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Finance Aspect-cost of capital

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Long run and short run cost functions
In the long run, the firm can vary all its inputs. In the short run, some of these inputs are
fixed. Since the firm is constrained in the short run, and not constrained in the long run, the
long run cost TC(y) of producing any given output y is no greater than the short run cost
STC(y) of producing that output:
TC(y) STC(y) for all y.
Now consider the case in which in the short run exactly one of the firm's inputs is fixed. For
concreteness, suppose that the firm uses two inputs, and the amount of input 2 is fixed at k.
For many (but not all) production functions, there is some level of output, say y
0
, such that the
firm would choose to use k units of input 2 to produce y
0
, even if it were free to choose any
amount it wanted. In such a case, for this level of output the short run total cost when the firm
is constrained to use k units of input 2 is equal to the long run total cost: STC
k
(y
0
) = TC(y
0
).
We generally assume that for any level at which input 2 is fixed, there is some level of output
for which that amount of input 2 is appropriate, so that for any value of k,
TC(y) = STC
k
(y) for some y.
(There are production functions for which this relation is not true, however: see the example
of a production function in which the inputs are perfect substitutes.)
Example: a production function in which the inputs are perfect substitutes
Consider the production function F (z
1
, z
2
) = z
1
+ z
2
, in which the inputs are perfect substitutes. The
long run total cost function for this production function is given by
w
1
y
if w
1
< w
2
TC(y,w
1
,w
2
) =
wy
if w
1
= w
2
= w
w
2
y
if w
1
> w
2
Its short run total cost of production when the amount of input 2 is fixed at k is
STC
k
(y) =
w
2
k
if y k
w
1
(y k) + w
2
k
if y > k.
Note that in this case if w
1
< w
2
then there is no level of output for which short run total cost is equal
to the long run total cost. In this case the firm would like to use only input 1 to produce any output,
so if k > 0 there is no output for which the short run total cost is equal to the long run total cost. The
short and long run cost functions in this case are shown in the following figure.

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Example: a Cobb-Douglas production function
Consider the production function F (z
1
, z
2
) = z
1
1/2
z
2
1/2
. The long run total cost function for this
production function is given by
TC(y,w
1
,w
2
) = 2y(w
1
w
2
)
1/2
.
Its short run total cost of production when the amount of input 2 is fixed at k is
STC
k
(y) =w
1
y
2
/k + w
2
k.
Note that TC is a linear function of y while STC is a quadratic function. (Remember that w
1
and w
2
are
fixed.) The functions are shown in the following figure.
For a total cost function with the typical shape, the following figure shows the relations
between STC and TC.

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Long run and short run cost functions In the long run, the firm can vary all its inputs. In the short run, some of these inputs are fixed. Since the firm is constrained in the short run, and not constrained in the long run, the long run cost TC(y) of producing any given output y is no greater than the short run cost STC(y) of producing that output: TC(y)  STC(y) for all y. Now consider the case in which in the short run exactly one of the firm's inputs is fixed. For concreteness, suppose that the firm uses two inputs, and the amount of input 2 is fixed at k. For many (but not all) production functions, there is some level of output, say y0, such that the firm would choose to use k units of input 2 to produce y0, even if it were free to choose any amount it wanted. In such a case, for this level of output the short run total cost when the firm is constrained to use k units of input 2 is equal to the long run total cost: STCk(y0) = TC(y0). We generally assume that for any level at which input 2 is fixed, there is some level of output for which that amount of input 2 is appropriate, so that for any value of k, TC(y) = STCk(y) for some y. (There are production functions for which this relation is not true, however: see the example of a production function in which the inputs are perfect substitutes.) Example: a production function in which the inputs are perfect substitutes Consider the production function F (z1, z2) = z1 + z2, in which the inputs are perfect sub ...
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