Transcript Introduction to Production and Resource Use

Profit Maximization
Beattie, Taylor, and Watts
Sections: 3.1b-c, 3.2c, 4.2-4.3,
5.2a-d
1
Agenda




Generalized Profit Maximization
Profit Maximization with One Input and
One Output
Profit Maximization with Two Inputs
and One Output
Profit Maximization with One Input and
Two Outputs
2
Defining Profit

Profit can be generally defined as total
revenue minus total cost.



Total revenue is the summation of the
revenue from each enterprise.
The revenue from one enterprise is defined
as price multiplied by quantity.
Total cost is the summation of all fixed and
variable cost.
3
Defining Profit Cont.

Short-run profit () can be defined
mathematically as the following:
  TR  TC
n
m
i 1
j 1
 ( y1 , y2 ,..., yn , x1 , x2 ,..., xm )   pi yi   w j x j  TFC
y1  f1 ( x11 , x12 ,..., x1m )
y2  f 2 ( x21 , x22 ,..., x2 m )

yn  f 2 ( xn1 , xn 2 ,..., xnm )
x1  x11  x21    xn1
x2  x12  x22    xn 2

xm  x1m  x2 m    xnm
4
Revenue


In a perfectly competitive market
revenue from a particular enterprise can
be defined as p*y.
When the producer can have an effect
on price, then price becomes a function
of output, which can be represented as
p(y)*y.
5
Marginal Revenue


Marginal Revenue (MR) is defined as the change in
revenue due to a change in output.
In a perfectly competitive world, marginal revenue
equals average revenue which equals price.
TR  py
dTR
MR 
p
dy
6
Marginal Revenue Cont.

When the market is not perfectly competitive,
then MR can be represented as the following:
TR  p ( y ) * y
dTR
MR 
 p' ( y ) * y  p( y )
dy
 p' ( y ) * y 
MR  p ( y )
 1
 p( y )

 1


1
MR  p ( y )  1  p( y )1 
 d

 d



7
Marginal Value Of Product


Marginal Value of Product (MVP) is defined as the
change in revenue due to a change in the input.
To find MVP, you need to substitute the production
function y=f(x) into the TR function.
TR( y )  py
TR( x)  pf ( x)
dTR( x)
MVP 
 pf ' ( x)  pMPP
dx
8
Cost Side of Profit
Maximization

Marginal Cost (MC) and Marginal Input Cost
(MIC) can be derived from the cost side of
the profit function.

Marginal cost is defined as the change in cost due
to a change in output.


From the cost minimization problem, it was shown the
different forms that marginal cost could take.
Marginal Input Cost is the change in cost due to a
change in the input.

MIC is equal to the price of the input.
9
Standard Profit Maximization
Model
n
Max
m
 p y w x
x1 , x2 ,...,xm
y1 , y 2 ,...,y n i 1
i
i
j 1
j
j
 TFC
subject to:
y1  f1 ( x1 , x2 ,..., xm )
y2  f 2 ( x1 , x2 ,..., xm )

yn  f n ( x1 , x2 ,..., xm )
10
Profit Maximization with One
Input and One Output


Assume that we have one variable input (x)
which costs w.
Assume that the general production function
can be represented as y = f(x).
Max py  wx  TFC
x, y
subject to: y  f ( x)
11
Examining Results of Profit Maximization
with One Input and One Output
( x, y,  )  py  wx  TFC   ( f ( x)  y )
d
  w  f ' ( x )  0
dx
d
 p  0
dy
d
 f ( x)  y  0
dy
w
 p   and

f ' ( x)
w
 p
f ' ( x)
 pf ' ( x)  w
 pMPP  w
 MVP  MIC
12
Notes on Profit Maximization

By solving the profit maximization
problem, we get the optimum decision
rule where MVP=MIC.

With minor manipulation we can transform
the result from the previous slide using the
production function into the other form of
the optimum decision MR = MC.
13
Notes on Profit Maximization
Cont.

There are two primary ways to solve
the profit maximization problem.


