Answer to Question #162361 in Macroeconomics for SYEDA ZAINAB

Question

The demand and Supply functions for three goods are given as follows:

D_x=100-3P_x+P_y+3P_z

D_y=80 +P_x-2P_y-2P_z

D_z=120+3P_x-P_y-4P_z

S_x=-10+P_x

S_y=-20=3P_y

S_z=-3+2P_z

Q1: Determine the equilibrium prices and quantities of all three goods.

·         The government decides to:

a.       Impose a 25% tax on X

b.      Impose a 5Rs unit Tax on Y

c.      Gives a 10% subsidy on good Z

·         Analyze the impact of each of these three policies separately on equilibrium prices and Quantities.

·         Also calculate changes in consumer and producer surpluses and the amount of revenue earned by government.

Q2: Repeat this exercise when polices (a, b),(b,c) & (a, b,c) are jointly implemented. Which policy choice is best? Why?

Q: 3 Provide theoretical justification (using diagrams) of all results obtained.

 

Answer 1

$$solution$$

At equilibrium the quantity demanded is equal to quantity supplied i.e.

$Qd=Qs$

$Dx=Sx\\ Dy=Sy\\ Dz=Sz\\$

$100-3Px+Py+3Pz=-10+Px\\ 110-4Px+Py+3Pz=0\\ 4Px-Py-3Pz=110......[1]$

$80+Px-2Py-2Pz=-20-3Py\\ 100+Px+Py-2Pz=0\\ Px-Py+2Pz=100.....[2]$

$120+3Px-Py-4Pz=-3+2Pz\\ 123+3Px-Py-6Pz=0\\ Py+6Pz-3Px=123.......[3]$

therefore we have the following set of equations with variables $Px,Py,Pz$ .

$\begin{pmatrix} 4 & -1 &-3 \\ -1 & -1 & 2\\ -3 & 1 & 6 \end{pmatrix}$ $\begin{pmatrix} Px\\ Py\\ Pz\end{pmatrix}$ = $\begin{pmatrix} 110\\ 100\\ 123\end{pmatrix}$

lets solve the equation using crammers rule

$Px=\frac{Dx}{D}\\ Py=\frac{Dy}{D}\\ Pz=\frac{Dz}{D}\\$

Here ;

$D=$ $\left|\begin{matrix} 4 & -1 & -3 \\ -1 & -1 & 2 \\ -3 & 1 & 6 \end{matrix}\right|$

$4[-6-2]+1[-6+6]-3[-1-3]\\ 4[-8]+1[0]-3[-4]\\ -32+12\\ =-20$

$cramers \ rule:$

$DPx=\begin{vmatrix} 110 & -1 & -3\\ 100 & -1 & 2\\ 123 & 1 & 6 \end{vmatrix}$ =-1195

$DPy=\begin{vmatrix} 4 & 110 & -3\\ -1 &100 & 2\\ -3 & 123 & 6 \end{vmatrix}$ =885

$DPz=\begin{vmatrix} 4 & -1& 110\\ -1 &-1 & 100\\ -3 & 1 & 123 \end{vmatrix}$ =-1155

$DPx=-1195\\ DPy=885\\ DPz=-1155\\ substituted:$

$Px= \frac{-1195}{-20}=\frac{239}{4}\\ Py=\frac{885}{-20}=\frac{177}{4}\\ Pz=\frac{-1155}{-20}=\frac{231}{4}\\ after \ finding \ the \ values \ of \ Px, Py\ and \ Pz ,\ substitute \ in \ the \ original\\ \ equations \ to \ find:$

$Dx=100-{3\times\frac{239}{4}}+\frac{177}{4}+3\times\frac{231}{4}=\frac{553}{4}\\ Dy=80+\frac{239}{4}-2\times\frac{177}{4}-2\times\frac{231}{4}=\frac{-719}{4}\\ Dz=120+3\times\frac{239}{4}-\frac{177}{4}-4\times\frac{231}{4}=24$

B. impact on equilibrium prices and quantities when:

a] imposes 25% tax on X

The production cost will increase leading to increase in equilibrium prices .

The equilibrium quantities produced will be reduced due to reduced demand and supply as prices have gone high.

b]imposes a 5Rs unit tax on Y

shift in equilibrium prices downwards or no impact .

the equilibrium quantities supplied will meet demand due to minimal or reduced cost of production hence increase or remain fixed

c] gives 10% subsidy on good Z

reduction in equilibrium prices

increase in equilibrium quantity

AB

The production cost will increase leading to increase in equilibrium prices .

The equilibrium quantities produced will be reduced due to reduced demand and supply as prices have gone high.

shift in equilibrium prices downwards or no impact .

the equilibrium quantities supplied will meet demand due to minimal or reduced cost of production hence increase or remain fixed.

BC

shift in equilibrium prices downwards or no impact .

the equilibrium quantities supplied will meet demand due to minimal or reduced cost of production hence increase or remain fixed reduction in equilibrium prices

increase in equilibrium quantity

ABC

The production cost will increase leading to increase in equilibrium prices .

The equilibrium quantities produced will be reduced due to reduced demand and supply as prices have gone high.

shift in equilibrium prices downwards or no impact .

the equilibrium quantities supplied will meet demand due to minimal or reduced cost of production hence increase or remain fixed

reduction in equilibrium prices

increase in equilibrium quantity

SOLUTION B

In the sample market shown in the graph, equilibrium price is $10 and equilibrium quantity is 3 units. The consumer surplus area is highlighted above the equilibrium price line. This area can be calculated as the area of a triangle.

Let’s apply the calculation for the area of a triangle to our example market to see the added value that consumers will get for this item at the equilibrium price in our sample market.

Step 1: Define the base and height of the consumer surplus triangle.

The base of the consumer surplus triangle is 3 units long. Be careful when you define the height

,it is tempting to say it is 25, can you see why it isn’t? The height is determined by the distance from the equilibrium price line and where the demand curve intersects the vertical axis. The height of the triangle begins at $10 and ends at $25, so it will be $25 – $10 = $15

$b=3\\ h=15\\$

Step 2: Apply the values for base and height to the formula for the area of a triangle.

$A = \displaystyle \frac{1}{2}b\times h ​ ​\\ A=\frac{1}{2}[3\times15]\\ A=\frac{1}{2}[45]\\ A=\frac{45}{2}\\ A=22.5 ​ ​ ​$

therefore consumer surplus =22.5

for producer surplus :

$Ps=\frac{1}{2}b\times h\\ Ps=\frac{1}{2}[3\times 5]\\ Ps=\frac{1}{2}[15]\\ Ps=\frac{15}{2}\\ Ps=7.5$

the producer surplus =7.5

SOLUTION C