Transcript Chapter Nineteen - University of Minnesota

The Firm’s Decision in Space
Production theory
• A firm is characterized by it’s technology
represented by the production function
– Y=f(x1, x2)
• It is a price taker in the products and inputs
markets and it faces prices py w1 and w2.
• It chooses how much to produce given these
prices in order to maximize profits
• Two step process
– minimize the cost of producing any given output
– Choses output optimally given prices and it’s marginal
cost function
Cost Minimization
• A firm first computes how to produce any
given output level y  0 at smallest possible
total cost.
• c(y) denotes the firm’s smallest possible total
cost for producing y units of output.
• c(y) is the firm’s total cost function.
• When the firm faces given input prices w =
(w1,w2,…,wn) the total cost function will be
written as c(w1,…,wn,y).
The Cost-Minimization Problem
• Consider a firm using two inputs to make one
output.
• The production function is
y = f(x1,x2).
• Take the output level y  0 as given.
• Given the input prices w1 and w2, the cost of
an input bundle (x1,x2) is
w1x1 + w2x2.
The Cost-Minimization Problem
• For given w1, w2 and y, the firm’s costminimization problem is to solve
subject to
min w1 x1  w2 x2
x1 , x2  0
f ( x1 , x2 )  y.
The Cost-Minimization Problem
• The levels x1*(w1,w2,y) and x2*(w1,w2,y) in
the least-costly input bundle are the firm’s
conditional demands for inputs 1 and 2.
• The (smallest possible) total cost for
producing y output units is therefore
c( w1 , w2 , y )  w x ( w1 , w2 , y )
*
1 1
 w2 x ( w1 , w2 , y ).
*
2
Iso-cost Lines
x2
c”  w1x1+w2x2
c’  w1x1+w2x2
c’ < c”
x1
Iso-cost Lines
x2
Slopes = -w1/w2.
c”  w1x1+w2x2
c’  w1x1+w2x2
c’ < c”
x1
The y’-Output Unit Isoquant
x2
All input bundles yielding y’ units
of output. Which is the cheapest?
f(x1,x2)  y’
x1
The Cost-Minimization Problem
x2
All input bundles yielding y’ units
of output. Which is the cheapest?
f(x1,x2)  y’
x1
The Cost-Minimization Problem
x2
All input bundles yielding y’ units
of output. Which is the cheapest?
f(x1,x2)  y’
x1
The Cost-Minimization Problem
x2
All input bundles yielding y’ units
of output. Which is the cheapest?
f(x1,x2)  y’
x1
The Cost-Minimization Problem
x2
All input bundles yielding y’ units
of output. Which is the cheapest?
x 2*
f(x1,x2)  y’
x 1*
x1
The Cost-Minimization Problem
x2
At an interior cost-min input bundle:
* *
(a) f ( x1 , x 2 )  y 
x 2*
f(x1,x2)  y’
x 1*
x1
The Cost-Minimization Problem
x2
At an interior cost-min input bundle:
* *
(a) f ( x1 , x 2 )  y  and
(b) slope of isocost = slope of
isoquant
x 2*
f(x1,x2)  y’
x 1*
x1
The Cost-Minimization Problem
x2
At an interior cost-min input bundle:
(a) f ( x , x )  y and
(b) slope of isocost = slope of
w
MP

 TRS  
at ( x , x ). isoquant; i.e.
w
MP
*
1
*
2
1
1
2
2
*
1
*
2
x 2*
f(x1,x2)  y’
x 1*
x1
A Cobb-Douglas Example of Cost
Minimization
• A firm’s Cobb-Douglas production function is
y  f ( x1 , x2 )  x11/ 3 x22 / 3.
• Input prices are w1 and w2.
• What are the firm’s conditional input demand
functions?
A Cobb-Douglas Example of Cost
Minimization
At the input bundle (x1*,x2*) which minimizes
the cost of producing y output units:
(a) y  ( x* )1/ 3 ( x* ) 2 / 3
and
1
(b)
2
w1
 y /  x1
(1 / 3)( x ) ( x )



w2
 y /  x2
(2 / 3)( x ) ( x )
* 2 / 3
1
* 1/ 3
1
*
2
*
1
x

.
2x
* 2/3
2
* 1 / 3
2
A Cobb-Douglas Example of Cost
Minimization
(a)
y  (x ) (x )
* 1/ 3
1
* 2/3
2
(b)
w1
x2*

.
*
w2
2 x1
A Cobb-Douglas Example of Cost
Minimization
*
* 1/ 3 * 2/ 3
(a) y  ( x1 ) ( x 2 )
2w 1 *
*
x1 .
From (b), x 2 
w2
w1 x 2

