Course 2-MAE (1)

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Mathematics Applied to Economics
2023-2024
Prof. dr. Paula Curt
Faculty of Economics and Business Administration, UBB Cluj-Napoca
Statistics, Forecasts and Mathematics Department
paula.curt@econ.ubbcluj.ro
Course 2
Real Functions of Several Variables
Real Functions of Several Variables
In many practical situations, the value of one quantity depens on the values
of two or more others (the output of a factory depends on the amount of
capital invested and on the size of labor force, etc.)
R2 = R× R = {(x, y)|x, y ∈ R} -set of orders pairs (x, y) ∈ R2 (Cartesian
plane)
Algebraic Operations in R2
I adition: (x1, y1) + (x2, y2) = (x1 + x2, y1 + y2) ∈ R2
I scalar multiplication: (x, y) ∈ R2, α ∈ R→ α(x, y) = (αx, αy) ∈ R2
Prof. dr. Paula Curt (UBB FSEGA) Mathematics Applied to Economics Course 2 2 / 17
Real Functions of Several Variables
Real Functions of Several Variables
I A function f of the two variables x and y is a rule that assigns
to each ordered pair (x, y) of real numbers in some set, one and only
one real number denoted f(x, y).
I The domain of the function f is the set of all ordered pairs (x, y)
for which f(x, y) can be evaluated
Prof. dr. Paula Curt (UBB FSEGA) Mathematics Applied to Economics Course 2 3 / 17
Real Functions of Several Variables
Real Functions of Several Variables
Example 1 Suppose f(x, y) = 3x
2+5y
x−y . Find the domain of f. Compute
f(1,−2).
Similar notions can be introduced for more than two variables: R3;
Rn, n ≥ 3
Prof. dr. Paula Curt (UBB FSEGA) Mathematics Applied to Economics Course 2 4 / 17
Partial derivatives
1st order partial derivatives
In many economic problems involving functions of several variables, the goal is to find the
rate of change of the function with respect to one of its variables when the others are held
constant. This process is known as partial differentiation and the resulting derivative
is said to be partial derivative of the function
1st Order Partial Derivatives=Partial Derivatives
Let D ⊂ R2, (a, b) ∈ D a̧nd f : D −→ R. Partial Derivatives of the
function f with respect to x, respectively y, at the point (a, b) are
f ′x(a, b) =
∂f
∂x
(a, b) = lim
x→a
f(x, b)− f(a, b)
x− a
,
f ′y(a, b) =
Similarly we can define partial derivatives for functions with more than 2
variables.
Prof. dr. Paula Curt (UBB FSEGA) Mathematics Applied to Economics Course 2 5 / 17
Partial derivatives
1st order partial derivatives
Prof. dr. Paula Curt (UBB FSEGA) Mathematics Applied to Economics Course 2 6 / 17
Partial derivatives
1st order partial derivatives
Remark 1. The partial derivative of f with respect to x is denoted
by f ′x or
∂f
∂x and is the function obtained by differentiating f with respect
to x treating y as a constant.
Example 2. Compute 1st order partial derivatives of the function
f : R2 → R, f(x, y) = x2 + 3xy − 4y at the point (a, b) = (5,−3).
Prof. dr. Paula Curt (UBB FSEGA) Mathematics Applied to Economics Course 2 7 / 17
Partial derivatives
1st order partial derivatives
Prof. dr. Paula Curt (UBB FSEGA) Mathematics Applied to Economics Course 2 8 / 17
Partial derivatives
Economic Applications of Partial Derivatives
Let Q = Q(L,K) be a production function where Q represents the output, L
represents the labor input (worker-hours or cost of labor) and K represents
the capital input (cost of capital equipment).
Marginal Productivity of Labor:
Q′L(a, b) = lim
L→a
Q(L, b)−Q(a, b)
L− a
∼=
∆Q
∆L
Q′L(a, b)
∼= Q(a+ 1, b)−Q(a, b) is the change in total output due to
one unit increase in labor at the same level of capital
Marginal Productivity of Capital :
Q′K(a, b) = lim
K→b
Q(a,K)−Q(a, b)
K − b
∼=
∆Q
∆K
Q′K(a, b)
∼= Q(a, b+ 1)−Q(a, b) is the change in total output due to
one unit increase in capital at the same level of labor
Prof. dr. Paula Curt (UBB FSEGA) Mathematics Applied to Economics Course 2 9 / 17
Partial derivatives
Economic Applications of Partial Derivatives
Example 3. The production for a certain country is described by the
function Q(L,K) = 30L2/3K1/3. Compute Q′L, Q
′
K . What is the marginal
productivity of labor (capital) when L = 125 and K = 27? In this case
should the government encourage capital or labor investment in order to
increase the total output of that country?
