Download Math Review1 1. Variables, Constants and Functions A variable is a

Antoine Gervais – University of Notre Dame
Math Review1
1. Variables, Constants and Functions
A variable is a mathematical abbreviation for a concept. For example in economics, the variable Y usually
represents the level of output of a firm or the GDP of an economy, while the variables K and L usually
represent the quantity of capital and labor, respectively. Although the specific value is not known the range of
value is usually restricted, either by assumption or economic intuition or logic. For example all of the above
variables will have non negative value, i.e. a number greater than or equal to zero.
A function is a mathematical equation that relates two or more variables together. There are two types of
functional forms: general functional form and specific functional form.
A general functional form relates the variables together in a non-specific format. For instance, the general
functional form for the production function is Y = F(K , L ) which reads Y is a function of K and L. In this
case, it is known that both capital and labor affect the level of output but not the exact manner in which they
do so.
A specific functional form relates the variables in a precise format. There can be many different specific
functions for a general function. For example, the Cobb-Douglas production function Y = K α L1−α , where α
(alpha) is a constant between 0 and 1, is one specific functional form for the general production function
Y = F( K , L ) .
In the context of an economic model, we make a distinction between two kinds of variables: endogenous
variables and exogenous variables. An endogenous variable is a variable that an economic model tries to
explain (determined within the model). An exogenous variable, on the other hand, is a variable that an
economic model takes as given (determined outside the model). To illustrate, consider the following simple
production function:
Y = KL .
This function summarizes all the feasible production patterns available; it will give the value of output Y
(exogenous) as a function of K and L (endogenous). For instance if K=10 and L=5 then the firm will
produce 50 units of output.
A constant is a mathematical abbreviation for a number that does not change. A constant can be represented
by a Greek letter like α, β (beta) or (more simply) by a letter like a, b or A, B. For example, let's introduce a
constant representing a (Hicks neutral) technology parameter in our production function:
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Y = A F (K , L ) .
It is clear that the greater the value of A the greater the output of the firm for any given level of capital and labor.
2. Power Functions
A power function is a function where a variable is raised to a constant power. A power function takes the
following general form:
f (x ) = kx ε ,
where k and ε are constants. The constant ε is called the exponent of the function. For example, the variable
x could be raised to the second, third or 1/2 power.
There are many rules of exponents that are very useful to simplify some expression.
Rule 1:
x0 = 1
Rule 3:
x−p =
Rule 5:
(xa )b = x a ⋅b
Rule 6:
Rule 7:
x a y a = (xy )a
Rule 8:
Rule 2:
1
xp
Rule 4:
x1 = x
xa
= xa −b
xb
xa x b = xa + b
x a x
=  
ya
y
a
3. Natural Logarithms
The natural logarithm is a logarithmic function that takes the Euler constant e = 2.7182818…, as its base.
In other words, the natural logarithm y = ln (x ) is the inverse of the exponential function x = e y . For many
reasons, economists often employ a logarithmic transformation to an equation. First, a logarithmic
transformation converts a product of two or more variables into a sum of those same variables. A sum is
often easier to deal with than a product. Second, a logarithmic transformation is strictly monotonic in the
sense that all the peaks (maximums) and valleys (minimums) of the original function are retained. Therefore,
the transformed equation preserves important properties of the original equation. Third, the growth rate of a
variable can also be approximated by taking the log difference of the variable (see section 4 below).
The rules for natural logarithms are as follows. For any constant r and variables x and y:
Rule 1:
ln e r = r
Rule 2:
2
e ln x = x
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Rule 3:
ln (xy ) = ln x + ln y
Rule 5:
ln x r = r ln x
Rule 4:
ln (x y ) = ln x − ln y
For example, the logarithmic transformation of the function y = Ae x is simply ln y = ln A + x . While the
first is not a linear function the second clearly is. Let A=10 and x range from 0 to 2.5, then graphically:
lnY
Y
140.0
6
120.0
5
100.0
4
80.0
3
60.0
2
40.0
1
20.0
0.0
0
0
0.5
1
1.5
2
2.5
3
0
0.5
Y
1
1.5
2
2.5
3
lnY
4. Levels vs. Growth Rates
A level variable records an amount. For instance if we let Y denote the level of real GDP then is the amount
of goods and services produced in a year. A growth rate variable records the percentage change across a
specific time period. To calculate a growth rate, you need to have the level at the beginning of the time period
and at the end of the period. For instance, the annual growth rate of Y for 2006 is given by:
g ( Y ) ≡ % ∆Y =
Y2006 − Y2005
,
Y2005
where g ( Y ) represents the annual growth rate, Y2006 is the level of real GDP in 2006 and Y2005 is the level
of real GDP in 2005.
The growth rate can also be approximated by taking the log difference of the variable. Therefore, the growth
rate of real GDP in 2002 can also be calculated as:
g ( Y ) ≅ ln Y2006 − ln Y2005 .
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ln Y1 − ln Y0 ≅
Proof2:
∂ ln Y ∂ ln Y ∂Y ∂Y ∂t ∆Y
=
=
≅
= g(Y )
∂t
∂Y ∂t
Y
Y
To calculate an average annual growth rate, you need to divide the log difference by the number of years.
