12.11 Productive efficiency with calculus In Example 12.3 we showed how a Pareto efficiency exchange equilibrium can be described as the solution to a constrained maximum problem. In this problem we provide a similar illustration for an economy involving production. Suppose that there is only one person in a two good economy and that his or her utility function is given by U(x, y). Suppose also that this economy’s production possibility frontier can be written in implicit form as T (x, y) = 0.
a.
What is the constrained optimisation problem that this economy will seek to solve if it wishes to make the best use of its available resources?
b.
c.
d.
What are the irst-order conditions for a maximum in this situation?
How would the eficient situation described in part

(b) be brought about by a perfectly competitive system in which this individual maximises utility and the irms underlying the production possibility frontier maximise proits.
Under what situations might the irst-order conditions described in part

(b) not yield a utility maximum?