I could not understand how to analyse the effect of the given situation in this question on government's expenditure and income. If the budget is balanced, the use of calculus technique also didn't help me. Any views on the solution to this question are welcome.
Answer is (d).
Due to the transfer, increase in income of workers = decrease in income of capitalists. Since the tax schedules on both persons are the same, the increase in taxation from workers offset the loss from capitalists, thus effect on budget (gov income - gov expenditure) is zero. Note that the tax is levied before any consumption decision, so things like propensity are irrelevant.
Since the worker has higher marginal propensity (say 0.5, and capitalist's is 0.3), then aggregate expenditure increases because 0.5T-0.3T > 0, where T is the transfer.
Actually I think the wording means "individual income" and "individual expenditure" rather than aggregate amount, because aggregate income does not change. Then the question is even more straightforward: both individual income and individual expenditure change.