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Round your answers to two digits after the decimal. Given the utility function U(x, y) = 6 /I + 2y. Consider the utility maximization problem: max U(x, y) s.t. 4x + 8y = 65 1. Using the Lagrange method, find the optimal quantities of the ri, yl, and the Lagrange multiplier A The optimal value at equals X . The optimal value y; equals The value of the Lagrange multiplier X equals 2. Suppose income increases from 65 to 66. What are the new optimal quantities of the x3, y;, and A? The new optimal value ag equals . The new optimal value y; equals . The new value of the Lagrange multiplier X equals 3. What is the increase in the realized value of AU = U(x}, y; ) - U(r), y; ) due to the increase in income? The realized value of U(x], y; ) equals The realized value of U(x;, y; ) equals The value of AU equals 4. Compare the change in the realized utility function with the Lagrangian multiplier: In this case, the two values (rounded to two digits) are Oexactly the same. Oapproximately the same. Osomewhat different. Overy different Mark 0.00 out of 1.00 The correct answer is: exactly the same.Round your answers to two digits after the decimal. Consider the following optimization problem: min 2x- - 80x -4ry + 2y" - 80y s.t. 8x + 6y =67 Express the Lagrangian of this optimization problem as L = ar' - br - cry + dy' - ey - X(kx + my" - T) where . the value of coefficient a equals X . the value of coefficient b equals X the value of coefficient c equals the value of coefficient d equals X the value of coefficient e equals X . the value of coefficient g equals X the value of coefficient k equals the value of coefficient / equals the value of coefficient m equals X the value of coefficient n equals . the value of coefficient T equals X Use Lagrange's method to find the optimal values of a*, y*, and A . The value p* equals X . The value y* equals . The value of Lagrange multiplier X equals