Page B-13 Question 4

Discuss how you might go about validating the nuclear arms race model. What data would you collect? Is it possible to obtain the data?

Page B-13 Answer 4

To validate nuclear arms race model, one can examine its performance in simulating the three historical great power arms races:
(a) The great power arms race from 1870 to 1913 which culminated in World War I;
(b) The arms race in Europe from 1925 to 1939 prior to World War II; and
(c) The global arms race between China, NATO and the Warsaw Pact from 1952 to 1973.
There were three reasons for choosing these particular arms races. First of all, they are of greater contemporary salience and historical importance than smaller regional arms competitions. Second, in each case, arms competition is almost exclusively endogenous to the system under consideration; i.e. the actors were scarcely concerned with the arms levels of actors other than those being modelled. Thirdly, the data on armed forces levels for the actors in these arms races are readily available and of high quality.

To simplify the model, the theoretical assumption is that all states or actors have access to complete and current information regarding the distribution of military strength, and that no errors are introduced in processing this information. The data needed will be about missile locations, technological advances, missile strength, missiles inventory, military personnel and their strengths, locations, and even data about their daily habits.
However, it is likely that in the real world information will be subject to delay: even the best intelligence service is likely to report, not the current situation, but one that existed many months or years prior. Information may be incomplete; there may simply exist no good estimate of effective enemy strength. Finally, information is subject to systematic bias for cultural or ideological reasons; the American experience in Vietnam gives ample evidence of this process.
Nevertheless, for model simplicity is advised to assume be perfect information for two reasons: first, the desire to use the balance of power assumptions; second, to avoid introducing excessive complexity into the model at an early stage in the analysis.

Page B-17 Question 1

Build a numerical solution to Equations (15.8).

  1. Graph your results.
  2. Is an equilibrium value reached?
  3. Try other starting values. Do you think the equilibrium value is stable?
  4. Explore other values for the survival coefficients of Countries X and Y. Describe your results.

Page B-17 Answer 1

First recall Equations (15.8).

\[ y_{n+1} = 120 + \frac 1 2 x_n \] \[x_{n+1} = 60 + \frac 1 3 y_n \]

with \[x_0= 100\] \[y_0=200\]

  1. Graph your results.

Let: n = stage (years, decades, fiscal periods, etc.)

\(x_n\) = number of weapons possessed by X in stage n

\(y_n\) = number of weapons possessed by Y in stage n

\(x_0\) and \(y_0\) are called initial values.

library(ggplot2)
## Warning: package 'ggplot2' was built under R version 3.2.3
library(reshape2)
x0 <- 100
y0 <- 200
xs <- 1/3
ys <- 1/2


model1 <- function(x0,y0,xs=(1/3),ys=1/2,period=30){
  stage <- 0
  data <- data.frame(X=x0,Y=y0,stage=stage)
  repeat{
    Y <- 120 + ys*tail(data$X,1)
    X <- 60 + xs*tail(data$Y,1)
    stage <- tail(data$stage,1) + 1
    data <- rbind(data,c(X,Y,stage))
    if(stage == period) break
  }
  return(data)  
  
}

data<- model1(100,200)
x_equil <- signif(tail(data$X,1))
y_equil <- signif(tail(data$Y,1))
data <- melt(data,id.vars="stage",variable.name="Country",value.name="Arms")
ggplot(data,aes(x=stage,y=Arms,color=Country)) + geom_line() +
  ggtitle(paste("X0 = ",x0," and Y0 = ",y0,"\n",
                "X survival param = ",signif(xs,1)," ",
                "Y survival param = ",signif(ys,1),"\n",
                "X equilibrium = ",x_equil," ",
                "Y equilibrium = ",y_equil,sep=""))

  1. Is an equilibrium value reached?
    Yes, the X equilibrium = 120, and Y equilibrium = 180. The equilibrium started happening around year or stage 8 and stabilized starting year or stage 16.
  2. Try other starting values. Do you think the equilibrium value is stable?

