A recycling company makes paper from a combination of two raw materials A and B where A is wood pulp from a timber company and B is waste paper from a recycling depot. The production function is Q = (x,y) where Q is the quantity of paper the company can make using x units of A and y units of B. The cost of acquiring these materials is given by the cost function C = (x,y) = px +qy, where p is the unit price of A and q is the unit price of B.

a) If the company decides to reduce the amount of A that it buysm then it can use the money saved to buy additional B. The economic rate of substitutuion ERS of B for A tells how much additional B can be bought for the cost of a unit of A. Show that the ERS is gx/gy = p/q.

b) If the company decides to reduce slightly the amount of A that it buys then it must buy additional B in order to maintain a constant level of production. The technical rate of substitution TRS of B for A tells how much additional B to buy per unit reduction in A. Show that the TRS is fx/fy.

c) Show that to maximize the quantity of paper produced with a fixed budget the company should use raw materials A and B in quantities such that ERS = TRS

d) Show that to minimize the cost of producing a fixed quantity of paper the company should use raw materials A and B in quantities such that ERS = TRS.