Question

7. Find the sequence whose 8th term is 38 and 22nd term is 108.

10. Given the market demand schedule of a commodity
Qx = 100 – Px + 0.75Pm – 0.25Pz + 0.00750075Y. If Px=10, Pm =20, Pz =40 and Y = 10000
Where; Px = price of the commodity
Pm= price of another commodity M
Pz = price of another commodity Z
Y = level of income
Calculate:
i) The different cross elasticities of demand and indicate the types of commodities ii) The income elasticity of demand for commodity X and indicate the type of commodity
iii) The own-price elasticity of demand and comment on the stabilityresult.

Answer 1

7.

Given term 22 and 8,

So their difference = 22–8 = 14

Given the difference between those terms

$=108–38 = 70$

Now we get,

$\frac{70}{14}=5$ (Difference divided by number of terms)

So the common difference is 5.

8th term is 38, so the sequence is :

$...,38,43,48,53,58,63,68,73,78,83,88,93,98,103,108,113,…$

The pattern is $5x-2.$

In 8 th term $= 5\times8–2= 40–2= 38$

In 22 th term $= 5\times22–2=110–2= 108$

So the whole sequence is:

$3,8,13,18,23,28,33,38,…$

10.

Given:

$Qx = 100 – Px + 0.75Pm – 0.25Pz + 0.00750075Y$

When $Px=10, Pm =20, Pz =40 \space and\space Y = 10000$

$Qx=100-10+0.75(20)-0.25(40)+0.00750075(10000)$

$Qx=170.0075$

Px = price of the commodity

Pm= price of another commodity M

Pz = price of another commodity Z

Y = level of income

 

i) Cross price elasticity measures the percentage change in the quantity demanded of good1 due to the percentage change in the price of the good 2. Here good 1 and good 2 are related goods( substitutes/complements)

1) Cross price elasticity of X and M

$XED=\frac{Change \space in\space Qx}{Price \space of\space good \space M}\times\frac{ Initial\space price\space of\space good\space M}{ Initial\space Quantity\space of\space good \space X}$

$XED=\frac{dQx}{dPm}\times\frac{Pm}{Qx}$

$Qx = 100 – Px + 0.75Pm – 0.25Pz + 0.00750075Y$

$\frac{dQ_x}{dP_m}=0.75$

When Pm=20 , Qx=170.005

$XED=0.75\times\frac{20}{170.005}$

$XED=0.08$

Positive cross price elasticity means Substitutes.

2) Cross price elasticity of X and Z

$XED=\frac{Change\space in\space Q_x}{Price\space of\space good\space Z}\times\frac{ Initial\space price\space of\space good\space Z}{ Initial\space Quantity\space of\space good\space X}$

$XED=\frac{dQx}{dPz}\times\frac{Pz}{Qx}$

$Qx = 100 – Px + 0.75Pm – 0.25Pz + 0.00750075Y$

$\frac{dQx}{dPz}=-0.25$

When Pz=40 , Qx=170.005

$XED=[0.25\times \frac{40}{170.005}$

$XED=-0.05$

Negative cross price elasticity means Complements.

3) Income Elasticity of Demand

$YED=\frac{Change\space in\space Qx}{Income\space of\space the\space consumer}\times\frac{ Initial\space income\space of\space the\space consumer}{Initial\space Quantity\space of\space good X}$

$YED=\frac{dQx}{dY}\times\frac{Y}{Qx}$

$Qx = 100 – Px + 0.75Pm – 0.25Pz + 0.00750075Y$

$\frac{dQx}{dY}=0.00750075$

When Y=10000 , Qx=170.005

$YED=0.00750075\times\frac{10000}{170.005}$

$YED=0.44$

Positive Income elasticity means normal goods.

 4) Own price elasticity of X and M

$PED=\frac{Change\space in\space Qx}{Price\space of\space good X}\times\frac{Initial\space price\space of\space good\space X}{ Initial\space Quantity\space of\space good\space X}$

$PED=\frac{dQx}{dPx}\times\frac{Px}{Qx}$

$Qx = 100 – Px + 0.75Pm – 0.25Pz + 0.00750075Y$

$\frac{dQx}{dPm}=-1$

When Px=10 , Qx=170.005

$PED=-1\times\frac{10}{170.005}$

$PED=-0.05$

Price elasticity of demand is less than 1, good is inelastic.