A consumer's utility function is
u = xx
-a
where 1 and 2 are the quantities consumed of goods one and two, respectively, and a is a parameter. Let P1 and p2 denote the prices of goods one and two, respectively, so that the total expenditure in goods one and two is P11+ P2x2. The minimum expenditure that can be achieved subject to the constraint that the consumer achieves the level of utility u is given by
e = papaa
(this expression can be obtained by solving the corresponding constrained optimization problem). Calculate an approximation to the change in e due to increasing the level of utility by a small quantity d.