Because of the "great recession", applications to the Supplemental Nutritional Assistance Program (SNAP) soared.
The program distributes debit cards that can only be used to buy food. At times, SNAP recipients sell part of the money loaded on their cards on the black market. Some of them do so at less than 80 cents to the SNAP card dollar.
In this exercise we use a consumption diagram to analyze why this happens.
Consider a family that spends one fourth of its total monthly income (M) on food.

a) On a graph with food measured along the horizontal axis and other products along the vertical axis, draw the family budget constraint. Assume that the unit price of food and that of other items are both $1. Then, add an indifference curve to illustrate the family's optimal bundle.
The family applies to the SNAP program and receives an electronic card worth a third of the family total monthly income. The money on the card can only be spent on food.

b) In the diagram, show how participation to the SNAP alters the family's budget constraint. Show also the family's budget constraint if, instead of the SNAP card, it received $M/3 in cash (that can be spent on any product).

c) Using algebra, show that if the family has homothetic preferences (i.e., if at every level of income it devotes the same income share to food) and is given $M/3 in cash (that can be spent on any product) it wants to spend exactly $M/3 on food.

d) In the diagram, add an indifference curve to illustrate the family's optimal bundle after receiving the SNAP card. Assume the family has homothetic preferences. Label this bundle with the letter H for homothetic.
At bundle H, a family with homothetic preferences has a MRSO,F equal to negative one. If there were a black market for SNAP card swipes, this family would sell SNAP card dollars at a one to one rate.

e) Most families, however, consider food a necessity (the higher their income, the smaller the share of income they want to spend on food). How would the indifference curves of this type of family differ from the indifference curves of a family that has homothetic preferences?

f) In your diagram, show that a family that considers food a necessity, given $M/3 in cash would spend less than $M/3 on food. What is the optimal bundle of this type of family if instead of receiving cash it received the SNAP card?

g) At bundle H, is the MRSO,F of this type of family higher or lower than 1? What does this imply for the price at which the family is willing to sell SNAP card dollars on the black market?