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


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 $I$.



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