Producer Theory
Econ 2100
Fall 2015
Lecture 9, September 30
Outline
1
Properties of Supply and Pro…t
2
Hotelling and Shephard Lemmas
3
Cost Minimization
4
Aggregation
From Last Class
De…nition
A production set is a subset Y
De…nition
Given a production set Y
Y = fy 2 Y : F (y )
Rn .
Rn , the transformation function F : Y ! R is
0 and F (y ) = 0 if and only if y is on the boundary of Y g ;
the transformation frontier is fy 2 Rn : F (y ) = 0g.
A …rm’s objective is to choose an output vector in its production set so as to
maximize pro…ts.
De…nition
Given a production set Y
Rn , the supply correspondence y : Rn++ ! Rn is:
y (p) = arg max p y :
y 2Y
De…nition
Given a production set Y
Rn , the pro…t function
(p) = max p y :
y 2Y
: Rn++ ! R is:
Properties of Supply and Pro…t Functions
Proposition
Suppose Y is closed and satis…es free disposal. Then:
( p) =
(p) for all
> 0;
is convex in p;
y ( p) = y (p) for all
> 0;
if Y is convex, then y (p) is convex;
if jy (p)j = 1, then
Lemma).
is di¤erentiable at p and r (p) = y (p) (Hotelling’s
if y (p) is di¤erentiable at p, then Dy (p) = D 2 (p) is symmetric and
positive semide…nite with Dy (p)p = 0.
The Pro…t Function Is Convex
Proof.
Let p,p 0 2 Rn++ and let the corresponding pro…t maximizing solutions be y and y 0 .
) p 0 and let y be the pro…t maximizing
For any 2 (0; 1) let p = p + (1
output vector when prices are p.
By “revealed preferences”
p y
p y
and
p0 y 0
p0 y
why?
multiply these inequalities by
p y
p y
and 1
and
(1
) p0 y 0
) p0 y
(1
summing up
p y + (1
) p0 y 0
[ p + (1
) p0] y
using the de…nition of pro…t function:
(p) + (1
proving convexity of
(p).
) (p 0 )
( p + (1
) p0)
The Supply Correspondence Is Convex
Proof.
Let p 2 Rn++ and let y ; y 0 2 y (p).
We need to show that if Y is convex then
) y 0 2 y (p)
y + (1
for any
2 (0; 1)
By de…nition:
p y
p y
multiplying by
p y
for any y 2 Y
and 1
p y0
and
p y
for any y 2 Y
we get
p y
and
) p y0
(1
(1
Therefore, summing up, we have
p y + (1
) p y0
[ + (1
Rearranging:
p [ y + (1
proving convexity of y (p).
) y 0]
p y
)] p y
)p y
Hotelling’s Lemma
if jy (p)j = 1, then
is di¤erentiable at p and r (p) = y (p)
Proof.
Suppose y (p) is the unique solution to max p y subject to F (y ) = 0.
The Envelope Theorem says
Dq (x (q); q) = Dq (x; q)jx =x
(q);q=q
[
>
(q)] Dq F (x; q)jx =x
(q);q=q
In our setting,
(x ; q) = p y ,
F (x ; q) = F (y ), and
(x (q); q) = (p).
Thus, by the envelope theorem:
r (p) = Dp (p y )jy =y
(p)
[
>
(p)] Dp F (y )jy =y
because p y is linear in p and Dp F (y ) = 0.
Therefore
as desired.
r (p) = y (p)
(p)
= y jy =y
(p)
[
(p)]
>
0
Law of Supply
Remark
If y (p) is di¤erentiable at p, then Dy (p) = D 2 (p) is positive semide…nite.
Write the Lagrangian
L=p y
By the Envelope Theorem:
@ (p)
@L
=
@pi
@pi
Therefore, we have
F (y )
= yi (p)
y =y
@ 2 (p)
@y (p)
= i
@pi
@pi
0
where the inequality follows from convexity of the pro…t function.
This is called the Law of Supply: quantity responds in the same direction as
prices.
Notice that here yi can be either input or output.
What does this mean for outputs?
What does this mean for inputs?
Factor Demand, Supply, and Pro…t Function
The previous concepts can be stated using the production function notation.
