To lower tax rates mean a larger deficit? Why or Why not?

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.