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\( m \) Miles driven over vehicle life
\( p_\text{gas} \) Price of gas
\( \eta \) Vehicle efficiency (mpg)
\( V \) Vehicle purchase price

How much should we pay for better gas mileage?

If we are choosing between two vehicles on the basis of cost and gas mileage, we want the following to be true if we pick vehicle 1. We may plan to drive one car less depending on how many miles are currently on the odometer so we will distinguish between the expected vehicle lives in miles (\(m_1\) and \(m_2\)).

$$ \begin{align} V_1 + \frac{p_\text{gas}}{\eta_1} m_1 &< V_2 + \frac{p_\text{gas}}{\eta_2}m_2\\ (V_1-V_2) &< p_\text{gas} \left( \frac{m_2}{\eta_2}- \frac{m_1}{\eta_1} \right)\\ \left. \Delta V \right\rvert_{1,2} &< -p_\text{gas} \left.\Delta \left( \frac{m}{\eta}\right)\right\rvert_{1,2} \end{align} $$

For two vehicles with the same expected life \(m\), the break even point occurs where

$$ \label{eq:gas_dif} \left. \Delta V \right\rvert_{1,2} < p_\text{gas}m\left( \frac{\eta_1-\eta_2}{\eta_1\eta_2}\right) $$

Consider two vehicles, one at 28 mpg and one at 60 (my Honda Civic and an optimistic guess on a Prius). If they last for 100,000 miles and gas is $3 per gallon, the break-even point is where the Prius costs $5,700 more than the Civic. I have purchased gas for over $5 per gallon and as little as $1.40 (between 2012 and 2017 mostly in TX and CA). Unfortunately, we cannot know what gas prices will do in the future but we can use our formalized analysis to know the extent to which gas prices justify higher vehicle costs. We know that a more fuel-efficient vehicle is not worth a $10,000 premium unless gas prices take a horrible unexpected turn.

To better illustrate this as a ratio we can easily plot, move everything that is not a fuel efficiency to the left.

$$ \label{eq:gas_dif_nd} \frac{\left. \Delta V \right\rvert_{1,2}}{p_\text{gas}m} < \left( \frac{\eta_1-\eta_2}{\eta_1\eta_2}\right) $$ price difference justified by improved fuel economy

[Caption] The price difference justified by improved fuel economy between two vehicles with mileages \(\eta_1\) and \(\eta_2\). The left axis shows non-dimensionalize cost, \(\frac{\left. \Delta V \right\rvert_{1,2}}{p_\text{gas}m}\), and the right axis shows the cost difference assuming \(m = \) 100,000 miles and gasoline \(p_\text{gas} = \) $2.50 per gallon. ($2.50/gal was the average price I paid for gas between 2013 and 2019, mostly in Texas.)

© MC Byington