First I must say that I have little experience in implementing the bootstrap procedure in practice, so please correct me if I am wrong.

Now you want to test

\( H_0: \frac {-\alpha_1} {2\alpha_2} = \frac {-\beta_1} {2\beta_2} \)

which is equivalent to

\( H_0: \alpha_1\beta_2 - \beta_1\alpha_2 = 0 \)

The basic idea is that as the estimators \( \hat{\alpha}_1, \hat{\alpha}_2, \hat{\beta}_1, \hat{\beta}_2 \) are consistent estimators, we may use

\( T = \hat{\alpha}_1\hat{\beta}_2 - \hat{\beta}_1\hat{\alpha}_2 \)

as the test statistic and reject \( H_0 \) when it is significantly different from 0.

To determine whether it is significant or not, you need to determine the distribution of \( T \) under \( H_0 \) and find out the corresponding quantiles (with the given significance level) and use that to give the rejection/acceptance region.

The steps could be like the following:

1. Suppose you have \( m, n \) pairs of data for each situation. Now you re-sample from the original sample with replacement with the same sample size for each situation.

2. Using the generated sample, now estimate all the parameters under the \( H_0 \) constraint - which you need to jointly estimate for both situation and you may need to use the Lagrange multiplier if you are seeking a closed-form solution.

3. Calculate the test statistic \( T \) in this sample, and record it.

4. Return to step 1 and repeat for \( B \) times. Use the sample percentile of these recorded \( T_1, T_2, \ldots, T_B \) to construct the acceptance region. E.g. if your significance level is \( 5\% \) then you will use the 2.5 and 97.5 percentile as the pair of end-points for the acceptance region (interval).

Once you obtain the acceptance region, you can calculate \( T \) for the original sample again and make the decision.