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ECON 200
Macroeconomic Theory
Before proving the claim that in a competitive economy with an aggregate CRS
production function, the payments to factors of production equal total output, recall from
multivariable calculus that the total derivative of a function such as
is given by
1ÐDß AÑ
.1 œ
.D
.A
`1
`D
`A
where I have assumed that all the required derivatives exist.
Recall now that if the
production function
has constant returns to scale, we can write
]œJÐO
ßP
Ñ


ß P
Ñ
for any
As this is an identity in
we may different both sides of the expression
!
Þ
with respect to
and retain the equality.
Differentiating the LHS we have

.Ð ] Ñ
.
œ]


.
To differentiate the RHS use the result on total derivatives to write
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 Spring '08
 Staff
 Microeconomics

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