Given a Cobb-Douglas utility function given by

U(X,Y) X Y

, where for convenience we assume

1

(a) Form the relevant Lagrangian expression if X and Y have prices PX, PY and the consumers income

is given by I

(b) Derive the first-order conditions

(c) Solve for the utility maximizing values of X* and Y*

(d) Explain why an individual whose utility function is given by the equation above will always

choose to allocate

percent of his or her income to buying good X and

percent to buying

good Y, i.e. show that

P_xX/I= Alfa

PyY/I=Beta