Economic Interpretation of Dual

       Suppose the primal problem is that of maximization of the total net revenue with give cost outlay. The dual would be that of minimization of costs for the given output.

       Consider the following production problem.

       A furniture firm specializes in the production of three products chairs tables and bookshelves. The three products require in main wood and labour whose supplies are limited. The technological requirement of production and resource availabilities are shown in the following table.

                                               Chair                        table                   Bookshelf             Available

                                                                                                                                        Resources

Units of woods required            a₁₁                        a₁₂                        a₁₃                       r₁

Units of labour required            a₂₁                        a₂₂                        a₂₃                      r₂

     Price/unit                              p₁                        p₂                         p₃

Let x₁,x₂,x₃ denote the number of chairs, table and boodkshelves to be produced by a furniture firm. The mathematical formulation of the above problem is :

 A₁₁x₁ +a₁₂x₁₂ +a₁₃x₁₃ < r₁; a₂₁x₁ +a₂₂x₂ + a₂₃x₃ < r₂

    X_j > 0; j = 1,2,3.

In other words, we wish to answer the resource allocation problem: how should the limited resources of wood and labour, given as r₁ and r₂ be allocated among the three products so as to produce certain quantities of those products, x₁,x₂ and x₃ whereby at the given prices p₁,p₂and p₃ of the products, the total revenue from the sale of the products would be maximized?

To dual of the above problem would be:

Minimize Z* = r1y1+ r2y2 subject to the constraints:

      A₁₁y₁ +a₂₁y₂ > p₁; a₁₂y₁> p₂; a₁₃ y₃ +a₂₃y₃> p₃

      Y₁> 0; I = 1,2.

To interpret this dual problem, we must try to understand what interpretation is to be given to the variables y₁ and y₂. Consider the first inequality.

                 A₁₁y₁ + a₂₁y₂ > p₁

P₁ is the price or unit value of a chair. A₁₁ and a₂₁ are the quantities of wood and labour required to produce one chair. Since p₁ is expressed in money terms and a₁₁ and a₂₁ are in physical terms, to make both sides of the inequality comparable, we must convert the left-hand side of the inequality also in money terms. Y₁ and y₂ may thus be regarded as the price or unit value of wood and labour respectively. A11y1 thus represents the value of wood used to produce a single chair and a21 y2 the value of labour used for producing a single chair. The left-hand side of the first inequality is thus the total value of the resources required to produce a chair.

The first inequality therefore means that the total value of the inputs required to produce a chair must be a least equal to the price of the chair, a similar interpretation can be given to the second and third inequalities corresponding to the table and bookshelf.

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