The demand function for two commodities A and B in a market are given as follows. QA=96-2PA-3PB, QB=30-25PA+0.32Y. Where PA and PB are prices of commodity A and B respectively, and Y is consumers average money income, given PA=#2, PB=#4, and Y=#1000. Required, (a) calculate the price elasticity of demand for commodity A. (b) what is the income elasticity of demand for commodity B. (c) determine the cross-price elasticity of demand for commodity A with respect to the price of commodity B.
Hudu
Yes
(a) To calculate the price elasticity of demand for commodity A, we need to use the following formula:
Price Elasticity of Demand (E) = (% Change in Quantity Demanded) / (% Change in Price)
Given:
QA = 96 - 2PA - 3PB
PA = $2
To find the % change in quantity demanded, we need to calculate the quantity demanded at two different prices. Let's take PA = $2 and PA' = $4.
Therefore, QA = 96 - 2(2) - 3(4) = 96 - 4 - 12 = 80
and QA' = 96 - 2(4) - 3(4) = 96 - 8 - 12 = 76
Now, let's calculate the % change in quantity demanded:
% Change in Quantity Demanded = (QA' - QA) / QA * 100%
= (76 - 80) / 80 * 100%
= -4 / 80 * 100%
= -5%
Now, let's calculate the % change in price. Here, ΔPA = PA' - PA = $4 - $2 = $2.
% Change in Price = (ΔPA / PA) * 100%
= (2 / 2) * 100%
= 100%
Now, let's substitute the values in the formula for price elasticity of demand for commodity A:
E = (% Change in Quantity Demanded) / (% Change in Price)
= (-5%) / (100%)
= -0.05
Therefore, the price elasticity of demand for commodity A is -0.05.
(b) To calculate the income elasticity of demand for commodity B, we use the formula:
Income Elasticity of Demand (Ey) = (% Change in Quantity Demanded) / (% Change in Income)
Given:
QB = 30 - 25PA + 0.32Y
Y = $1000
To find the % change in quantity demanded, we need to calculate the quantity demanded at two different income levels. Let's take Y = $1000 and Y' = $1200.
Therefore, QB = 30 - 25(2) + 0.32(1000) = 30 - 50 + 320 = 300
and QB' = 30 - 25(2) + 0.32(1200) = 30 - 50 + 384 = 364
Now, let's calculate the % change in quantity demanded:
% Change in Quantity Demanded = (QB' - QB) / QB * 100%
= (364 - 300) / 300 * 100%
= 64 / 300 * 100%
= 21.33%
Now, let's calculate the % change in income. Here, ΔY = Y' - Y = $1200 - $1000 = $200.
% Change in Income = (ΔY / Y) * 100%
= (200 / 1000) * 100%
= 20%
Now, let's substitute the values in the formula for income elasticity of demand for commodity B:
Ey = (% Change in Quantity Demanded) / (% Change in Income)
= (21.33%) / (20%)
= 1.0665
Therefore, the income elasticity of demand for commodity B is 1.0665.
(c) To determine the cross-price elasticity of demand for commodity A with respect to the price of commodity B, we use the formula:
Cross-Price Elasticity of Demand (Exy) = (% Change in Quantity Demanded of A) / (% Change in Price of B)
Given:
QA = 96 - 2PA - 3PB
PB = $4
To find the % change in quantity demanded for commodity A, we need to calculate the quantity demanded at two different prices of commodity B. Let's take PB = $4 and PB' = $5.
Therefore, QA = 96 - 2(2) - 3(4) = 96 - 4 - 12 = 80
and QA' = 96 - 2(2) - 3(5) = 96 - 4 - 15 = 77
Now, let's calculate the % change in quantity demanded of commodity A:
% Change in Quantity Demanded of A = (QA' - QA) / QA * 100%
= (77 - 80) / 80 * 100%
= -3 / 80 * 100%
= -3.75%
Now, let's calculate the % change in price of commodity B. Here, ΔPB = PB' - PB = $5 - $4 = $1.
