solution

Kate will be better off with food stamps but will consume more food with the excise subsidy. Her optimal point with the food stamps is E on budget line AA′Z′, with the $100 cost shown by the distance ET. Her optimal point with the excise subsidy is E′ on budget line AZ1 with the $100 cost equal to E′T′. The trick is to locate the size of the excise subsidy per unit that costs exactly $100, and this implies that the optimal point with the excise subsidy is where Kate’s price-consumption curve intersects A′Z′. Note that with a larger excise subsidy per unit (lower price), total cost would exceed $100, and vice versa