.
(b)
w 2 2x*1
A Cobb-Douglas Example of Cost
Minimization
*
* 1/ 3 * 2/ 3
(a) y  ( x1 ) ( x 2 )
2w 1 *
*
x1 .
From (b), x 2 
w2
w1 x 2

.
(b)
w 2 2x*1
Now substitute into (a) to get
2/ 3
* 1/ 3  2w 1 * 
y  ( x1 ) 
x1 
 w2 
A Cobb-Douglas Example of Cost
Minimization
*
* 1/ 3 * 2/ 3
(a) y  ( x1 ) ( x 2 )
2w 1 *
*
x1 .
From (b), x 2 
w2
w1 x 2

.
(b)
w 2 2x*1
Now substitute into (a) to get
2/ 3
2/ 3
 2w 1 
* 1/ 3  2w 1 * 
*
y  ( x1 ) 
x1 

x1 .

 w2 
 w2 
A Cobb-Douglas Example of Cost
Minimization
(a) y  ( x )
* 1/ 3
1
*
2
(x )
2/3
(b)
2 w1 *
From (b), x  w x1 .
2
w1
x2*

.
*
w2
2 x1
*
2
Now substitute into (a) to get
2/3
2/3
 2w1 *   2w1  *
y  ( x )  x1     x1 .
 w2   w2 
* 1/3
1
So
 w2
x 
 2w
1

*
1




2/3
y
is the firm’s conditional demand for input 1.
 2 w1
x2*  
 w
 2
1/ 3




y
Is the conditional demand for
input 2
A Cobb-Douglas Example of Cost
Minimization
So the cheapest input bundle yielding y
output units is
x (w , w , y), x (w , w , y)
*
1
1
*
2
2
 w 
2



 
  2 w1 

2/3
1
2
1/ 3
 2 w1 

y, 
 w2 


y .


A Cobb-Douglas Example of Cost
Minimization
So the firm’s total cost function is
c( w1 , w2 , y )  w1 x1* ( w1 , w2 , y )  w2 x2* ( w1 , w2 , y )
2/3
 w2 

 w1 
 2 w1 
1
 
2
1/ 3
 2 w1 

y  w2 
 w2 
y
2/3
w11/ 3 w22 / 3 y  21/ 3 w11/ 3 w22 / 3 y
 w1w
 3
 4
2
2
1/ 3



y.
Output Decision
• Marginal cost equals marginal revenue
• In the case of price takers the marginal
revenue is the price of output
• Marginal cost is the derivative of C(y) with
respect to y
• Marginal cost curve is also known as the
supply curve
Taking Space into Account
• The firm is now characterized by the
technology and it is still a price taker in all
markets
• It can buy inputs at constant location-specific
prices.
• It sells at a fixed output price
• Locations are spatially separated and the firm
incurs linear transportation costs
Two Inputs and One Market
• consider the decision of a locational unit with
two transferable inputs (x1 located at S1 and x2
located at S2) and one transferable output
with a market located at M.
• Limit consideration to locations I and J, which
are equidistant from the market
• The arc IJ includes additional locations at that
same distance from the market
Visually:
Incorporating Distance Into Prices
• Their delivered prices are respectively
p’1=p1 + r1d1
and p’2=p2 + r2d2
– where p1 and p2 are the prices of each input at is
source,
– r1 and r2 represent transfer rates per unit distance
for these inputs.
• The distance from each source to a particular
location such as I or J is given by d1 and d2.
• Location I is closer than J to the source of x1, but
farther away from the source of x2.
• So x1 is relatively cheaper at I and x2 is relatively
cheaper at J.
Iso-outlay lines
• The total outlay (TO) of the locational unit on
transferable inputs is
TO=p’1x1 +p’2x2
(2)
This equation may be reexpressed as
• x1=(TO / p’1) – (p’2 / p’1)x2
(3)
• For any given total outlay (TO), the possible
combinations of the two inputs that could be
bought are determined by equation (2),
• These can be plotted by equation (3) as an isooutlay line
Iso-Outlay lines (contd.)
• The iso-outlay line is linear. It has the form
x1=a + ßx2, where the slope (ß) is - (p'2/p'1),
and the vertical intercept (a ) is (TO/p'1)
• Locations have different sets of delivered
prices, so the combinations of inputs x1 and x2
that any given outlay TO can buy vary by
location
Location Decision and Inputs
The choice
• Consider the isoquant Q0, it indicates all
possible combinations of inputs that produce
that quantity.
• It is clear here that the cheapest way to
produce Q0 is to locate in I