cost function and it derivation

2.1. Objectives

  • To define the cost function and understand its properties.
  • To learn how to derive the cost function.
  • To learn the duality between the cost function and the production function.

2.2. The Cost Function

It is defined as the minimum cost of producing a given output level during a given period of time. The cost function is expressed as a function of input prices and output


Where  W is a vector of strictly positive input prices.

X is a vector of strictly positive inputs and,

The cost function depends on technology since the only constraint to the minimization problem expressed above is that  must be capable of producing at least output.

2.2.1. Properties of a Cost Function

  1. Non-negativity property i.e.  ,

It’s not possible to produce a positive output at zero cost. As long as input prices are all strictly positive, the cost of producing a positive output must also be positive

  1. Non-decreasing in input prices i.e.  if  and  are two vectors of input prices and that , then;
  2. The cost function is non-decreasing in output; if   i.e. increasing output cannot reduce costs.
  3. Fixed cost in the long run are zero
  4. If the cost function is differentiable in, then there exists a unique vector of cost minimizing demand given by the first derivative of the cost function with respect to the input price.


2.2.2. Derivation of cost function.

Given the firms production function as

Let  be the price of labour and  the price of capital. So that the firm’s expression of the cost equation is given as

The firm seeks the minimum cost of producing the level of output.

Rewriting equations (i) and (ii)

Dividing two equations
Substituting equation (v) into equation (iii)

and is the combination of inputs that minimizes the cost of producing y. They are referred to as the conditional factor demands.

The minimum cost of producing is given by

2.2.3 Duality between production function and cost function

We can use exclusively the cost phenomena to reconstruct and to study the properties of the technology. i.e. we can describe the technology entirely in terms of the cost function.

The implication here of being able to describe the technology using the cost function is that the specification of a well cost function is equivalent to the specification of a well behaved production function.

We could therefore say that the cost function is a sufficient statistics for the technology since all the economically relevant information about the technology can be obtained from the cost function.

The decision therefore to use either the direct function (production function) or the indirect function (cost function) is a matter of convenience. This is the most important aspect of duality theorem.

Given a production function we can obtain the cost function by solving the constrained cost minimization problem as shown above. We wish now to see how, given a cost function, we could recover the form of the underlying technology i.e. the production function.

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