tau'}$
\end_inset
, where
\begin_inset Formula $\tau'$
\end_inset
is the technology of the imitated firm.
\end_layout
\begin_layout Standard
At the end the firm compares the
\begin_inset Formula $m_{\tau}^{rad}$
\end_inset
new techniques discovered through imitation, and chooses the technique
and the technology
\begin_inset Formula $\tau'$
\end_inset
that yields the highest profit using current economic data (average prices
and wages, see
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:profit-rate-technique"
\end_inset
).
\end_layout
\begin_layout Standard
(Possibility of taking into account public research as way to guide firms'
voyage through the paradigm space).
\end_layout
\begin_layout Section
Adaptive behavior and expectations
\end_layout
\begin_layout Subsection
Firms behavior
\end_layout
\begin_layout Standard
Nominal wage is based on the desired growth rate of the wage given by
\begin_inset Formula
\[
g_{w}=max\left\{ 0,g_{N^{d}}-u^{S}\right\}
^{\phi_{1}}\left(1+g_{\Pi}-\pi\right)^{\phi_{2}}\left(1+\pi\right)^{\phi_{3}}
\]
\end_inset
which gives
\begin_inset Formula $w_{t}=(1+g_{w})w_{t-1}$
\end_inset
.
\end_layout
\begin_layout Standard
Setting the the number of researchers
\begin_inset Formula
\[
g_{R}=\rho_{1}log\left(\frac{\bar{P_{L}}}{P_{L}}\right)+\rho_{2}max\left\{
0,g_{\bar{P_{L}}}-g_{P_{L}}\right\}
\]
\end_inset
which gives
\begin_inset Formula $R_{t}=(1+g_{R})R_{t-1}$
\end_inset
.
Where
\begin_inset Formula $\bar{P_{L}}$
\end_inset
is the weighted average.
\end_layout
\begin_layout Standard
The share of radical R&D depends on the relative productivities of the firm
and the relative diffusion of its technology:
\begin_inset Formula
\[
\nu_{t}=\nu_{t-1}+\zeta_{1}log\left(\frac{\bar{P_{L}}}{P_{L}}\right)+\zeta_{2}log\left(\frac{\bar{P_{K}}}{P_{K}}\right)+\zeta_{3}ms_{\tau_{j}}
\]
\end_inset
where
\begin_inset Formula $ms_{\tau_{j}}$
\end_inset
is the share of firms using firm j's technology in the population.(Check!)
\end_layout
\begin_layout Standard
Markup rate of the firm is based
\end_layout
\begin_layout Standard
\begin_inset Formula $Q^{d}$
\end_inset
...
\end_layout
\begin_layout Subsection
Households behavior
\end_layout
\begin_layout Subsection
Bank's behavior
\end_layout
\begin_layout Subsection
Government's behavior
\end_layout
\begin_layout Standard
\begin_inset Newpage clearpage
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
small
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Flex Multiple Columns
status open
\begin_layout Plain Layout
\begin_inset Tabular