For instance, the average annual growth rate from 1975 to 2000 is:
g(Y ) ≅
ln Y2000 − ln Y1975
.
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There exists a couple of “Math Tricks” that use the rules for natural logarithms and derivatives to obtain the
two simple rules of converting levels into growth rates:
Rule 1: The growth rate of a sum is the sum of the growth rates,
g ( XY ) = g ( X ) + g ( Y ) .
Rule 2: The growth rate of a ratio is the difference of the growth rates,
g (X Y ) = g ( X ) − g ( Y ) ,
where X and Y are two variables measured in levels, g ( XY ) is the growth rates of the product XY,
while g ( X ) and g ( Y ) are the growth rates of X and Y, respectively.
Proof of 1:
ln ( XY )1 − ln ( XY )0 = (ln X 1 − ln X 0 ) + (ln Y1 − ln Y0 ) ≅
Proof of 2:
similar
∂ ln X ∂ ln Y
+
≅ g (X ) + g ( Y )
∂t
∂t
5. Derivatives
If we define some function y = f (x ) , then for two distinct points x and c we can form the difference quotient,
∆≡
f ( x ) − f (c )
.
x−c
The derivative of the function y = f (x ) with respect to x is the limit of that quotient when x becomes very
close to c. Formally,
f ′(x ) = lim ∆ .
x →c
Hence the derivative provides the rate of change in f(x) as the change in x goes to 0.
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Do not panic! The proofs are included for completeness only. You will not be asked to reproduce them on the exam.
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There are many notations used to represent a derivative. Let y = f (x ) , then the derivative of y with respect to
x can be represented by all the following:
f ′(x ),
f x (x ),
or
dy
.
dx
A number of simple rules exist to compute derivatives. Here we list a subset of these rules:
Rule 1: The Constant rule:
∂[kf (x )]
∂[f (x )]
=k
∂x
∂x
Rule 2: The exponent rule:
∂ xε
= εx ε −1
∂x
Rule 3: The Product rule:
∂[f (x ) ⋅ g (x )]
= f ′(x ) ⋅ g (x ) + f (x ) ⋅ g ′(x )
∂x
Rule 4: The Ratio rule:
∂[f (x ) g (x )] f ′(x ) ⋅ g (x ) − f (x ) ⋅ g ′(x )
=
∂x
[g (x )]2
Rule 5: The Log rule:
∂[ln (x )] 1
=
∂x
x
Rule 6: The Chain rule:
∂{f [g (x )]}
= f ′[g (x )] ⋅ g′(x )
∂x
[ ]
The partial derivative of some function y = f (x , z ) with respect to x reveals how much f (x, z ) changes
when x changes by a very small amount holding z constant. Similarly, the partial derivative of y = f (x , z ) with
respect to z reveals how much f (x, z ) changes when z changes by a very small amount holding x constant.
Fortunately, the rules for partial derivatives are the same as those of the "regular" derivative. Again, there are
many notations used to represent a partial derivative. Let y = f (x , z ) , then the derivative of y with respect to x
can be represented by all the following:
f x′ (x, z ),
f1 (x, z ), or
∂y
.
∂x
For example, taking the partial derivative of the production function Y = A F(K , L ) with respect to L gives
the change in output Y when the quantity of labor L changes by a very small amount, holding capital K
constant. This derivative is defined as the marginal product of labor (MPL) while the partial derivative of Y with
respect to K is defined as the marginal product of capital (MPK).
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6. Optimization
Many problems in economics take the form of an optimization problem: consumers maximize utility or
minimize expenditure and firms maximize profits or minimize costs. Generally speaking, a maximization
problem is one in which the agent selects those values of the choice variable(s) that provides the highest
value of the objective function (like Profit or Utility) potentially facing some constraints (like income,
technology, or input availability).
In general the objective function of a maximization problem will be hump-shaped such that the maximum
will be attained at the peak of the function (see graph below). Therefore, the optimal values can be obtain by
taking the first derivative of the function with respect to the choice variable(s) and setting them equal to zero,
these are called the First-Order Conditions (FOC).
For example, the competitive firm (price taker) profit (π) maximization problem is a situation in which the
firm selects values for the quantity of labor (L) and stock of capital (K) that provides the highest level of
(economic) profits Π = PY − RK − WL , subject to the technology constraint that Y = AF(K, L ) .
π
FL′ (K , L *) = 0
π*
L*
L
π (K , L )
Max Π = PY − RK − WL
( K ,L )
s.t.
Y = AF(K, L )
FOC( K ) :
∂Π
= PAFK′ (K , L ) − R = 0
∂K
⇒
AFK′ (K , L ) =
R
P
FOC( L ) :
∂Π
= PAFL′ (K , L ) − W = 0
∂L
⇒
AFL′ (K , L ) =
W
P
The first-order conditions tell us that the firm hires labor until the marginal product of labor is equal to the
real wage ( w = W P ) and rents capital until the marginal product of capital is equal to the real rental rate on
capital ( r = R P ).
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