Let $x$ a TVL farming at weekly APR $X$ the reward token of a ve(3,3) offering on average an weekly APR $Y$ of voting reward. We assume weekly APRs are already divided by 100 (not percentages but factors).

Each week the farming TVL offers a value of $x \times X$ farming rewards. Assuming constant prices and APR, after $N$ weeks we have a value of $N \times x \times X$ veTokens by locking them (assuming no rebase).

After $N$ weeks our veTokens allows us to earn a value of $N \times x \times X \times Y$ voting rewards.

We are searching for the bootstrap period $N$ such that the voting rewards of a week are the same as the farming rewards in terms of value:

$N \times x \times X \times Y = x \times X$

The $x \times X$ factors cancels and we get the equation:

$N \times Y = 1$

So $N = \frac{1}{Y}$

Let’s test with some values. Assume the yearly voting APR is 100%, then:

$Y = \frac{100}{52 \times 100} = \frac{1}{52}$

So $N = 52$ weeks in that case, we need a year to farm enough veToken to incentivize the same TVL as farming.

Assume the yearly voting APR is 200%, then:

$Y = \frac{200}{52 \times 100} = \frac{2}{52} = \frac{1}{26}$

So $N = 26$ weeks in that case, we need 6 months to farm enough veToken to incentivize the same TVL as farming.

This mathematical demonstration does not reflect reality, as APRs and token prices always vary. However, it helps to understand that Autobribes would be more powerful if deployed early in the lifetime of a ve(3,3) DEX, since voting APRs tend to be higher early on and decrease over time.