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Question 1169744: The general linear supply and demand for one-commodity market model are given by
P = a๐๐ +b (a>0, b>0)
P = - c๐๐ท+d (c>0, d>0)
a. Show that in matrix notation the equilibrium price, P, and quantity, Q satisfy.
[1 โ๐] [๐]=[๐]
1 ๐ ๐ ๐
b. Solve this system to express P and Q in terms of a, b, c and d.
c. Write down the multiplier for Q due to changes in b and deduce that an increase in b lends to an decrease in Q.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Absolutely, let's solve this supply and demand problem.
**Understanding the Model**
We have a one-commodity market with linear supply and demand equations:
* Supply: P = aQs + b (a > 0, b > 0)
* Demand: P = -cQd + d (c > 0, d > 0)
Where:
* P = Price
* Qs = Quantity supplied
* Qd = Quantity demanded
At equilibrium, Qs = Qd = Q.
**a) Show the Matrix Notation**
1. **Rewrite the Equations:**
* Supply: P - aQ = b
* Demand: P + cQ = d
2. **Matrix Form:**
We can represent these equations in matrix form:
```
[ 1 -a ] [ P ] = [ b ]
[ 1 c ] [ Q ] = [ d ]
```
Thus, we have shown the required matrix equation.
**b) Solve for P and Q**
1. **Matrix Equation:**
Let:
```
A = [ 1 -a ]
[ 1 c ]
X = [ P ]
[ Q ]
B = [ b ]
[ d ]
```
We have AX = B, so X = AโปยนB.
2. **Inverse of A:**
* det(A) = (1 * c) - (-a * 1) = c + a
* Aโปยน = (1 / (c + a)) * [ c a ]
[ -1 1 ]
3. **Solve for X (P and Q):**
```
[ P ] = (1 / (c + a)) * [ c a ] [ b ]
[ Q ] [ -1 1 ] [ d ]
```
```
P = (1 / (c + a)) * (cb + ad) = (cb + ad) / (c + a)
Q = (1 / (c + a)) * (-b + d) = (d - b) / (c + a)
```
Therefore:
* P = (cb + ad) / (c + a)
* Q = (d - b) / (c + a)
**c) Multiplier for Q due to Changes in b**
1. **Q in terms of b:**
* Q = (d - b) / (c + a)
2. **Multiplier:**
The multiplier for Q due to changes in b is the derivative of Q with respect to b:
* dQ/db = -1 / (c + a)
3. **Deduction:**
Since a > 0 and c > 0, (c + a) > 0. Therefore, -1 / (c + a) is always negative.
This means that an increase in b (the y-intercept of the supply curve) leads to a decrease in Q (the equilibrium quantity).
**Final Answers**
* **(a)** The matrix notation is: [ 1 -a ] [ P ] = [ b ]
[ 1 c ] [ Q ] = [ d ]
* **(b)** P = (cb + ad) / (c + a)
Q = (d - b) / (c + a)
* **(c)** The multiplier for Q due to changes in b is -1 / (c + a). Since this is negative, an increase in b leads to a decrease in Q.
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