Parashar's Digital Notebook!: October 2013

Friday, October 18, 2013

Post 12: Matrix Operation

Matrix Equality: Two matrices $A = [a_{ij}]$ and $B = [b_{ij}]$ are said to be equal if and only if they have the same dimension and have identical elements in the corresponding locations in the array.

Addition and Subtraction of Matrices:
Two matrices can be added if and only if they have the same dimension. The addition rule can be stated as:

                               $[a_{ij}] + [b_{ij}] = [c_{ij}] \: \: \: \: \: \: \: where c_{ij} = a_{ij} + b_{ij}$

The subtraction operation A - B can be similarly defined if and only if a and B have the same dimension. The operation entails the result:

                               $[a_{ij}] - [b_{ij}] = [d_{ij}] \: \: \: \: \: \: \: where d_{ij} = a_{ij} - b_{ij}$

Scalar Multiplication:
To multiply a matrix by a number - or in matrix algebra terminology, by a scalar - is to multiply every element of that matrix by the given scalar.

Multiplication of Matrices:
Suppose that, given two matrices A and B, we want to find the product of AB. The conformability condition for manipulation is that the column dimension of A (the "lead" matrix the expression AB) must be equal to the row dimension of B (the "lag" matrix).

In general, if A is of dimension m x n and B is of dimension p x q, the matrix product AB will be defined i and only if n = p. If defined, moreover, the product matrix AB will have the dimension m x q - the same number of rows as the lead matrix A and the same number of columns as the lag matrix B.

An example of matrix multiplication is given below:

                               $A_{1x2} = \begin{bmatrix} a_{11} & a_{12} \end{bmatrix}$                  $B_{2x3} = \begin{bmatrix} b_{11} & b_{12} & b_{13}\\ b_{21} & b_{22} & b_{23} \end{bmatrix}$

For the metrics given, AB will be 1 x 3. So let's consider,

                               $AB = C = \begin{bmatrix} c_{11} & c_{12} & c_{13} \end{bmatrix}$

The placeholder values of $c_{11}, c_{12}, \; and \: c_{13}$ , mentioned above, can be calculated as:

                               $c_{11} = a_{11}b_{11} + a_{12}b_{21}$
                               $c_{12} = a_{11}b_{12} + a_{12}b_{22}$
                               $c_{13} = a_{11}b_{13} + a_{12}b_{23}$

Given two vectors u and v with n elements each, say, $(u_{1}, u_{2}, ..., u_{n})$ and $(v_{1}, v_{2}, ..., v_{n})$ , arranged either as two rows or as two columns, or as one row and one column, their inner product, written as u . v, is defined as:

                               $u . v = u_{1}v_{1} + u_{2}v_{2}+... + u_{n}v_{n}$

Note: A square matrix with 1s in its principal diagonal and 0s everywhere else - examplifies the important type of matrix known as identity matrix.

Division:
It is not possible to divide one matrix by another.

$\sum$ Notation:

The summation shorthand makes use of the Greek letter $\sum$ (sigma). To express the sum of $x_{1}, x_{2}, \: and \: x_{3}$ , for instance, we may write:

                               $x_{1} +x_{2} + x_{3} = \sum_{j=1}^{3}x_{j}$

Extending this to the multiplication of an m x n matrix $A = [a_{ik}]$ and an n x p matrix $B = [b_{kj}]$ , we may now write the elements of the m x p product matrix AB = C = $[c_{ij}]$ as,

                                   $c_{11} = \sum_{k = 1}^{n} a_{1k}b_{k1}\: \: \: \: \: c_{12} = \sum_{k=1}^{n}a_{1k}b_{k2}\; \; \; ...$

or more generally,

                                   $c_{ij} = \sum_{k=1}^{n}a_{ik}b_{kj}\: \: \: \: \: \bigl(\begin{smallmatrix} i = 1, 2, ..., m\\ j = 1, 2, ..., p \end{smallmatrix}\bigr)$

Thursday, October 17, 2013

Post 11: Matrices and Vectors

Matrix algebra can enable us to do many things.

1) It provides a compact way of writing an equation system, even an extremely large one.
2) It leads to a way of testing the existence of solution by evaluation of a determinant
3) It gives a method of finding that solution (if it exists)

However, one slight catch is that matrix algebra is applicable only to linear-equation system. In many cases, even if some sacrifice of realism is entailed by the assumption of linearity, an assumed linear relationship can produce sufficiently close approximation to an actual nonlinear relationship to warrant its use.

In other cases, while preserving the non-linearity in the model, we can effect a transformation of variables so as to obtain a linear relation to work with. For example, the nonlinear function,

                                         $y = ax^b$

can be readily transformed, by taking the logarithm on both sides as:

                                      log y = log a + b logx

Matrices and Vectors
In general, a system of m linear equations in n variables $(x_{1}, x_{2}, ..., x_{n})$ can be arranged into such a format:

                                $a_{11}x_{1}+a_{12}x_{2}+...+a_{1n}x_{n}=d_{1}$
                                $a_{21}x_{1}+a_{22}x_{2}+...+a_{2n}x_{n}=d_{2}$                                 (11.1)
                                ...............................................................
                                $a_{m1}x_{1}+a_{m2}x_{2}+...+a_{mn}x_{n}=d_{m}$

There are essentially three types of ingredients in the equation system 11.1. The first is set of coefficients $a_{ij}$ ; the second is the set of variables $(x_{1}, x_{2}, ..., x_{n})$ ; and the last is the set of constant terms $d_{1}, d_{2},..., d_{m}$ . If we label them, respectively, as A, x, and d, then we have:

                $A = \begin{bmatrix} a_{11} & a_{12} & ... & a_{1n}\\ a_{21} & a_{22} & ... & a_{2n}\\ ... & ... & ... & ...\\ a_{m1} & a_{m2} & ... & a_{mn} \end{bmatrix}$         $x = \begin{bmatrix} x_{1}\\ x_{2}\\ .\\ .\\ .\\ x_{n} \end{bmatrix}$                   $d = \begin{bmatrix} d_{1}\\ d_{2}\\ .\\ .\\ .\\ d_{m} \end{bmatrix}$

As a shorthand device, the array in matrix A can be written more simply as:

               $A = [a_{ij}] \; \; \; \; \; \; \begin{pmatrix} i = 1, 2, ...., m\\ j = 1, 2, ...., n \end{pmatrix}$

Some matrices may contain only one column, such as x and d above. Such matrices are given the special name column vectors. If we arrange the vector x in horizontal way, though, there would result a 1 x n matrix, which is called row vector. A row vector is often distinguished from a column vector by the use of a prime symbol, X'.

A vector, whether row or column, is merely an ordered n-tuple, and as such it may sometimes be interpreted as a point in an n-dimensional space. With the matrices defined above, we can express the equation system in 11.1 as,

                                                     Ax = d

However, the equation Ax = d prompts at least two questions. How do we multiply two matrices A and x? What is meant by the equality of Ax and d? Since matrices involve whole block of numbers, the familiar algebraic operations defined for single numbers are not directly applicable, and there is a need for a new set of operational rules, which would be discussed in Post 12 onwards.

Post 10: Equilibrium in National-Income Analysis

The static analysis can be applied to other areas of economics, like Keynesian national-income model too. If we consider the simplest Keynesian national-income model below:

                                   $Y = C + I_{0} + G_{0}\; \; \; \; \; \; \; (a> 0, \; \; \; 0<b<1)$
                                   $C = a+bY$

where Y = Endogenous variable National Income
           C = Endogenous variable Planned Consumption Expenditure
           $I_{0}$ = Exogenous variable Investment
          $G_{0}$ = Exogenous variable Government Expenditure
           a = Parameter variable Autonomous Consumption
           b = Parameter variable Marginal Propensity to Consume

The equations above give us the Y* and C* values as:

                                   $Y^* = (a + I_{0} + G_{0}) / (1 - b)$
                                   $C^* = (a + b(I_{0} + G_{0})) / (1 - b)$

Both Y* and C* have the expression (1 - b) in the denominator, thus the restriction b $\neq$ 1 is necessary, to avoid division by zero.

Wednesday, October 16, 2013

Post 9: General Market Equilibrium

The equilibrium condition of an n-commodity market model will involve n equations, one for each commodity, in the form:

                                   $\mathbf{E_{i} \equiv Q_{di} - Q_{si} = 0 \; \; \; \; (i = 1, 2, ..., n)}$                          (9.1)

If a solution exists, there will be a set of prices $P_{i}^*$ and corresponding quantities $Q_{i}^*$ such that all n equations in the equilibrium condition will be simultaneously satisfied.

Derivation with Two-Commodity Model
To illustrate the general market equilibrium equation, let us consider a simple 2 commodity model. For simplicity, the demand and supply functions of both commodities are assumed to be linear. In parametric terms, such a model can be written as:

                                   $Q_{d1} - Q_{s1} = 0$

                                   $Q_{d1} = a_{0} + a_{1}P_{1} + a_{2}P_{2}$
                                   $Q_{s1} = b_{0} + b_{1}P_{1} + b_{2}P_{2}$                                                      (9.2)
                                   $Q_{d2} - Q_{s2} = 0$
                                   $Q_{d2} = \alpha _{0} + \alpha _{1}P_{1} + \alpha _{2}P_{2}$

                                   $Q_{s2} = \beta _{0} + \beta _{1}P_{1} + \beta _{2}P_{2}$

where the a and b coefficients pertain to the demand and supply functions of the first commodity, and $\alpha$ and $\mathbf{\beta }$ coefficients are assigned to those of the second, and P1 and P2 are the prices of the two commodities in consideration.The restrictions and the other signs for the coefficients would be considered later.

From 9.2,
   $(a_{0}-b_{0}) + (a_{1}-b_{1})P_{1} + (a_{2}-b_{2})P_{2} = 0$                   (9.3)
                                 $(\alpha _{0}-\beta _{0}) + (\alpha _{1}-\beta _{1})P_{1} + (\alpha _{2}-\beta _{2})P_{2} = 0$

There are as many as 12 parameters involved in equations 9.3. Let us therefor define the shorthand symbols,
                                 $c_{i} \equiv a_{i} - b{i}$          (i = 0, 1, 2)
                                 $\gamma _{i} \equiv \alpha _{i} - \beta {i}$
gives,
                                 $c_{1}P_{1} + c_{2}P_{2} = -c_{0}$                                                                  (9.4)
                                 $\gamma _{1}P_{1} + \gamma _{2}P_{2} = -\gamma _{0}$

which may be solved further to get

                                 $P_{1}^* = (c_{2}\gamma _{0} - c_{0}\gamma _{2})/ (c_{1}\gamma _{2} - c_{2}\gamma _{1})$
                                 $P_{2}^* = (c_{0}\gamma _{1} - c_{1}\gamma _{0})/ (c_{1}\gamma _{2} - c_{2}\gamma _{1})$

For these two values to make sense, however, certain restrictions should be imposed on the model. First, since division by zero is undefined, so $c_{1}\gamma _{2}$ should be $\neq c_{2}\gamma _{1}$ . Second, to assure positivity, the numerator must have the same sign as the denominator.

Also, a negative sign in coefficient before the prices would suggest that the two commodities are substitutes for each other.

Derivation with n-Commodity
If all commodities in an economy are included in a comprehensive market model, the result will be a Walrasian type of general-equilibrium model, in which the excess demand for every commodity is considered to be a function of the prices of all the commodities in the economy.

Some of the prices may, of course, carry zero coefficients when they play no role in the determination of the excess demand of a particular commodity. In general, however, with n-commodities in all, we may express the demand and supply functions as follows:

                                $Q_{di} = Q_{di}(P_{1}, P_{1}, ..., P_{n})$         (i = 1, 2, ..., n)
                                $Q_{si} = Q_{si}(P_{1}, P_{1}, ..., P_{n})$

The equilibrium condition iss itself composed of a set of n equations:

                                $Q_{di} - Q_{si} = 0$      (i = 1, 2, ..., n)

So we there have a 3n equations.

Upon substitution, however, the model can be reduced to a set of n simultaneous equations only:

                        $Q_{si}(P_{1}, P_{1}, ..., P_{n}) - Q_{si}(P_{1}, P_{1}, ..., P_{n}) = 0 \! \; \: \: \: \: \: \: \:(i = 1, 2,..., n)$

Alternatively,
                        $E_{i}(P_{1}, P_{2}, ..., P_{n}) = 0 \: \: \: \: \: \: (i = 1, 2, ..., n)$

Solved simultaneously, these n equations can determine the n equilibrium prices $P_{i}^*$ , if a solution indeed exists. And then the $Q_{i}^*$ may be derived from the demand or supply functions.

Solution of a General-Equation System
For a general-function model containing, say, a total of m parameters () - where m is not necessarily equal to n - the n equilibrium prices can be expected to take the general analytical form of

                      $P_{i}^* = P_{i}^*(a_{1}, a_{2}, ...., a_{m})\: \: \: \: \: \: (i = 1, 2, ..., n)$

But an important catch exists: the expression can be justified if and only if a unique solution does exist. Yet, there is no a priori reason to presume that every model will automatically yield a unique solution. So it is important to mention the consistency and functional independence as the two prerequisites for application of the process of counting equations and unknowns. In general, in order to apply that process, make sure that:
1) the satisfaction of any one equation in the model will not preclude the satisfaction of another and
2) no equation is redundant

For simultaneous-equation model, there exists systematic methods of testing the existence of unique solution. These would involve, for linear models, an application of the concept of determinants. In the non-linear models, such a test would also require a knowledge of so-called partial derivatives and a special type of determinant, called Jacobian determinant. Those will be discussed in later posts.