The question is "If the government of country Z increases spending by \$12 million dollars and raises tax collections by the same amount, then what will be the overall impact of theses moves on real GDP in country Z." The official solution is that real GDP will increase by \$12 million because it is a balanced budget move.
However, I think it depends. Increased tax collection will discourage production while the spending would encourage production. Any suggestions will be appreciated.
Your professor want probably that you compute the keynesian multiplicator (hint: once you have the general formula, replace dT by dG and see what you get).
Intuition: if you raise G, you will raise your GDP. As Y increases, your consumption increases. But the taxes (T, lump sum) increase also, so the consumption decreases (it depends of the disposable income). The overall effect is a 1 to 1 increase of your GDP.
This question is too vague to be answered, there are many factors. One, for example:
This is relevant because if the households receive utility from what the government is doing with the money, they will "feel richer" - the expenditure has a wealth effect (and is neutral in case of no other rigidities). In the other case, even with lump sump taxation, households will feel poorer and respond differently.
If you want to look more into this, solve the following simple optimization problem:
$$ \max_{c,l} U(c + G(T),l) \text{ s.t. } c = w(1-l) + a - T $$
where $a$ is assets, $T$ is the lump sump taxation, and $c$ and $l$ are consumption and leisure. Look at the impact of different functional forms of $G(T)$ onto the optimal choice. For example: $G(T) = T$, $G(T) = 0.5 T$, and $G(T) = 0$.
