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Mathematical Methods in Economics - II

BECC-104

For Bachelor of Arts (Honors) [Economics]

Based on New Syllabus

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athematics allows economists to form meaningful, testable propositions about wide-ranging and complex subjects which could less easily be expressed informally. Further, the language of mathematics allows economists to make specific, positive claims about controversial or contentious subjects that would be impossible without mathematics. Formal economic modeling began in the 19th century with the use of differential calculus to represent and explain economic behavior, such as utility maximisation, an early economic application of mathematical optimisation. Economics became more mathematical as a discipline throughout the first half of the 20th century, but introduction of new and generalised techniques in the period around the Second World War, as in game theory, would greatly broaden the use of mathematical formulations in economics. In the book, Mathematical Methods in Economics - II [BECC-104] we have tried to solve all possible questions from the exams’ point of view to help students in preparing for examination in a short period of time. The book is enriched with useful and to-the-point matter. Sample papers and guess papers have also been included to help students to understand the unique examination structure and practice. We hope that this book would not only be a favourite study material for the students but also can be a nice resource for teaching. An attempt has been carefully made to present this book more useful and meet the requirement and challenges of the course prescribed by IGNOU University. We hope that this effort will fulfill the expectations of the readers and help them to excel in exams. We wish you a successful and rewarding career. Feedback in this regard is solicited. – GPH Panel of Experts

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Block 1

Functions of Several Variables

Unit 1 Unit 2

Multivariate Calculus-I Multivariate Calculus -II

Block 2

Differential Equations

Unit 3 Unit 4

First-Order Differential Equations Second-Order Differential Equations

Block 3

Linear Algebra

Unit 5 Unit 6 Unit 7

Vectors and Vector Spaces Matrices and Determinants Linear Economic Models

Block 4

Multivariate Optimisation

Unit 8 Unit 9 Unit 10

Unconstrained Optimisation Constrained Optimisation with equality Constraints Duality

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Chapter-1

Multivariate Calculus-I.............................................…………....…1

Chapter-2

Multivariate Calculus-II.........................................……....…….....41

Chapter-3

First-Order Differential Equations.................................................65

Chapter-4

Second-Order Differential Equations..…………………….....119

Chapter-5

Vectors and Vector Spaces......................…………....….............133

Chapter-6

Matrices and Determinants.........................................……..........143

Chapter-7

Linear Economic Models................................…….......................155

Chapter-8

Unconstrained Optimisation.……............................................171

Chapter-9

Constrained Optimisation with equality Constraints....…....199

Chapter-10

Duality.........................................……....……...…...........................211

(1) Sample Paper-I (Solved)............................................................................................225 (2) Sample Paper-II (Solved)...........................................................................................227 (3) Guess Paper-I...............................................................................................................229 (4) Guess Paper-II.......................................................................................................231 (5) December, 2020 (Solved)................................................................................233

Multivariate Calculus-I

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INTRODUCTION

M

ultivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving several variables, rather than just one. A study of limits and continuity in multivariable calculus yields many counterintuitive results not demonstrated by single-variable functions. For example, there are scalar functions of two variables with points in their domain which give different limits when approached along different paths.

Mathematical Methods in Economics-II [BECC-104]

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1.1 FUNCTION OF SEVERAL VARIABLES 1.1.1 Functions of Two Independent Variables We know that almost exclusively functions of one variable [y = f(x)] that is, functions where a variable is dependent on one independent variable. However, in real life one comes across cases where more than one independent variables influence one dependent variable. For simplicity, let us begin by considering a function which shows an independent variable, say z, being a function of two independent variables, x and y. In notation z = f (x, y) We can define the above function in the following way. A function f of two real variables x and y with domain D is a rule that assigns a specified number f(x, y) to each point (x,y) in domain D. In the above function, x and y are called the independent variables, or arguments of the function f. The variable z is called the dependent variable. The domain of the function f is the set of all possible ordered pairs of the independent variables, while the range is the set of corresponding values of the dependent variable. In some contexts, z is called the endogenous variable while x and y are called exogenous variables. Apart from simplicity, one reason for beginning with functions of only two variables is that we are able to draw diagrams. We can depict f (x, y) diagrammatically in three dimensional space by drawing three mutually perpendicular axes Ox, Oy and Oz. Figure 1.1 below shows such a diagram depicting a surface in three-dimensional space, and a point (x0, y0, z0) on that surface. z

z0 (x0, y0, z0)

y0

x0

y

x

Fig. 1.1 Now let us suppose that this surface is traced out by the function z = f(x, y), and that z is traced out as (x, y) varies over the xy-plane. Then z0 = f(x0, y0).

Multivariate Calculus-I

3

1.1.2 Level Curves If we have z = f(x, y), the graph of this function in three-dimensional space can visualised as being cut by horizontal planes that are parallel to the x y – plane. The intersections between the planes and the graph can be projected onto the x y – plane. If the intersecting plane is at z = k then the projection onto the x y – plane is known as the level curve or contour at height k for the function f. the contour or level will consist of points that satisfy the equation f (x, y)= k Map shows the geographical location of a place (in terms of latitudes and longitudes, for example). To show altitude or height, what we can do (in maps showing physical properties of places), is to draw a set of contours or level curves connecting those points on the map that lie at the same altitude, or elevation above sea level. So on a two-dimensional map (basically representing direction, distance, area, and so on) we can show the third dimension (altitude) by drawing a set of level curves. It is the same idea that we apply to the diagrammatical representation of functions of two variables. You can think of the graph of a function in three dimensional space (as shown on a two-dimensional diagram) as being represented by horizontal planes that are parallel to the x-y plane. These intersections between the planes and the graph, we project onto the x-y plane. At this point, think back to the unit on coordinate geometry. The equation of a line parallel to the y-axis is x = c , while the equation of a line parallel to the x-axis is y = c. In the present context, we are talking of the equation of a plane (because it is in three dimensions x-y-z). Hence the equation of the plane parallel to the xy - plane will be, in an analogous fashion, z = c . This shows the projection of the intersection of this plane with the graph at height c for z. since z = f (x, y ), this projection is called the level curve for f at height c. This level curve then consists of all points that satisfy the equation f (x, y) = c The level curve thus connects points whose functional values are equal. In the above, the level curve connects all the points for which the value of f is c. it is the locus of all points, that is, the combination of x and y, for which the value of z is equal to c. 1.1.3 General Multivariate Functions We can extend the discussion to functions in which the dependent variable is a function of several independent variables. Let us denote a list

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Mathematical Methods in Economics-II [BECC-104]

of n variables by (x1, x2, ….xn) We say that the variable x is indexed by i, where i = 1, 2,…,n. This collection of n-variables (each xi is a real number) is called a vector. A vector is an ordered n-tuple. One point that presents itself immediately is that now we are in n+ 1 dimension. Suppose n = 2 where n is the number of independent variable. There n + 1 was = 3 and we could depict f on a diagram. When n is greater than 2, we end up with more than 3 dimensions, and the function cannot be depicted diagrammatically. So we have to think of the function conceptually and in an abstract manner. When a line is generalised to two-dimensions, it is called a plane. Above that it is called a hyperplane. A general surface in higher dimensions is called a hypersurface. Just remember that the function z = f (x, y), which can as well be depicted z = f (x1, x2) is simply a special case of z = f (x1, x2, …..xn), where n = 2. 1.2 FIRST ORDER PARTIAL DERIVATIVES Definition 1: Let z = f(x, y) be a function of two variables x and y. The ∂z partial derivative of z w.r.t. x denoted by is defined as: ∂x f ( x + h, y ) − f ( x, y ) ∂z provided the limit exist = lim h ∂x h → 0

Definition 2: The partial derivative of z w.r.t. y denoted by as:

∂z is defined ∂y

f ( x,y + h ) − f ( x,y ) ∂z = lim provided the limit exist. h → 0 ∂y h

∂z ∂z are also called first order partial derivatives of z. The , ∂x ∂y ∂z ∂z other notation of are fx and fy. , ∂x ∂y

Remark:

Rule: (1)

To obtain

∂z , differentiate z = f(x, y) w.r.t. x treating y as ∂x

constant. (2)

To obtain

∂z , differentiate z = f(x, y) w.r.t. y treating x as ∂y

constant. Geometric Interpretation of Partial Derivatives:

∂f ( a, b ) or ∂∂xz ( a, b ) ∂x

gives the slopes of the tangent at the point (a, b, c) to the curve C1 , which is the intersection of the surface z = f(x, y) and the plane y = b. Similarly,

Multivariate Calculus-I

5

∂z ( a, b ) gives the slope of the tangent at the point (a, b, c) to the curve C 1 , ∂y

which is the intersection of the surface z = f ( x,y ) and the plane x = a. Z

C1 P1 (a, b, c)

Y

X

Fig. 1.2

Relation between Continuity and Partial Derivative: Suppose f(x, y) is a real-valued function having partial derivatives at a point (a, b). Then for f(a + h, b) − f(a, b) h ≠ 0, f(a + h,b) − f(a, b) = × h and therefore, h f(a + h, b) − f(a, b) lim f(a + h, b) − f(a, b)  = lim . lim h = fx (a, b).0 = 0. h→0 h→0 h→0 h Therefore, we can say that f(a + h, b) → f(a, b) as h → 0 , i.e.

f(x, y) → f(a, b) as (x, y) approaches (a, b) along a line parallel to the

x-

axis. Similarly, the existence of the other partial derivative shows that yf(x, y) → f(a, b) as (x, y) approaches (a, b) along a line parallel to the axis. The existence of fx (a, b) and fy (a,b) does not give us any further information. So, we do not know whether the limit of f(x, y) exists or not if (x, y) → (a, b) along any other path. Thus, it is clear that the mere existence of partial derivatives need not ensure the continuity of the function at that point. 1.3 DIFFERENTIABILITY OF FUNCTIONS FROM R2 toR Let f be a real-valued function defined in a neighbourhood N of point (a, b). We say that the function f is differentiable at (a, b), if f(a + h, b + k) − f(a, b)= Ah + Bk + hφ(h, k) + kψ(h, k), where, (1) h and k are real numbers such that (a + h, b + k) ∈ N,

Mathematical Methods in Economics-II [BECC-104]

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(2) A and B are constants independent of h and k but dependent on the function f and the point (a, b), (3) φ and ψ are two functions tending to zero as (h, k) → (0,0). Differential of Function: Let f(x, y) be a real-valued function defined in a neighbourhood of the point (a, b). If f(x, y) is differentiable at (a, b), then the linear function T : R 2 → R defined by T ( h,k= ) hfx ( a + b ) + kf y ( a, b ) is called the differential of f at (a, b). It will be denoted by df(a, b) Relation between Existence of Partial Derivatives, Differentiability, and Continuity of a Function: A real-valued function f of two variables is said to be continuously differentiable at a point (a, b) if both the first order partial derivatives exist in a neighbourhood of (a, b) and are continuous at the point (a, b). Differentiability

Existence of Partial derivatives

Continuity

Fig. 1.3

1.4 DIFFERENTIABILITY OF FUNCTIONS FROM R n → R, n > 2 Let f be a real-valued function defined in a neighbourhood of the point a= ( a1 ,a 2 ,...........a n ). The function f is said to be differentiable at the point a if

there exist constants A1 ,A2 ,.........An (depending on the function and the point a) such that f(a 1 + h1 ........a n + h n ) − f(a 1 ......,a= n)

n

n

∑ h A + ∑ h φ (h .....h i =1

i

i

i =1

i i

1

n

where each φi → 0 as (h1 ,h 2 ......h n ) → (0,0,.....0). As in the case of two variables, we have the following results: (1)

Ai =

∂f at (a1 ,....an ). ∂x i

(2) If f is differentiable at a, then f is continuous at a. (3) F is differentiable at a if and only if f(a + h) − f(a) =

n

∑h A i =1

i

i

+ h φ(h1 , h 2 ,...h n )

where φ(h1 ,h 2 ,.....h n ) → 0 as h → 0 , h = (h1 ,h 2 ,.......h n ) and h =

∑h

2 i

)

Multivariate Calculus-I

7

(4) If f is differentiable at a, then f has all the partial derivatives at a. (5) If the partial derivatives of f are continuous at a, then f is differentiable at a. 1.5 HIGHER ORDER PARTIAL DERIVATIVES If f(x, y) is a real-valued function defined in a neighbourhood of (a, b) having both the partial derivatives at all the points of the neighbourhood. fx (a + h, b) − fx (a, b) h fx (a, b + k) − fx (a,b) fxy (a,b) = lim k→0 k fy (a + h, b) − fy (a,b) fyx (a,b) = lim h →0 h fy (a, b + k) − fy (a,b) fyy (a, b) = lim k →0 k

Then, fxx (a, b) = lim h →0

provided each one of these limits exists. We also denote the second order partial derivatives of f by ∂ 2f ∂2f = f xx = ;f xy ∂x 2 ∂x∂y = f yy

∂ 2f ∂ 2f = ;f yx 2 ∂y ∂y∂x

If we want to indicate the particular point at which the second order partial derivatives are taken, then we write  ∂ 2f   ∂ 2f  ∂ 2 f (a, b) , , f (a, b) ,    2 xx ∂x 2  ∂x (a , b)  ∂x ∂y (a , b)

∂ 2 f (a,b) , fxy (a, b) , and so on. ∂x ∂y

1.6 CHAIN RULE It is a rule, which enables us to calculate the derivative of functions of several variables where each variable itself is a function of an independent variable. Rule 1: If f :R → R,g : R 2 → R and φ = f ° g, then the partial derivatives of the = φ f ° g:R 2 → R are given by composite function

φx (a,b) = g'(f(a, b))fx (a, b)

φy (a,b) = g'(f(a, b))fy (a, b). f ° g, then Rule 2: If f :R 2 → R,g : R → R 2 and φ =

φ '(t0 ) = f'(t 0 ) φx (f(t 0 ),g(t 0 ))+g'(t 0 )φy (f(t 0 ),g(t 0 ))

f ° g, then Rule 3: If f :R 2 → R 2 ,g : R 2 → R 2 and φ =

Mathematical Methods in Economics-II [BECC-104]

8 GPH Book

φx (a)= D1 φ(a) = D1f(g 1 (a),g 2 (a)) D1 g 1 (a)+ D 2 f(g 1 (a),g 2 (a)) D1 g 2 (a) φ y (a) = D 2 φ(a) = D1f(g 1 (a),g 2 (a))D 2g 1 (a) + D 2f(g 1 (a),g 2 (a))D 2g 2 (a)

1.7 IMPLICIT FUNCTION Many times, we come across equation of the type x + e xy + 8 xy = 0. given any value of x, there exist a unique value of y such that the above equation is satisfied. Thus, y is a function of x, but we cannot express it explicitly, i.e. we cannot express it in the form y = f(x) . In such a situation, we say that y is an implicit function of x. In such a situation, using chain ∂f dy = − ∂x rule, ∂f dx ∂y 1.8 HOMOGENEOUS FUNCTION Let D be a subset of R n such that if

(x1 ,x 2 ,...,x n ) ∈ D,

(tx1 ,tx 2 ,...,tx n ) ∈ D for all t > 0. A function f : D → R

then

is said to be a

homogeneous function of degree k, k being a real number, if f(tx1 ,tx2 ,...,tx n )=t kf(x1 ,x 2 ,...,x n ) for all points (x1 ,x 2 ,...,x n ) ∈ D and all t > 0. 1.9 DIRECTIONAL DERIVATIVES Let f(x,y) be a real-valued function defined on an open disc S(a,r) with centre a = (a 1 ,a 2 ) in R 2 and let v = (cos θ,sin θ) be a unit vector. If lim t→0

( a 1 + t cos θ , a 2 + t sin θ ) − f(a 1 ,a 2 ) t

exists,

Then, we say that f has a directional derivative at a in the direction of v and the value of the limit is called the directional derivative of f at a in the direction of v. We denote the directional derivative of f at the point a in the direction v = (cos θ , sin θ) by fv (a) or D θ f(a).

SOME IMPORTANT THEOREMS Theorem 1: Let f be a real-valued function defined on a neighbourhood N of (a, b). If fy exist at all points of N and fx exists at (a, b), then for real h and k such that

(a + h, b + k) ∈ N,

f ( a + h, b= + k ) f ( a,b ) + kf y ( a, b + θ'k ) + h ( fx ( a,b ) + η' )

where

we have

θ ’ depends

upon h and k, 0 < θ ' < 1,η ' is a function of h and tends to zero as h → 0. Theorem 2: Let f be a real-valued function of two variables defined in a neighbourhood N of a point (a, b) such that one of the first order partial derivatives exists at all points (x, y) ∈ N and is bounded on N, whereas the other partial derivative exists at the point (a, b). Then, the function f is

Multivariate Calculus-I

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continuous at the point (a, b). Theorem 3: (Continuity of a Function): Let f be a real-valued function defined in a neighbourhood N of a point (a, b). If f is differentiable at (a, b), then f is continuous at (a,b) Proof: Let h and k be two real numbers such that (a + h, b + k) ∈ N. Thus, differentiability of f at (a, b) implies that there exist two constants A, B and two function φ(h, k), ψ(h, k) such that f(a + h, b + k) − f(a , b)= Ah + Bk + hφ(h, k) + k ψ (h,k)

…(i)

where φ ( h ,k ) → 0, ψ ( h,k ) → 0 as ( h ,k ) → ( 0,0 ) . Now, taking the limit on both sides of (i) as (h, k) → (0,0), we get lim

( h ,k ) → ( 0 ,0 )

0, or ( f ( a + h, b + k ) − f ( a , b ) ) =

lim

( h ,k ) → ( 0,0 )

f ( a + h, b + k ) = f ( a,b )

This shows that the function f is continuous at (a, b). Theorem 4: Let f be a real-valued function defined in a neighbourhood N of the point (a, b). If f is differentiable at (a, b), then f possesses both the partial derivatives at (a, b). Proof: Let h, k be real numbers such that (a + h, b + k) ∈ N. Since f is differentiable at the point (a, b), it follows that f(a + h, b + k) − f(a , b)= Ah + Bk + h φ(h,k) + k ψ (h, k).

…(i)

where A and B are constants,

φ(h, k) → 0, ψ(h, k) → 0 as (h, k) → (0, 0).

Now, if (a + h, b + k) ∈ N, then (a + h, b) and (a, b + k) also belong to N. So, if we let k = 0 in equation (i), then we get, f(a + h, b) − f(a, b)= Ah + hφ(h, 0) f ( a + h,b ) − f ( a, b ) = A + φ ( h,0 ) for h ≠ 0. h f ( a + h, b ) − f ( a, b ) Therefore, lim = A, h→0 h

i.e.

i.e., fx (a, b) = A.

Similarly, by setting h = 0 and proceeding as above, we can prove that f y ( a, b ) = B. Theorem 5: If f is a real-valued function defined in a neighbourhood of (a, b) such that (1) fx is continuous at (a, b), and (2)

f y exists at (a, b),

Then f is differentiable at the point (a, b). Similarly, the statement that f is differentiable at (a, b) if fx exists at (a, b) and f y is continuous at (a, b) is true. Thus, the continuity of one of