Lower tax rates need not mean a larger deficit if the lowering of tax rates encourages more people to start paying taxes. This is especially true in nations where the number of tax defaulters is very high.
If the tax rates are extremely high, prospective tax payers do not report their earnings and try all means to avoid paying taxes. A large percentage of money here is driven into black money, with people not reporting income and other transactions. It has been seen that a reduction in tax rates actually boosts tax revenue collections by a large extent and hence contributes to decreasing the deficit.
Deficit in the question is, referring to the shortfall in revenues over total expenditure of a government. Thus:
Deficit = Expenditure - Revenue.
For a fixed expenditure, lower the revenue, higher will be the deficit.
Revenue itself can be expressed as:
Revenue = (Income)x(Tax rate as proportion of income)
If we assume that the income remains same irrespective of the rate of tax, the revenue will be directly proportional to the rate of tax. This will result in deficit increasing with fall in tax rate. However, in reality an increase in tax rate may lead to some reduction in total income. Because of this the the total revenue may increase, decrease, or remain constant with reduction in tax rate.
When the percentage rise in income matches the percentage reduction in tax rate the revenue will remain constant. When the percentage rise in income is lower than the percentage reduction in tax rate the revenue will reduce. Finally, hen the percentage rise in income is higher than the percentage reduction in tax rate the revenue will increase.