(a) Solve the problem to maximise the utility, the constraint by the budget using Lagrange multipliers. Step 1: Initialize the Lagrange function: L(x, y, ) = (13x + 15y - 560) x(y + 1) Step 2: Take the partial derivatives and multiply them by zero: ¿L/Öx = y + 1 - 13A = 0 (1) ¿L/By = x - 15A = 0 (2) 13x + 15y - 560 = 0 (3) Step 3: Concurrently solve the systems of equations (1), (2), and (3): x = 15à in equation (2). When we substitute x in equation (3), we get: 13(15) + 15y - 560=0 195A+15y - 5600 15y=560-195 (4) Substituting x = 15 and equation (4) into equation (1) yields: y +1-13=0y=13-1 (5) Substituting equation (5) for equation (4): 15(13-1)=560-195 195A-15=560-195 3901=575-575/390 A = 1.474358974 1.474 Substituting 1.474 into equation (5) yields: y=13(1.474)-1 y=18.16666667 18.167 Substituting x = 15A into equation (2) yields: x = 15(1.474) x=22.11538462 = 22.11 As a result, the best values are x = 22.11 & y = 18.167. (b) Show that the solution found does actually maximise the utility function (amongst feasible x and y choices). To demonstrate that the solution identified maximises the utility function, we must first determine whether it meets the second- order requirements for a maximum. This necessitates computing and evaluating the second-order partial derivatives of the Lagrang function at optimum values. #Liêx* =Ũ ởLiêyt = 0 ởL/©xây=1 We may deduce that the solution maximises the utility function since the second-order partial derivatives are constant and do not rely on x or y. (e) Calculate the optimal values for x, y, à, and U. Determine the best values for x, y, and U: We get x = 22.11538462 22.11 y = 18.16666667 18.167. λ=1474358974 1.474 U( x, y) = x (y+1 ) = 22.11( 18.167 +1) = 423.88 (d) Consider now the dual optimisation problem of minimising the cost (Budget, B(x, y)) to achieve the utility U as calculated in (c). Relate the optimal values of this problem to those found in (e) and also B.. The goal of the dual optimisation problem is to minimise the cost (budget) B(x, y) in order to maximise the utility U as estimated in (c). Because we now know the best values for x and y, we can plug them into the budget constraint equation:B 13(22.11538462) + 15(18.16666667) B 287.43+272.43-B B=560 The best solution for the dual optimisation issue is B=560, which equals the entire budget.