# Does this conditional increase in income affect a budget line in the same way as an unconditional increase in income would?

Note: It's been awhile since I've taken introductory microeconomics. I remember increases in income move budget line outward. What if the increases have some condition?

The problem:

Jill has $$I$$ to spend on food and clothing. One unit of food costs $$p_F$$ while one unit of clothing costs $$p_C$$.

Budget line is

$$I=p_FF+p_CC$$

Say Jill receives a \$V voucher for food. What is the new budget line?

If the voucher was instead cash, we obviously have

$$I+V=p_FF+p_CC \tag{*}$$

If the voucher was really for food, we might not have $$(*)$$

However, I think we still have $$(*)$$ if:

$$I > C' := (I+V)/p_C$$ (the max that can be spent on clothing in $$(*)$$) because:

If the voucher is for food, then no more than $$I$$ is allowed to be spent on clothing.

However, such constraint is redundant if $$I>C'$$ because we know we won't exceed $$C'$$ and thus we won't exceed $$I$$.