A consumer has the following utility function: U(x,y)=x(y+1), where x and y are quantities of two consumption goods whose prices are Px and Py, respectively. The consumer also has a budget of B. Therefore, the Lagrangian for this consumer is

x(y+1)+(BPxxPyy)

(a) Verify that this is a maximum by checking the second-order conditions. By substituting x* and y* into the utility function, find an expression for the indirect utility function

U*=U(Px,Py,B)and derive an expression for the expenditure function

E=E(Px,Py,U*)

(b) This problem could be recast as the following dual problem

Find the values of x and y that solve this minimization problem and show that the values of x and y are equal to the partial derivatives of the expenditure function,