Solve the constrained profit max problem
w.r.t. x and y.
Transform the constrained profit max
problem into an unconstrained problem by
substituting the production function or its
inverse into the profit max problem and
solve w.r.t. to the appropriate variable.
14
Solving the Profit Maximization
Problem W.R.T. Inputs


Assume that we have one variable input (x)
which costs w.
Assume that the general production function
can be represented as y = f(x).
Max pf ( x)  wx  TFC
x
15
Solving the Profit Maximization
Problem W.R.T. Inputs Cont.
( x)  pf ( x)  wx
d
 pf ' ( x)  w  0
dx
 pf ' ( x)  w
 pMPP  w
w
 MPP 
p
16
Solving the Profit Maximization
Problem W.R.T. Outputs


Assume that we have one variable input (x)
which costs w.
Assume that the general production function
can be represented as y = f(x) with an output
price of p.
1
Max py  wf ( y )  TFC
y
17
Solving the Profit Maximization
Problem W.R.T. Outputs Cont.
( y )  py  wf 1 ( y )  TFC
d
w
 p
0
dx
MPP
w
 p
MPP
w
 MPP 
p
18
Profit Max Example 1

Suppose that you would like to
maximize profits given the following
information:




Output Price = 10
Input Price = 200
TFC = 100
y=f(x)=50x-x2
19
Profit Max Example 1:
Lagrangean
Max 10 y  200x  100 s.t . y  f ( x)  50x  x 2
x, y
( x, y,  )  10 y  200x  100  (50x  x 2  y )
d
 200  (50  2 x)  0
dx
d
 10    0
dy
d
 50x  x 2  y  0
d
Solut ion will be done in class.
20
Profit Max Example 1:
Unconstrained W.R.T. Input


Max 10 50x  x 2  200x  100
x


( x)  10 50x  x 2  200x  100
d
 10(50  2 x)  200  0
dx
10(50  2 x)  200
(50  2 x)  20
 2 x  30
x  15
y  f (15)  525
  2250
21
Profit Max Example 1: Solving
Using MC=MR
y  f ( x)  50x  x 2
 x  f 1 ( y )  25  625 y
w  200, p  10, TFC  100
TC ( y )  wx  TFC  200(25  625 y )  100  5100 200 625 y
MR  p  10
MC 
dTC
1  200 
100
 0 
* (1) 
dy
2  625 y 
625 y
MR  MC
 10 
100
625 y
1
1

10
625 y
10  625 y
100  625- y
y  525
22
Profit Max Example 1: Solving
Using MIC=MVP
w  200
p  10
TR( x)  10(50x  x 2 )  500x  10x 2
MIC  w  200
dTR
MVP 
 500 20x
dx
MVP  MIC
 500 20x  200
 20x  300
x  15
23
Profit Max Example 1: Solving
Using MPP=w/p
w  200
p  10
y  f ( x)  50x  x 2
dy
MPP 
 50  2 x
dx
w
MPP 
p
200
 50  2 x 
10
 2 x  30
x  15
24
Question: How would you find
the loss in profit (π) if you were a
revenue maximizer instead a
profit maximizer?
Loss = ππ-Max - πRevenue-Max
25
Graph of Profit and Production
3000
2000
1000
Production
Profit
0
0
5
10
15
20
25
30
35
-1000
-2000
26
Graph of Profit and Total
Revenue
7000
6000
5000
4000
Total Revenue
Profit
3000
2000
1000
0
0
5
10
15
20
25
30
35
27
Graph of Marginal Revenue
and Marginal Cost
50
40
30
M arginal Revenue
M arginal Cost
20
10
0
0
100
200
300
400
500
600
700
28
Graph of Marginal Value of Product
and Marginal Input Cost
29
Profit Max Example 2

Suppose that you would like to
maximize profits given the following
information:




Output Price = 20
Input Price = 200
TFC=100
y=f(x)=50x-x2
30
Profit Max Example 2:
Lagrangean
Max 20 y  200x  100 s.t . y  f ( x)  50x  x 2
x, y
( x, y,  )  20 y  200x  100  (50x  x 2  y )
d
 200  (50  2 x)  0
dx
d
 20    0
dy
d
 50x  x 2  y  0
d
Solut ion will be done in class.
31
Profit Max Example 2:
Unconstrained W.R.T. Input


Max 20 50x  x 2  200x  100
x


( x)  20 50x  x 2  200x  100
d
 20(50  2 x)  200  0
dx
20(50  2 x)  200
(50  2 x)  10
 2 x  40
x  20
y  f (20)  600
  8000
32
Profit Max Example 2:
Unconstrained W.R.T. Output
y  f ( x)  50x  x 2
 x  f 1 ( y )  25  625 y


max 20 y  200 25  625 y  100
y


( y )  20 y  200 25  625 y  100
d
 1  200
 20   
0
dy
 2  625 y
 20 
100
625 y
2
1

10
625 y
5  625 y
25  625- y
y  600
 x  f -1 (600) 20
33
Profit Max Example 2: Solving
Using MC=MR
y  f ( x)  50x  x 2
 x  f 1 ( y )  25  625 y
w  200, p  20, TFC  100
TC ( y )  wx  TFC  200(25  625 y )  100  5100 200 625 y
MR  p  20
MC 
dTC
1  200 
100
 0 
* (1) 
dy
2  625 y 
625 y
MR  MC
 20 
100
625 y
2
1

