The general linear demand for good Xis estimated to be Q= 18,000-175P+0.35M-16PR
ID: 1217386 • Letter: T
Question
The general linear demand for good Xis estimated to be Q= 18,000-175P+0.35M-16PRwhere P is the price of good X, Mis average income of consumers who buy good X, and PR is the price of related good R. The values of P, M, and PR are expected to be $65, $52,000, and $100, respectively. Use these values at this point on demand to make the fo1lowing computations. (a) Compute the quantity of good X demanded for the given values of P, M, and PR.
(b) Calculate the price elasticity of demand E. At this point on the demand for,¥, is demand elastic, inelastic, or unitary elastic? How would increasing the price of X affect total revenue? Explain.
( c) Calculate the income elasticity of demand EM. Is good X nonnal or inferior? Explain how a 1.75 percent decrease in income would affect demand for,¥, all other factors affecting the demand for X remaining the same.
(D) Calculate the cross-price elasticity ExR· Are the goods X and R substitutes or complements? Explain how a 2.5 percent increase in the price of related good R would affect demand for,¥, all other factors affecting the demand for X remaining the same.
Explanation / Answer
a.
The price of good X = P = $65
The average income of the consumers who buy good X = M = $52,000
The price of the related good R = Pr = $100
Given the linear demand for good X is:
Q= 18,000 – 175P + 0.35M – 16PR
Now substitute the given values in the equation:
Q= 18,000 – (175 × 65) + (0.35 × 52,000) – (16 × 100)
Q = 18,000 – 11375 + 18200 – 1600
Q = 23,225
Therefore, the quantity of good X demanded for the given values of P, M and Pr is 23,225.
b.
as per the question, for price of X = P1 = $65, the demand is Q1 = 23,225.
Assume that the price of X increased to $75 (which is P2) then the demand will be:
Q= 18,000 – (175 × 75) + (0.35 × 52,000) – (16 × 100)
Q = 18,000 – 13125 + 18200 – 1600
Q = 21,475
Thus, change in demand with change in price will be:
Q/ P = (Q2 – Q1)/(P2 – P1)
=(21,475 – 23225) / (75 – 65)
=(– 1750) / 10 = – 175
Price elasticity of demand = Ep = %of change in quantity demanded ÷ %change in price
= Q/ P × P / Q
= – 175 × 65 / 52000 = – 175 × 0.00125 = – 2.1875
Therefore, as the price elasticity of demand is – 2.1875, the demand is price elastic. The reason behind is the percentage decline in the demand is greater than the percentage increase in price.
Total revenue = Price × Quantity = 65 × 52,000 = 3,380,000
With a change in price to $75, the total revenue is
Total revenue = Price × Quantity = 75 × 21,475 = 1,610,625
c.
Income elasticity of demand = of change in quantity demanded ÷ %change in income
= Q/ I × I / Q
Assume that the income increase from $52,000 to $62,000, keeping everything constant. Then the demand for X will change as:
Q= 18,000 – (175 × 65) + (0.35 × 62,000) – (16 × 100)
Q = 18,000 – 11375 + 21700 – 1600
Q = 26725
Q/ I = (Q2 – Q1)/(I2 – I1)
= (26,725 – 23225) / (62,000 – 52,000) = 3500 / 10,000 = 0.35
Thus, income elasticity of demand = 0.35 × (52,000/23225) = 0.35 × 2.2389 = 0.7836
If the income is decreased to 1.75%, then the new income will be 52,000 × 1.75 = 91,000 – 52,000 = 39,000
Therefore the quantity will be:
Q= 18,000 – (175 × 65) + (0.35 × 39,000) – (16 × 100)
Q = 18,000 – 11375 + 13650 – 1600
Q = 18,675.
This implies that the quantity has decreased.
d.
Cross elasticity = %change in the demand for good X / %change in the price of good Y
= Qx/ Py × Py/Qx => Qx/ Py = - 16
Hence,
Cross elasticity = –16 × (100/23225) = –16 × 0.004306 = –0.068891
If price of the related good is increased by 2.5%, then the new price = Pr = 100 × 2.5 = 250
So, the quantity will be:
Q= 18,000 – (175 × 65) + (0.35 × 39,000) – (16 × 2500)
Q = 18,000 – 11375 + 13650 – 4000
Q = 16,275.
Hence, the quantity decreased.
As the price of the related good increases, there is a decrease in the quantity of X. the related good is compliment of good X. therefore, Good X and good R are complements.
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