De…nition
Given p 2 R++ and w 2 Rn++ and a production function f : Rn+ ! R+ , the …rm’s
factor demand is
x (p; w ) = arg max fpy
x
w x subject to f (x) = y g = arg max pf (x)
x
w x:
De…nition
Given p 2 R++ and w 2 Rn++ and a production function f : Rn+ ! R+ , the …rm’s
supply function y : Rn+ ! R is de…ned by
y (p; w ) = f (x (p; w )) :
De…nition
Given p 2 R++ and w 2 Rn++ and a production function f : Rn+ ! R+ , the …rm’s
pro…t function : R++ Rn++ ! R is de…ned by
(p; w ) = py (p; w )
w x (p; w ) :
Factor Demand Properties
Given these de…nitions, the following results “translate” the results for output
sets to production functions.
Proposition
Given p 2 R++ and w 2 Rn++ and a production function f : Rn+ ! R+ ,
1
2
(p; w ) is convex in (p; w ).
y (p; w ) is non decreasing in p (i.e.
increasing in w (i.e.
@xi (p;w )
@w i
Proof.
Problem 2a,b; Problem Set 5.
@y (p;w )
@p
0) and x (p; w ) is non
0) (Hotelling’s Lemma).
Cost Minimization
Cost Minimizing
Consider the one output case and suppose the …rm wants to deliver a given
output quantity at the lowest possible costs. The …rm solves
min w x
subject to
f (x) = y
This has no simple equivalent in the output vector notation.
De…nition
Given w 2 Rn++ and a production function f : Rn+ ! R+ , the …rm’s conditional
factor demand is
x (w ; y ) = arg min fw x subject to f (x) = y g ;
De…nition
Given w 2 Rn++ and a production function f : Rn+ ! R+ , the …rm’s cost function
C : Rn++ R+ ! R is de…ned by
C (w ; y ) = w x (w ; y ) :
Properties of Cost Functions
Proposition
Given a production function f : Rn+ ! R+ , the corresponding cost function C (w ; y )
is concave in w .
Proof.
Question 2c; Problem Set 5. (Hint: use a ‘revealed preferences’argument)
Shephard’s Lemma
Write the Lagrangian
L=w x
[f (x)
y]
by the Envelope Theorem
@C (w ; y )
@L
=
= xi (w ; y )
@wi
@wi
Conditional factor demands are downward sloping
2
@xi (w ;y )
(w ;y )
Di¤erentiating one more time: @C
=
@w i @w i
@w i
inequality follows by Question 2c in Problem Set 5.
0 where the
Aggregation
Back to our abstract analysis of …rms/producers.
Next, we compare the pro…t maximizing production choices individually made
by …rms with those that would be made by a “central planner” that tries to
maximize the sum of the pro…ts of all …rms simultaneously. This compares
‘centralized’decision making with ‘decentralized’choices.
We can show that the planner cannot do better: individual maximizing behavior
cannot be improved upon.
We can also show that any ‘e¢ cient’aggregate output level can be achieved
when each …rm maximizes its own pro…ts.
Later, we will use these results to prove the two fundamental theorems of
welfare economics.
Roughly, the …rst theorem says that a competitive equilibrium cannot be
improved upon by a central planner.
Decentralized choices cannot be improved upon.
Roughly, the second theorem says that any outcome chosen by a central planner
can be achieved in a competitive equilibrium.
Centralized choices can be achieved in a decentralized manner.
Aggregate Supply and Aggregate Pro…ts
Notation
There are J …rms, each with a non empty closed production set Yj satisfying
free disposal.
Denote …rm j supply correspondence and pro…t function by yj and
respectively.
j
De…nitions
agg
The aggregate supply correspondence y (p) is the sum of all …rms’supply
correspondences:
8
9
J
<X
=
agg
y (p) =
yj : yj 2 yj (p) ;
:
;
j =1
the aggregate pro…t function
agg
(p) is the sum of all …rms’pro…t functions:
agg
(p) =
J
X
j (p):
j =1
agg
y (p) is what results from individual pro…t maximization given p.
