Production
Production Theory
Technology 1
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Consider an economy with L goods.
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A …rm is seen as a black box which uses these goods to serve
as inputs and/or outputs.
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The production plan is a vector
y = (y1 ; :::; yL ) 2 <L
that describes the (net) output of the L goods of the
production process of the …rm.
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By convention,
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a positive yi indicates a good which is a net output and
a negative yi indicates a good which is a net input.
y = ( 5; 2; 6; 3; 0): 2 and 3 units of commodities 2 and 4 are
produced, 5 and 6 units of commodity 1 and 3 are used,
commodity 5 is neither produced nor used as an input
Technology 2
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The existing technology, taken as primitive datum, de…nes the
production set Y
<, which is the set of the
feasible production plans
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Can describe the production set using the transformation
function F (:), which is such that:
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Y = fy 2 <L : F (y) 0g; (intuition: producing only output
from nothing is out of the production process)
F (y) = 0 i¤ there is no y0 2 Y such that y0 y.
,! F (y) = 0 i¤ y is (technically) e¢ cient, that is there is no
way to produce more of a least one output with the same
inputs or the same output with less of at least one input.
Intuition: necessity of technological constraints.
The set fy 2 <L : F (y) = 0g is de…ned as the
transformation frontier.
Production set and transformation frontier
Marginal rate of transformation
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Provided that F (:) is di¤erentiable and F (y) = 0, then the
marginal rate of transformation (at y) is given by:
MRTlk (y) =
@F (y)
@yl
@F (y)
@yk
obtained simply by totally di¤erentiating F (:) and evaluating
it at y.
how much the (net) output of good k can
increase if the …rm decreases marginally
the (net) output of good `
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Graphically, this is simply the slope of the transformation
frontier.
Technology 3
Sometimes we’ll use two simpli…cations
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separation between inputs and outputs.
With L M inputs (always negative) and M outputs (always
positive),
Y
= f( z1 ; :::; zL
(z1 ; :::; zL
and F (:)
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M ; q1 ; :::; qM )
M ); (q1 ; :::; qM )
:
0
0g;
single-output technology. With L 1 inputs and 1 output, we
can make use of the production function q = f (z1 ; :::; zL 1 )
de…ned as
Y
= f( z1 ; :::; zL
q
f (z1 ; :::; zL
and (z1 ; :::; zL
1 ; q)
1)
1)
:
0
0g:
Single output technology
Useful concepts in the single-output case:
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input requirement set
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isoquant
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marginal rate of technical substitution
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elasticity of substitution
Single output technology
Useful concepts in the single-output case:
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,! input requirement set
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isoquant
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marginal rate of technical substitution
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elasticity of substitution
Input requirement set
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the input requirement set V (q) the set of all input bundles
that produce at least q, given by:
V (q) = f(z1 ; :::; zL
(z1 ; :::; zL
1)
1)
: ( z1 ; :::; zL
0g:
1 ; q)
2 Y and
Single output technology
Useful concepts in the single-output case:
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input requirement set
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,! isoquant
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marginal rate of technical substitution
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elasticity of substitution
Isoquant
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the isoquant Q(q) is the set of all input bundles that produce
exactly q, given by:
Q(q) = f(z1 ; :::; zL
1)
(z1 ; :::; zL
1)
(z1 ; :::; zL
1)
: (z1 ; :::; zL
2 V (q) and
1)
0;
2
= V (q 0 ) if q 0 > qg:
Single output technology
Useful concepts in the single-output case:
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input requirement set
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isoquant
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,! marginal rate of technical substitution
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elasticity of substitution
Marginal rate of technical substitution
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if f (:) is di¤erentiable, the marginal rate of technical
substitution of input k for input l (MRTSkl ), (holding output
…xed at q = f (z)) is the given by:
MRTSkl =
@f (z)
@zl
@f (z)
@zk
simply obtained by totally di¤erentiating f (:).
The MRTSkl is simply the slope of the isoquant Q(q) and it
is the analogue of the MRTkl (when k and l are inputs).
Single output technology
Useful concepts in the single-output case:
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input requirement set
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isoquant
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marginal rate of technical substitution
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,! elasticity of substitution
Elasticity of substitution
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if the marginal rate of technical substitution gives the
slope of an isoquant, the elasticity of substitution measures
the curvature of an isoquant.