Prof. dr. Paula Curt (UBB FSEGA) Mathematics Applied to Economics Course 2 10 / 17
Partial derivatives
Economic Applications of Partial Derivatives
Prof. dr. Paula Curt (UBB FSEGA) Mathematics Applied to Economics Course 2 11 / 17
Partial derivatives
Second order partial derivatives
2nd order partial derivatives
Let D ⊂ R2, (a, b) ∈ D a̧nd f : D −→ R
2nd order partial derivatives of the function f at the point (a, b) are
f ′′x2 =
∂2f
∂x2
= (f ′x)
′
x; f
′′
y2 =
∂2f
∂y2
= (f ′y)
′
y
f ′′xy =
∂2f
∂x∂y
= (f ′x)
′
y; f
′′
yx =
∂2f
∂y∂x
= (f ′y)
′
x
Remark 2. f ′′xy and f
′′
yx are called mixed partial derivatives
Prof. dr. Paula Curt (UBB FSEGA) Mathematics Applied to Economics Course 2 12 / 17
Partial derivatives
Second Order Partial Derivatives
Example 4. Compute the 1stand 2nd order partial derivatives for the
function
f : R2 → R, f(x, y) = x3y2 − 4y2x.
Prof. dr. Paula Curt (UBB FSEGA) Mathematics Applied to Economics Course 2 13 / 17
Partial derivatives
Higher Order Partial Derivatives
Prof. dr. Paula Curt (UBB FSEGA) Mathematics Applied to Economics Course 2 14 / 17
Partial derivatives
Higher Order Partial Derivatives
Remark 4.
I Mixed partial derivatives are equal in the following way:
f ′′xy = f
′′
yx
f ′′′x2y = f
′′′
xyx = f
′′′
yx2 , f
′′′
xy2 = f
′′′
yxy = f
′′′
y2x
f ′′′xyz = f
′′′
xzy = f
′′′
yxz = f
′′′
yzy = f
′′′
zxy = f
′′′
zyx
I It doesn’t matter the differentiation order, it only matters how many
times we differentiate and with respect to which variables we
differentiate, so we can reverse the order of any two successive
differentiations
Prof. dr. Paula Curt (UBB FSEGA) Mathematics Applied to Economics Course 2 15 / 17
Partial derivatives
1st Order Approximation Formula
Review: Approximations-one variable
f(x) ≈ f(a) + f ′(a)(x− a); x close to a
1st Order Approximation Formula; n = 2
Fie D ⊂ R2, (a, b) ∈ D and f : D −→ R.
1st Order Approximation of function f around the point (a, b) is
f(x, y) ≈ f(a, b) + f ′x(a, b)(x− a) + f ′y(a, b)(y − b); (x, y) close to (a, b)
1st Order Approximation Formula; n = 3
Fie D ⊂ R3, (a, b, c) ∈ D and f : D −→ R.
1st Order Approximation of function f around the point (a, b, c) is
f(x, y, z) ≈ f(a, b, c) + f ′x(a, b, c)(x− a) + f ′y(a, b, c)(y − b) + f ′z(a, b, c)(z − c)
(x, y, z) close to (a, b, c)
Prof. dr. Paula Curt (UBB FSEGA) Mathematics Applied to Economics Course 2 16 / 17
Partial derivatives
1st Order Approximation Formula
Example 5. Let Q be the production function defined by
Q(L,K) = 30L2/3K1/3.
I Write down the 1st order approximation of function Q around the
point (125, 27)
I Use the the previous formula to estimate Q(124, 28).
Prof. dr. Paula Curt (UBB FSEGA) Mathematics Applied to Economics Course 2 17 / 17
	Real Functions of Several Variables
	Partial derivatives