Question: a consumer faces the following utility function uxm with...
. A consumer faces the following utility function: U=xM, with M representing dollars spent on all goods other than good x (therefore PM º 1). Assume that Px =$1 and I = $100.
a. Find the optimal consumption bundle and the level of utility at that bundle. Show the result from this part on a graph. Place x on the horizontal axis and M on the vertical axis.
b. Suppose the government provides the consumer with $20 worth of X-stamps. Find the new optimal consumption bundle. HINT: To find the solution you should assume that the consumer received a gift of $20 cash. (QUESTIONS TO PONDER: Why can make we make this assumption—after all, the consumer received food stamps not cash? Can we always make this assumption?). Show this result on the same graph as used in part (a).
c. Suppose the government replaces its food stamp program with a per-unit subsidy program. The per-unit subsidy is selected so as to allow the consumer to achieve the same level of utility as under the food stamp program. Using the indirect utility function, find the per-unit subsidy that would be required to achieve this result. (NOTE: The per-unit subsidy equals $1 minus price of X under the per-unit subsidy. Notice that we are implicitly assuming that the supply of X is perfectly elastic and therefore the entire subsidy is passed on to consumers).
Find x, M, and the cost to the government of providing this subsidy. Show this outcome on the same graph as used in parts (a) and (b). On your graph, indicate the cost to the government of each program.