Hot Wheels has a monopoly in the manufacturing of wheels. Each one of its wheels requires exactly one sprocket as an input plus other variable costs of $5 per unit of output. Sprockets are made by Spacely Space Sprockets which is also a monopoly. The variable costs required to produce sprockets is $5 per unit. Suppose the inverse demand for wheels is pw = 50 − yw where pw is the price of wheels and yw is the quantity of wheels sold by Hot Wheels. a) Write down and solve the profit maximization problem for Hot Wheels assuming both firms act as independent monopolists. b) Use your answer above to compute the demand of sprockets and write down the profit maximization problem for Spacely Space Sprockets Inc. c) Compute the equilibrium profits for the two firms. d) Suppose the two firms decide to merge and form Hot Wheels. Compute the post-merger profits for the new firm. e) Show that welfare has increased. f) What is the largest price that the owners of Spacely Space Sprockets would be willing to pay to purchase Hot Wheels? What is the largest price that the owners of Hot Wheels would be willing to pay to purchase Spacely Space Sprockets?