Hi Khanh, this is the answer to your question.
In order to close a recessionary gap, it is always more efficient to use an increase in government expenditures G than a reduction in taxes T. The government expenditure multiplier is 1/MPS, where MPS is the marginal propensity to save. The tax multiplier is, however, -MPC/MPS, where MPC is the marginal propensity to consume. (MPC + MPS = 1). The tax multiplier is thus 1 less than the government expenditure multiplier.
Let us take an example with MPC = 0.8. MPS = 1 - 0.8 = 0.2. The government expenditure multiplier is thus 1/0.2 = 5. The tax multiplier is - MPC/MPS = - 0.8/0.2 = - 4. Also, suppose we need to close a recessionary gap of $40 billion.
This will require an increase in government spending of ($40 billion)/5 = $8 billion.
If the government reduces taxes to close the gap, the amount by which taxes have to be reduced is ($40 billion)/4 = $10 billion. Therefore, the reduction in taxes have to be more than the increase in government spending. When taxes are reduced, real GDP rises, but not by the full amount of the tax decrease. People save some of the increase in their disposable income, which is a leakage from the multiplier process. Also, if households view the tax cut as temporary, then spending may not increase much. This will also reduce the strength of the multiplier process.