3.15 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$ð Þ x, y : Suppose also that the elementary consumption bundle is given by ð Þ x0, y0 . Then the value of the benefit function, b U$ ð Þ, is that value of a for which Uð Þ¼ ax0, ay0 U$.

a. Suppose utility is given by U xð Þ¼ , y xby1%b. Calculate the benefit function for x0 ¼ y0 ¼ 1.

b. Using the utility function from part (a), calculate the benefit function for x0 ¼ 1, y0 ¼ 0. Explain why your results differ from those in part (a).

c. The benefit function can also be defined when an individual has initial endowments of the two goods. If these initial endowments are given by x, y, then b U$ ð Þ , x, y is given by that value of a which satisfies the equation Uð Þ x þ ax0, y þ ay0 ¼ U$. In this situation the ‘‘benefit’’ can be either positive (when Uð Þ x, y < U$) or negative (when Uð Þ x, y > U$). Develop a graphical description of these two possibilities, and explain how the nature of the elementary bundle may affect the benefit calculation.

d. Consider two possible initial endowments, x1, y1 and x2, y2. Explain both graphically and intuitively why bðU$, x1 þ x2 2 , y1 þ y2 2 Þ < 0:5b U$

, x1, y1

% & þ 0:5b U$

, x2, y2

% &. (Note: This shows that the benefit function is concave in the initial endowments.)