Question

Q10. The demand for good X is given by the following equation:
      

QX = 325 - PX – 1.5 PW + 1.25 PG + 0.8 PY - 0.1 M

where  

QX is the number of X sold per week; PX, PW, PG, PY are the prices of the respective goods and M is the average monthly income.

Currently PX = 200, PW = 50, PG = 80, PY= 125, and M = 2000.

(a) Should PX be increased or decreased to maximize revenue? How do you know?
(b) Calculate the elasticity of demand for good X with respect to PW, PG, PY and M. In less than two sentences, explain precisely what each elasticity measure means.
(c) Calculate the price range over which the demand for good X is elastic.
(e) By how much must the price of X change if there is a 1% increase in the average monthly income (M) and the goal is to keep QX constant?

Answer 1

(a)

Plugging in given values,

QX = 325 - PX – 1.5 x 50 + 1.25 x 80 + 0.8 x 125 - 0.1 x 2000

QX = 325 - PX - 75 + 100 + 100 - 200

QX = 250 - PX

PX = 250 - QX

Total revenue (TRX) = PX.QX = 250QX - QX²

Revenue is maximized when dTRX/dX = 0

dTRX/dX = 250 - 2QX = 0

2QX = 250

QX = 125

PX = 250 - 125 = 125

Therefore PX has to be decreased from 200 to 125.

(b)

When PX = 200, QX = 250 - 200 = 50

(i) Elasticity of demand with respect to PW = (QX/PW) x (PW/QX) = - 1.5 x (50/50) = - 1.5

It signifies that as PW increases (decreases) by 1 unit, demand for X (QX) decreases (increases) by 1.5 units, and since Elasticity of demand with respect to PW is negative, goods X and W are complements in consumption.

(ii) Elasticity of demand with respect to PG = (QX/PG) x (PG/QX) = 1.25 x (80/50) = 2

It signifies that as PG increases (decreases) by 1 unit, demand for X (QX) increases (decreases) by 2 units, and since Elasticity of demand with respect to PG is positive, goods X and G are Substitutes in consumption.

(iii) Elasticity of demand with respect to PY = (QX/PY) x (PY/QX) = 0.8 x (125/50) = 2

It signifies that as PY increases (decreases) by 1 unit, demand for X (QX) increases (decreases) by 2 units, and since Elasticity of demand with respect to PY is positive, goods X and Y are Substitutes in consumption.

(iv) Elasticity of demand with respect to M (Income elasticity) = (QX/M) x (M/QX) = 0.1 x (2000/50) = 4

It signifies that as M increases (decreases) by 1 unit, demand for X (QX) increases (decreases) by 4 units, and since Income Elasticity of demand is positive and higher than 1, goods X is a normal and luxury good.

(c)

Own Price elasticity (E) = (dQX/dPX) x (PX/QX) = - 1 x [PX / (250 - PX)]

Demand is elastic when absolute value of E is higher than 1.

1 x [PX / (250 - PX)] > 1

PX > (250 - PX)

2PX > 250

PX > 125

(e)

New value of M = 2000 x 1.01 = 2020

From demand function with QX = 50,

50 = 325 - PX – 1.5 x 50 + 1.25 x 80 + 0.8 x 125 - 0.1 x 2020

50 = 325 - PX - 75 + 100 + 100 - 202

50 = 248 - PX

PX = 198

Therefore, Price has to be decreased by (200 - 198) = 2.