Review Class Seven
Producer theory
 Key sentence: A representative , or say,
typical firm will maximize his profit under
the restriction of technology and market
structure.
 Hey, guys! What we are going to do is
only to translate this key sentence.
 Translation is an easy job for you, Ok?
Market structures




Perfect competition
Monopoly
Oligopoly (Duopoly)
Monopolistic competition
 This term we will introduce above market
structures, and now we focus on the
simplest case of Perfect Competition.
Ch 18:Technological constraints
 We can consider the firm as a BLACK
BOX, which means we do not think about
the Production Process ,(That is the
matter of the operating managers and
workers.) but we care the capability that
transforms the inputs to outputs facing the
typical firm――Technological constraint.
Technological constraints
 Nature imposes the constraint that there are
only certain feasible ways to produce outputs
from inputs: There are only certain kinds of
technological choices that are possible.
 Precisely speaking, only certain combinations of
inputs are feasible ways to produce a given
amount of output, and the firm must limit itself to
technologically feasible production plans.
 So we can describe the technological
constraints by listing the feasible bundle of
inputs and outputs.
Technological constraints
 Def.1 of the production set: The set of all
combinations of inputs and outputs that
comprise a technologically feasible way to
produce is called a production set.
 Def.2 of the production function: The function
describing the boundary of this set is known as
the production function. It measures the maximum
possible output that you can get from the given
amount of input.
 Warning: production function is the cardinal
function.
One-input to one-output case
Technological constraints
 We often consider the two-input and one-output
case. (Two inputs are often enough.)
 Arithmetically, Y  f ( x1 , x2 )
 Geometrically, we can use the definition of
isoquants to describe the technology. An isoquant
is the set of all possible combinations of inputs 1
and 2 that are just sufficient to produce a given
amount of output.
Technological constraints
Q  f ( X ,Y )
等高线
投影
Y
X
Q
O
Examples of technology
(isoquants analysis):
 Fixed proportions,
 Perfect substitutes,
 Cobb-Douglas.
Fixed proportion
x2
Isoquants
x1
Perfect subsitutes
x2
Isoquants
x1
Cobb Douglas
Y
等产量曲线
O
X
Well-behaved isoquants
 Monotonic (free disposal)
To avoid the positive slope of any Isoquant
 Convex: Given any two possible input
bundles to produce the certain level of
production, the weighted average of these
two bundles can produce a higher level of
output.
To avoid the concavity case
Three Indicators of
Technology
 Marginal Product (MP)
 Technical Rate of Substitution (TRS)
 Returns to Scale (RS)
 Relationship?
Marginal product:
f xi  x, x j   f xi , x j 
y

xi
x
Warning: To keep other factors
constant
MP
The Law of Diminishing MP
The Law of Diminishing MP：
It’s the assumption we
usually apply.
It’s a short-run concept.
Technical Rate of Substitution
 Def 3: It measures the rate at which the firm will
have to substitute one input for another in order
to keep output constant.
x2
MP1 ( x1 , x2 )
TRS ( x1 , x2 )  

x1
MP2 ( x1 , x2 )
 Geometrically, TRS is just the slope of the given
isoquant.
Technical Rate of Substitution
 Roughly speaking, the assumption of
diminishing TRS means that the slope of an
isoquant must decrease in absolute value as we
move along the isoquant in the direction of
increasing x1, and it must increase as we move
in the direction of increasing x2. Do you know
how to prove it?
 It is based on the assumption of well-behaved
isoquant.
Returns to Scale (RS)
tf ( x1 , x2 )  f (tx1 , tx2 ), t  1
tf ( x1 , x2 )  f (tx1 , tx2 ), t  1
tf ( x1 , x2 )  f (tx1 , tx2 ), t  1
Ch 19
Short-run and Long-run
 In Microeconomics, Short-run and Longrun are based on the capability of
adjustment of factors of production.
 So when we analyze the problem of firms,
we should think about both short-run and
long-run.
 In Macroeconomics, Short-run and Longrun are based on the capability of
adjustment of price level.
Profit-maximization
(short-run)
 A representative , or say, typical firm will
maximize his profit under the restriction of
technology and market structure.

