If income elasticity is given as Y=dlogq÷dlogy=y÷q×dq÷dy proof that the sum of the income elasticity weighted by total expenditure is proportionally equal to unity
To prove that the sum of the income elasticity weighted by total expenditure is proportionally equal to unity, we need to utilize the definition and properties of elasticity and total expenditure.
First, let's define some terms:
- Y: Income elasticity of a good (Y = dlogq / dlogy, where q is the quantity of the good and y is the income).
- q: Quantity of the good.
- y: Income.
- E: Total expenditure on the good (E = q * p, where p is the price of the good).
- Yx: Income elasticity of good x.
- Ex: Expenditure on good x.
Now, let's calculate the sum of the income elasticities weighted by total expenditure:
Σ(Yx * Ex) / E (Equation 1)
To simplify the equation, we can express the expenditure Ex in terms of the quantity q and the price p:
Ex = q * p
Substituting this into Equation 1, we get:
Σ(Yx * (q * p)) / E
Since we are given that Y = Yx, we can replace Yx with Y:
Σ(Y * (q * p)) / E
Expanding the sum, we get:
(Y1 * (q1 * p1) + Y2 * (q2 * p2) + ... + Yn * (qn * pn)) / E
Now, let's consider the total expenditure E:
E = (q1 * p1) + (q2 * p2) + ... + (qn * pn)
Dividing the numerator and denominator of the original equation by E, we get:
Σ((Y1 * (q1 * p1) / E) + (Y2 * (q2 * p2) / E) + ... + (Yn * (qn * pn) / E)) / (E / E)
By rearranging the terms, we can rewrite this as:
Σ(E1 / E + E2 / E + ... + En / E)
Since the total expenditure E cancels out, we are left with:
Σ(1) (Equation 2)
The sum of the income elasticity weighted by total expenditure is therefore equal to the sum of ones, which is the number of goods considered (n). Hence, the sum of the income elasticity weighted by total expenditure is proportionally equal to unity (n = 1).
Note: This proof assumes that the prices of goods are constant and have no effect on the income elasticity. Additionally, it assumes that the income elasticity and total expenditure for each good are known.