% Change in Price of B = (ΔPB / PB) * 100%
= (1 / 4) * 100%
= 25%
Now, let's substitute the values in the formula for cross-price elasticity of demand for commodity A:
Exy = (% Change in Quantity Demanded of A) / (% Change in Price of B)
= (-3.75%) / (25%)
= -0.15
Therefore, the cross-price elasticity of demand for commodity A with respect to the price of commodity B is -0.15.
To calculate the price elasticity of demand for commodity A, we will use the formula:
Price Elasticity of Demand (ε) = (% change in quantity demanded) / (% change in price)
(a) Calculate the price elasticity of demand for commodity A:
Given:
PA = $2
PB = $4
QA = 96 - 2PA - 3PB
To calculate the percentage change in quantity demanded, we need to find the initial quantity demanded (Q1) and the new quantity demanded (Q2).
If we substitute the given prices into the demand function, we can find Q1 and Q2:
Q1 = 96 - 2(2) - 3(4) = 96 - 4 - 12 = 80
Q2 = 96 - 2(2) - 3(4) = 96 - 4 - 12 = 80
Notice that the quantity demanded is the same for both initial and new values. Therefore, the percentage change in quantity demanded is 0%.
Next, we need to find the percentage change in price. For this, we compare the initial price (P1) with the new price (P2):
P1 = $2
P2 = $4
The percentage change in price can be calculated as follows:
(% change in price) = (P2 - P1) / P1 * 100
= (4 - 2) / 2 * 100
= 2/2 * 100
= 100%
Now, substitute the values into the price elasticity formula:
ε = (% change in quantity demanded) / (% change in price)
= 0% / 100%
= 0
Therefore, the price elasticity of demand for commodity A is 0.
(b) To calculate the income elasticity of demand for commodity B, we use the same formula:
Income Elasticity of Demand (ε) = (% change in quantity demanded) / (% change in income)
Given:
Y = $1000
QB = 30 - 25PA + 0.32Y
Again, we need to find the initial quantity demanded (Q1) and the new quantity demanded (Q2).
Substituting the given income and prices into the demand function, we can find Q1 and Q2:
Q1 = 30 - 25(2) + 0.32(1000) = 30 - 50 + 320 = 300
Q2 = 30 - 25(2) + 0.32(1000) = 30 - 50 + 320 = 300
Since the quantity demanded is the same for both initial and new values, the percentage change in quantity demanded is 0%.
To find the percentage change in income, compare the initial income (Y1) with the new income (Y2):
Y1 = $1000
Y2 = $1000
The percentage change in income is calculated as follows:
(% change in income) = (Y2 - Y1) / Y1 * 100
= (1000 - 1000) / 1000 * 100
= 0%
Substitute the values into the income elasticity formula:
ε = (% change in quantity demanded) / (% change in income)
= 0% / 0%
= undefined
Therefore, the income elasticity of demand for commodity B is undefined.
(c) To determine the cross-price elasticity of demand for commodity A with respect to the price of commodity B, we use the same formula:
Cross-Price Elasticity of Demand (ε) = (% change in quantity demanded of A) / (% change in price of B)
Given:
PA = $2
PB = $4
QA = 96 - 2PA - 3PB
To find the percentage change in quantity demanded of A, we need to find the initial quantity demanded (Q1) and the new quantity demanded (Q2).
Substituting the given prices into the demand function, we can find Q1 and Q2:
Q1 = 96 - 2(2) - 3(4) = 96 - 4 - 12 = 80
Q2 = 96 - 2(2) - 3(4) = 96 - 4 - 12 = 80
Since the quantity demanded is the same for both initial and new values, the percentage change in quantity demanded is 0%.
To find the percentage change in the price of B, compare the initial price (P1) with the new price (P2):
P1 = $4
P2 = $4
The percentage change in the price of B is calculated as follows:
(% change in price) = (P2 - P1) / P1 * 100
= (4 - 4) / 4 * 100
= 0%
Substitute the values into the cross-price elasticity formula:
ε = (% change in quantity demanded of A) / (% change in price of B)
= 0% / 0%
= undefined
Therefore, the cross-price elasticity of demand for commodity A with respect to the price of commodity B is undefined.