This preview shows page 1. Sign up to view the full content.
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 20122013) 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 20122013) 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 20122013) 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 20122013) 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 20122013) 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...
View Full
Document
 Fall '14

Click to edit the document details