demand function for kerosene: Kd=a0+a1Pk
supply function for kerosene: Ks=b0+b1Pk+b2Pg
demand function for gasoline: Gd=c0+c1Pg+u1 (ul here shows a exogenous random shock to the demand of gasoline)
supply function for gasoline: Gs=d0+d1Pk+d2Pg
a) if gasoline and kerosene are joint products,what are the signs of b2 and d2 ?
ok,for this question I assumes the signs of b2 and d2 are the same because they are generated simultaneously using a common input, so Gs=Ks
but I am not sure how to find out the signs of b2 and d2 as I am not sure of the signs of b0,b1,d0,d1??
b)if there is an exogenous increase in the demand for gasoline, (delta ul >0) what is the effect on the price of kerosene and quantity of kerosene produced?
for this question I had quantity of kerosene increases, and price of kerosene decreases, but not sure about the exact workingout.
c)assume Kd=Ks
and Gd=Gs
we can write:
(a1-b1)Pk-b2Pg=(b0-a0) and
-d1Pk+(c1-d2)Pg=(d0-c0)-ul
use cramer's rule to solve this system of equations for Pk and Pg.
this is the question that I am most confused about.by cramer's rule, using matrix and determinant I got Apk=(b0-a0)(c1-d2)-(-b2((d0-c0)-ul)
but this answer seemed so complicated and strange that I think I have done something wrong...
can somebody please help? thanks.
You're on the right track, but you need to complete the process.
Your answer for x is partially correct, namely it consists of the denominator (ce-bf, see text below), but you have not divided by the denominator.
Cramer's rule for a 2x2 system of equations works as follows:
ax+by=c
dx+ey=f
then δ=ae-bd (≠0)
and
x=(ce-bf)/δ
y=(af-cd)/δ
Using the given equations,
x=Pk
y=Pg
a=(a1-b1)
b=-b2
c=(b0-a0)
d=-d1
e=(c1-d2)
f=(d0-c0)-ul
For your information, the denominator δ would be as follows:
δ
= ae-bd
= (a1-b1)*(c1-d2)-b2*d1
and it does not look strange to me.
Continue the calculations for x and y, post your answer for checking if you wish.
my answer:
x=(b0-a0)*(c1-d2)-(-b2((d0-c0)-ul) divided by(a1-b1)*(c1-d2)-b2*d1
y=[(d0-c0)-ul]*(a1-b1)+[d1(b0-a0)]
divided by(a1-b1)*(c1-d2)-b2*d1
but I can't simplify it further...
Also for question (a) I somehow got a negative sign for b2, is it correct? how do I work out the sign for d2?
And can you please give me some hints on how to do the working out for question (c)? thanks!
The answer for x is correct.
There is a mistake of the sign in the answer for y:
y=[(d0-c0)-ul]*(a1-b1) - [d1(b0-a0)]
divided by(a1-b1)*(c1-d2)-b2*d1
I'm not sure if I understand your question "how to do the working out for question (c)".
Do you mean where those formulae came from, or how to derive those formulae? These are the standard equations for Cramer's rule for a system of two equations, which you can find in your textbook, or if you don;t have one, you can find it on the Internet.
For questions related to (a) and (b), I suggest you repost with a subject of macroeconomics.
a) If gasoline and kerosene are joint products, meaning they are derived from a common input, then the signs of b2 and d2 should be the same. This is because an increase in the price of the common input (Pg) would affect the supply of both gasoline and kerosene in the same direction. However, without specific information about the individual supply functions (b0, b1, d0, d1), it is difficult to determine the signs of b2 and d2.
b) If there is an exogenous increase in the demand for gasoline (delta ul >0), the effect on the price and quantity of kerosene can be analyzed as follows:
From the kerosene demand function Kd = a0 + a1Pk, an increase in the demand for gasoline (represented by ul) would not directly impact the price of kerosene (Pk) as there is no direct relationship between ul and Pk in the kerosene demand function. Therefore, the price of kerosene may not change in response to the increased demand for gasoline.
However, the exogenous increase in the demand for gasoline could indirectly affect the quantity of kerosene produced. Given that gasoline and kerosene are joint products, an increase in the demand for gasoline would require a higher input of the common input (Pg) in the production process. This would reduce the availability of the common input for kerosene production, potentially leading to a decrease in the quantity of kerosene produced.
c) Assuming Kd = Ks and Gd = Gs, we can write the following system of equations:
(a1 - b1)Pk - b2Pg = (b0 - a0) -- (Equation 1)
-d1Pk + (c1 - d2)Pg = (d0 - c0) - ul -- (Equation 2)
To solve this system of equations using Cramer's Rule, we need to create a determinant matrix. Let's define the determinant matrix D as:
D = | (a1 - b1) -b2 |
| -d1 (c1 - d2) |
Now, create matrices Dx, Dy, and Dz by replacing the respective columns of D (column 1 for Dx, column 2 for Dy, and column 3 for Dz) with the constants from Equations 1 and 2:
Dx = | (b0 - a0) -b2 |
| (d0 - c0) (c1 - d2) |
Dy = | (a1 - b1) (b0 - a0) |
| -d1 (d0 - c0) |
Dz = | (a1 - b1) -b2 |
| -d1 (c1 - d2) |
Now, calculate the determinants of Dx, Dy, and Dz:
Det(Dx) = ((b0 - a0) * (c1 - d2)) - ((-b2) * (d0 - c0))
Det(Dy) = ((a1 - b1) * (d0 - c0)) - ((-d1) * (b0 - a0))
Det(Dz) = ((a1 - b1) * (c1 - d2)) - ((-d1) * (-b2))
Next, calculate the solution for Pk and Pg using Cramer's Rule:
Pk = Det(Dx) / Det(D)
Pg = Det(Dy) / Det(D)
This should give you the values of Pk and Pg in the system of equations when using Cramer's Rule. Make sure to substitute the appropriate values for a0, a1, b0, b1, b2, c0, c1, d0, d1, d2, ul in the determinants.