Question 1 Consider the minimal cost for obtaining a specified level of utility problem between two goods, Goodx and Good . 2 Let x be the quantity of Good x whose price is $13 per unit and y be the quantity of Good y whose price is $15 per unit. The associated cost is therefore C(x,y)=13x+15y The targeted utility value is 178.4 where the utility function is given by U(x,y)=y(x+1) (a) Solve the problem to minimise the cost, constrained by the desired utility level using Lagrange Multipliers. (b) Show that the solution found does actually minimise the cost function (amongst feasible x and y choices). (c) Calculate the optimal values for x,y, and C (d) Consider now the dual optimisation problem of maximising the utility for the same money spent (cost) as calculated in part (c). Relate the optimal value of this problem to found in part (c). Question 2 A young family is looking to enter the property market. They plan to save for a 10% deposit and take out a loan (for the 90% balance). The deposit is for 10% of the property's value at the time the loan is taken out. The property is valued at $890,000 today, and is modelled to increase in value by 0.5% each month. The family can set aside $5,000 each month-end till the 10% deposit is made. The savings account pays interest at a rate of 0.5% each month. When the deposit is reached, a 30 -year loan is taken out at an interest rate of 0.6% per month. (a) Write the value, P(t) for the value of the property at the end of the tth month. (b) Formulate a difference equation for the future value of the savings plan. (c) Detail the finite difference solution for S(t), the future value of the savings plan. (d) Solve for t, the time when the 10% deposit may be made and calculate the loan amount (to the nearest dollar). (e) Detail the solution to calculate the monthly repayment amount (to the nearest cent)