. Richard and Mary Johnson have fallen on hard times, but remain rational consumers. They are making do on $100 a week. Food (x) costs $2 per unit. Draw a budget line for the couple on the (x, y) coordinates, where x is food and y is the money for other goods (money is a numeraire whose price is 1).

a. Assume that their utility is U = x 0.5y 0.5 . Figure out the dollar amounts they spend on x and y[[ans: Given that their utility is Cobb-Douglass, they spend $50 (i.e., x=25) on food and $50 on all other goods (i.e., y=50)]].

(a) The couple suddenly become eligible for food stamps. This means that they can go to the agency and buy coupons that can be exchanged for $2 worth of food. Each coupon costs them $1. However, the maximum number of coupons they can buy per week is 10. On the graph, draw their new budget line.. [[ans: The budget line slopes down from the vertical intercept (100) at a rate of -1/1 until x reaches 10. Then, it then carries a slope of -2/1 until it crosses the x-axis.]]

(b) How many units of x and y will they buy once they enter the food stamp program? How many more units of x will they purchase?[[ans: With food coupons, their budget lines consist of two line segments with different slopes. The new budget line lies outside of the original one. Note that for x>10, the horizontal distance between two budget lines is 5. Given that the initial consumption is (25, 50), this will lead them to consume 2.5 more units of food. Essentially, if they buy the maximum allowable coupons, the money they save is $10. Out of $10, $5 are spent on food and $5 on other goods, which means that x increases by 2.5 units since food costs $2 per piece. ]]

(c) If the maximum coupons they can buy increase to c units (where c>10), what will the effect be on their consumption? Characterize the effect in details as c continues to increase. Obtain the maximum units of coupon they are willing to buy. [[ans: As c increases from 10 units, the coupon constraint remains binding as long as c < 50. The effect per additional unit of c on x is 1/4 (specifically, 0.5/px) for c < 100/3 and 1/2 for 100/3 < c < 50. For c > 50, the effect is zero. That means they will not buy more than 50 units of coupons.]]]