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Course: Mathematics for Economists (803)
Semester: Spring, 2022
Assignment No. 1
Q.1 Differentiate among variables, constants and parameters. Also define endogenous as
well as exogenous variables.
A constant is something like a "number". It doesn't change as variables change. For example 3 is
a constant as is π.
A parameter is a constant that defines a class of equations.
(xa)2+(yb)2=1
is the general equation for an ellipse. aa and bb are constants in this equation, but if we want to
talk about the entire class of ellipses then they are also parameters -- because even though they
are constant for any particular ellipse, they can take any positive real values.
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2. A variable is an element of the domain or codomain of a relation. Remember that functions are
just relations so the input and output of functions are variables. For example, if we talk about the
function x↦ax+3x↦ax+3, then xx is a variable and aa is a parameter -- and thus a constant. 33 is
also a constant but it is not a parameter.
A "known" variable is typically a value that the conditions of the problem dictate the
variable must take. For example if we are discussing an object an free fall, then acceleration is a
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variable. But physics puts a constraint on the value that that variable may take -- acceleration in
free fall is a=g≈9.8. Thus, though aa may be defined as the input of a function, it must take a
"known" value. Thus it is a known variable.
The Pythagorean theorem states that a2+b2=c2 for sides a,b and hypotenuse cc of a right triangle.
These are parameters -- thus they are also constants.
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They are all the same sort of thing on different levels of abstraction/generalization. Setting a
value creates a more specialized (less general) version of the mathematical object (function,
optimization problem, etc.), and replacing a formerly exactly defined value by a symbol creates a
generalized problem (covering a whole family of the specific problems).
 If you set aa to some value in ax+3ax+3, you get a more specific version, for example 5x+3.
If you further set xx to some value, you get a specific number out, like 5⋅6+3.
 In the other direction, if you turn ax+3 into ax+tax+t, you can represent a whole family of
(parameterized) functions including ax+8 and ax+1.
tt is the highest level parameter, aa is one lower and xx is the lowest. Since we usually only use a
few such levels at a time, we like to use names for them instead of just saying "higher level"
parameter. Variables are usually those that get adjusted on the lowest level, parameters are a level
above and constants are those that we don't change or adjust in our current task, but they could be
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turned into even higher-level parameters (called hyperparameters) if we wanted to further
generalize our problem or object.
Any function with multiple parameters can be turned into a higher-level function that just takes
one parameter and gives you a new function which now takes one less parameter than the
original. This is called currying. So your f(a,x)=ax+3f(a,x)=ax+3 can be turned into a function
which gives a new function for each aa:
F=(a↦(x↦(ax+3)))
So F(7) would be a function itself, 7x+3.
If you are familiar with programming it is also similar to variable scoping, i.e. that values are
defined in nested contexts. Functional programming uses these concepts even more heavily.
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Which parameter you put on which level depends on the current problem at hand and the same
problem can often be analyzed in multiple ways, i.e. by swapping parameters across levels (like
in our example, interpreting aa as the lowest and xx as the higher-level parameter).
A variable is, of course, a quantity that is allowed to vary over its range of definition. For
example, f(x)=3x+5f(x)=3x+5 is a function, where x ranges over the real numbers.
Now, I think the difference between constants and parameters is a bit more subtle. First,
constants:
A constant is just something that doesn't vary. 3 is a constant value, π is a constant value. But
then, in the function f(x)=ax+b. a and b are arbitrary constants. So, for whatever reason, say we
want to study functions of the form f(x)=ax+b, where aa and bb are some fixed values, but we
don't really care if those values are 3, 42, or π, so we say aa and bb are constants. I think in that
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sense there's a distinction between specific constants, like π, and arbitrary ones, like a,b in the
previous formula.
Now, with parameters, in my experience, there's always some notion of partial application going
on. I think statistical distributions are a really good example of this. For example, take the normal
distribution, if we wanted to, we could think of it as a function of three variables, x,μ and σ, but
that's not what a normal distribution is! A normal distribution is the particular single-variable
function of x you get when you choose a particular σ and μ, as opposed to being arbitrary,
like aa and bb we talked about above.
Parameter=para+meter=against+measure to measure something against some other thing"
(against an object treated as a unit). So basicly a "parameter" is something which could be
measured with a ruler.