First let’s make sure we start \(x_0=0\) and \(y_0=0\) to make sure we have the same result as the example in the book..

x0 <- 0
y0 <- 0
xs <- 1/3
ys <- 1/2
data<- model1(0,0)
x_equil <- signif(tail(data$X,1))
y_equil <- signif(tail(data$Y,1))
data <- melt(data,id.vars="stage",variable.name="Country",value.name="Arms")
ggplot(data,aes(x=stage,y=Arms,color=Country)) + geom_line() +
  ggtitle(paste("X0 = ",x0," and Y0 = ",y0,"\n",
                "X survival param = ",signif(xs,1)," ",
                "Y survival param = ",signif(ys,1),"\n",
                "X equilibrium = ",x_equil," ",
                "Y equilibrium = ",y_equil,sep=""))

Now lets set \(x_0=0\) and \(y_0=300\)

x0 <- 0
y0 <- 300
xs <- 1/3
ys <- 1/2
data<- model1(0,300)
x_equil <- signif(tail(data$X,1))
y_equil <- signif(tail(data$Y,1))
data <- melt(data,id.vars="stage",variable.name="Country",value.name="Arms")
ggplot(data,aes(x=stage,y=Arms,color=Country)) + geom_line() +
  ggtitle(paste("X0 = ",x0," and Y0 = ",y0,"\n",
                "X survival param = ",signif(xs,1)," ",
                "Y survival param = ",signif(ys,1),"\n",
                "X equilibrium = ",x_equil," ",
                "Y equilibrium = ",y_equil,sep=""))

Now let set \(x_0=0\) and \(y_0=1000\)

x0 <- 0
y0 <- 1000
xs <- 1/3
ys <- 1/2
data<- model1(0,1000)
x_equil <- signif(tail(data$X,1))
y_equil <- signif(tail(data$Y,1))
data <- melt(data,id.vars="stage",variable.name="Country",value.name="Arms")
ggplot(data,aes(x=stage,y=Arms,color=Country)) + geom_line() +
  ggtitle(paste("X0 = ",x0," and Y0 = ",y0,"\n",
                "X survival param = ",signif(xs,1)," ",
                "Y survival param = ",signif(ys,1),"\n",
                "X equilibrium = ",x_equil," ",
                "Y equilibrium = ",y_equil,sep=""))

Now let set \(x_0=1000\) and \(y_0=-100\)

From the graphical results above, it looks like the equilibrium for both X and Y are stable at 120 and 180 respectively regardless if we change the initial values \(x_0\) and \(y_0\).

  1. Explore other values for the survival coefficients of Countries X and Y. Describe your results.

We will make our model simpler by assuming that the initial values are \(x_0\) and \(y_0\) are the same for both X and Y at \(x_0\)=\(y_0\) = 0  

Let’s start with sx = sy = 0

x0 <- 0
y0 <- 0
xs <- 0
ys <- 0
data<- model1(0, 0,0,0)
x_equil <- signif(tail(data$X,1))
y_equil <- signif(tail(data$Y,1))
data <- melt(data,id.vars="stage",variable.name="Country",value.name="Arms")
ggplot(data,aes(x=stage,y=Arms,color=Country)) + geom_line() +
  ggtitle(paste("X0 = ",x0," and Y0 = ",y0,"\n",
                "X survival param = ",signif(xs,1)," ",
                "Y survival param = ",signif(ys,1),"\n",
                "X equilibrium = ",x_equil," ",
                "Y equilibrium = ",y_equil,sep=""))

Let’s make sx = 2 and sy = 0

x0 <- 0
y0 <- 0
xs <- 2
ys <- 0
data<- model1(0, 0,2,0)
x_equil <- signif(tail(data$X,1))
y_equil <- signif(tail(data$Y,1))
data <- melt(data,id.vars="stage",variable.name="Country",value.name="Arms")
ggplot(data,aes(x=stage,y=Arms,color=Country)) + geom_line() +
  ggtitle(paste("X0 = ",x0," and Y0 = ",y0,"\n",
                "X survival param = ",signif(xs,1)," ",
                "Y survival param = ",signif(ys,1),"\n",
                "X equilibrium = ",x_equil," ",
                "Y equilibrium = ",y_equil,sep=""))

Let’s make sx = 0 and sy = 2

x0 <- 0
y0 <- 0
xs <- 0
ys <- 2
data<- model1(0, 0,0,2)
x_equil <- signif(tail(data$X,1))
y_equil <- signif(tail(data$Y,1))
data <- melt(data,id.vars="stage",variable.name="Country",value.name="Arms")
ggplot(data,aes(x=stage,y=Arms,color=Country)) + geom_line() +
  ggtitle(paste("X0 = ",x0," and Y0 = ",y0,"\n",
                "X survival param = ",signif(xs,1)," ",
                "Y survival param = ",signif(ys,1),"\n",
                "X equilibrium = ",x_equil," ",
                "Y equilibrium = ",y_equil,sep=""))

