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Macromath 122
These are some important mathematical concepts used in
macroeconomics (and in other parts of economics). They are not a
substitute for the serious study of mathematics; rather, they serve as a
reminder on important concepts. Any comments or corrections are
welcome.
I.
Single variable calculus:
Farmer estimates an agricultural production function from
experiments:
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.5
1.0
1.5
2.0
2.5
N
CORN
C = corn production = 2N – N
2
, where N is nitrogen fertilizer
Marginal product of nitrogen is:
dC
dN
= 2 – 2N, which is positive for N < 1.
d
2
C
dN
2
=
-2, which shows diminishing returns.
Note that dC/dN = 0 at N = 1, this is maximum output.

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II.
Multivariate calculus
With multivariate calculus, we have multiple variables in our
function. For example, we now have capital as well as nitrogen:
C = 12 K
α
(2N – N
2
)
The marginal product of nitrogen is:
∂
C/
∂
N = 12 K
α
(2 – 2N) , holding K fixed.
A partial derivative uses
∂
instead of d. Like economics, it “holds
other things constant.”
III.
Powers and logs
Power functions are widely used in sciences, finance, and economics:
y = a
x
As for example in
10,000 = 10
4
A logarithm is a base for representing a power function:
x = log
a
y
as for example
y = log
10
(10,000)
Since 1 = b
0
,
log
b
(1) = 0 for any base.

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A plot of the log to the base 10 is shown below. Note that it
tends to minus infinite as x goes to 0, and is undefined at 0 or
negative numbers:
-1.2
-0.8
-0.4
0.0
0.4
0.8
1.2
0
2
4
6
8
10
12
X
log10(x)
IV.
Rules for logs for any base b
Two important rules are the following:
log (xy) = 1og(x)
+ log (y)
log (x
y
) = y
log (x)

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