At a price of $1.88 per pound, the supply for cherries in a large city is 16,000 pounds, and the demand is 10,600 pounds. When the price drops to $1.46 per pound, the supply decreases to 10,000 pounds, and the demand increases to 12,700 pounds. Assume that the price–supply and price–demand equations are linear. Use the matrix method to find the equilibrium price and equilibrium demand.
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Matrix Equation
- Thread starter chumaason
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At a price of $1.88 per pound, the supply for cherries in a large city is 16,000 pounds, and the demand is 10,600 pounds. When the price drops to $1.46 per pound, the supply decreases to 10,000 pounds, and the demand increases to 12,700 pounds. Assume that the price–supply and price–demand equations are linear. Use the matrix method to find the equilibrium price and equilibrium demand.
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This is what I have done. I got stocked on how to apply the matrix method on itPlease show us what you have tried and exactly where you are stuck.Please follow the rules of posting in this forum, as enunciated at:Please share your work/thoughts about this assignment.
When the price is $1.88 the supply is 16, 000, so we have the point (16, 1.88); and when the price is $1.46 the supply is 10, 000, giving (10, 1.46). When the price is $1.88 the demand is 10, 600, so we have (10.6, 1.88); and when the price is $1.46 the demand is 12, 700, giving the point (12.7, 1.46)
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Now use the fact:This is what I have done. I got stocked on how to apply the matrix method on it
When the price is $1.88 the supply is 16, 000, so we have the point (16, 1.88); and when the price is $1.46 the supply is 10, 000, giving (10, 1.46). When the price is $1.88 the demand is 10, 600, so we have (10.6, 1.88); and when the price is $1.46 the demand is 12, 700, giving the point (12.7, 1.46)
Assume that the price–supply and price–demand equations are linear. (given in OP)
Write equations using the given facts as listed above.
Let us first calculate the price(P)-supply(S) equation. We have two points (1.46,10) and (1.88, 16)
So the equation the straight line through these equations is:
\(\displaystyle \frac{P - 1.46}{S - 10} \ = \ \frac{1.88 - 1.46}{16 - 10} \)
Similarly calculate the equation of the line for Price-Demand.
What happens at equilibrium condition?
HallsofIvy
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Let P be the price of cherries in "dollars per pound", D the demand in pounds, and S the supply in pounds.
16000= a(1.88)+ b
10000= a(1.46)+ b
Eliminate b by subtracting the second equation from the first: 6000= a(0.42) and then a= 6000/0.42= 14285.71. Then 10000= (14285.71)(1.46)+b= 20857+ b so b= 10000- 20857= -10857
S= 14285.71P- 10857.
10600= a(1.88)+ b
12700= a(1.46)+ b
Now eliminate b by subtracting the first equation from the second:
-2100= a(0.42) so a= =2100/0.42= =5000. Then 10600= (-5000)(1.88)+ b= -9400+ b so b= 10600+ 9400= 30000.
D=-5000P+ 30000.
The equilibrium price is the unique price, P, where the demand is the same as the supply. Calling that common supply and demand "M", we must have M= -5000P+ 30000 and M= 14285.7P- 10857.
Personally, I wouldn't write that in matrix form. I would just write M= -5000P+ 30000= 14285.7P- 10857 and solve that equation for P. But since you ask it, we can write the two equations as
M+ 5000P= 30000 and
M-14285.7P= -10857
and write that as the matrix equation
\(\displaystyle \begin{bmatrix}1 & 1 \\ 5000 & -14285.7\end{bmatrix}\begin{bmatrix}M \\ P \end{bmatrix}= \begin{bmatrix}30000 \\ -10857\end{bmatrix}\).
Since the supply is a linear function of price, We can write S= aP+ b for some numbers, a and b.At a price of $1.88 per pound, the supply for cherries in a large city is 16,000 pounds,. When the price drops to $1.46 per pound, the supply decreases to 10,000 pounds.
16000= a(1.88)+ b
10000= a(1.46)+ b
Eliminate b by subtracting the second equation from the first: 6000= a(0.42) and then a= 6000/0.42= 14285.71. Then 10000= (14285.71)(1.46)+b= 20857+ b so b= 10000- 20857= -10857
S= 14285.71P- 10857.
As before we can write D= aS+ b.At a price of $1.88 per pound, the demand is 10,600 pounds. When the price drops to $1.46 per pound, the demand increases to 12,700 pounds.
10600= a(1.88)+ b
12700= a(1.46)+ b
Now eliminate b by subtracting the first equation from the second:
-2100= a(0.42) so a= =2100/0.42= =5000. Then 10600= (-5000)(1.88)+ b= -9400+ b so b= 10600+ 9400= 30000.
D=-5000P+ 30000.
The equilibrium price is the unique price, P, where the demand is the same as the supply. Calling that common supply and demand "M", we must have M= -5000P+ 30000 and M= 14285.7P- 10857.
Personally, I wouldn't write that in matrix form. I would just write M= -5000P+ 30000= 14285.7P- 10857 and solve that equation for P. But since you ask it, we can write the two equations as
M+ 5000P= 30000 and
M-14285.7P= -10857
and write that as the matrix equation
\(\displaystyle \begin{bmatrix}1 & 1 \\ 5000 & -14285.7\end{bmatrix}\begin{bmatrix}M \\ P \end{bmatrix}= \begin{bmatrix}30000 \\ -10857\end{bmatrix}\).
) Supply function is P = 0.07X + 0.76, but P – 0.07X = 0.76
Demand function is P = -0.2X + 4, but P + 0.2X = 4
Using Matric Method, P – 0.07X = 0.76
P + 0.2X = 4
By Cramer Rule, P= DP and X = DX D D
2 by 2 matrices D = {1 -0.07} = 1*(0.2) – 1(-0.07) = 0.27
{1 0.2)
DP = {0.76 -0.07} = 0.76(0.2) – 4(-0.07) =0.432
{ 4 0.2)
Dx = {1 0.76} = 1*(4) – 1(0.76) = 3.24
{1 4)
P = Dp = 0.432 =$1.6 D 0.27
X = Dx = 3.24 = 12 D 0.27
This is the solution. Solved
Demand function is P = -0.2X + 4, but P + 0.2X = 4
Using Matric Method, P – 0.07X = 0.76
P + 0.2X = 4
By Cramer Rule, P= DP and X = DX D D
2 by 2 matrices D = {1 -0.07} = 1*(0.2) – 1(-0.07) = 0.27
{1 0.2)
DP = {0.76 -0.07} = 0.76(0.2) – 4(-0.07) =0.432
{ 4 0.2)
Dx = {1 0.76} = 1*(4) – 1(0.76) = 3.24
{1 4)
P = Dp = 0.432 =$1.6 D 0.27
X = Dx = 3.24 = 12 D 0.27
This is the solution. Solved