Handout Lecture 1, ECON 4230/35
Kjell Arne Brekke
August 19, 2010
1. Di¤erent ways to express a Technology
A production plan is a vector y = (y1 ; :::; yn ) where yi > 0 means net output while
yi < 0 means net output.
We often chose the indexes such that output are listed …rst while inputs last,
thus
y = (y0 ; x) with y 2 Rm ; y0 2 Rk ; x 2 Rn with m = n + k
The most general way to specify a Production possibility set: The set Y of all
feasible production plans. Thus if y 2 Y then the production plan y is feasible.
Problem 1. A …rm is producing pommes frites from potatos. (We assume that
they need no other inputs!!) From 10 kg potatoes they produce 5 kg pommes
frites. Draw the …rms production possibility set.
The technology exhibit free disposal if the …rm can dispose of unused inputs
for free. Formally:
y = (y0 ; x) 2 Y then y
~ = (y0 ; x
~) 2Y for all x
~
(If you are unsure about what x~
textbook - or come to class.)
x
x means when these are vectors, check the
1
An alternative way to specify technology is to specify the input requirement
set V (y) denote what is required to produce y, and is de…ned as
V (y) = fx 2 Rn : (y; x) 2 Y )
Finally, when the output is one-dimensional technologies can be given by a
production function
y = f (x)
Assume free disposal (see textbook), then we can derive Y and V (y) :
Y
= f(y; x) 2 Rn+1 : y
V (y) = fx 2 Rn : y
f (x)g
f (x)g
We can also de…ne the isoquant
Q(y) = fx 2 Rn : f (x) = yg
Problem 2. Consider a …rm with technology (type Cobb-Douglas)
y=
p
x1 x2
What is Y for this …rm, with and without free disposal?
What is V (1), with and without free disposal?
Derive and expression for an isoquant.
1.1. Convex technologies
A set A is convex if for any x 2 A and x0 2 A then tx + (1
0 t 1
An assumption that will be important in the following is:
Axiom 3. The input requirement set is convex.
t)x0 2 A for all
Problem 4. With the technology
y=
p
x1 x2
Is V (1) convex, with and without free disposal?
Two counter examples:
A car-dealer sells whole cars. Half a car have no value. If inputs are blue cars
and red cars, and output is measured in number of sold cars. He sell all whole
cars he gets in. Is the technology convex?
A …rm with given production need 5 PC’s or 5 Mac; but 2 PC and 3 Mac will
not do the same job (e.g. extra software cost). Is the technology convex?
Still, convexity is often a reasonable assumption:
Suppose A and B produce the same (NB!!)
A: 5 persons & 1 PC
B: 1 person & 5 PC
C: 12 A+ 12 B = 3 Persons and 3 PC, will usually be better.
1.2. Substitution
Recall from earlier courses (or listen to me in class) that the technical rate of
substitution (slope of the isoquant) is
y = f (x1 ; x2 )
dx2
T RS =
jalong isoquant =
dx1
@f @f
=
@x1 @x2
and express how much more of input to do we need to produce the same, if we
have one unit more of input 1. (Note, it is negative). Recall also that in an
equilibrium
@f @f
w1
=
=
@x1 @x2
w2
but we do not have prices yet, as we only discuss technology, so we study the
e¤ect of changing TRS instead, but we can think of it as changing relative prices.
1.3. Elasticity of substitution
Consider production with capital and labour as input. What happend to capital intensity nk if capital becomes relatively cheaper ( wq falls, q cost of capital,
w wage)? Put this in terms of an elasticity and use TRS rather than relative
prices, and we get the elasticity of substitution.
If TSR shifts 1%, how many % will the relative input use shift?
Formally
ij
Where
(fj =fi ):
d(xi =xj )
(xi =xj )
d(xi =xj )
(xi =xj )
d(fj =fi )
(fj =fi )
=
d(xi =xj ) (fj =fi )
d ln(xi =xj )
=
(xi =xj ) d(fj =fi )
d ln(fj =fi )
is percentage change in (xi =xj ) and
d(fj =fi )
percentage
(fj =fi )
change in
1.4. Examples
Consider …rst Cobb-Douglas
y = f (x1 ; x2 ) = xa1 xb2
the textbook does some arithmetic …nd (as we will in class) that
x2
b
= T RS
x1
a
And we can compute the elasticity = 1
The textbook uses logarithms, I’ll discuss that in class.
Problem 5. What is the Elasticity of substitution for the ’Leontief’production
function
y = min(x1 ; x2 )
Other examples in the textbook and in class.
1.5. Constant return to scale
A technology given by a production function exhibit constant return to scale if
for all t > 0 :
f (tx) = tf (x)
and increasing returns to scale if
f (tx) > tf (x) for t > 1
but decreasing returns to scale if
f (tx) < tf (x) for t > 1
How do we write constant returns to scale with production possibility sets and
input requirement sets? Like this:
x 2 V (y) ) tx 2 V (ty)
y 2 Y ) ty 2 Y
Problem 6. A technology has the property
y 2 Y ) ty 2 Y for all t
1
is this increasing or decreasing returns to scale?
1.6. Convexity
Some useful results about convexity:
Theorem 7. Convex Y implies convex V (y)
f (x) is quasiconcave if for any y the set fx : f (x) yg is convex
"V (y) is convex for all y" equivalent to "f (x) is quasiconcave"
Proposition 8. If g is increasing and f is concave then
h(x) = g(f (x))
is quasiconcave