Centralized Supply and Centralized Pro…ts
De…nition
The aggregate production set Y is the sum of the …rms’production sets:
8
9
n
<X
=
Y =
yj : yj 2 Yj :
:
;
j =1
No individual …rm has access to points in Y , but if a central planner tells each
…rm what to do, and …rms do as they are told, then anything in Y can be
produced.
De…nitions
cent
The centralized supply correspondence y (p) is de…ned as
cent
y (p) = arg max p y ;
the centralized pro…t function
cent
y 2Y
(p) is de…ned as
cent
(p) = max p y :
y 2Y
cent
y (p) is what results from centralized pro…t maximization given p.
Individual and Joint Pro…t Maximization
Proposition
The following hold:
1
Aggregate supply equals centralized supply:
2
Aggregate pro…t equal centralized pro…t:
agg
cent
y (p) = y (p);
cent
(p) =
agg
(p).
In words: given a price vector:
any production vector that maximizes individual pro…ts also maximizes
centralized pro…ts;
viceversa, any production vector that maximizes centralized pro…ts also
maximizes individual pro…ts.
therefore, aggregate pro…ts are the same as centralized pro…ts.
Pro…t maximizing …rms individually choose the same output that would be
chosen by a central planner that controls their production choices and
maximizes joint pro…ts.
There is no bene…t, in terms of pro…ts, to centralized decision making.
Individual and Joint Supply and Pro…ts
Proof.
agg
cent
First, show that y (p)
y (p) by contradiction.
cent
Suppose y 2 y (p). Since y 2 Y , there exist yj 2 Yj such that y =
Suppose that some particular yk 2
= yk (p).
Then p yk0 > p yk for some yk0 2 Yk .
P
P
cent
Then j 6=k yj + yk0 2 Y and p ( j 6=k yj + yk0 ) > p y =
(p).
P
j
yj .
cent
But this contradicts the fact that y (p) is centralized supply.
agg
So we conclude that each yj 2 yj (p), i.e. y 2 y (p).
agg
Next, show that y (p)
cent
y (p) also by contradiction.
P
cent
Select yj 2 yj (p) for each j, and suppose
yj 2
= y (p).
P
P
Thus, there exist yj0 2 Yj such that p ( j yj0 ) > p ( j yj ).
This implies there exists at least one k such that p yk0 > p yk .
But since yk0 2 Yk , this contradicts the fact that yk (p) is individual supply.
agg
cent
y (p).
Therefore y (p)
agg
cent
Finally, since y (p) = y (p):
Hence,
cent
(p) =
agg
cent
(p) = p
(p) as desired.
cent
y (p)
and
agg
(p) = p
agg
y (p)
Aggregate Choices Are E¢ cient
De…nition
A production vector y 2 Y is e¢ cient if there exists no y 0 2 Y such that y 0
and y 0 6= y .
y
E¢ ciency means there is no waste. This has nothing to do with prices.
Proposition
cent
If y 2 y (p), then y is e¢ cient.
An output vector that maximizes joint pro…ts is e¢ cient.
Proof.
cent
Suppose not: y 2 y (p) but it is not e¢ cient;
Hence 9y 0 2 Y such that y 0 6= y and y 0 y .
Thus, p y 0 > p y and therefore y does not maximize aggregate pro…ts. This is
a contradiction.
Remark
Using the earlier Proposition, we conclude output that maximizes …rm’s
individual pro…ts yields aggregate production that is also e¢ cient.
E¢ cient Production Maximizes Centralized Pro…t
Proposition
Assume Y is convex. If y^ 2 Y is e¢ cient, then there exists p 2 Rn+ n f0n g such
cent
that y^ 2 y (p).
This goes in the opposite direction of the previous result: if we have an
e¢ cient output vector, there are prices that make this vector the solution to
the joint pro…t maximization problem.
Also, since joint pro…t maximization and individual pro…t maximization are the
same, if a production vector is e¢ cient, there are prices that make the …rm
level pieces of this vector the solution to each …rm’s individual pro…t
maximization problem.
Anything e¢ cient can be achieved by decentralized choices of pro…t
maximizing …rms, if one chooses the ‘right’prices.
