Consider the standard utility maximization problem: subject to max U = U(x, y) P
ID: 1117538 • Letter: C
Question
Consider the standard utility maximization problem: subject to max U = U(x, y) Pxx + Pyy = B, where Px and Py denote the price of goods x and y, respectively, and B represents the consumer’s available income. (a) Write the Lagrangian function and obtain the first-order conditions. (6 marks) (b) Assume that U = xy,Px = 1,Py = 4 and B = 120. Find along with the optimal levels of purchase x and y Calculate U (10 marks) (c) Using the Bordered Hessian matrix, verify that the second-order conditions for a maximum are satisfied. (9 marks
Explanation / Answer
Maximize U = xy
subject to Pxx + Pyy = B, Here Px = 1, Py = 4 and B = 120
i.e. x + 4 y = 120
Setting up langarangian function
(a) Z = x + 4y -120 + ( U - xy)
(b) The optimal level of
x* = 1/1+1 (120/1) = 60
y* = 1/1+1(120/4) = 15
U* = xy= 60*15 = 900
(c) The value of bordered heasian is negative which means maximum. So the second order conditionfor a maximum are satisfied.