A parent isdeciding on an allowance for a teenager. The parent's concernis with how many books (x1) the teenager buys, whereas the teenage is also concerned with how much he has to expend on all other goods (x2). Assume fractional amounts of books are obtainable. The teenager's preferences are given by the following utility function (U X1, X2) = 10( X 1)1/2 + 2
a) Suppose the allowance is initially set at m = $30 when p1 = $6/unit, what will the optimal choices of the teenager be?
b) If the parents doubled the teenager's allowance, how would the number of books chosen change when p1 = $6/unit? Explain the result by reference to the type of preferences.
c) Suppose the parents agree to subsidize books (pay half the cost of books purchased) with allowance fixed at m = $30. What bundle does the teenager choose with the subsidy?
d) Construct a sketch of the choices in a) and c), by drawing the budget constraints, optimal choices and indifference curves through them in graph with x1 measured horizontally and x2 measured vertically.
e) How much does the subsidy in c) cost the parents? What would the outcome be if the parents simply used the subsidy expenditure to buy books for the teenager (an in-kind gift), and then left him to decide with m = $30 when p1 = $6/unit? Is the subsidy in c) or the direct purchase of books preferable in the eyes of the parents? Which arrangement does the teenager prefer?