10
625 y
5  625 y
25  625- y
y  600
34
Profit Max Example 2: Solving
Using MIC=MVP
w  200
p  20
TR( x)  20(50x  x 2 )  1000x  20x 2
MIC  w  200
dTR
MVP 
 1000 40x
dx
MVP  MIC
 1000 40x  200
 40x  800
x  20
35
Profit Max Example 2: Solving
Using MPP=w/p
w  200
p  20
y  f ( x)  50x  x 2
dy
MPP 
 50  2 x
dx
w
MPP 
p
200
 50  2 x 
20
 2 x  40
x  20
36
Profit Maximization with Two
Inputs and One Output


Assume that we have two variable inputs (x1 and x2)
which cost respectively w1 and w2. Also, let TFC
represent the total fixed costs.
Assume that the general production function can be
represented as y = f(x1,x2), where y sells at a price
of p.
Max py  w1 x1  w2 x2  TFC
x1 , x2 , y
subject to: y  f ( x1 , x2 )
37
First Order Conditions for the Constrained Profit
Maximization Problem with Two Inputs
( y, x1 , x2 ,  )  py  w1 x1  w2 x2  TFC   ( f ( x1 , x2 )  y )

 p  0
y

f
  w1  
0
x1
x1
 MPPx1  w1
 
w1
MPPx1

f
  w2  
0
x2
x2
 
w2
MPPx2

 f ( x1 , x2 )  y  0

38
First Order Conditions for the Unconstrained
Profit Maximization Problem with Two Inputs
 ( x1 , x2 )  pf ( x1 , x2 )  w1 x1  w2 x2  TFC

f
x1
p
x1
 w1  0
 pMPPx1  w1
 p
w1
MPPx1

f
p
 w2  0
x2
x2
 pMPPx2  w2
 p
w2
MPPx2
39
Summary of Profit Max Results


At the optimum, each input selected will
cause the MPP with respect to that
input to equal the ratio of input price to
output price.
For example:


MPPx1= w1/p
MPPx2= w2/p
40
Summary of Profit Max Results
Cont.

From the profit max problem you will get a
relationship between the two inputs.


This relationship is called the expansion path.
Once you selected a certain output, your
revenue becomes trivially given to you when
output price is fixed.

Hence, you are just minimizing cost.
41
Example 1 of Profit Maximization with
Two Variable Inputs

Suppose you have the following production
function:



y = f(x1,x2) = 40x1½ x2½
Suppose the price of input 1 is $1 and the
price of input 2 is $16. Let the total fixed
cost equal $100.
What is the optimal amount of input 1 and 2
if you have a price of 20 for the output and
you want to produce y units?

What is the profit?
42
Example 1 of Profit Max with Two
Variable Inputs Cont.

Summary of what is known:



w1 = 1, w2 = 16
y = 40x1½ x2½
p = 20
Max 20 y  x1  16x2  100
x1 , x2
1
2
1
1
2
2
subject to: y  40x x
43
Example 1 of Profit Max with Two
Variable Inputs Cont.
1 1


( y, x1, x 2,  )  20 y  1x1  16x2  100   40x12 x22  y 



 20    0
y
1 1

 1  2 2
 1   40  x1 x2  0
x1
2

 1  2 2
 16   40  x1 x2  0
x2
2
1
1
1 1

 y  40x12 x22  0

Solution done in class
44
Example 2 of Profit Max with Two
Variable Inputs Cont.

Summary of what is known:



w1 = 1, w2 = 16
y = 40x11/4 x21/4
p = 20
Max 20 y  x1  16x2  100
x1 , x2
1
4
1
1
4
2
subject to: y  40x x
45
Example 2 of Profit Max with Two
Variable Inputs Cont.
1 1


( y, x1, x 2,  )  20 y  1x1  16x2  100   40x14 x24  y 



 20    0
y
3 1

 1  4 4
 1   40  x1 x2  0
x1
4


1
 16   40  x14 x2 4  0
x2
4
1
3
1 1

 y  40x14 x24  0

Solution done in class
46
Example 2: Finding the Profit Max Inputs Using the
Production Function and MPPxi=wi/p
1
4
y  40x1 x 2
1
4
p  20, w1  1, w2  16, TFC  100
 1   34 14
MPPx1  40  x1 x 2
4
 1  14  34
MPPx2  40  x1 x 2
4
w
1
Set MPPx1  1 
p 20
 1   34 14 1
 40  x1 x 2 
20
4
w 16
Set MPPx2  2 
p 20
 1   3 1 16
 40  x1 4 x 2 4 
20
4
Solution done in class.
47
Profit Maximization with Two
Outputs and One Input


Assume that we have two production
functions (y1 and y2) which have a price
of p1 and p2 respectively.
Assume that you have one input X that
can be divided between production
function 1 (y1=f1(x1)) and production
function 2 (y2=f2(x2)).
48
Profit Maximization with Two
Outputs and One Input Cont.