More technically, the elasticity of substitution of input k for
input l (with output …xed at q) is given by :
kl
=
(zl =zk )
(zl =zk )
MRTS kl
MRTS kl
or, for in…nitesimal variations
kl
=
MRTSkl d(zl =zk )
d ln(zl =zk )
=
(zl =zk ) dMRTSkl
d ln jMRTSkl j
Intuitively, the more the factor input ratio changes for a given
change in the slope of the isoquant, the larger the elasticity of
substitution.
Properties of ALL production sets 1
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Y is closed.
If y n ! y and y n 2 Y , then y 2 Y : the production set
contains its own boundary;
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Y is no empty:
At least one productions plan is always possible;
Properties of ALL production sets 2
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Y satis…es no free lunch:
not possible to produce something from nothing (i.e. if y 2 Y
and y 0, then y = 0), Geometrically Y \ RL+ f0g
Properties of ALL production sets 4
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Y satis…es free disposal (monotonicity in technology)
Is possible: 1) to produce the same amount of output with
more inputs, 2) to produce less output with the same inputs
If y 2 Y and y0 y, then y0 2 Y : always possible to throw
away (at no cost) some inputs or outputs
(absorption-disposal-of additional inputs at no cost) ;
Y RL+ Y
(Possible) properties of production sets 1
Not always satis…ed, sometimes mutually exclusive
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Y satis…es possibility of inaction:
0 2 Y : …rm can shut down production.
It holds before any production decision is made. Otherwise,
sunk costs or …xed factors of production may make it invalid.
(Possible) properties of production sets 2
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Y is convex.
If y; y0 2 Y and 2 [0; 1], then y + (1
nonincreasing return to scale.
)y0 2 Y ;
(Possible) properties of production sets 3
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nonincreasing returns to scale:
y 2 Y ) y 2 Y ; 8 2 [0; 1]. Any feasible production plan
can be scaled down.
(Possible) properties of production sets 4
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nondecreasing returns to scale:
y 2 Y ) y 2 Y ; 8 2 [1; 1]. Any feasible production plan
can be scaled up.
(Possible) properties of production sets 5
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constant returns to scale: y 2 Y ) y 2 Y ; 8 2 [0; 1].
Scale in the single-output case
In the single-output case, 8 t > 1, then
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f (tz) < tf (z) ) nonincreasing RS;
f (tz) = tf (z) ) constant RS;
f (tz) > tf (z) ) nondecreasing RS;
Pro…t maximisation 1
Assume now
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L-dimensional vector of prices p = (p1 ; :::; pL ) > 0,
independent from the choices of the …rm:
) …rm is price taker in input and output markets
…rm maximises pro…ts
Y is not empty, closed and satis…es free disposal
The …rm’s problem can be stated as
maxy p y
s.t
y2Y
PMP
or, equivalently
max
p y
s.t
F (y)
y
0
Pro…t maximisation 2
If F (:) is di¤erentiable, necessary condition for pro…t maximisation
are
p=
rF (y )
0
FOC-PMP
) y is chosen so that p and rF (y ) are proportional (
If Y is convex, FOC-PMP is not only necessary but also su¢ cient
for pro…t maximisation (convexity =) nonincreasing return to
scale)
More than simply a technical point!! For instance, if L = 2 (with
good 1 being a net input and 2 a net output), Y shows CRS or
IRS, then y2 = 1 when p1 < p2 , and y2 = 0 otherwise.
Pro…t maximisation 3
FOC-PMP can be rewritten as follows, for any k; l = 1; ::; L and
k 6= l:
pl
=
pk
@F (y )
@yl
@F (y )
@yk
= MRTkl
FOC-PMP2
pro…t-maximization plan (existence)
Single-output technology
maxpf (z)
z 0
FOC for z for all l = 1; :::; L
p
@f (z )
@zl
wz
1
wl ; with equality if zl > 0
or in matrix:
prf (z )
w
and
[prf (z )
w]
z =0
the marginal productivity of every input l must equalize its price in
terms of output ( wpl )
Pro…t function and supply correspondence
Two fundamental functions/correspondences ONLY deriving from
the pro…t maximising behaviour hypothesis are:
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the pro…t function
(p) = p y
which associates to every p the maximum value of p y;
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the supply correspondence
y(p) = fy 2 Y : p y = (p)g
which associates to every p the pro…t maximising production
plan y .
