Relationship Between Long-Run and Short-Run Total Cost Curves
Long-run expansion path
Short-run expansion path
k
$
Long-run total cost
𝑘�
d
b
Short-run total cost
c
e
e
c
a
d
x ’’’
x ’’
x’
l
𝑘� 𝑝𝑘
a
x’
b
x ’’
x ’’’
x
In the input space of the left-hand diagram there are three isoquants denoted by the
outputs produced along them, namely, 𝑥 ′ , 𝑥 ′′ , and 𝑥′′′. Each is tangent to an iso-cost line,
whose slope is the negative of the fixed input price ratio 𝑝𝑙 ⁄𝑝𝑘 , at capital and labor input
combinations a, b, and, c respectively. The curve on which a, b, and, c lie is the long-run
expansion path. In the short run, with capital or firm size fixed at 𝑘�, outputs 𝑥 ′ , 𝑥 ′′ , and 𝑥′′′ are
produced with respective capital and labor input combinations d, b, and, e. The straight line on
which d, b, and, e appear is the short-run expansion path.
The long-run cost of producing the three outputs is, in each case, the cost of the least-cost
basket of capital and labor inputs for producing those outputs, that is, the cost of baskets a, b,
and, c in the left-hand diagram, respectively. These costs are associated with their outputs and
identified in the right-hand diagram at points a, b, and, c on the long-run total cost curve. Since,
in the left-hand diagram, basket a is cost-minimizing and basket d is not, the short-run total cost
of producing output 𝑥 ′ with basket d has to be larger than the long-run total cost of producing
that output with basket a. Hence, in the right-hand diagram, the short-run total cost of producing
output 𝑥 ′ lies at point d on the short-run total cost curve which is necessarily above the long-run
total cost of producing that output at point a on the long-run curve. The same argument (that c in
the left-hand diagram is cost-minimizing while e is not) implies that the total cost of producing
output 𝑥′′′ on the short-run total cost curve at point e has to be above the long-run total cost of
producing that output at point c on the long-run curve. The same basket of inputs b in the lefthand diagram is used to produce output 𝑥 ′′ in both long and short runs. It follows that the longrun and short-run total costs of producing that output are identical, and that the two curves are
tangent at point b in the right-hand diagram. Therefore, except where the two curves are tangent
at b, the short-run total cost curve always lies above the long-run total cost curve. (Note that, in
the right-hand diagram, when output falls to zero, so does long-run total cost. But in the short
run, the firm still has to cover its fixed cost 𝑘� 𝑝𝑘 even if it is not producing anything.)