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For the sake of modelling some real life system we make up a mathematical object. Now let's
imagine that we are modelling our solar system. After many hours of mental labor we came up
with a model:
x=f(t)=at−bx=f(t)=at−b
where
 x - a variable, position of the 3rd planet in the solar system under consideration,
 t - a variable, time,
 a - a parameter of the system, speed of the planet rotation,
 b - a parameter of the system, planet's initial position relative to some point,
thus by plugging tt into f(t) we should get the future position of the planet in the solar system.
Now f(t)=at−b is a model of the system but it is a general model, not a model of our particular
solar system, but a framework for modelling any solar system there is! To
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render f(t)=at−bf(t)=at−b into OUR home system we should measure our
system's parameters: a,b. By measureing them we transform model of general solar system into
particular model of our solar system.
Now let's imagine that after taking measurments by using telescopes we get values for our
parameters: aa=500 km/h and bb=100 000 km. So f(t)=at−bf(t)=at−b transforms
into f(t)=500t−100000f(t)=500t−100000. Now we can calculate the variable x.
In the example above you should see that:
1.We do not measure variables! We calculate them or plug them in the model (function). We
calculate variables from measured parameters.
2.We do not calculate parameters, instead we measure them as the etymology of the word
suggest.
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3. By measuring parameters we select particular model f(t)=500t−100000 from the class of
models f(t)=at−b.
So in the end: If you measure something - it's a parameter. If you calculate something - it's a
variable.
In the field of mathematics, a variable defines as an element connected to a number, known as an
estimation of the variable that is self-estimated, not completely determined, or ambiguous. The
expression “variable” originates from the way that, when the argument (additionally called the
“variable of the Function”) changes, then the estimate changes accordingly.
2. Consider the market model.
Qd = 20-3P
Qs = 10P-2
a) Calculate the equilibrium price and quantity.
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equilibrium price and quantity after tax?
Qd = 20-3P
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Qd=Qs
20-3P = 10P-2
20+2 = 10P +13P
22 = 13P
P = 22 / 13 = 1.692
Qd = 20-3P
Qd = 20-3(22 / 13)
Qd = 20- 66/13
Qd = 194 / 13 = 14.92
b) Suppose government impose Rs 16 per unit tax on consumers. What will be the

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3P = 20 – Qd
P = 20/3 – Qd / 3
Qs = 10P-2
10P = Qs+2
P = Qs / 10 + 1 / 5 + 16
3. a) Define matrix, vector and scalar.
Scalars
 These are direction independent quantities that can be fully described by a single number, and
are unaffected by rotations or changes in co-ordinate system. Examples of physical properties
that are scalars: Energy, Temperature, Mass.
 For this TLP scalars will be written in italics.
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Vectors
 These are objects that possess a magnitude and a direction, and are referenced to a particular
set of axes known as a basis. A basis is a set of unit vectors (vectors with a magnitude of 1)
from which any other vector can be constructed by multiplication and addition.
 The vector is referenced to the basis by its components. If possible, the maths is simplified by
using an orthonormal base with orthogonal (mutually perpendicular) unit vectors. Examples
of physical properties that are described by vectors: Mechanical force, Heat flow, Electric
field.
 Vectors will be written in bold and components of a vector, say x, will be written as xi
Matrices
 A matrix is a mathematical object that contains a rectangular array of numbers that can be
added and multiplied (according to matrix multiplication rules). They are very useful in many
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applications, for example in reducing a set of linear equations into a single equation, storing
the coefficients of linear transformations (e.g. rotations), and as we shall see, in describing
tensors.
 The components of matrix A are written aij where i refers to the row element and j refers to
the column element.
Scalar products
 For two vectors: a = (a1, a2, a3) and b = (b1, b2, b3) The scalar product (also known as the dot
product)
and so, for
is defined as: a.b = a1b1 + a2b2 + a3b3
example, the vectors (1, 4, −3) and (2, 5, 1) have a scalar product
of 1×2 + 4×5 − 3×1 = 19.
 The scalar product is related to θ, the angle between the two vectors, and can equivalently be
written as: a.b = |a||b|cosθ.
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 For vectors of unit length, we can see that the scalar product is equal to the cosine of the
angle between them.
Matrix multiplication
If we have two matrices,
A =
a11 a12 a13
a21 a22 a23
a31 a32 a33
and B =
b11 b12 b13
b21 b22 b23
b31 b32 b33
Then the product C = AB is found by ∑3
k=1 aikbkj where i, j and k are indices that represent the
position of the element in the matrix.
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15. C = AB =
a11 a12 a13
a21 a22 a23
a31 a32 a33
b11 b12 b13
b21 b22 b23
b31 b32 b33
=
a11b11 + a12b21 + a13b31 a11b12 + a12b22 + a13b32 a11b13 + a12b23 + a13b33
a21b11 + a22b21 + a23b31 a21b12 + a22b22 + a23b32 a21b13 + a22b23 + a23b33
a31b11 + a32b21 + a33b31 a31b12 + a32b22 + a33b32 a31b13 + a32b23 + a33b33
Don't bother trying to remember the above result, remember the rule:
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ROW COLUMN
× = RC = "Race Car" or "Really Cool!" or make up your own acronym to
remember it. This is also useful to remember the conventional order of suffices, where the first
suffix indicates the row and the second indicates the column.
You can use the following activity to practice more matrix multiplication.
b) Using Cramer’s rule find the value of ‘x’, ‘y’ and ‘t’
4x + 2y+t= 2
5x + y+2t= 1
3x + 4y+6t= 4
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20. Q.4 Write a detailed note on Jacobean determinants, also give economic interpretation of
total differentiation.
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In vector calculus, the Jacobian matrix of a vector-valued function of several variables is
the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the
function takes the same number of variables as input as the number of vector components of its
output, its determinant is referred to as the Jacobian determinant. Both the matrix and (if
applicable) the determinant are often referred to simply as the Jacobian in literature.
The Jacobian of a vector-valued function in several variables generalizes the gradient of a scalar-
valued function in several variables, which in turn generalizes the derivative of a scalar-valued
function of a single variable. In other words, the Jacobian matrix of a scalar-valued function in
several variables is (the transpose of) its gradient and the gradient of a scalar-valued function of a
single variable is its derivative.
At each point where a function is differentiable, its Jacobian matrix can also be thought of as
describing the amount of "stretching", "rotating" or "transforming" that the function imposes
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locally near that point. For example, if (x′, y′) = f(x, y) is used to smoothly transform an image,
the Jacobian matrix Jf(x, y), describes how the image in the neighborhood of (x, y) is
transformed.
If a function is differentiable at a point, its differential is given in coordinates by the Jacobian
matrix. However a function does not need to be differentiable for its Jacobian matrix to be
defined, since only its first-order partial derivatives are required to exist.
If f is differentiable at a point p in Rn, then its differential is represented by Jf(p). In this case,
the linear transformation represented by Jf(p) is the best linear approximation of f near the
point p, in the sense that
where o(‖x − p‖) is a quantity that approaches zero much
the distance between x and p does as x approaches p. This approximation
faster than
specializes to the
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approximation of a scalar function of a single variable by its Taylor polynomial of degree one,
namely
In this sense, the Jacobian may be regarded as a kind of "first-order derivative" of a vector-valued
function of several variables. In particular, this means that the gradient of a scalar-valued function
of several variables may too be regarded as its "first-order derivative".
Composable differentiable functions f : Rn → Rm and g : Rm → Rk satisfy the chain rule,
namely for x in Rn.
If m = n, then f is a function from Rn to itself and the Jacobian matrix is a square matrix. We can
then form its determinant, known as the Jacobian determinant. The Jacobian determinant is
sometimes simply referred to as "the Jacobian".
The Jacobian determinant at a given point gives important information about the behavior
of f near that point. For instance, the continuously differentiable function f is invertible near a
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point p ∈ Rn if the Jacobian determinant at p is non-zero. This is the inverse function theorem.
Furthermore, if the Jacobian determinant at p is positive, then f preserves orientation near p; if it
is negative, f reverses orientation. The absolute value of the Jacobian determinant at p gives us
the factor by which the function f expands or shrinks volumes near p; this is why it occurs in the
general substitution rule.
The Jacobian determinant is used when making a change of variables when evaluating a multiple
integral of a function over a region within its domain. To accommodate for the change of
coordinates the magnitude of the Jacobian determinant arises as a multiplicative factor within the
integral. This is because the n-dimensional dV element is in general a parallelepiped in the new
coordinate system, and the n-volume of a parallelepiped is the determinant of its edge vectors.
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25. The Jacobian can also be used to determine the stability of equilibria for systems of differential
equations by approximating behavior near an equilibrium point. Its applications include
determining the stability of the disease-free equilibrium in disease modelling.
Q.5 Find the derivatives of the following:
a) Y= (24x2+4) (23x+11)
b) Y= (12x2 + 14) (4x-1 -3)
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