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main.Rmd
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---
title: "FamiliaR with Dynamics"
output:
flexdashboard::flex_dashboard:
orientation: column
social: menu
source_code: embed
theme: spacelab
runtime: shiny
---
<style>
.form-control {
width: 90%;
}
</style>
```{r global, include=FALSE}
# load data in 'global' chunk so it can be shared by all users of the dashboard
library(datasets)
library(ggplot2)
library(bslib)
library(shinycssloaders)
options(shiny.maxRequestSize = 30*1024^2)
data(faithful)
Solow1 <- function(s,k,a){
y <- s*k^a
return <- y
}
Solow2 <- function(d,k){
y <- d*k+k^-1
}
solow_de <- function(t,y,parameters){
s <- parameters[1]
a <- parameters[2]
del <- parameters[3]
#k <- parameters[4]
rhs <- s*y^a
lhs <- del*y+(1/y)
dy = rhs-lhs
list(dy)
}
goodwin2<-function(t, state, parameters) {
with(as.list(c(state, parameters)),{
sigma <- parameters[1]
alpha <- parameters[2]
beta <- parameters[3]
gamma <- parameters[4]
rho <- parameters[5]
A <- 1/sigma - alpha - beta
B <- 1/sigma
C <- gamma+alpha
D <- rho
dV <- (A-B*U)*V
dU <- (-C+D*V)*U
dQ <- ((1-U)/sigma)*Q
list(c(dV, dU, dQ))})}
goodwin <- function(t, y, parameters) {
x <- y[1]
y <- y[2]
sigma <- parameters[1]
alpha <- parameters[2]
beta <- parameters[3]
gamma <- parameters[4]
rho <- parameters[5]
A <- 1/sigma - alpha - beta
B <- 1/sigma
C <- gamma+alpha
D <- rho
dy <- numeric(2)
dy[1] <- A*x - B*x*y
dy[2] <- -C*y +D*y*x
list(dy)
}
ramsey <- function(t, y, parameters) {
x <- y[1] #capital dc/dt
y <- y[2] #consumption dk/dt
theta <- parameters[1]
rho <- parameters[2]
delta <- parameters[3]
alpha <- parameters[4]
dy <- numeric(2)
r <- alpha*x^(alpha-1)
dy[1] <- x^alpha-y-delta*x
dy[2] <- (y*(r-theta-delta))/delta
list(dy)
}
shone <- function(t, y, parameters){
ry <- y[1] # real output
pie <- y[2] # expected inflation
c0 <- parameters[1]
i0 <- parameters[2]
G <- parameters[3]
b <- parameters[4]
k <- parameters[5]
d <- parameters[6]
h <- parameters[7]
tau <- parameters[8]
alpha <- parameters[9]
beta <- parameters[10]
mdot <- parameters[11]
yn <- parameters[12]
a0 <- (c0+i0+G)/(1-b*(1-tau)+((h*k)/d))
a1 <- (h/d)/(1-b*(1-tau)+((h*k)/d))
a2 <- h/(1-b*(1-tau)+((h*k)/d))
dy <- numeric(2)
dy[1] <- a1*mdot-alpha*(a1-a2*beta)*(ry-yn)-a1*pie
dy[2] <- beta*alpha*(ry-yn)
list(dy)
}
```
Dynamics {data-navmenu='Models' data-icon='ion-levels'}
=======================================================================
Column {.sidebar}
-----------------------------------------------------------------------
Select your model and the value of each parameters.
