1. THE FIRM
1.1. Basic settings
A) Technology:
We shall express the technological possibilities for a firm in terms of a fundamental inequality
specifying the relationship between a single output and a vector of m inputs: q ≤ φ(z).
This allows us to handle multiple inputs and consider the possibility of inefficient production (if =
holds we shall call the production ‘technically efficient’).
B) Environment:
Assume pure competition (the price ‘p’ for its output and ‘w’ for its inputs is given)
C) Motivation:
The firm aims to maximise profits. (Note that it could also be: maximise managerial utility or sales)
We will assume that φ(0) = 0. (Note that it could be the case that
the firm makes losses if we introduced time, uncertainty or some
return from the firm which is note measured in money.
1.1.1. Properties of the production function
Main types (Cobb-Douglas, Leontief, Perfect substitutes, CES, Non-convex to origin, Quasi linear)
We will refer to a particular vector of inputs as a technique.
• Pick some arbitrary level of output q: then the input-requirement set for the specified value q
is the following set of techniques:
• The q-isoquant of the production function φ is que contour of φ in the space of inputs
Clearly the q-isoquant is the boundary of Z(q). In a two-input version of the model Figure 2.1.
illustrates Z(q) corresponding to different assumptions about the production function:
• An isoquant can touch the axis if one input is not
essential.
• An isoquant may have flat segments. We can interpret
this as locally perfect substitutes in production.
• The convexity of Z(q) implies that production processes
are, in some sense, divisible (take any two vectors z0 and
z00 that lie in Z(q); draw the straight line between them;
any point on this line clearly also belongs to Z(q) and
such a point can be expressed as tz0 +[1t]z00 where
0<t<1). What you have established is that if the
production techniques z0 and z00 are feasible for q, then
so too is a mixture of them.
• An isoquant may have kinks corners
,1.1.2. Marginal rate of technical substitution (MRTS)
If we assume that φ is differentiable then its partial derivative is well-defined.
It reflects the ‘relative value’ of one input in terms
of another from the firm’s point of view (internal
price): if one unit of input j were to be lost, how
much of input i would be needed to compensate
s
for it so as to maintain q constant.
1.1.3. Elasticity of substitution
How responsive is the firm’s production technology to a change in its relative valuation of its
inputs? We can see it this way:
WHICH MEANS
Higher values of elasticity mean
that the production function is
more flexible in that there is a
proportionately larger change in
the production technique in
response to a given proportionate
change in the implicit relative
valuation of the factors
, 1.1.4. Homothetic and homogenous production functions
Homothetic contours: each isoquant appears like a photocopied enlargement; along any ray
through the origin all the tangents have the same slope so that the MRTS depends only on the
relative proportions of the inputs used in the production process.
Homogenous contours: are a special of the
homothetic contours, the map looks the
same but the labelling of the contours has to
satisfy the following rule: for any scalar t > 0
and any input vector z>= 0:
where r>0
1.1.5. Returns to scale
In each case the set of points on or underneath the tent-like shape
represent feasible input-output combinations. Take a point on the
surface such as the one marked in each of the three figures:
• Its vertical coordinate gives the maximum amount of output that
1.1. Basic settings
A) Technology:
We shall express the technological possibilities for a firm in terms of a fundamental inequality
specifying the relationship between a single output and a vector of m inputs: q ≤ φ(z).
This allows us to handle multiple inputs and consider the possibility of inefficient production (if =
holds we shall call the production ‘technically efficient’).
B) Environment:
Assume pure competition (the price ‘p’ for its output and ‘w’ for its inputs is given)
C) Motivation:
The firm aims to maximise profits. (Note that it could also be: maximise managerial utility or sales)
We will assume that φ(0) = 0. (Note that it could be the case that
the firm makes losses if we introduced time, uncertainty or some
return from the firm which is note measured in money.
1.1.1. Properties of the production function
Main types (Cobb-Douglas, Leontief, Perfect substitutes, CES, Non-convex to origin, Quasi linear)
We will refer to a particular vector of inputs as a technique.
• Pick some arbitrary level of output q: then the input-requirement set for the specified value q
is the following set of techniques:
• The q-isoquant of the production function φ is que contour of φ in the space of inputs
Clearly the q-isoquant is the boundary of Z(q). In a two-input version of the model Figure 2.1.
illustrates Z(q) corresponding to different assumptions about the production function:
• An isoquant can touch the axis if one input is not
essential.
• An isoquant may have flat segments. We can interpret
this as locally perfect substitutes in production.
• The convexity of Z(q) implies that production processes
are, in some sense, divisible (take any two vectors z0 and
z00 that lie in Z(q); draw the straight line between them;
any point on this line clearly also belongs to Z(q) and
such a point can be expressed as tz0 +[1t]z00 where
0<t<1). What you have established is that if the
production techniques z0 and z00 are feasible for q, then
so too is a mixture of them.
• An isoquant may have kinks corners
,1.1.2. Marginal rate of technical substitution (MRTS)
If we assume that φ is differentiable then its partial derivative is well-defined.
It reflects the ‘relative value’ of one input in terms
of another from the firm’s point of view (internal
price): if one unit of input j were to be lost, how
much of input i would be needed to compensate
s
for it so as to maintain q constant.
1.1.3. Elasticity of substitution
How responsive is the firm’s production technology to a change in its relative valuation of its
inputs? We can see it this way:
WHICH MEANS
Higher values of elasticity mean
that the production function is
more flexible in that there is a
proportionately larger change in
the production technique in
response to a given proportionate
change in the implicit relative
valuation of the factors
, 1.1.4. Homothetic and homogenous production functions
Homothetic contours: each isoquant appears like a photocopied enlargement; along any ray
through the origin all the tangents have the same slope so that the MRTS depends only on the
relative proportions of the inputs used in the production process.
Homogenous contours: are a special of the
homothetic contours, the map looks the
same but the labelling of the contours has to
satisfy the following rule: for any scalar t > 0
and any input vector z>= 0:
where r>0
1.1.5. Returns to scale
In each case the set of points on or underneath the tent-like shape
represent feasible input-output combinations. Take a point on the
surface such as the one marked in each of the three figures:
• Its vertical coordinate gives the maximum amount of output that
