5. A perfectly competitive firm is trying to adjust to a recentdecrease in consumer demand by adjusting their levels ofproduction. The firm has more than enough existing capital to meetany reasonable production levels, so must only consider how muchlabour to hire. Assume the cost of using this capital is zero. Theprice of a unit of labour (the wage rate) is z, the quantity oflabour used is O, and the price of a unit of output is s. Labouruse is transformed into output at a rate of { = ln(O), where { isthe quantity of output.
(a) What are the firm’s total cost, total revenue, marginalcost, and marginal revenue functions?
(b) What is the profit maximising level of labour use? How do weknow for sure this is the profit maximising level?
(c) How does the profit maximising level of labour use changewith wages and prices?
(d) Write down the indirect profit function of the firm, andinvestigate the eect of increases in s and z upon the value ofoptimal profits. Are the signs of those eects what you wouldexpect?
(e) Is the indirect profit function concave, convex or linear inprice? Is it concave, convex or linear in the wage rate? From youranswers to these questions, can you conclude whether the indirectprofit function is concave or convex in the vector (s> z)? (f)Assuming that the indirect profit function is convex in the vector(s> z), sketch a contour of the indirect profit function in thespace of the output price and the wage rate, and indicate thecorresponding better set.