The amount of input allocated to y1 is
defined as x1 and the amount of input
allocated to y2 is x2.
The summation of x1 and x2 have to
sum to X, i.e., x1+x2=X.
The price of the input is w.
49
Profit Maximization with Two
Outputs and One Input Cont.
Max p1 y1  p2 y2  wx1  wx2  TFC
x1 , x2 , y1 , y2
subject to:
y1  f1 ( x1 )
y2  f 2 ( x2 )
x1  x2  X
50
First Order Conditions for the Constrained Profit
Maximization Problem with Two Outputs
( y1 , y2 , x1 , x2 , 1 , 2 , 3 ) 
p1 y1  p2 y2  wx1  wx2  TFC  1 ( f1 ( x1 )  y1 )  2 ( f 2 ( x2 )  y2 )  3 ( X  x1  x2 )

 p1  1  0
y1

 p2  2  0
y2

f
  w  1 1  3  0
x1
x1

f
  w  2 2  3  0
x2
x2

 f1 ( x1 )  y1  0
1

 f 2 ( x2 )  y 2  0
2

 X  x1  x2  0
3
Solution discussed in class.
51
Summary of Profit Max Results

At the optimum, the marginal value of
product of the first production function with
respect to input 1 (MVPy1) is equal to the
marginal value of product of the second
production function (MVPy2).


This gives you the optimal allocation of inputs.
For example:

MVPy1= MVPy2
52
Summary of Profit Max Results
Cont.

With some manipulation of the previous
fact, the optimum rule for output
selection occurs where the slope of the
PPF, i.e., MRPT, is equal to the negative
of the output price ratio.


This gives you the optimal allocation of
outputs.
MRPT=-p1/p2
53
Example 1 of Profit Maximization with
Two Outputs and One Input

Suppose you have the following production functions:





y1 = f1(x1) = 300x11/3
y2 = f2(x2) = 300x21/3
Suppose the price of output 1 is $4 and the price of
output 2 is $1.
The price of the input w is 1 and the total fixed cost
is 1000.
What is the optimal amount of output 1 and 2 if you
have 9000 units of input X to allocate to both
productions?

What is the profit?
54
Example 1 of Profit Max with Two
Outputs and One Input Cont.

Summary of what is known:



w=1, p1=4, p2=1, X=9000, TFC=1000
y1 = 300x11/3
y2 = 300x21/3
Max 4 y1  y2  x1  x2  1000
x1 , x2
subject to:
1
3
1
y1  300x
y2  300x
1
3
2
9000 x1  x2
55
Example 1 of Profit Max with Two
Outputs Cont.
( y1 , y2 , x1 , x2 , 1 , 2 , 3 ) 
1
1
4 y1  y2  x1  x2  1000 1 (300x1 3  y1 )  2 (300x2 3  y2 )  3 (9000 x1  x2 )

 4  1  0
y1

 1  2  0
y2


 1  1100x1 3  3  0
x1
2


 1  2100x2 3  3  0
x2
2
1

 300x1 3  y1  0
1
1

 300x2 3  y2  0
2

 9000 x1  x2  0
3
Solution discussed in class.
56
Example 1: Finding the Profit Max
Outputs Using MRPT = p1/p2
2
y1
y1  300x1  x1 
3003
3
1
y2
y2  300x 2 3  x2 
3003
p1  4, p2  1
1
3
9000 x1  x2
 9000* 3003  y1  y 2
3
MRPT 
3
dy2
dy1
Solution done in class.
57
Example 1: Finding the Profit Max Inputs
Using MVPy1 = MVPy2
2
dy1

y1  300x1  MPPy1 
 100x1 3
dx1
1
3
y2  300x 2
1
3
2
dy2

 MPPy 2 
 100x 2 3
dx2
p1  4, p2  1
9000 x1  x2
MVPy1  p1 * MPPy1  4100x1

MVPy 2  p2 * MPPy 2  1100x 2
2
3

2
3
 400x1

 100x 2
2
3

2
3
MVPy1  MVPy 2
 400x1

2
3
 100x 2

2
3
 x1  8 x2
Solution finishedin class.
58