```{r}
selectInput("model", label = "Economic Model:",
choices = c("Solow", "Ramsey", "Shone", "Goodwin"), selected = "Solow")
conditionalPanel("input.model == 'Solow'",
sliderInput("k", label = "Chart length (k):", min = 1, max = 100, value = 20, step = 1),
sliderInput("s", label = "Saving rate:", min = 0, max = 1, value =0.2, step = 0.01),
sliderInput("a", label = "Elasticty:", min = 0, max = 1, value =0.3, step = 0.01),
sliderInput("d", label = "Depreciation:", min = 0, max = 1, value =0.02, step = 0.01),
sliderInput("n", label = "Population growth rate:", min = 0, max = 1, value =0.5, step = 0.01),
sliderInput("t", label = "Time", min = 0, max = 100, value =20, step = 5)
)
conditionalPanel("input.model == 'Ramsey'",
sliderInput("tr", label = "Time", min = 0, max = 200, value = 50, step = 5),
sliderInput("cr", label = "Consumption, dc/dt",min = 0, max = 5, value = 3, step = 0.05),
sliderInput("kr", label = "Capital, dk/dt", min = 0, max = 50, value = 30, step = 0.5),
sliderInput("thetar", label = "Elasticity, tendendy of consumers to smooth their consumption over time:", min = 0, max = 1, value = 0.03, step = 0.01),
sliderInput("rhor", label = "Rate at which consumers discount their future consumption", min = 0, max = 1, value = 0.2, step = 0.01),
sliderInput("deltar", label = "Depreciation rate of capital", min = 0, max = 1, value = 0.1, step = 0.01),
sliderInput("alphar", label = "Elasticity parameter (measuring the responsiveness of the output production to changes in the input capital):", min = 0, max = 1, value = 0.3, step = 0.01),
)
conditionalPanel("input.model == 'Goodwin'",
sliderInput("alphag", label = "Technical progress growth rate", min = 0, max = 1, value = 0.0196, step = 0.0001),
sliderInput("betag", label = "Labour force growth rate",min = 0, max = 1, value = 0.016, step = 0.0001),
sliderInput("gammag", label = "Real Phillips curve intercept", min = 0, max = 1, value = 0.13, step = 0.0001),
sliderInput("rhog", label = "Real Phillips curve slope", min = 0, max = 1, value = 0.96, step = 0.0001),
sliderInput("sigmag", label = "Capital-output ratio", min = 0, max = 5, value = 3, step = 0.0001),
sliderInput("tg", label = "Time", min=1, max=1000, value=130, step=1)
)
conditionalPanel("input.model == 'Shone'",
sliderInput("c0s", label = "Initial level of cons.", min = 0, max = 50, value = 10, step = 1),
sliderInput("i0s", label = "Initial level of invest.", min = 0, max = 50, value = 5, step = 1),
sliderInput("Gs", label = "Government spendings", min = 0, max = 50, value = 5.5, step = 1),
sliderInput("bs", label = "b par. in the IS", min = 0, max = 1, value = 0.8, step = 0.05),
sliderInput("ks", label = "k par. in the LM", min = 0, max = 1, value = 0.05, step = 0.01),
sliderInput("ds", label = "d parameter in the LM", min = 0, max = 1, value = 0.05, step = 0.01),
sliderInput("hs", label = "h par. in the i", min = 0, max = 1, value = 0.1, step = 0.01),
sliderInput("taus", label = "Exogenous tax rate", min = 0, max = 1, value = 0.25, step = 0.01),
sliderInput("alphas", label = "Weight par. Philips curve", min = 0, max = 1, value = 0.1, step = 0.01),
sliderInput("betas", label = "Beta parameter", min = 0, max = 10, value = 1, step = 0.5),
sliderInput("mdots", label = "Dynamics of the M. market", min = 0, max = 0.5, value = 0.01, step = 0.01),
sliderInput("yns", "Long-term output", min = 0, max = 10, value = 1, step = 0.5)
)
```
Column
-----------------------------------------------------------------------
### Initial values {data-height=350}
```{r}
####################################################
#Conditional panel for initial values for Solow model#
####################################################
conditionalPanel("input.model == 'Solow'",
sliderInput("initial_values", label="initial values", min=1, max=5, value=1, step=1),
conditionalPanel("input.initial_values == 1",
numericInput('aa1', 'a', 1 )),
conditionalPanel("input.initial_values == 2",
fillRow(numericInput('aa2', 'a', 1),
numericInput('bb2', 'b', 1))),