Properties of the pro…t and supply
functions/correspondences
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If Y is convex, y(p) is a convex set for all p. If Y is strictly
convex, y(p) is single-valued.
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If Y is convex, then
Y = fy 2 <L : p y
(p) for all p
0g.
The pro…t function is a complete description of the technology.
Properties of the pro…t and supply
functions/correspondences 2
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(:) is convex in prices.
Let p00 = tp + (1 t)p0 for all 0
(p00 ) t (p) + (1 t) (p0 ).
t
1. Then,
Properties of the pro…t and supply
functions/correspondences 3
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When (:) is di¤erentiable, can obtain the
supply correspondence from the pro…t function, using the
Hotelling’s lemma
r (p) = y(p)
or, equivalently,
@ (p)
= yi (p) for i = 1; ::; L:
@pi
(when i is an input, yi (p) is usually referred to as factor
demand function).
Hotelling’s lemma is simply an application of the envelope theorem
(see previous picture).
Proof
Suppose y is the pro…t-maximization net output vector at p , in
other words this implies (p ) = p y (p ).
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Let’s de…ne another price vector p with p 6= p and
maximizing (p) by inducing y (p).
We then de…ne a new function
g (p) =
(p)
py
(g (p) 6= 0 di¤erent from zero by de…nition of p 6= p ).
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The FOC for the minimization of g (p) (to its minimum
g (p ) = 0) is:
@g (p )
@ (p )
=
@pi
@pi
yi = 0 for all i = 1; :::; L
this is true for all choices of p . Hotelling’s Lemma holds.
Properties of the pro…t and supply
functions/correspondences 4
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Dy(p) is positive semide…nite.
Because of Hotelling’s lemma, Dy(p) = D 2 (p). Since (:) is
convex, its Hessian matrix must be positive semide…nite, so
that also Dy(p) must be positive semide…nite.
Positive semide…niteness of Dy(p) implies...
Properties of the pro…t and supply
functions/correspondences 5
1. Dy(p) is symmetric: cross-substitution e¤ects are symmetric
@ 2 (:)
@ 2 (:)
@y` (:)
@yk (:)
=
:
=
=
@p` @pk
@pk
@p`
@pk @p`
0
1
!
@ 2
@ 2
@
@p 12
@ 2
@p 1 @p 2
@p 1 @p 2
@ 2
@p 22
A=
@y1
@p 1
@y2
@p 1
@y1
@p 2
@y2
@p 2
for `; k = 1::; L
very little intuition...
2. law of supply: own-price e¤ects are nonnegative
@y` (:)
@p`
0 for ` = 1::; L:
,! optimal amount of output increases with its price and
optimal amount of input decreases with its price
3. the principal-minor determinants have non-negative alternate
sign, starting from positive.
Properties of the pro…t and supply
functions/correspondences 6
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(:) is homogenous of degree one
y(p) is homogenous of degree zero
For all t > 0; (tp) = t (p) and y(tp) = y(p).
A proportional change of all prices changes (optimal) pro…ts
by the same proportion but does not change the (optimal)
production plan.
The relationship between these two results follows from
Hotelling’s lemma, being the factor demands the derivative of
the pro…t function.
Cost minimisation
A choice of inputs that minimises the cost of producing a given
output is a necessary (but not su¢ cient) condition for pro…t
maximisation.
Result on costs of interest because
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often more useful than results on technology (ex. in applied
works)
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require only price-taking assumption in input markets (no
assumption about the output-market power)
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better accomodate constant or nondecreasing returns to scale
Focus on single-output technology (restrictive assumption).
Cost minimisation problem
To minimise costs, a …rm solves the problem
min w z
CMP
z
s.t
q
f (z)
Necessary condition for z(q; w) to be the solution to CMP are, for
some
0 and for ` = 1; ::; L 1,
@f (z )
(with = when z` > 0) FOC-CMP
@z`
w`
or, equivalently,
w
rf (z ) and [w
rf (z )] z = 0
If f (:) is concave (Y is convex), these conditions are also su¢ cient
for cost minimization.
Cost minimisation problem 2
In case of interior solutions, FOC-CMP can be re-written as
follows, for inputs w ; ` = 1; ::; L and w 6= l,
w`
=
wk
@f (z` ;zk )
@zk
@f (z` ;zk )
@z`
= MRTS`k
(FOC-CMP2)
which is clearly a special case of the condition FOC-PMP2 for
pro…t maximisation and which has a nice graphical interpretation.
