Midterm control

Midterm control

Exercise 1

The table contains information on the balance sheet of three branches of industry for a certain period of time. If the final work produced by the industry is increased to 60, 70, and 30 conditional monetary units, respectively, find the gross output of each product.

№	Networks	Consumotion			Final product		Gross production
		1	2	3		4		5
1	Cotton cultivation and processing	5	35	20		40		100
2	Energetics	10	10	20		60		100
3	Mechanical engineering	20	10	10		10		50

Solution: We write the vectors of gross production and final consumption and the matrix of coefficients of direct costs.

Matrix A satisfies both efficiency criteria. When final consumption increases in a given volume, a new vector of final consumption will appear.

It is required to find a new gross output vector X * that satisfies the equilibrium relations under the assumption that the matrix A is unchanged. In this case, the 1 x , 2 x , 3 x components of the unknown vector X * are found from the system of equations in the matrix form Х *  AХ * У* or (Е  A)Х * У*

The matrix of this system has the following form.

Solving in the given vector of the right-hand side of the system of linear equations (for example, by the Gaussian method) gives a new vector 𝑋 ∗, the solution of the interbranch balance equations:

Thus, in order to ensure the increase of the final consumption vector components in the given volume, it is necessary to increase the corresponding gross production: cotton cultivation and processing by 52.1%, energy level by 35.8%, and machinery production by 41.5% compared to the initial data shown in the table.

Example 2.
у  a₀*x1^a1 *x2^a2 Average of resources for Cobb-Douglas production function (KDICHF).