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A firm can allocate its labour force between two factories that produce two different goods. The production function of the first factory, which produces good A, is qA=L. The production function of the second factory, which produces good B, is qB=2L. The firm can sell qA units of good A at a price of pA=15qA and they can sell qB units of good B at a price of pB=302qB. The firm has a total labour force of L=100 available. (a) How much revenue does the firm earn if they allocate LA units of labour to the first factory and the remaining LB units to the other factory? 5 marks (b) What constraint is the firm facing? 2 marks (c) Write down a Lagrangian that represents the firm's constrained optimization problem if their goal is to allocate their labour force to maximize their revenues. 2 marks (d) What are the optimal quantities of labour LA and LB for the firm to allocate to their two factories? Show that this is a global maximum. 6 marks A firm can allocate its labour force between two factories that produce two different goods. The production function of the first factory, which produces good A, is qA=L. The production function of the second factory, which produces good B, is qB=2L. The firm can sell qA units of good A at a price of pA=15qA and they can sell qB units of good B at a price of pB=302qB. The firm has a total labour force of L=100 available. (a) How much revenue does the firm earn if they allocate LA units of labour to the first factory and the remaining LB units to the other factory? 5 marks (b) What constraint is the firm facing? 2 marks (c) Write down a Lagrangian that represents the firm's constrained optimization problem if their goal is to allocate their labour force to maximize their revenues. 2 marks (d) What are the optimal quantities of labour LA and LB for the firm to allocate to their two factories? Show that this is a global maximum. 6 marks