Copy of The demand function for good 1 is given by 01 = 401- 3021/2 where p1 is the price of good 1, p2 is the price of good 2, and y is income. i. are goods 1 and 2 complements or substitutes? ii. what is the income elasticity of demand for good 1?

The answers are:

 

i.) Substitutes

ii.) Income elasticity of demand for good 1 is 2.

Step-by-step explanation

Given a demand function for good 1:

 

q₁ = 4p₁^-3 p₂y²

q₁ = 4p₂y² / p₁³

 

i. Are goods 1 and 2 complements or substitutes?

 

Assuming p₁ > 0, p₂ > 0, and y > 0.

 

We need to derive q₁ with respect to p₂ to answer this question.

 

∂q₁/∂p₂ = 4(1p₂^1-1)y² / p₁³

∂q₁/∂p₂ = 4(1)y² / p₁³

∂q₁/∂p₂ = 4y² / p₁³

∂q₁/∂p₂ = 4y² / p₁³ > 0

 

Since ∂q₁/∂p₂ > 0, we can conclude that goods 1 and 2 are substitutes. Since y > 0, and p₁ > 0, then ∂q₁/∂p₂ should be greater than 0.  The partial derivate that we just solved entails that an increase in the price of good 2 will lead to an increase in the demand for good 1.

 

ii. Solving for the income elasticity of demand for good 1.

 

There are two ways.

 

Solution 1:

 

Step 1: Take the natural log of the equation.

 

lnq₁ = ln4 - 3lnp₁ + lnp₂ + 2lny

 

Step 2: Take the total differential of the transformed demand equation.

 

(1/q₁) dq₁ = -3(1/p₁) dp₁ + (1/p₂) dp₂ + (2/y) dy

(1/q₁) dq₁ = (-3/p₁) dp₁ + (1/p₂) dp₂ + (2/y) dy

 

Step 3: Let dp₁ = dp₂ = 0

 

(1/q₁) dq₁ = (-3/p₁) (0) + (1/p₂) (0) + (2/y) dy

(1/q₁) dq₁ = (2/y) dy

(dq₁ / dy) * (y/q₁) = 2

E_y = (dq₁ / dy) * (y/q₁) = 2

 

The income elasticity of demand for good 1 is 2.

 

Solution 2:

 

Step 1. Derive the original demand function with respect to y.

 

∂q₁/∂y = 4p₁^-3 p₂ (2y^2-1)

∂q₁/∂y = 8p₁^-3 p₂ y

 

Step 2: Set up the income elasticity of demand equation.

 

Substitute ∂q₁/∂y = 8p₁^-3 p₂ y , q₁ = 4p₁^-3 p₂y² to the elasticity equation.

 

E_y = (∂q₁/∂y) * (y/q₁)

E_y = (8p₁^-3 p₂ y) * (y / 4p₁^-3 p₂y²)

E_y = 8p₁^-3 p₂ y² / 4p₁^-3 p₂y²

 

Cancel out p₁^-3 p₂ y² in both numerator and denominator. So, we are left with

 

E_y  = 8 / 4

E_y = 2

 

Hence, the income elasticity of demand is 2.