Handout 4th Lecture, ECON 5200, Production
Kjell Arne Brekke
September 14, 2010
1
Production set
Describes a production plan
y 2 RL
where yl < 0 means input and yk > 0 means output.
The production set y 2 Y may also be described by a transformation
function
Y = fy 2 RL : F (y)
0g
with
F (y) describing the trasformation frontier
The
M RT =
1.1
@F (y)=@yl
@F (y)=@yk
Distinct input and output
y = ( z; q) with q 2 RM and z 2 RL
M
with a single output we usually use a production function
Y = f( z; q) : q
f (z)
1
0 and z
0g
Properties of Y
1.2
1. Y is non-empty
2. Y
is closed. (mostly technical)
3. No free luch: y
0 =) y = 0
4. Possibility of inaction: 0 2 Y . (Depends on when we consider the
production desision.)
5. Free disposal: y 2 Y and y 0
y =) y 0 2 Y
6. Irreversibility: If y 2 Y , then
y2
= Y.
7. Returns to scale
Non-increasing, y 2 Y; 2 (0; 1) =) y 2 Y
Non-decreasing: y 2 Y; > 1 =) y 2 Y
Constant: y 2 Y; > 0 =) y 2 Y
8. Additivity (or free entry) y; y 0 2 Y then y + y 0 2 Y . (What does this
imply about returns to scale?)
9. Convexity y; y 0 2 Y then
)y 0 2 Y for any
y + (1
2 (0; 1):(Together
with 4, what does this imply about returns to scale?)
10. Y is a convex cone. Formally y; y 0 2 Y then
y + y 0 2 Y for any
;
>
0.(Which propertise above are implied by this?)
Proposition 1 Y is additive and satisifes non-increasing returns if and only
if it is a convex cone.
Decreasing return can be seen as a scarcity of some underlying resource:
Proposition 2 For any convex Y
technology Y 0
RL there is a constant return convex
RL+1 such that
Y = fy : (y; 1) 2 Y 0 g
2
2
Pro…t Maximization and Cost Minimization
The pro…t function
(p) = max py = max py
y2Y
F (y) 0
If F is di¤erentiable at y , then
max py
F (y)
yields …rst order condition
pl = Fl0 (y )
[F (y) may be non-di¤erentiable when yl = 0 for some l]
The case with a production function yields familiar …rst order conditions.
Proposition 3 The pro…t function
and the associated sypply correspon-
dence, has the following properties (assuming nonempty, closed Y with free
disposal)
1. Homogenous of degree 1:
2.
( p) =
(p)
is convex
3. If Y is convex then Y = fy 2 RL : py
(p) for all p
0g
4. y(p) is homogenous of degree zero
5. If Y is convex then y(p) is convex for all p and if Y is strictly convex
then y(p) is a singleton.
6. Hotelling: If y(p) consist of a single point then
is di¤erentiable at p
and
y(p) = r (p)
7. If y( ) is a function that is di¤erentiable at p, then
Dy(p) = D2 (p)
3
is positive semide…nite with
Dy(p)p = 0 (the zero vector)
2.0.1
Cost minimization
We similarly de…ne a cost minimization problem. This is the minimum
cost to produce an output, so we need a speci…cation that distinguishes
input and output. Usually this is with a production function.
c(w; q) = min wz
f (z) q
Proposition 4 Let c(w; q) be a cost function and z(w; q) the corresonding fac-
tor demand correspondence. Then we have the following properties
1. c( ) is homogenous of degree 1 in w and non-decreasing in q
2. c( ) is a concave function of w
3. If the input requirement sets fz
0 : f (z)
qg are convex for every q
then
Y = f( z; q) : wz
c(w; q) for all w
0g
4. z( ) is homogenous of degree zero in w
5. If the input requirement set fz
0 : f (z)
qg is convex, then z(w; q)
is convex, moreover, if the input requirement set fz
0 : f (z)
qg is
strictly convex, then z(w; q) is a singleton.
6. If z(w; q) consists of single points, then c( ) is di¤erentiable with respect
to w at w and
rw c(w; q) = z(w; q):
7. If z( ) is di¤erentiable at w, then
Dw z(w; q) = Dw2 c(w; q)
4
is a symmetric and negative semide…nite matrix, with
Dw z(w; q)w = 0
8. If f ( ) is homogenous of degree one (constant return) then c and z are
homogenous of degree one in q .
9. If f ( ) is concave then c is convex.
3
The geomety of cost and Supply
Assumed to be known.
4
Aggregation
Aggregation is much simpler in production theory.
For all p
1.
(p) =
2. y (p) =
5
0
P
j
P
j (p)
yj (p) (Adding correspondences:
P
P
yj (p) = f yj : yj 2 yj (p); 8jg)
E¢ cient production
De…nition 5 A production plan y 2 Y is e¢ cient if there is no y 0 2 Y such
that y 0
y and y 0 6= y
Proposition 6 If p
0 any pro…t maximizing plan y 2 Y is e¢ cient.