conditionalPanel("input.initial_values == 3",
fillRow(numericInput('aa3', 'a', 1),
numericInput('bb3', 'b', 1),
numericInput('cc3', 'c', 1))),
conditionalPanel("input.initial_values == 4",
fillRow(
numericInput('aa4', 'a', 1),
numericInput('bb4', 'b', 1),
numericInput('cc4', 'c', 1),
numericInput('dd4', 'd', 1))
),
conditionalPanel("input.initial_values == 5",
fillRow(
numericInput('aa5', 'a', 1),
numericInput('bb5', 'b', 1),
numericInput('cc5', 'c', 1),
numericInput('dd5', 'd', 1),
numericInput('ee5', 'e', 1))
),
actionButton("go_sol","(Re)start Numeric Simulation Solow", style="background-color: #ec7063 ;
border-radius: 10px !important; border: none; position:absolute; bottom: 5px;
margin: auto;"
))
######################################################
#Conditional panel for initial values for Goodwin model#
#####################################################
#y0 <- matrix(c(0.2, 1, 0.6, 1, 1.1, 1), ncol = 2, nrow = 3, byrow = TRUE)
conditionalPanel("input.model == 'Goodwin'",
fluidPage(
fluidRow(
sliderInput("initial_values_g",
label="initial values",
min=1,
max=3,
value=1,
step=1)
),
fluidRow(
conditionalPanel("input.initial_values_g == 1",
numericInput('gaa1', 'a', 0.2 ),
numericInput('gbb1', 'b', 1)),
conditionalPanel("input.initial_values_g == 2",
column(6,
numericInput('gaa2', 'a1', 0.2),
numericInput('gbb2', 'b1', 1)),
column(6,
numericInput('gcc2', 'a2',0.6),
numericInput('gdd2', 'b2',1))),
conditionalPanel("input.initial_values_g == 3",
column(4,
numericInput('gaa3', 'a1', 0.2),
numericInput('gbb3', 'b1', 1)),
column(4,
numericInput('gcc3', 'a2', 0.6),
numericInput('gdd3', 'b2', 1)),
column(4,
numericInput('gee3', 'a3', 0.8),
numericInput('gff3', 'b3', 1)))),
fluidRow(
actionButton("go_good",
"(Re)start Numeric Simulation",
style="background-color: #ec7063 ; border-radius: 10px !important; border: none; position:absolute; bottom: 5px; margin: auto;"))))
######################################################
#Conditional panel for initial values for Ramsey model#
#####################################################
conditionalPanel("input.model == 'Ramsey'",
fluidPage(
fluidRow(
sliderInput("initial_values_r",
label="initial values",
min=1,
max=3,
value=1,
step=1)
),
fluidRow(
conditionalPanel("input.initial_values_r == 1",
numericInput('raa1', 'a', 0.2 ),
numericInput('rbb1', 'b', 1)),
conditionalPanel("input.initial_values_r == 2",
column(6,
numericInput('raa2', 'a1', 0.2),
numericInput('rbb2', 'b1', 1)),
column(6,
numericInput('rcc2', 'a2',0.6),
numericInput('rdd2', 'b2',1))),
conditionalPanel("input.initial_values_r == 3",
column(4,
numericInput('raa3', 'a1', 0.2),
numericInput('rbb3', 'b1', 1)),
column(4,
numericInput('rcc3', 'a2', 0.6),
numericInput('rdd3', 'b2', 1)),
column(4,
numericInput('ree3', 'a3', 0.8),
numericInput('rff3', 'b3', 1)))),
fluidRow(
actionButton("go_ram",
"(Re)start Numeric Simulation for RCK model",
style="background-color: #ec7063 ; border-radius: 10px !important; border: none; position:absolute; bottom: 5px; margin: auto;"))))
######################################################
#Conditional panel for initial values for Shone model#
#####################################################
conditionalPanel("input.model == 'Shone'",
fluidPage(
fluidRow(
sliderInput("initial_values_s",
label="initial values",
min=1,
max=3,
value=1,
step=1)
),
fluidRow(
conditionalPanel("input.initial_values_s == 1",
numericInput('saa1', 'a', 0.05 ),
numericInput('sbb1', 'b', 0.15)),
conditionalPanel("input.initial_values_s == 2",
column(6,
numericInput('saa2', 'a1', 0.05),
numericInput('sbb2', 'b1', 0.15)),
column(6,
numericInput('scc2', 'a2',0.1),
numericInput('sdd2', 'b2',0.2))),
conditionalPanel("input.initial_values_r == 3",
column(4,
numericInput('saa3', 'a1', 0.05),
numericInput('sbb3', 'b1', 0.15)),
column(4,
numericInput('scc3', 'a2', 0.1),
numericInput('sdd3', 'b2', 0.2)),
column(4,
numericInput('see3', 'a3', 0.8),
numericInput('sff3', 'b3', 1)))),
fluidRow(
actionButton("go_shone",
"(Re)start Numeric Simulation for Shone model",
