Suppose that a consumer has the following utility function: U(x,y) = sqrt(x) + sqrt(y). The price of x is Px = $1 and the price of y is py = $1. Suppose that the objective of this consumer is to minimise his expenditure subject to attaining utility U (x,y) ≥ 4.
a) Write down the consumer's expenditure minimisation problem.
b) Construct the Lagrangian of this problem and derive the first-order conditions.
c) Solve for the optimal amounts of x and y.
d) What is the lowest income this consumer needs to attain utility U(x,y) = 4.