Proposition 7 If Y is convex, then every e¢ cient production plan y 2 Y is
pro…t maximizing production for some nonzero price vector p
0.
Problem 8 Production in j pro…t maximizing …rms is just one feasible way of organizing
production. Can the society do better with alternative organization?
5
Problem 9 It has been argued that the reason why there is no vaccine or other e¢ cient
medicine agains malaria is that the victims of malaria are mostly poor with low willingness
to pay. Does this indicate that medical production violates the assumptions above?
Problem 10 Oscar Lange argued that state owned production could do at least as good
P
as the market. One possible production plan is the market outcome y (p) = yj (p), and
the state can choose this one, unless a feasible alternative is preferred. What is implicitly
assumed in this argument?
6
Objectives of the …rm
We can introduce ownership of …rms into the consumption theory. If individual i owns a share
ij
of …rm j , then the budget constraint would
be
pxi
j
X
wi +
ij pyj
j=1
and clearly the owners want to maximize pyj for all …rms.
7
General equilibrium and Pareto Optimality (Chapter 10)
An economic allocation (x1 ; ::; xI ; y1 :::; yJ ) is feasible if
X
xli
!l +
I
X
ylj
J
An allocation (x; y) is Pareto e¢ cient if there is no alternative allocation (x0 ; y 0 )
such that
ui (x0i )
ui (xi ) for all i with ui (x0i ) > ui (xi ) for some i
A competitive equilibrium is an allocation (x ; y ) and a vector of prices p such
that
(i) For all …rms j
yj solves max p yj
yj 2Yj
6
(ii) For all consumers i
xi solves max ui (xi )
X
p !i +
ij (p yj )
s.t.p xi
J
(iii) Market clearing: For each good l we have
X
xli = ! l +
i
Note here that
ij
X
ylj
J
is consumer i0 s ownership of …rm j. Clearly, any …rm j is totally
owned by consumers
X
ij
=1
i
To solve this model for all commodities is left for a later chapter. Now we consider
the case where one good is small in the economy, so we can ignore income e¤cts.
7.1
Partial equilibrium with quasi linear preferences
Consumers preferences are given by the utility functions (m the numeraire)
ui (mi ; xi ) = mi +
i (xi )
Where we normalize the price of commodity 1 to 1 and set the price of x as p:
Firms produce qj using the numerair as input, and the amount used is c(qj ) they thus
maximize
Note that the budget condition for the consumer is
mi + p xi
! mi +
X
ij (p
qj
cj (qj ))
j
Where it is assumed that consumer only have endowment in the m commodity.
Equilibrium conditions
p
p
X
I
xi
c0 (qj ) equality if qj > 0
0
(xi ) with = if xi > 0
X
=
qj
J
7
Problem 11 What about market clearing in the m commodity?
Some important things to note:
The equilibrium allocation and price is independent of the inital allocation of endowment and ownership!
Theorem 12 (The First Fundamental Theorem of Welfare Economics) If the price
p and the allocation (x ; y ) constitutes a competitive equilibrium, then this allocation is
Pareto optimal.
Theorem 13 For any pareto e¢ cient levels of utility u = (u1 ; :::; uI ), there are transfers
P
Ti with
Ti = 0, such that the competitive equilibrium with endowments ! i + Ti yield
utility u .
7.2
Marshallian surplus
Can show:
X
ui
!+
De…ne the Marshallian surpus
X
i (xi )
S(x1 ; :::xI ; q1 ; :::qJ ) =
Follows that
S=
Z
X
X
cj (qj )
i (xi )
X
cj (qj )
x
(P (s)
C 0 (s))ds
0
which yields the standard measurement of consumer and producer surplus. Note that as
there is no income e¤ect, the walrasian demand yield the correct consumer surplus.
7.3
Free entry and exit
We focus on the supply side, and take aggregate demand as given, x(p), but this can be
as above. There is a potentially in…nite number of …rms that can ente, all with the same
technology and hence same cost function. The free entry equilibrium is then given as
8
(i) Pro…t maximization
q solves max p q
c(q)
(ii) Market clearing
x(p ) = J q
(iii) Free entry condition (no incentives to enter/exit)
pq
c(q ) = 0
With constant returns to scale, q is indeterminate, and hence the equilibrium can
arise for any number J of …rms.
Proposition 14 If c0 (q) > 0 for all q
0 and c is strictly convex, then there is no free
entry equilibrium provided x(c0 (0)) > 0.
Proof: If p > c0 (0) pro…t is positive and supply in…nite.
(p) =
Z
q
(p
0
c (s))ds >
0
If p
Z
0
c0 (0); then q = 0 while x(p) > 0:
9
q
(p
c0 (q ))ds = 0