style="background-color: #ec7063 ; border-radius: 10px !important; border: none; position:absolute; bottom: 5px; margin: auto;"))))
```
### Numerical Simulations
```{r}
######################################################
#Initial Values Solow Model#
#####################################################
initial_val <- eventReactive(input$go_sol, {
print(input$initial_values)
print(typeof(input$initial_values))
nval <- input$initial_values
print(paste("nval=", nval))
if(nval==1){return(c(input$aa1))}
else if (nval==2){return(c(input$aa2,input$bb2))}
else if(nval==3){return(c(input$aa3,input$bb3,input$cc3))}
else if(nval==4){return(c(input$aa4,input$bb4,input$cc4,input$dd4))}
else
{return(c(input$aa5,input$bb5,input$cc5,input$dd5,input$ee5))}})
######################################################
#Initial Values Goodwin Model#
#####################################################
initial_val_g <- eventReactive(input$go_good, {
return(c(input$gaa2,input$gbb2))})
######################################################
#Initial Values Ramsey Model#
#####################################################
initial_val_r <- eventReactive(input$go_ram, {
return(c(input$raa2,input$rbb2))})
######################################################
#Initial Values Shone Model#
#####################################################
initial_val_s <- eventReactive(input$go_shone, {
return(c(input$saa2,input$sbb2))})
conditionalPanel("input.model == 'Solow'",
absolutePanel(
width = "100%", height = "100%",
renderPlot({
ini <- initial_val()
phaseR::flowField(solow_de,
xlim = c(0.0001, input$t),
ylim = c(0.0001, input$k),
parameters = c(input$s,input$a,input$d),
points=10,
system = 'one.dim',
add = F,
xlab = 't',
ylab = 'dk/dt')
phaseR::nullclines(solow_de,
xlim = c(0.0001, input$t),
ylim <- c(0.0001,input$k),
parameters = c(input$s,input$a,input$d),
system = "one.dim",
state.names = 'k')
phaseR::trajectory(solow_de,
y0 = ini,
tlim = c(0, input$t),
parameters = c(input$s,input$a,input$d),
system = "one.dim",
add = T)})))
conditionalPanel("input.model == 'Goodwin'",
absolutePanel(
width = "100%", height = "100%",
renderPlot({
ini_g <- y_in()
parameters <- c(input$sigmag, input$alphag, input$betag, input$gammag, input$rhog)
state <- c(V = ini_g[1,1], #0.5
U = ini_g[2,1], #1
Q = 1)
times <- seq(0, input$tg, by = 0.01)
out <- deSolve::ode(y = state, times = times, func = goodwin2, parms = parameters)
par(oma = c(0, 0, 3, 0))
plot(out, xlab = "time", ylab = "-")})))
conditionalPanel("input.model == 'Ramsey'",
absolutePanel(
width = "100%", height = "100%",
((
renderPlot({
ramsey.numericalSolution <- phaseR::numericalSolution(
ramsey,
y0 = c(0.5,0.1),
tlim = c(0,input$tr),
tstep = 0.01,
add.grid = T,
type = "one",
parameters = c(input$thetar,input$rhor,input$deltar,input$alphar),
ylab = "x,y",
xlab = "t",
ylim = c(0,5))})))))
```
Column {.tabset}
--------------------------------------------------------------------
### Dynamics
```{r}
#renderPrint(input$model)
#uiOutput("dynamic")
fluidPage(
conditionalPanel("input.model == 'Solow'",
absolutePanel(
width = "100%", height = "100%",
renderPlot({
k <- 1:input$k
ggplot()+
geom_line(aes(x = k, y = Solow1(input$s,k, input$a), color = "blue"))+
geom_line(aes(x=k,y=Solow2(input$d, k), color = "red"))+
theme_bw()
#plot(k,Solow(input$s,k, input$a), type="l")
#lines(k,Solow(input$s,k, -input$a), type="l")
}))))
```
### Phase Portrait/Phase Diagram
```{r}
y_in <- eventReactive(input$go_good, {
if(input$initial_values_g==1){
return(matrix(c(input$gaa1, input$gbb1), ncol = 1, nrow = 2, byrow = TRUE))}
else if(input$initial_values_g==2){
return(matrix(c(input$gaa2, input$gbb2, input$gcc2, input$gdd2), ncol = 2, nrow = 2, byrow = TRUE))}
else{return(matrix(c(input$gaa3, input$gbb3, input$gcc3, input$gdd3, input$gee3, input$gff3), ncol = 3, nrow = 2, byrow = TRUE))}})
y_in_r <- eventReactive(input$go_ram, {
if(input$initial_values_r==1){
return(matrix(c(input$raa1,input$raa2), ncol=1, nrow = 2, byrow = TRUE))
