Outline the structure of an input-output model (including assumptions about supply and demand). What is an inverse matrix? Why is inverting a matrix significant in terms of input-output analysis?C.2. Describe a Linear Programming (LP) Problem. Specifically, describe (you can use an example):Primal Linear Programming ProblemDual Linear Programming ProblemInterpretation of the Primal Linear Programming ProblemInterpretation of the Dual Linear Programming Problem

Explanation:

C.1 Input-Output Model

It is a formal model that divides the economy into 2 sectors and traces the flow of inter-industry purchases and sales. This model was developed by Wassily Leontief in 1951. In simpler terms, the inter-industry model is a quantitative economic model that defines how the output of one industry becomes the input of another industrial sector. It is an interdependent economic model where the output of one becomes the input of another. For Eg: The Agriculture sector produces output using the inputs from the manufacturing sector.

The 3 main elements are:

Concentrates on an economy which is in equilibrium

Deals with technical aspects of production

Based on empirical investigations and assumptions

Assumptions

2 sectors - " Inter industry sector" and "final sector"

Output of one industry is the input for another

No 2 goods are produced jointly. i.e each industry produces homogenous goods

Prices, factor suppliers and consumer demands are given

No external economies or diseconomies of production

Constant returns to scale

The combinations of inputs are employed in rigidly fixed proportions.

Structure of IO model

See image 1

Quadrant 1: Flow of products which are both produced and consumed in the process of production

Quadrant 2: Final demand for products of each producing industry.

Quadrant 3: Primary inputs to industries (raw materials)

Quadrant 4: Primary inputs to direct consumption (Eg: electricity)

The model can be used in the analysis of the labor market, forecast economic development of a nation and analyze economic developments of various regions.

Leontief inverse matrix shows the output rises in each sector due to a unit increase in final demand. Inverting the matrix is significant since it is a linear system of equations with unique solutions. Thus, the final demand vector for the required output can be found.

C.2 Linear programming problems

Linear programming problems are optimization problems in which objective function and the constraints are all linear. It is most useful in making the best use of scarce resources during complex decision makings.

Primal LP, Dual LP, and Interpretations

Primal linear programming: They can be viewed as a resource allocation model that seeks to maximize revenue under limited resources. Every linear program has associated with it a related linear program called dual program. The original problem in relation to its dual is termed as a primal problem. The objective function is a linear combination of n variables. There are m constraints that place an upper bound on a linear combination of the n variables The goal is to maximize the value of objective functions that are subject to the constraints. If the primal linear programming has finite optimal value, then the dual has finite optimal value, and the primal and dual have the same optimal value. If the optimal solution to the primal problem makes a constraint into a strict inequality, it implies that the corresponding dual variable must be 0. The revenue-maximizing problem is an example of a primal problem.

Dual Linear Programming: They represent the worth per unit of resource. The objective function is a linear combination of m values that are the limits in the m constraints from the primal problem. There are n dual constraints that place a lower bound on a linear combination of m dual variables. The optimal dual solution implies fair prices for associated resources. Stri=ong duality implies the Company’s maximum revenue from selling furniture = Entrepreneur’s minimum cost of purchasing resources, i.e company makes no profit. Cost minimizing problem is an example of dual problems

See image 2

n - economic activities

m - resources

cj - revenue per unit of activity j

answers are discussed in the explanation section.

Explanation:

a) This model is used to track the flow of purchases and sales between industries. Developed by Wassily Leontief in 1951. This is an economic model in which production in one industry becomes the entrance to another specific industrial sector. It is a model in which the production of one company becomes the input of another company. Its 3 main characteristics are:

-economy that is in balance.

-It has to do with the technical aspects of production.

-based on empirical research and assumptions

some of the assumptions are as follows:

-as already said, the production of one industry is the entrance to another industry.

-each industry is capable of producing homogeneous products

-Yields are considered constant.

the structure of IO for this model is observed in the image:

-1: products produced and consumed in the production process.

-2: final demand for goods from the industry in question.

-3: chain of primary inputs for the industry.

-4: chain of primary inputs into direct consumption by industry.

b) In primary linear programming: the resource allocation model is observed looking for the maximization of income from limited resources. These programs are associated with a program known as the dual program. The objective function is defined as a linear combination of n variables. There are also more restrictions for these variables. Its objective is to maximize the value of the objective functions that are subject to the restrictions.

Regarding dual linear programming, the objective function is defined as a linear combination of n values, which are the limits on the constraints m. The problem is the minimization of costs as an example of dual programming.

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explanation: