Computational Tools for Economics and Management
Assignment 1
The User-guide
Valentina Antoniazzi (832424)
Nora Grammar Piros (832595)
Eder Belluzzo (832416)
Calculating equilibrium Price and Quantity,
Producer and Consumer Surplus
Suppose the market demand for cranberries is given by the equation
Qd=500-4P, while the market supply curve for cranberries (when P is ≥50)
is described by the equation Qs=-100+2P, where P is the price of
cranberries expressed in dollars per barrel, and quantity (Qd or Qs) is in
thousands of barrels per year.
Problem.
• At what price and quantity is the market for cranberries in equilibrium?
• Consider the two non-linear functions, P=-Q2+20 and P=-Q2+4Q+16 find
their intersections.
• Find then the consumer surplus and the producer surplus
Solution Since we are interested in the equilibrium price and quantity, we then need to look for the intersection
between the demand and supply curve. Since it is a system of two linear equations, one possible way to find it is by
constructing a matrix by first rearranging the original demand and supply curve in the following way:
Qd +4P = 500
Qs -2P = -100
1
4 500
1 -2 - 100
A
b
What we do here is to create a matrix , where in the left side (A), we find the
values corresponding to and Q, and on the right side (b) we find the constant of the
two functions.
Calculating it with R…
A<-matrix(c(1,4,1,-2),2,2,byrow=T)
A
[,1] [,2]
[1,] 1 4
[2,] 1 -2
b<-c(500,-100)
solve(A,b)
100 100 =>
Q*=100 and P*=100
In letters…
Dear R, we need to solve a
system of two equations and
we need your help. Please
create a matrix of two rows
and two columns, with the
data I provide you,
remember to read it by
rows. We also give you the
vector b. Then use “solve “
for A and b to find the
result.
R responds: it’s 100, 100 !
From these computations we find out that the equilibrium price, P* is equal to 100, as well as
the equilibrium quantity Q* is also equal to 100.
Note: you can use this procedure only if it is a system of two linear equations
We only believe in what we see!
To make sure that the result is correct and to see the patterns of the demand and the supply
curve we need to make a graph so to explain it also visually.
Calcualting it with R
demand<-function(Q)-Q/4+500/4
curve(demand,0,550,xlab="Quantity",ylab="Price
",main=”Demand and Supply",col=1)
text(500,20,"Demand")
supply<-function(Q)Q/2+100/2
curve(supply,col=4,add=T)
text(200,120,"Supply")
text(100,110,"E")
abline(h=100,v=100,lty=2)
R kindly responds with a graph
Comparing the graph and the results given by the matrix
we can see that there is indeed consistency, also in the
graph both P* and Q* are equal to 100
!
Note! If you use the function plot, the x axes will go from 0 to 1.
If you want the real values of the function instead of a
percentage, you should use the function “curve” and specify
the interval you want to have displayed.
In letters….
Dear R, we also need you to show us graphically
those two equations.
This is our demand function: -Q/4+500/4 we
need you to graph it, and call it “Demand”,
writing on the x axis “Quantity”, on the y axis
“Price” and we also need a nice title that says
“Demand and Supply”.
Please do the same with the supply function
adding it to the previous graph and calling it
“Supply”.
we would also like to see the ablines passing
through the equilibrium point (100.100).
Thanks
Non-Linear Equations
F <-function(Q) -(Q)**2+20
curve(F,0,2,col=2)
G <-function(Q) -Q**2 +4*Q+16
curve(G,0,2,add=T)
H <-function(Q) F(Q)-G(Q)
curve(H,0,2,add=T)
Note! also H is a function! Do not forget to say
F<-function(Q)…….
uniroot(H,c(0,2))
curve(H,0,2)
abline(h=19,v=1,lty=2)
points(1,19,pch=19)
text(0.95,19.2,"E")
Plugging x =1 in the function
F <-function(Q) -(Q)**2+20
F(1)
[1] 19
To have a further proof we plug it in the other function as well
G <-function(Q) -Q**2 +4*Q+16
G(1)
[1] 19
If the equations are not linear, as we have already
said, it is not possible to use the matrix to find the
intersection point of the two equations, but the
following procedure can help us finding it!
Having the two equations F(Q) and G(Q), we call
“H(Q)” the difference between F(Q) and G(Q)
H(Q)=F(Q)-G(Q)
With the function “uniroot” we find where the
two equations equal out. Uniroot gives us the
value of Q for which this happens but we can
easily find the value of P plugging it into
one of the two
functions.
Finding Producer and Consumer Surplus
We now have to find the consumer and producer surplus. The consumer surplus is the area BEC,
while the producer surplus is the area enclosed in the triangle ABE.
From the demand equation: d <-function(Q) -(Q)**2+20, and the supply equation:
s <-function(Q) -(Q-2)**2+20, we draw a graphical representation using the R command “ curve”
curve(s,0,2,col=2,ylab="Price",xlab="Quantity",lwd=3,main="Supply and Demand
Equilibrium")
curve(d,0,2,col=4,lwd=3,add=T)
This is our initial graph. Now we add our abline and grid, and this will improve our graph,
and show better the area of consumer and producer surplus. A common mistake in this step,
is on the abline, we should write it as a formula because in this way we would be able to color
the areas. Also, we should add points and name it.
line <- function(Q)19+0*Q)
curve(line,lwd=3,add=T)
grid(col=1)
text(0.05,16,"A")
text(1,19.2,"E")
text(0.05,19.1,"B")
text(0.05,19.9,"C")
points(1,19,pch=19)
points(0,16,pch=19)
points(0,19,pch=19)
points(0,20,pch=19)
Using the function “polygon”
In order to color, we will be using the R command “polygon”. To do so, we must create a vector, in this case x, that would be
a sequence of points, from a specific point to another and length. This vector, will create determine and unify the points
between 2 lines using a reverse path. On our case, we define the vector x, a starting point, the reverse function and the
final point.
x <- seq(0,1,len=10)
polygon(c(x,0),c(d(x),19),col="pink")
polygon(c(x,0),c(s(x),19),col="light blue")
And, our final touch will be to reinforce the line since they
fade once we color the areas. Also, we can name the
specific areas.
curve(g,0,2,col=4,lwd=3,add=T)
curve(line,lwd=3,add=T)
curve(f,0,2,col=2,lwd=3,add=T)
text(0.4,19.4,"Consumer Surplus")
text(0.4,18.4,"Producer Surplus")
Calculating the producer and consumer surplus
The final step, will be to calculate the surplus areas. The R function for such is “integrate”.
We must define a function, and the interval area to be calculated. R will actually calculate
the whole area under the specific curve, and this is a possible common mistake. The
graph we have, have the y axis from 16 to 20. So, once we calculate the integrals, R will give
values from zero up the number given by the function. This is why to must subtract the
undesired area values given the rectangular area 19.(1,19) In the case of supply function,
the value would be negative, and that's why we must multiply the value by (-1) and then
add 19 to get a precise absolute value for the area. Another common mistake in this
process, is that we can't just add or subtract the value, we must extract it from the
integrate area value.
integrate(d,0,1)$value-19
integrate(s,0,1)$value*(-1)+19
Consumer Surplus area= 19.6666667-19=0.6666667
Producer Surplus area= 17.66667*(-1)+19=1.33333
The End
Dear User,
We hope our guide has been useful.
We thank you for your attention.
Best wishes
Valentina,
Eder,
Nora