Unformatted text preview: market price, a competitive firm’s quantity supplied (“supply curve”) is given by:
{ However, as we are about to show, these “rules” apply to competitive firms with decreasing or constant returns, but not
increasing returns (and if the firm is in the long run, these rules are for competitive firms with decreasing or constant
returns to scale but not increasing returns to scale). This is because, as we are about to show: even if
, it
may be optimal for the firm to produce up to 100% capacity (of course if the firm has ample capacity then it should keep
expanding output forever – an economic black hole, so to speak); moreover,
output isn’t necessarily profit
maximizing. Let’s first see this graphically. Take a look at these graphs: 13
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. Competitive Firm with Increasing Returns in Various Scenarios Consider the leftmost graphs: here, revenue is always less than cost so that the firm will always lose money by
producing a positive output. In fact, the loss minimizing output is to produce nothing. Notice in particular that the
output where
(i.e. the output where revenue and cost functions have the same slope) doesn’t maximize
profits and in fact maximizes losses. If you look at the graphs on the right then you’ll see that even if
a
competitive firm with increasing returns should produce output, and keep producing past
(because at
the firm is maximizing losses), until it reaches capacity where it in fact maximizes profits.
This might worry some of you: after all, we have put out trust in the mathematics of KuhnTucker cases and their
conditions and we have found that situations satisfying conditions one or more of the KT cases and yet these don’t
maximize profits. How can this be? It turns out that the “conditions” for KT cases are valid under certain conditions, a
full treatment of which is beyond the scope of this course (those of you who’ll make it to U of T’s MA and MFE programs
will see this full treatment with me in ECO 1010 ). For a single good PMP, these conditions require that the Lagrangian
function to be concave (you’ll see a different argument below for why profit maximization requires increasing returns): ⏟ ⏟ Since the sum of concave functions is concave, we see that for the Lagrangian function to be concave, each of the
following individual terms must be concave:
⏟ ⏟
⏟ [
⏟ [
⏟ 14
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. Now if
require: is convex then we know that
⏟ Now is convex, so that for the Lagrangian function to be concave we ⏟
⏟ ⏟ ⏟ is convex since:
[ [ [ [ For the profit function to be concave we require:
⏟ Or that the cost function must be strictly convex or linear; put another way, the cost function must have decreasing or
constant returns (pretty cool, eh?).
If you didn’t find this argument convincing let’s try another approach. Let’s show that for a competitive firm with
increasing returns
doesn’t maximize profits – we do this by checking the second order conditions for case D
when
:
Case D
is where The FOC for case D is:
⏟ 15
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. ⏟ ⏟ ⏟ ⏟ ⏟
The second order condition for a maximum requires that4: As before this implies that the cost function must be strictly convex or linear; put another way, the cost function must
have decreasing or constant returns.
In light of our discussion we can state that: { The following graphs illustrate these rules for a competitive firm with decreasing returns (left) and constant returns
(right):
A competitive firm with decreasing returns A competitive firm with constant returns Case B
If Case C
If Case D
If
found by setting 4 This second order condition for constrained optimization problem works for when there is one “x” variable (in this case, we are
solving for one “x” variable
)
16
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. Combining these cases gives us a competitive firm’s supply curve: given a price, the firm’s output is “read off” the supply
curve:
A competitive firm with decreasing returns Firm’s Supply A competitive firm with constant returns Firm’s Supply Notice that competitive firm’s supply curve for
coincides with its marginal cost curve (this is will
be useful when we analyze the HBS case The Aluminum Industry in 1994). Let’s do some examples.
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Example: Suppose there are firms (A, B)...
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 Fall '14
 Supply And Demand, Competitive Firm, S. Ajaz Hussain, Sayed Ajaz Hussain

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