the firm’s decision in space. production theory a firm is characterized by it’s technology...

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 isy = 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 cost-minimization problem is to solve

2211

21 0,min xwxwxx

subject to

.),( 21 yxxf

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

).,,(

),,(),,(

21*22

21*1121

ywwxw

ywwxwywwc

Iso-cost Lines

c’ w1x1+w2x2

c” w1x1+w2x2

c’ < c”

x1

x2

Iso-cost Lines

c’ w1x1+w2x2

c” w1x1+w2x2

c’ < c”

x1

x2 Slopes = -w1/w2.

The y’-Output Unit Isoquant

x1

x2 All input bundles yielding y’ unitsof output. Which is the cheapest?

f(x1,x2) y’

The Cost-Minimization Problem

x1

x2 All input bundles yielding y’ unitsof output. Which is the cheapest?

f(x1,x2) y’

The Cost-Minimization Problem

x1

x2 All input bundles yielding y’ unitsof output. Which is the cheapest?

f(x1,x2) y’

The Cost-Minimization Problem

x1

x2 All input bundles yielding y’ unitsof output. Which is the cheapest?

f(x1,x2) y’

The Cost-Minimization Problem

x1

x2 All input bundles yielding y’ unitsof output. Which is the cheapest?

f(x1,x2) y’

x1*

x2*

The Cost-Minimization Problem

x1

x2

f(x1,x2) y’

x1*

x2*

At an interior cost-min input bundle:(a) f x x y( , )* *

1 2

The Cost-Minimization Problem

x1

x2

f(x1,x2) y’

x1*

x2*

At an interior cost-min input bundle:(a) and(b) slope of isocost = slope of isoquant

f x x y( , )* *1 2

The Cost-Minimization Problem

x1

x2

f(x1,x2) y’

x1*

x2*

At an interior cost-min input bundle:(a) and(b) slope of isocost = slope of isoquant; i.e.

yxxf ),( *2

*1

).,( *2

*1

2

1

2

1 xxatMP

MPTRS

w

w

A Cobb-Douglas Example of Cost Minimization

• A firm’s Cobb-Douglas production function is

• Input prices are w1 and w2.

• What are the firm’s conditional input demand functions?

.),( 3/22

3/1121 xxxxfy

A Cobb-Douglas Example of Cost Minimization

At the input bundle (x1*,x2*) which minimizesthe cost of producing y output units:(a)

(b)

3/2*2

3/1*1 )()( xxy and

.2

)())(3/2(

)())(3/1(

/

/

*1

*2

3/1*2

3/1*1

3/2*2

3/2*1

2

1

2

1

x

x

xx

xx

xy

xy

w

w

A Cobb-Douglas Example of Cost Minimization

3/2*2

3/1*1 )()( xxy .

2 *1

*2

2

1

x

x

w

w(a) (b)

A Cobb-Douglas Example of Cost Minimization

y x x( ) ( )* / * /11 3

22 3 w

wx

x1

2

2

12

*

*.(a) (b)

From (b), xww

x21

21

2* * .

A Cobb-Douglas Example of Cost Minimization

y x x( ) ( )* / * /11 3

22 3 w

wx

x1

2

2

12

*

*.(a) (b)

From (b), xww

x21

21

2* * .

Now substitute into (a) to get

y xww

x

( )* / */

11 3 1

21

2 32

A Cobb-Douglas Example of Cost Minimization

y x x( ) ( )* / * /11 3

22 3 w

wx

x1

2

2

12

*

*.(a) (b)

From (b), xww

x21

21

2* * .

Now substitute into (a) to get

y xww

xww

x

( ) .* / */ /

*11 3 1

21

2 31

2

2 3

12 2

A Cobb-Douglas Example of Cost Minimization

3/2*2

3/1*1 )()( xxy .

2 *1

*2

2

1

x

x

w

w(a) (b)

From (b), .2 *

12

1*2 x

w

wx

Now substitute into (a) to get2/3 2/3

* 1/3 * *1 11 1 1

2 2

2 2( ) .

w wy x x x

w w

y

w

wx

3/2

1

2*1 2

So

is the firm’s conditional demand for input 1.

yw

wx

3/1

2

1*2

2

Is the conditional demand for

input 2

A Cobb-Douglas Example of Cost Minimization

So the cheapest input bundle yielding y output units is

.2

,2

),,(),,,(3/1

2

1

3/2

1

2

21*221

*1

y

w

wy

w

w

ywwxywwx

A Cobb-Douglas Example of Cost Minimization

.4

3

22

1

2

2

),,(),,(),,(

3/1221

3/22

3/11

3/13/22

3/11

3/2

3/1

2

12

3/2

1

21

21*2221

*1121

yww

ywwyww

yw

wwy

w

ww

ywwxwywwxwywwc

So the firm’s total cost function is

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 iso-outlay 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