Question

The U.S. government spends over$\backslash (33$billion on its Food Stamp program to provide millions of Americans with the means to purchase food. These stamps are redeemable for food at over$160,000$store locations throughout the nation, and they cannot be sold for cash or used to purchase non-food items. The average food stamp benefit is about$\backslash )284$per month. Suppose that, in the absence of food stamps, the average consumer must divide$\backslash (600$in monthly income between food and “all other goods” such that the following budget constraint holds:$\backslash )600+\backslash (12A+\backslash )4F$,where Ais the quantity of “all other goods” and Fis the quantity of food purchased. Using the vertical axis for “all other goods,” draw the consumer’s budget line in the absence of the Food Stamp program. What is the market rate of substitution between food and “all other goods”? On the same graph, show how the Food Stamp program alters the average consumer’s budget line. Would this consumer benefit from illegally exchanging food stamps for cash? Explain.

Short answer

Illustration is represented via two graphs.

Step-by-step solution

Find the horizontal and vertical intercept 

Given, the budget constraint,

$\$600=\$12A+\$4F$

Here,

A is the quantity of “all other goods”

F is the quantity of food purchased.

Use the equation;

$\frac{M}{P_F}$

$M=\$600$is consumer’s income

$P_F=4$is the price of food purchased

By substituting we will get,

$\begin{array}{rcl}\frac{M}{P_F}& =& \frac{\$600}{4}\\ & =& 150\\ & & \end{array}$

Therefore, horizontal intercept$=150$

Using the equation;

$\frac{M}{P_A}$

$M=\$600$is consumer’s income

$P_A=12$is the price of all other goods

By substituting we will get,

$\begin{array}{rcl}\frac{M}{P_A}& =& \frac{\$600}{12}\\ & =& 50\\ & & \end{array}$

Therefore, the vertical intercept$=50$

Plotting the graph 

Here, the horizontal intercept$=150$

The vertical intercept $=50$

Finding the Market rate of substitution

Market rate of substitution is equal to the ratio of the prices.

Using the following equation to get the Market Rate of Substitution:

$MRS=\frac{\Delta A}{\Delta F}$

$\Delta A$is the vertical concept

$\Delta F$is the horizontal concept

So,

$\begin{array}{rcl}MRS& =& -\frac{\Delta A}{\Delta F}\\ & =& -\frac{50}{150}\\ MRS& =& -\frac{1}{3}\\ & & \end{array}$

Therefore, the Market Rate of Substitution is$-\frac{1}{3}$

Finding the new value of  and  axis intercept

Given that the average food stamp benefit is about 284 per month, the monthly income will be,

$\$600+\$284=\$884$

The monthly income isper month. So, the new budget line will be,

$\begin{array}{rcl}\frac{M}{P_F}& =& \frac{\$884}{4}\\ & =& 221\\ & & \end{array}$

Therefore, the new value of F- axis intercept is 221 .

Again,

$\begin{array}{rcl}\frac{M}{P_A}& =& \frac{\$884}{12}\\ & =& 73.7\\ & & \end{array}$

Therefore, the new value of A- axis intercept is$73.7$ .

Plotting the graph

With the new values the graph will be: