Econ 407: Further application of matrices.. In Economics, tools from matrix algebra are used to solve systems of equations, to find soulutions for input-output models, and in regression analysis and quantative systems that can be made linear through proper lineraization methods. The following example shows how we can solve for the B matrix (the matrix of coefficients) using the Y and the X matrix. We know that Y= X*B + E, where X is the data matrix, Y is the observed dependent variable and E is the amount of error not explained by the regression equation. Please pay careful attention to the procedure used. The solution for the matrix of coefficeints is:

B=inverse of [X(transposed)*X]*[X(transposed)]*[Y].

> with(linalg):
Y:=linalg[matrix]([[9],[13],[17],[21],[25],[29],[32],[37]]);

> X:=linalg[matrix]([[1,1],[1,2.0],[1,3.0],[1,4],[1,5],[1,6],[1,7],[1,8]]);

> Xt:=transpose(X);

> C:=evalm(Xt&*X);

> Cin:=inverse(C);

> Bcoef:=evalm(Cin&*Xt&*Y);

> PY:=evalm(X&*Bcoef);

> Error:=evalm(Y-PY);

The estimated equation is Y= 5.14 + 3.94X. We can plot the regression line as shown below.

> y:=5.14 +3.94*x;

> plot(y, x=1..9);

Applications to input-output model. An input output model consisits of a matrix of technical coefficients A, and a final demand matrix B, total demand, gross output and value added for sectors or several industries in the economy. , shows the value or the amount of inputs(i) required to produce a dollar or a unit worth of output of a given product. The total demand for product i, can be expressed as the sum of intermediate demand and the final demand. The intermediate demand is what is used up by the industries as input in the production process, and final demand is what avilabale for purchase by consumers towards final consumption. Prior knowledge about the matrix of technical coefficients A, enables one to compute total demand for a product as shown below:

xi= a11*x1 + a12*x2 + a13*x3+...a1n*xn + bi.

In matrix notation we can express this relationship as X=AX +B, where X=[x1,x2,x3...xn], A is a matrix of tehcnical coefficients as defined above, and B is the matrix showing the amount of

final demand for different products. We can solve for total demand--X using: X-AX=B or (I-A)X=B, . The equations are solved by dividing B by (I-A) or multiplying B by the inverse of (I-A). See page 286, and 308 of the text. The Maple instructions for finding total demand are given below. Reember that A is the matrix of technical coefficients, and B is the matrix showing final demand.

> with(linalg):
A:=linalg[matrix]([[0.2,0.3,0.2],[0.4,0.1, 0.3],[0.3,0.5,0.2]]);

> with(linalg):
B:=linalg[matrix]([[150],[200],[210]]);

> In:=linalg[matrix]([[1,0,0],[0,1,0],[0,0,1]]);

> C:=evalm(In-A);

> Cin:=inverse(C);

> X:=evalm(Cin&*B);

Total demand is shown in Matrix X above. Note that total demand is greater than final demand because part of the output is used up by the industry that produces the product or other industries that produce other products.

Do problems 12.36, 12.37, and 12.38