max pf ( x1 , x2 )  1 x1  2 x2

 pMP1 ( x , x2 )  1
*
1
 The value of the marginal product of a factor should equal
its price.
 Geometrically, find highest isoprofit line in the
production set!
Profit-maximization
(short-run)
Profit-maximization
(short-run)
 Comparative statics: change w and p and see how
x1 , y and  respond?
Comparative statics:
 Increasing p increases x1 and then y.
产品价格
f(x1)
Low w1
Low p
High p
x1
Profit-maximization
(long-run)
max pf ( x1 , x2 )  1 x1  2 x2 
x1 , x2
 p1MP1 ( x , x )  1 , p2 MP2 ( x , x )  2
*
1
*
2
*
1
*
2
Profit-maximization
(long-run)
 Profit-maximization, together with perfect
competition and constant returns to scale
implies 0 economic profit in the long run!
 How to prove it ?
Deeper sight
 Profit maximization problem facing a typical firm
can be divided into two parts:
 Part 1: Given any production level of y ,how
to choose the optimal inputs bundle to minimize
cost? (Ch 20 and Ch 21)
 Part 2: How to choose the proper output level of
y, such that the profit can be maximized? (Ch
22 and Ch 23 are referred to perfect
competition, while Ch 24 and Ch 25 are
referred to Monopoly.)
Deeper sight
 For part 1, you should remember that cost minimization
problem is not the objective or the end, the purpose for
us is to obtain the cost function, which measures the
minimum cost of producing y units of output when
factor prices are (w1,w2).
 For part 2, having got the cost function, we will return to
the profit maximization problem, and analyze the
behavior of firm supply. Especially in the perfectly
competitive market, we will see how the supply curve is
proved.
 In chapter 20 and 21, we will learn the cost analysis.
Cost minimization
(short-run)
min  x
1 1
x1


  2 x2 ,
s.t. y  f x1 , x2

 x1  x (1 , 2 , x2 , y ), x2  x2
s
1
 cs ( y, x2 )   x (1 , 2 , x2 , y )  2 x2
s
1 1
Cost minimization
(long-run)
min w x
1 1
 w2 x2 
x1 , x2
s.t. y  f x1 , x2 
*
1
*
2
*
1
*
2
MP1 ( x , x ) MP2 ( x , x )



w1
w2
 x  x ( w1 , w2 , y ), x  x ( w1 , w2 , y )
*
1
*
1
*
2
*
2
 c( y )  w x ( w1 , w2 , y )  w x ( w1 , w2 , y )
*
1 1
*
2 2
Cost minimization
(long-run)
Cost minimization
(long-run)
 Geometric solution: Find the lowest
isocost line along the given isoquant!
 Some examples:
 y = ax1 + bx2; Perfect substitutes
 y = min{ax1 , bx2};Fix proportion
 y = x1a x2b. Cobb-Douglas
The relationship between the
objective function and the cost
function (Take long-run for example)
*
y

y
 When
, the minimum of the objective
function f ( x1 , x2 )  w1 x1  w2 x2 at point x1* , x2* 
is c( y * ) , where x1*  x1* ( y* ) ,
x2*  x2* ( y * )
.
 Please distinguish the objective function and the
cost function seriously.
Three definitions of fixed costs
 (Common) Fixed costs are the costs associated with the
fixed factor: they are independent of the level of output,
and in particular, they must be paid whether or not the
firm produces output.
 Quiz: Are there any fix costs in the long run?
 Quasi-fixed costs are costs that are also independent of
the level of output, but only need to be paid if the firm
produces a positive amount of output.
 Quiz: Can there be any quasi-fixed cost in the long run?
 Sunk costs are the costs that can not be recovered.
 Of cause, generally we only consider the variable cost and the
common fixed cost.
Cost curves
c( y )  cv ( y )  F
c( y ) cv ( y ) F
AC ( y ) 

  AVC ( y )  AFC ( y )
y
y
y
dc( y )
MC 
dy
TC，VC and FC
C
C (Q)
VC (Q )  C (Q )  C (0)
TC（0）
C（0）
O
FC (Q )  C (0)
Q
MC
AC
AVC
MC
.
.
AC
AVC
y
Two examples
 Specific cost function:
c( y )  y  1
2
 How to obtain the AC, MC and AVC?
Two examples
 Marginal cost curves for two plants:
 Suppose that you have two plants that have
two different cost functions, c1 ( y1 ) and c2 ( y2 ).
 You want to produce y units of output in the
cheapest way.
 How much should you produce in each plant?
Two examples
 Arithmetically,
min c ( y )  c ( y )
1
1
2
2
y1 , y 2
s.t. y1  y2  y
 mc( y )  mc( y ), y  y  y
*
1
*
2
*
1
*
2
 Geometrically, seek for the horizontal sum of
MC curves, and solve for the value of MC at
the given output level of y. Then setting each
MC equal to that value yields to the desired
output of each plant.
Two examples
 The above arithmetic and geometric
solution methods are the common methods
for the case of any plants.
 In the case of two-plant case, one shortcut
can be used.
Two examples
Approach 1:Long-run and short-run
(average) cost curves (Please refer
to the Dissertation.)
 The long-run cost curve is the lower
envelope of the short-run cost curves.
 The long-run average cost curve is the
lower envelope of the short-run average
cost curves.
 Here I will prove the first conclusion, and
you should prove the second according to
my method.
Approach 1:Long-run and short-run
(average) cost curves
 1)Any one short-run cost curve is tangent
to the long-run cost curve at some y.
 2)short-run cost curves are above the
long-run cost curve.
Approach 2:Relationship between
returns to scale and LAC (Please
refer to the Dissertation.)
 Constant R.T.S ______constant LAC
 Increasing R.T.S ______downward –
sloping LAC
 Decreasing R.T.S _____upward-sloping
LAC
 How to prove?