Given a Cobb-Douglas utility function given by

U (X,Y) = X (alpha upper case) Y (beta uppercase 

, where for convenience we assume

alpha + beta =1

(a) Form the relevant Lagrangian expression if X and Y have prices P_X, P_Y 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

alpha

percent of his or her income to buying good X and

beta

percent to buying

good Y, i.e. show that

P_XX/I = ALPHA and P_YY/i=beta