Separating Hyperplane Theorem
To prove the previous proposition, we need the following result.
Theorem (Separating Hyperplane Theorem)
If A; B Rn are convex and disjoint, then there exist a p 2 Rn n f0n g and a k 2 R
such that
p a k
for all a 2 A
and
p b
k
for all b 2 B
Given two disjoint convex sets, there is an hyperplane that goes between them,
and the two sets lie on opoosite half spaces.
E¢ cient Production Maximizes Centralized
Pro…t
cent
If y^ 2 Y (convex) is e¢ cient, 9p 2 R+n n f0n g s.t. y^ 2 y (p) = arg maxy 2Y p y .
Proof.
Suppose y^ 2 Y is e¢ cient. Let Py^ = fy 2 Rn : yi > y^i for all i = 1; :::; ng.
Since y^ is e¢ cient, Py^ \ Y = ;.
Verify that Py^ is convex (Question 6, Problem Set 5).
By the Separating Hyperplane Theorem, 9p 2 Rn n f0n g and 9k 2 R such that
p y
Clearly, pi
k for all y 2 Y
and
p b
0 for all i = 1; :::; n.
k for all b 2 Py^
if pi < 0, then select x 2 Py^ with xi su¢ ciently large such that p x < k.
Choose a sequence b1 ; b2 ; : : : 2 Py^ such that bn ! y^ .
The set fz 2 Rn : p z
kg is closed and
But for all y 2 Y , p y
k by separation, hence k
bn 2 Py^
fz 2 Rn : p z
cent
Therefore p y^ = k, and y^ 2 y (p).
kg;
therefore
p y^
p y^
k
k:
E¢ cient Production Is Pro…t Maximizing
The last result is one building block of the Second Fundamental Theorem of
Welfare Economics.
Firms in the Second Fundamental Theorem of Welfare Economics
For every e¢ cient production vector, there exists some price vector that makes
that production vector a pro…t-maximizing choice for the planner and a pro…t
maximizing choice for the individual …rms.
Only one assumption is needed: convexity of production sets.
The complete version includes consumers, and a notion of e¢ ciency that
includes their preferences.
Econ 2100
Fall 2015
Problem Set 5
Due 5 October, Monday, at the beginning of class
1. Show that GARP is equivalent to the following: If xj %I xk then not xk
I
xj .
2. Suppose f(xj ; pj ; wj )gN
k=1 is a …nite set of demand data. Prove that if there exists a locally nonsatiated
utility function which rationalizes the data, then the data satisfy the Generalized Axiom of Revealed
Preference.
3. Consider the following data set of four demand observations for two commodities.
1
2
3
4
Find %R ,
R
, %I , and
I
x
(3; 9)
(12; 1)
(4; 2)
(1; 1)
p
(3; 3)
(1; 8)
(2; 3)
(4; 4)
w
36
20
14
8
for these observations. Check that the data satisfy GARP.
4. Prove that if Y satis…es non decreasing returns to scale either (p)
0 or (p) = +1.
5. Given p 2 R++ , w 2 Rn++ , and a production function f : Rn+ ! R+ , prove the following:
(a) the …rm’s pro…t function (p; w) is convex in (p; w);
(b) the …rm’s supply function y (p; w) is non decreasing in p (i.e.
increasing in w (i.e.
@xi (p;w)
@wi
@y (p;w)
@p
0) and x (p; w) is non
0) (Hotelling’s Lemma).
(c) The …rm’s cost function C(w; y) is concave in w (Hint: use a ‘revealed preferences’argument).
6. Derive cost function C (w; y) and conditional factor demand x (w; y), and then use them to determine
the pro…t function (w) and the supply function/correspondence y (w) for each of the following single
output production functions (remember to draw pictures).
(a) f (x1 ; x2 ) = x1 x2 .
(b) f (x1 ; x2 ) = x1 + x2 .
(c) f (x1 ; x2 ) = min fx1 ; x2 g.
1
(d) f (x1 ; x2 ) = (x1 + x2 ) with
1.
7. Suppose the aggregate production set Y is convex and y 2 Y is e¢ cient. Show that the set
Py = fx 2 Rn : xi > yi for all ig
is convex.
1