THE FIRM 3
3.0 Introduction
Having discussed the firm’s technology and the costs associated, in this lesson we
now turn to the firm’s solution to its optimization problem. The solution is expressed as a
profit function.
3.1. Objectives
At the end of this lesson, the learner should:
Understand the definition and properties of the profit function.
Understand how to derive the profit function.
3.2. The Profit Function
It represents the solution to the firm’s optimization problem. It’s expressed as a
function of output prices of input prices. i.e.
Direct function
Indirect function
Where is the output price which the producer takes as given.
is the vector of strictly the input prices.
is a vector of inputs
is the output
1
, 3.2.1. Properties of the profit function
(i) The profits are non-negative i.e. a producer never accepts negative profits in
the long run
(ii) Its non-decreasing in output price, if and are two output prices and that
the profits evaluated:
(iii) The profit function is non-increasing in input prices i.e. if and are two
vectors or sets of input prices and , then
(iv) The profit function is positively, linearly homogenous in both input and output
prices. i.e. where . If output and input prices is
double, then the profits will double. This property is a further consequence of
the principle that only relative prices matter in economics.
(v) If profit function is differentiable in & , then there exists a
unique profit maximizing supply and derived demand functions given as
follows:
Supply
Demand =
The Hotellings Lemma states that:
If the profit function is well behaved, the first partial positive derivative of
the profit function with respect to the output prices is the firms supply
function.
2
3.0 Introduction
Having discussed the firm’s technology and the costs associated, in this lesson we
now turn to the firm’s solution to its optimization problem. The solution is expressed as a
profit function.
3.1. Objectives
At the end of this lesson, the learner should:
Understand the definition and properties of the profit function.
Understand how to derive the profit function.
3.2. The Profit Function
It represents the solution to the firm’s optimization problem. It’s expressed as a
function of output prices of input prices. i.e.
Direct function
Indirect function
Where is the output price which the producer takes as given.
is the vector of strictly the input prices.
is a vector of inputs
is the output
1
, 3.2.1. Properties of the profit function
(i) The profits are non-negative i.e. a producer never accepts negative profits in
the long run
(ii) Its non-decreasing in output price, if and are two output prices and that
the profits evaluated:
(iii) The profit function is non-increasing in input prices i.e. if and are two
vectors or sets of input prices and , then
(iv) The profit function is positively, linearly homogenous in both input and output
prices. i.e. where . If output and input prices is
double, then the profits will double. This property is a further consequence of
the principle that only relative prices matter in economics.
(v) If profit function is differentiable in & , then there exists a
unique profit maximizing supply and derived demand functions given as
follows:
Supply
Demand =
The Hotellings Lemma states that:
If the profit function is well behaved, the first partial positive derivative of
the profit function with respect to the output prices is the firms supply
function.
2
