Here we are going to deal with some optimisation problems.

(a) In section 4.2 while dealing with average product and marginal product, we have shown graphically that when AP_{L} attains maximum, AP_{L} = MP_{L}.

**This can be shown by calculus. We define the production function as: **

The sufficient condition for maximisation holds. Therefore, when AP_{L} is maximum, AP_{L} = MP_{L}.

(b) Given the production function Q = 100(0.2 K^{0 5} + 0.8L^{0.5})^{2}, and prices of per unit capital and labour are 10 rupees and 4 rupees respectively. If the manager is further given a sum of Rs. 4,100 as expenditure for production, then find the optimal values of capital and labour which will maximise production.

The problem is to maximise Q subject to a given level of cost (Rs. 4,100).

**The cost equation: **

The partial derivatives are set equal to zero, which is the first order condition.

(c) A monopolist sells to products x and y, for which the demand is

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x = 25 — 0.5P_{x}, y = 30 = 30 – P_{y},

and the combined cost function is

c = x^{2} + 2xy + y^{2} + 20.

Find (a) the profit maximising level of output for each product (b) the profit maximising price for each product and (c) the maximum profit