}
else if(input$initial_values_r==2){
return(matrix(c(input$raa2, input$rbb2, input$rcc2, input$rdd2), ncol = 2, nrow = 2, byrow = TRUE))
}
else{return(matrix(c(input$raa3, input$rbb3, input$rcc3, input$rdd3, input$ree3, input$rff3), ncol = 3, nrow = 2, byrow = TRUE))
}
}
)
y_in_s <- eventReactive(input$go_shone, {
if(input$initial_values_s==1){
return(matrix(c(input$saa1,input$saa2), ncol=1, nrow = 2, byrow = TRUE))
}
else if(input$initial_values_s==2){
return(matrix(c(input$saa2, input$sbb2, input$scc2, input$sdd2), ncol = 2, nrow = 2, byrow = TRUE))}
else{return(matrix(c(input$saa3, input$sbb3, input$scc3, input$sdd3, input$see3, input$sff3), ncol = 3, nrow = 2, byrow = TRUE))
}
}
)
conditionalPanel("input.model == 'Solow'",
absolutePanel(
width = "100%", height = "100%",
renderPlot({
phaseR::phasePortrait(solow_de,
ylim <- c(0.5,input$k),
parameters = c(input$s,input$a,input$d),
col = 'black',
add.grid = T,
xlab='k',
ylab='dk/dt')})))
conditionalPanel("input.model == 'Goodwin'",
absolutePanel(
width = "100%", height = "100%",
renderPlot({
y0 <- y_in()
phaseR::flowField(goodwin,
xlim = c(0,0.6),
ylim = c(0,1.3),
parameters = c(input$sigmag, input$alphag, input$betag, input$gammag, input$rhog),
points = 30,
add = FALSE,
xlab = "a",
ylab = "b")
phaseR::nullclines(goodwin,
xlim = c(-1,3),
ylim = c(-1,3),
parameters = c(input$sigmag, input$alphag, input$betag, input$gammag, input$rhog),
points = 500)
phaseR::trajectory(goodwin,
y0 = y0,
tlim = c(0,input$tg),
parameters = c(input$sigmag, input$alphag, input$betag, input$gammag, input$rhog),
colour = rep("black", 3))})))
conditionalPanel("input.model == 'Ramsey'",
absolutePanel(
width="100%", height = "100%",
renderPlot({
y0 <- y_in_r()
print(input)
ramsey.flowField <- phaseR::flowField(ramsey,
xlim = c(0.01,input$kr),
ylim = c(0.01,input$cr),
parameters = c(input$thetar,input$rhor,input$deltar,input$alphar),
points = 10,
add=F)
ramsey.nullclines <- phaseR::nullclines(ramsey,
xlim = c(0,input$kr),
ylim = c(0.01,input$cr),
state.names = c('k','c'),
parameters = c(input$thetar,input$rhor,input$deltar,input$alphar),
points = 500,
col = c("blue","red"),
add.legend=T)
ramsey.trajectories <- phaseR::trajectory(
ramsey,
y0 = y0,
tlim = c(0,100),
parameters = c(input$thetar,input$rhor,input$deltar,input$alphar),
col = rep('black',3)
)
})
))
conditionalPanel("input.model == 'Shone'",
absolutePanel(
width = "100%", height = "100%",
renderPlot({
print(y_in_s())
y0 <- y_in_s()
print(input)
shone.flowField <- phaseR::flowField(
shone,
xlim = c(0,2),
ylim = c(-0.5,0.5),
parameters = c(input$c0s, input$i0s, input$Gs,input$bs, input$ks, input$ds, input$hs ,input$taus,
input$alphas, input$betas, input$mdots, input$yns),
add = F,
points = 20
)
shone.nullclines <- phaseR::nullclines(
shone,
xlim = c(0,2),ylim = c(-0.5,0.5),
state.names = c('dy/dt=0','dpi/dt=0'),
parameters = c(input$c0s, input$i0s, input$Gs,input$bs, input$ks, input$ds, input$hs,
input$taus, input$alphas, input$betas, input$mdots, input$yns),
points = 500,
col = c("blue","red"),
add.legend=T
)
shone.trajectories <- phaseR::trajectory(
shone,
y0 = y0,
tlim = c(0,100),
parameters = c(input$c0s, input$i0s, input$Gs,input$bs, input$ks, input$ds, input$hs,
input$taus, input$alphas, input$betas, input$mdots, input$yns),
col = rep('black',3)
)
})
)
)
```
PhaseR and Our Tool{data-navmenu='About this Project' data-icon="ion-cube"}
========================================================================
Link and short presentation of the package and the paper associated.
Future Perspectives{data-navmenu='About this Project' data-icon='ion-ios-paperplane-outline'}
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Possibility of a package, extending/generalizing the tool to all macro 1-ODE and 2-ODE...
Resources{data-navmenu='Models' data-icon='ion-merge'}
========================================================================
In that page we can propose short econ/math developments on the models and/or links to the papers and teaching courses