We shall assume that all relevant functions are “wellbehaved,” that we are con
cerned with profitmaximization within a single period, that there is only a single out
put (
X
, measured in kg) and a single variable input (
L,
measured in labourdays). The
price of the output (
P
X
) is in $/kg and the price of the input (
P
L
) is in $/labourday.
The condition for profitmaximization in the
output
market is that the firm’s Marginal
Revenue (the increment to Total Revenue from the last unit produced and sold) is equal
to its Marginal Cost (the increment to Total Cost from the last unit produced and sold):
MR
≡
∆TR/∆
Q
X
=
∆TC/∆
Q
X
≡
MC.
(M.4.3)
The condition for profitmaximization in the
input
market is that the firm’s Marginal
Revenue Product of Labour (the increment to Total Revenue from sale of the addition
al output resulting from the last labourday purchased) is equal to its Marginal Factor
M46
MATH MODULE 4: USING ECONOMIC UNITS
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MATH MODULE 4: USING ECONOMIC UNITS
M47
Cost of Labour (the increment to Total Cost resulting from the last labourday pur
chased):
MRP
L
≡
∆TRP
L
/∆
L =
∆TFC
L
/∆
L
≡
MFC
L
.
(M.4.4)
These two conditions look quite distinct, but we can show that they are in fact equiv
alent: two
different
ways of expressing the
same
profitmaximization condition. If we
define the Marginal Product of Labour (MP
L
= ∆
Q
X
/∆
L
) as the increment to output
resulting from the last labourday purchased, then (as outlined in Chapters 10 and 14)
the relations in Equations M.4.5 and M.4.6 hold:
MC
=
MFC
L
÷
MP
L
,
or
MFC
L
=
MC
MP
L
.
(M.4.5)
($/kg) = ($/
L
day) ÷ (kg/
L
day)
($/
L
day) = ($/kg)
(kg/
L
day)
In the lefthand version of this relation, the
labourdays
cancel, while in the righthand
version, the
kilograms
cancel.
Notice also that
MRP
L
=
MR
MP
L
,
or
MR
=
MRP
L
÷
MP
L
.
(M.4.6)
($/
L
day) = ($/kg)
(kg/
L
day)
($/kg)
=
($/
L
day) ÷
(kg/
L
day)
Here, in the lefthand version of this relation, the
kilograms
cancel, while in the right
hand version, the
labourdays
cancel.
Using Equations M.4.3 and M.4.5, we have as our profitmaximizing condition
MR = MC = MFC
L
/ MP
L
.
(M.4.7)
Rearranging terms in Equation M.4.7 and comparing with Equation M.4.6, we get
MR
MP
L
≡
MRP
L
= MFC
L
.
(M.4.8)
Thus, beginning with the profitmaximization condition in the
output
market, we have
generated the profitmaximization condition in the
input
market: the two conditions are
equivalent.
In the special case where the firm is a perfect competitor in the output market and
faces a horizontal demand curve for its output, MR =
P
X
. Hence in this case,
MRP
L
= MR x MP
L
=
P
X
MP
L
.
(M.4.9)
We give the term
P
X
x MP
L
the special name, Value of the Marginal Product of Labour
(VMP
L
). For a firm facing a
downward
sloping demand curve for its output, MR <
P
X
.
If the firm is a perfect competitor in the input market and faces a horizontal supply
curve for its input, then MFC
L
=
P
L
. Hence
MC = MFC
L
/MP
L
=
P
L
/MP
L
,
or
MC
MP
L
=
P
L
.
(M.4.10)
If the firm faces an upwardsloping input supply curve, then MFC
L
>
P
L
.
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 Fall '12
 Danvo
 Supply And Demand, Orders of magnitude, Qd, Kilometre, economic units

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