;q)
Note that is the marginal cost of production: = @c (w
@q .
FOC-CMP
From CMP
Two fundamental functions/correspondences deriving from cost
minimisation problem:
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the conditional factor demand correspondence
z(q; w)
associates to every q and w the cost minimising input demand
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the cost function
c(q; w) = w z(q; w)
which associates to every q and w the minimum production
cost
Properties of cost fct and conditional demand factor fct 1
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c(:) is concave in w.
Properties of cost fct and conditional demand factor fct 2
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If the sets fz > 0 : f (z) qg are convex for every q, then
Y = f( z; q) : w z c(w; q) for all w > 0g.
The cost function is a complete description of the technology.
Properties of cost fct and conditional demand factor fct 3
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Shepard’s Lemma: proof
Suppose z is the cost-minimization vector at w , in other words
this implies c (w ; q) = w z (w ; q).
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Let’s de…ne another price vector w with w 6= w that
minimizes c (w;q) by inducing z (w; q).
We then de…ne a new function
g (w) = c (w;q)
(g (w)
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wz
0, by de…nition of w 6= w and c (w ; q) ).
The FOC for the maximum of g (p) (to its maximum
g (p ) = 0 at w = w ) is:
@g (w)
@c (w ; q)
=0)
@wi
@wi
zi = 0 for all i = 1; :::; L
this is true for all choices of w and gives Shepard’s Lemma.
Properties of cost fct and conditional demand factor fct 4
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Dw z(w; q) is symmetric negative semide…nite.
Because of Shepard’s lemma,Dw z(w; q) = D 2 c(w; q).
Since c(:) is concave in w, its Hessiam matrix must be
negative semide…nite, so that also Dw z(w; q) must be
negative semide…nite.
Negative semide…niteness of Dw z(w; q) implies...
Properties of cost fct and conditional demand factor fct 4
1. Dw z(w; q) is symmetric:
@ 2 c(:)
@ 2 c(:)
@z` (:)
@zk (:)
=
:
=
=
@w` @wk
@wk
@w`
@wk @w`
,! very little intuition...
2. the conditional factor demand are (weakly) downward sloping:
@zi (:)
@ 2 c(:)
=
@wi
@wi
0 for i = 1::; L:
,! law of demand for inputs...
3. the principal-minor determinants have alternate sign, starting
from negative.
,! technical requirement for concavity of the cost function
Properties of cost fct and conditional demand factor fct 5
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c(:) is homogeneous of degree one in w:
c(q; w) = c(q; w);
z(q; w) is homogeneous of degree zero in w:
z(q; w) = z(q; w)
An equally proportional change of all input prices causes an
equal change in total cost but not a change in factor demands.
These two results depend on the Shepard’s lemma, being the
conditional factor demands the derivative of the cost function.
Properties of cost fct and conditional demand factor fct 6
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c(:) is nondecreasing in w: if w0 > w, then
c(q; w0 ) > c(q; w).
The total cost of producing q can only increase when at least
one of the input prices increases.
This again depends from the Shepard’s lemma, since
@c (w;q)
= zi (w; q) 0.
@w i
Using the cost function
Using the cost function, we can rewrite the pro…t maximisation
problem as follows
max p q
q 0
c(w; q)
When the technology is single-output, condition necessary for q to
be optimal is
p
@c(w; q )
@q
0 with strict equality if q > 0
Competitive …rms 1
The following …gures describe the optimal behaviour of a
competitive …rm under di¤erent technological conditions.
Let
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1 output
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p > 0 and w
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C (q) = c(q; w);
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AC (q) = C (q)=q;
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C 0 (q) = dC (q)=dq
0
Competitive …rms and strictly decreasing returns to scale
(convex)
Competitive …rms and constant returns to scale (convex)
Competitive …rms and non convex technology
Competitive …rms and strictly convex variable costs with
nonsunk setup costs
C (q) = Cv (q) + K for q > 0
Competitive …rms and constant returns variable costs with
nonsunk setup costs
with Cv (q) linear: C (q) = cq + K
Competitive …rms and strictly convex variable costs with
sunk setup costs
No shut down (pro…t would be negative anyway). Inaction is not
possible (look at the origin of the …gure (b)...)