How might the impact of an increase in government spending on the economy change if the increase in public expenditure is paid for by raising taxes?
This depends on the circumstances.
If the economy is in a recession and we are increasing government spending to act as a stimulus, then raising taxes will be a bad idea. In the event of a recession, an increase in government spending is meant to put more money in the pockets of consumers so as to increase aggregate demand. If, at the same time, we raise taxes, the impact of the spending increase is dramatically reduced. There will still be some impact, since the spending multiplier is larger than the tax multiplier, but it will be a much smaller impact. By taking money away from people at the same time that we give them money, we are reducing the impact of the government spending.
If, by contrast, the economy is not doing badly and we are trying to invest in the future, this is not such a bad course of action. Let us say our increase in spending is meant to build new infrastructure that will help with economic growth in the future. If this is the case, there is no need for an immediate stimulus. If we raise taxes a little bit, it will offset the spending and prevent inflation and/or crippling budget deficits.
Thus, the wisdom of such a policy depends on the situation.