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Unformatted text preview: s to scale
(CRS). In terms of the production function, this means that two times as much of each input gives two
times as much output. In general, if we scale all of the inputs up by some amount t > 1, constant returns to
scale implies that we should get t times as much output:
f (tx1 , tx2 ) = t f ( x1 , x2 ); t > 1 (5) We say that this is the likely outcome for the following reason: it should typically be possible for the ﬁrm to
replicate what it was doing before. If the ﬁrm has twice as much of each input, it can just set up two plants
side by side and thereby get twice as much output. With three times as much of each input, it can set up
three plants, and so on.
Note that it is perfectly possible for a technology to exhibit constant returns to scale and diminishing
marginal product to each factor. Returns to scale describes what happens when you increase all inputs,
while diminishing marginal product describes what happens when you increase one of the inputs and hold
the others ﬁxed. 5 Constant returns to scale is the most “natural” case because of the replication argument, but that isn’t
to say that other things might not happen. For example, it could happen that if we scale up both inputs
by some factor t, we get more than t times as much output. This is called the case of increasing returns to
scale (IRS). Mathematically, increasing returns to scale means that
f (tx1 , tx2 ) > t f ( x1 , x2 ); t > 1 (6) What would be an example of a technology that had increasing returns to scale? One nice example is that of
an oil pipeline. If we double the diameter of a pipe, we use twice as much materials, but the cross section of
the pipe goes up by a factor of 4. Thus we will likely be able to pump more than twice as much oil through
(Of course, we can’t push this example too far. If we keep doubling the diameter of the pipe, it will
eventually collapse of its own weight. Increasing returns to scale usually just applies over some range of
The other case to consider is that of decreasing returns to scale (DRS), where
f (tx1 , tx2 ) < t f ( x1 , x2 ); t > 1 (7) This case is somewhat peculiar. If we get less than twice as much output from having twice as much
of each input, we must be doing something wrong. After all, we could just replicate what we were doing
The usual way in which diminishing returns to scale arises is because we forgot to account for some
input. If we have twice as much of every input but one, we won’t be able to exactly replicate what we were
doing before, so there is no reason that we have to get twice as much output. Diminishing returns to scale
is really a short-run phenomenon, with something being held ﬁxed.
Of course, a technology can exhibit different kinds of returns to scale at different levels of production. It
may well happen that for low levels of production, the technology exhibits increasing returns to scale - as
you scale all the inputs by some small amount t, the output in...
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This document was uploaded on 09/21/2013.
- Summer '13