1. the benefit function In a 1992 article David G. Luenberger introduced what he termed the
benefit function
as a way of incorporating some degree of cardinal measurement into utility theory.11 The author asks us to specify a certain elementary consumption bundle and then measure how many replications of this bundle would need to be provided to an individual to raise his or her utility level to a particular target. Suppose there are only two goods and that the utility target is given by
U
*(x,
y). Suppose also that the elementary consumption bundle is given (x0,
y0). Then the value of the benefit function,
b(U
*) is that value of α for which
U
(αx0, αy0) =
U
*.
a. Suppose utility is given by
U(x,
y) =
x
βy1−β. Calculate the beneit function for
x0 =
y0 = 1.
b. Using the utility function from part (a), calculate the beneit function for
x0 = 1,
y0 = 0. Explain why your results differ from those in part (a).
c. The beneit function can also be deined when an individual has initial endowments of the two goods. If these initial endowments are given by
x,
y, then
b(U
*,
x,
y) is given by that value of α which satisies the equation
U(x
+ α
x0 ,
y
+ αy0) =
U
*. In this situation the ‘beneit’ can be either positive (when
U(x,
y)
U
*) or negative (when
U(x,
y) >
U
*). Develop a graphical description of these two possibilities, and explain how the nature of the elementary bundle may affect the beneit calculation.
d. Consider two possible initial endowments,
x1,
y1 and
x2,
y2. Explain both graphically and intuitively why
b(U
*,
x1 +
x2 2 ,
y1 +
y2 2 ) 0.5b(U
*,
x1,
y1) + 0.5b(U
*,
x
2,
y2). (Note: This shows that the beneit function is concave in the initial endowments.)