It is given that economy is at equilibrium at $ Y_0 $ = 1000. If Government undertakes a fiscal change so that tax rate $t$, increases by 0.05 and government spending increases by 50, will the budget surplus go up or down?

I tried doing it this way:

$$Y = C( Y - tY ) + I + G$$

Differentiating the above equation I got

$$dy = \frac{di+ dg - c'y dt}{1- c'(1-t) }$$

Where $c'$ is mpc, $t$ is tax rate.

Also,

$$BS = ty-G- TR$$

So,

$$dBS = tdy +ydt-dg-dtr$$

Now, $Y=1000, dt = 0.05, DG= 50$. Plugging these first into $dy$ and then using $dy$ in $dBS$, I got the following:

$$dBS = \frac{50t(1-c') }{(1-c'(1-t) }$$

Therefore the budget surplus goes up. Is my method correct? Also, what could be the economic intuition behind this? This is not a homework assignment. I am currently preparing for an upcoming entrance exam and was trying out a few back questions from dfs for conceptual clarity.

Your method is correct. This however does not have to be the case in general. Your have a budget surplus simply because your government spending does not rise too much relative to your tax revenue collection.

Consider the following with MPC = 0.3, t = 0.2 and Y = 1000 and dt = 0.5: enter image description here

If you increase spending too much, you will end up with a budget deficit.

There could be other factors as well, for example, the rate of taxes you have in the economy when the policy changes are made. For the same parameter set mentioned above, with dG and t free, you have (t on vertical axis and dG on horizontal axis):

enter image description here

There are some dG,t pairs which generate surplus/deficit/balanced budget.

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