Question

An analyst at the Green Car Co. Ltd. estimated the following demand function for the electric vehicles it sells:
\(Q_E=0.75-1.5P_{E} +2.5P_F - 0.5P_B+3.2l\)
where \(Q_E\) = Number of electric vehicles (in thousand per year), \(P_E\) = Unit price of electric vehicle (Rs. in Lakh), \(P_F\) = Average unit price of vehicle using fossil fuels (Rs. in Lakh), \(P_B\) = Unit price of battery used in electric vehicle (Rs. in Lakh), \(l\) = Personal disposable income (Rs. in Lakh).
Let \(P_E\)  = Rs. 6.5 Lakh, \(P_F\)= Rs. 4.5 Lakh,  \(P_B\)  = Rs. 0.5 Lakh and \(l\) = Rs. 10 Lakh. Then the income elasticity of demand (\(e_{Qgl}\)) and the cross price elasticity of demand (\(e_{Qgp_f}\)) satisfy

• A. 0.98 \(e_{Qgl}\)≤0.99 and 0.33  \(e_{Qgp_f}\) ≤ 0.34
• B. 0.94 \(e_{Qgl}\) ≤0.95 and 0.45 \(e_{Qgp_f}\)≤0.46
• C. 0.98 \(e_{Qgl}\) ≤0.99 and 0.45 \(e_{Qgp_f}\)≤0.46
• D. 0.94 \(e_{Qgl}\) ≤0.95 and 0.33 \(e_{Qgp_f}\) ≤ 0.34

Answer 1

The Correct Option is D

To solve the problem, we need to calculate the income elasticity of demand (\(e_{Qgl}\)) and the cross price elasticity of demand with respect to the fossil fuel vehicle price (\(e_{Qgp_f}\)) using the given demand function:

\(Q_E = 0.75 - 1.5P_E + 2.5P_F - 0.5P_B + 3.2l\)

Let us substitute the given values:

• \(P_E = 6.5\)
• \(P_F = 4.5\)
• \(P_B = 0.5\)
• \(l = 10\)

Substituting these into the demand function:

\(Q_E = 0.75 - 1.5 \times 6.5 + 2.5 \times 4.5 - 0.5 \times 0.5 + 3.2 \times 10\)

Compute each term:

• \(1.5 \times 6.5 = 9.75\)
• \(2.5 \times 4.5 = 11.25\)
• \(0.5 \times 0.5 = 0.25\)
• \(3.2 \times 10 = 32\)

Calculate \(Q_E\):

\(Q_E = 0.75 - 9.75 + 11.25 - 0.25 + 32 = 34\)

1. Income Elasticity of Demand (\(e_{Qgl}\)):

The formula for income elasticity is:

\(e_{Qgl} = \left(\frac{\partial Q_E}{\partial l} \right) \times \left(\frac{l}{Q_E}\right)\)

From the demand equation, \(\frac{\partial Q_E}{\partial l} = 3.2\).

Substitute the values:

\(e_{Qgl} = 3.2 \times \left(\frac{10}{34}\right)\)

\(e_{Qgl} \approx 0.941\)

2. Cross Price Elasticity of Demand (\(e_{Qgp_f}\)):

The formula for cross price elasticity is:

\(e_{Qgp_f} = \left(\frac{\partial Q_E}{\partial P_F} \right) \times \left(\frac{P_F}{Q_E}\right)\)

From the demand equation, \(\frac{\partial Q_E}{\partial P_F} = 2.5\).

Substitute the values:

\(e_{Qgp_f} = 2.5 \times \left(\frac{4.5}{34}\right)\)

\(e_{Qgp_f} \approx 0.331\)

Based on the above calculations:

• Income elasticity \(e_{Qgl}\) is approximately \(0.941\), which falls in the interval \(0.94 \leq e_{Qgl} \leq 0.95\).
• Cross price elasticity \(e_{Qgp_f}\) is approximately \(0.331\), which falls in the interval \(0.33 \leq e_{Qgp_f} \leq 0.34\).

Thus, the correct answer is:

0.94 \(e_{Qgl}\) ≤0.95 and 0.33 \(e_{Qgp_f}\) ≤ 0.34