In the following a consumer has a utility function ux y x2y

In the following, a consumer has a utility function u(x, y) = x^2y, where x is the quantity of good 1 and y is the quantity of good 2.. The price of good 1 is 1 and the price of good 2 is also 1. Budget is 10. (a) Solve the problem of maximizing u(x,y) subject to the budget constraint. What is the optimal values of x and y? (There are non-negativity constraints x,y Greaterthanorequalto 0 but we can ignore these so long as all points satisfying the necessary condition for maximum are positive.) What is the maximum utility? (b) Let (x,y) be the solution to problem a. Letting u = u(x,y) be the maximum utility you derived in a,solve the following minimization problem min x + y subject to u(x,y) Greaterthanorequalto u.

Solution

maximize x^2y

x+y <=10

this is maximum when x = 7 y = 3

maximum value = 49*3 = 147

b) given utility is more or equal to 147 we want to minimize x+y

this will give the same sol x+y = 10