ajaz_eco_204_2012_2013_chapter_15_Competition

How should a competitive firm choose lets start by

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Unformatted text preview: king firm will supply at that price (make sure you understand this concept: why doesn’t a monopolist have a supply curve?). Since a competitive firm has no control over the price, once operational, its only control variable is how much output it should produce which we denote by (another way to write this would be quantity supplied by the firm). How should a competitive firm choose ? Let’s start by examining a model that you never see in ECO 100 -- a competitive firm choosing output to maximize its revenue: ⏟ Competitive Firm’s Revenue Maximization Problem (RMP): Before jumping into algebra let’s see the intuition for why you never see models of competitive firms choosing output to maximize revenues. Since price is constant, the competitive firm’s revenue function is linear with a positive slope. In the following graph, revenues are maximized by 8 ECO 204 Chapter 15: Competitive Firms and Markets (this version 2012-2013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. produce as much as possible (i.e. ); this is an uninteresting and trivial solution which is why in ECO 100 you saw competitive firms choosing output to maximize profits (in ECO 100 firms had unlimited capacity and a competitive firm maximizing revenues would produce an infinite amount!). * Let’s prove this result formally: ⏟ ⏟ ⏟ The FOC and KT conditions are: ⏟ The KT cases are: 9 ECO 204 Chapter 15: Competitive Firms and Markets (this version 2012-2013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. First consider: Next consider: When is When is When is When is When is ? When is The cases can be summarized as: Case A Case B When is Case C ? When is When is When is ? Case D When is When is When is When is Let’s examine each case one by one: Case A When is When is Since this is impossible. Case B When is ? When is Since we see that Now, sub into the FOC: ⏟ As long as the market price is strictly positive we see that the firm will be in case B, i.e. produce at full capacity. ⏟ . That is, as long as there’s a market for the good 10 ECO 204 Chapter 15: Competitive Firms and Markets (this version 2012-2013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. Case C When is ? When is Since we see that Now, sub ⏟ ⏟ For into the FOC: we would need: But as long as there’s a market for this good we have which contradicts this condition. That is, case C is impossible and the firm will always produce output (so long as ). Case D When is When is Sub into the FOC: ⏟ ⏟ ⏟ Case D requires the price to be nil. This is impossible so long as long as there’s a market for the good (i.e. ). Thus, the only solution is Case B, where the firm produces at 100% capacity. Nice to see that math confirms our intuition. Since revenue maximization for competitive firms is a boring problem, we will model competitive firms’ choice of output as a Profit Maximization Problem (PMP). Competitive Firm’s Profit Maximization Problem (PMP): In the “Profit Maximization Problem Algebra” chapter we setup and solved this general PMP: ⏟ Since we have: ⏟ [ ⏟] We saw that the FOC and KT conditions were: 11 ECO 204 Chapter 15: Competitive Firms and Markets (this version 2012-2013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. ⏟ ⏟ We saw that the general PMP has 4 KT cases, one of which is impossible: Case A Impossible General Profit Maximization Problem: Three Possible Solutions Case B Case C Case D When When When neither cases B or C occur and found by Note: By the envelope theorem Please note that in tests and exams you will be asked to first prove these results and then use them for particular PMPs. We assume that the competitive firm is rational (i.e. will shut down when ) and noting that in perfect competition since here are the different possible “solutions” to a competitive firm’s PMP (make sure you know how we went from the table above to the one below): A Competitive Firm’s Profit Maximization Problem: Three Possible Solutions Case A Case B Case C Case D Impossible for Note: Note: Note: and When When When neither cases B or C occur found by (because ) Note: By the envelope theorem Combining these cases, we have a competitive firm’s supply curve (where the firm takes price as given): { 12 ECO 204 Chapter 15: Competitive Firms and Markets (this version 2012-2013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. To apply this in practice we need to know the competitive firm’s function and in particular at zero output and at capacity. Since capacity is a physical limit on production we can say once the firm reaches capacity it is prohibitively expensive to produce the next unit of output. That is, once capacity is reached, cost and marginal cost are both infinite. The following graph depicts the cost and marginal cost curves for firms with decreasing, constant, and increasing returns – notice how the cost and marginal cost spikes up at capacity: Decreasing Returns Constant Returns Increasing Returns Note: position and shapes of the curves not to be taken literally We just argued that given the...
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