If your are in a first-year macro economics course then...

1) yes government purchases can increase total income.
2) cutting taxes could achieve the same result,
3) But the tax cut would need to be larger because the tax multiplier is one less than the government multiplier.

If you are in a second year or higher, then then answer becomes murkier. Government purchases with out a tax increase can only occur by government borrowing or by printing more money. A tax cut, in addition to increasing spending, may change the labor supply.

In a first-year macroeconomics course, you learn about the basic concepts of government policies and their impact on the economy. Here's an explanation of the statements you mentioned:

1) Yes, government purchases can increase total income: In macroeconomics, it is recognized that government purchases, such as spending on infrastructure or public goods, can boost economic activity. By directly injecting money into the economy, government purchases can increase total income and stimulate various sectors.

2) Cutting taxes could achieve the same result: Cutting taxes can also have a positive effect on total income. When individuals or businesses pay fewer taxes, they have more disposable income. This increased income can lead to higher consumption and investment, thus boosting economic activity.

3) The tax cut would need to be larger because the tax multiplier is one less than the government multiplier: This statement is based on the concept of fiscal multipliers, which measure the ripple effect of changes in government spending or taxes on total income. The government multiplier represents how changes in government spending impact income, while the tax multiplier measures the impact of changes in taxes.

The tax multiplier is generally assumed to be one less than the government multiplier because part of any tax cut may be saved rather than spent. So, a larger tax cut would be needed to achieve the same increase in total income compared to a government purchase of the same magnitude.

Now, if you are studying in a second-year or higher macroeconomics course, the analysis becomes more complex. This is because you would likely dive deeper into the nuanced effects of government policies. For example:

- Government purchases without a tax increase can only occur by government borrowing or by printing more money: At an advanced level, you would consider the financing aspect of government purchases. Governments can fund their spending through borrowing (issuing bonds) or by monetizing debt (printing money). Each method has its own economic consequences and implications.

- A tax cut may change the labor supply: In more advanced macroeconomics, you would study how changes in taxes affect overall labor supply. A tax cut can incentivize individuals to work more, as they retain a higher share of their income. This, in turn, could impact the labor market and potentially lead to changes in overall economic output.

Overall, as you progress in your economics studies, you will encounter more intricate theories and models that will deepen your understanding of government policies and their impact on the economy.