Let’s make sx = 4 and sy = -2

x0 <- 0
y0 <- 0
xs <- 4
ys <- -2
data<- model1(0, 0,4,-2)
x_equil <- signif(tail(data$X,1))
y_equil <- signif(tail(data$Y,1))
data <- melt(data,id.vars="stage",variable.name="Country",value.name="Arms")
ggplot(data,aes(x=stage,y=Arms,color=Country)) + geom_line() +
  ggtitle(paste("X0 = ",x0," and Y0 = ",y0,"\n",
                "X survival param = ",signif(xs,1)," ",
                "Y survival param = ",signif(ys,1),"\n",
                "X equilibrium = ",x_equil," ",
                "Y equilibrium = ",y_equil,sep=""))

Let’s make sx = 4 and sy = 2

x0 <- 0
y0 <- 0
xs <- 4
ys <- 2
data<- model1(0, 0,4,2)
x_equil <- signif(tail(data$X,1))
y_equil <- signif(tail(data$Y,1))
data <- melt(data,id.vars="stage",variable.name="Country",value.name="Arms")
ggplot(data,aes(x=stage,y=Arms,color=Country)) + geom_line() +
  ggtitle(paste("X0 = ",x0," and Y0 = ",y0,"\n",
                "X survival param = ",signif(xs,1)," ",
                "Y survival param = ",signif(ys,1),"\n",
                "X equilibrium = ",x_equil," ",
                "Y equilibrium = ",y_equil,sep=""))

We have set x_0 and y_0 to 0. Certainly changing the survival parameters certainly changes the equilibrium values..

When we set sx = 0 and sy = 0, the equilibrium happens at 60 and 120 for X and Y. That means that
Both countries X and Y do not think the other side has arms. Hence they have the missiles they built when any knowing about the side has missiles.

As we increase the percentage of survivability, we see that the equilibrium changes in favor of the country with higher survival percentage. and one can that that a country which devotes its budget in making and hiding missiles sites has better changes of surviving and maintaining higher equilibrium.

Page B-21 Question 4

Verify the result that the marginal revenue of the q+1st unit equals the price of that unit minus the loss in revenue on previous units resulting from price reduction.

Page B-21 Answer 4

Marginal revenue is the change in revenue associated with given change in output. Marginal revenue has two components: the output effect which is the revenue generated by the additional units of output; and the price effect, which is the revenue lost on previous units due to the price reduction necessary to sell more units..

If the revenue generated by the additional units exceeds the lost revenue on the other units due to price reduction, marginal revenue is greater than zero. Increased production will lead to more revenue. If the revenue generated by the additional units is less than the lost revenue on the other units due to the price reduction, marginal revenue is less than zero. Increased production will cause total revenue to fall.

let’s try show mathematically using the relationship between marginal revenue and elasticity.

Given that \[R= pq\]

it followa that: \[\frac {dR} {dq} = p + q \frac {dp} {dq}\] \[ = p[ 1 + \frac {q} {p} \frac {dp} {dq}]\] \[ = p[1+ \frac {1} {\frac {p} {q} \frac {dq} {dp}}]\] \[ = p[1+ \frac {1} {\epsilon}] \]

If \(\epsilon\) = 1, an increase in quantity has no effect on revenue.
If \(\epsilon\) < 1, meaning demand is very responsive to price, you can cut the price just a bit to increase quantity; this means revenue goes up.
If \(\epsilon\) > 1, meaning demand is not very responsive to price, you have to cut price a lot to increase quantity; this means revenue will decline.

Page B-25: Question #1

Show that when the demand curve is very steep, a tax added to each item sold will fall primarily on consumers. Now show that when the demand curve is more nearly horizontal, the tax is paid mostly by the industry. What if the supply curve is very steep? What if the supply curve is nearly horizontal?

Page B-25: Question #1

Governments sometimes use the degree of responsiveness in supply or demand in relation to changes in price (elasticity) when determining the tax incidence or what portion of a tax is ultimately borne by the consumers and the producers.
Elasticities are used to determine how much of a tax is borne by the producer verses the consumer. For example, when a tax is imposed on cigarettes, producers are able to pass along most of the burden of the tax by raising the price. If the supply is more elastic than demand, then the producer will bear a greater burden of the tax.

\[Tax \ Burden \ on \ consumer = E_s / (E_d + E_s)\]

\[Tax \ Burden \ on \ supplier = E_d /(E_d +E_s)\]

The effect is best depicted by Brigham Young University - Idaho, in the Economic Principles and Problems course per the below figures from the tax effect on consumer and producer point view:

alt text

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