Answer to Question #191414 in Microeconomics for Gemechu

Question

The market demand for brand X has been estimated as

 

 

where Px is the price of brand X, I is per-capita income, Py is the price of brand Y, and Pz is the price of brand Z. Assume that Px = $2, I = $20,000, Py = $4, and Pz = $4.

 

a.      With respect to changes in per-capita income, what kind of good is brand X?

b.     How are brands X and Y related?

c.      How are brands X and Z related?

d.     How are brands Z and Y related?

e.      What is the market demand for brand X?

Answer 1

$Q X = 1,500 - 3 P {_X} - 0.05 I - 2.5 P{_Y} + 7.5 P{_Z}$

a)     Good X is a giffen good because its income elasticity of demand is negative. This means that as consumers' income increase, demand for good X decreases and when income decreases, demand for good X increases.

Income elasticity of demand$= \frac{∂Q}{∂I}\times \frac{I}{Q}$

$Q = 1,500 - 3 (2) - 0.05 (20,000) - 2.5 (4) + 7.5 (4)$

    $= 514$

$\frac{∂Q}{∂I} = - 0.05, I = 20,000 and Q = 514$

$IED = \frac{(-0.05 \times 20,000)}{514}$

    $= - 1.9455$

b)     The relationship between two goods is determined by finding cross price elasticity of demand. When cross price elasticity is negative, the two goods are complementary and when the cross price elasticity is positive, then goods are said to be supplementary

Cross price elasticity of demand $= \frac{∂Q}{∂ P{_Y}}\times\frac {P{_Y}}{Q}$

$\frac{∂Q}{∂P{_ Y}} = - 2.5, P{_Y} = 4 and Q = 514$

Cross price elasticity of demand $= - 2.5 \times \frac{4}{514}$

       $= - 0.01946$

Goods X and Y are complements because their cross price elasticity of demand is negative. This means that when the price of good Y increases by 1%, demand for X decreases by 0.01946%.

c)      Goods X and Z are supplements

Cross price elasticity of demand $= \frac{∂Q}{∂P{_Y}} \times\frac{P {_Y}}{Q}$

$\frac{∂Q}{∂ P{_Z}} = 7.5, P Z = 4 and Q = 514$

Cross price elasticity $= 7.5 \times\frac{ 4}{514}$

       = 0.05837

The two goods are supplementary because their cross price elasticity of demand is positive. This shows that when the price of Z increases by 1%, demand for X increases by 0.05837%.

d)     Complements.

Since X and Y are supplements and X and Z are complements, then Y and Z are complements. This is because good Y can be used in place of X.

e)     Market demand for brand X is determined by substituting the values of P _X, I, P _Y and P _Z in the demand function as shown below.

$Q = 1,500 - 3 (2) - 0.05 (20,000) - 2.5 (4) + 7.5 (4)$

    $= 514$