1

Profit Maximization

Beattie, Taylor, and WattsSections: 3.1b-c, 3.2c, 4.2-

4.3, 5.2a-d

2

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

3

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.

4

Defining Profit Cont. Short-run profit () can be defined

mathematically as the following:

nmmmm

n

n

nmnnn

m

m

m

jjj

n

iiimn

xxxx

xxxx

xxxx

xxxfy

xxxfy

xxxfy

TFCxwypxxxyyy

TCTR

21

222122

121111

212

2222122

1121111

112121

),...,,(

),...,,(

),...,,(

),...,,,,...,,(

5

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.

6

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.

pdy

dTRMR

pyTR

7

Marginal Revenue Cont. When the market is not perfectly competitive,

then MR can be represented as the following:

dd

ypypMR

yp

yypypMR

ypyypdy

dTRMR

yypTR

1

1)(11

)(

1)(

*)(')(

)(*)('

*)(

8

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.

pMPPxpfdx

xdTRMVP

xpfxTR

pyyTR

)(')(

)()(

)(

9

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.

10

Standard Profit Maximization Model

),...,,(

),...,,(

),...,,(

:subject to

21

2122

2111

11,...,,,...,,

21

21

mnn

m

m

m

jjj

n

iii

yyyxxx

xxxfy

xxxfy

xxxfy

TFCxwypMaxn

m

11

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).

)(:subject to

,

xfy

TFCwxpyMaxyx

12

Examining Results of Profit Maximization with One Input and One Output

MICMVP

wpMPP

wxpf

xf

wp

xf

wandp

yxfdy

d

pdy

d

xfwdx

d

yxfTFCwxpyyx

)('

)('

)('

0)(

0

0)('

))((),,(

13

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.

14

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.

15

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).

TFCwxxpfMaxx

)(

16

Solving the Profit Maximization Problem W.R.T. Inputs Cont.

p

)('

0)('

)()(

wMPP

wpMPP

wxpf

wxpfdx

d

wxxpfx

17

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.

TFCywfpyMaxy

)(1

18

Solving the Profit Maximization Problem W.R.T. Outputs Cont.

p

0

)()( 1

wMPP

MPP

wp

MPP

wp

dx

d

TFCywfpyy

19

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

20

Profit Max Example 1: Lagrangean

class.in done be illSolution w

050

010

0)250(200

)50(10020010),,(

50)( s.t. 10020010

2

2

2

,

yxxd

d

dy

d

xdx

d

yxxxyyx

xxxfyxyMaxyx

21

Profit Max Example 1: Unconstrained W.R.T. Input

2250

525)15(

15

302

20)250(

200)250(10

0200)250(10

1002005010)(

1002005010

2

2

fy

x

x

x

x

xdx

d

xxxx

xxxMaxx

22

Profit Max Example 1: Solving Using MC=MR

525y

y-625100

62510

625

1

10

1

625

10010

625

100)1(*

625

200

2

10

10

6252005100100)62525(200)(

100,10,200

62525)(

50)(1

2

y

y

y

MCMR

yydy

dTCMC

pMR

yyTFCwxyTC

TFCpw

yyfx

xxxfy

23

Profit Max Example 1: Solving Using MIC=MVP

15

30020

20020500

20500

200

10500)50(10)(

10p

200w

22

x

x

x

MICMVP

xdx

dTRMVP

wMIC

xxxxxTR

24

Profit Max Example 1: Solving Using MPP=w/p

15

30210

200250

250

50)(

10p

200w

2

x

x

x

p

wMPP

xdx

dyMPP

xxxfy

25

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

26

Graph of Profit and Production

-2000

-1000

0

1000

2000

3000

0 5 10 15 20 25 30 35

Production

Profit

27

Graph of Profit and Total Revenue

0

1000

2000

3000

4000

5000

6000

7000

0 5 10 15 20 25 30 35

Total Revenue

Profit

28

Graph of Marginal Revenue and Marginal Cost

0

10

20

30

40

50

0 100 200 300 400 500 600 700

Marginal Revenue

Marginal Cost

29

Graph of Marginal Value of Product and Marginal Input Cost

30

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

31

Profit Max Example 2: Lagrangean

class.in done be illSolution w

050

020

0)250(200

)50(10020020),,(

50)( s.t. 10020020

2

2

2

,

yxxd

d

dy

d

xdx

d

yxxxyyx

xxxfyxyMaxyx

32

Profit Max Example 2: Unconstrained W.R.T. Input

8000

600)20(

20

402

10)250(

200)250(20

0200)250(20

1002005020)(

1002005020

2

2

fy

x

x

x

x

xdx

d

xxxx

xxxMaxx

33

Profit Max Example 2: Unconstrained W.R.T. Output

20(600)fx

006y

y-62552

6255

625

1

10

2

625

10020

0625

200

2

120

1006252520020)(

1006252520020max

62525)(

50)(

1-

1

2

y

y

y

ydy

d

yyy

yy

yyfx

xxxfy

y

34

Profit Max Example 2: Solving Using MC=MR

006y

y-62552

6255

625

1

10

2

625

10020

625

100)1(*

625

200

2

10

20

6252005100100)62525(200)(

100,20,200

62525)(

50)(1

2

y

y

y

MCMR

yydy

dTCMC

pMR

yyTFCwxyTC

TFCpw

yyfx

xxxfy

35

Profit Max Example 2: Solving Using MIC=MVP

20

80040

200401000

401000

200

201000)50(20)(

20p

200w

22

x

x

x

MICMVP

xdx

dTRMVP

wMIC

xxxxxTR

36

Profit Max Example 2: Solving Using MPP=w/p

20

40220

200250

250

50)(

20p

200w

2

x

x

x

p

wMPP

xdx

dyMPP

xxxfy

37

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.

),(:subject to 21

2211,, 21

xxfy

TFCxwxwpyMaxyxx

38

First Order Conditions for the Constrained Profit Maximization Problem with Two Inputs

0),(

0

0

0

)),((),,,(

21

2

22

2

1

1

11

1

21221121

2

1

1

yxxf

MPP

w

x

fw

x

MPP

w

wMPP

x

fw

x

py

yxxfTFCxwxwpyxxy

x

x

x

39

First Order Conditions for the Unconstrained Profit Maximization Problem with Two Inputs

2

2

1

1

2

2

222

1

1

111

22112121

0

0

),(),(

x

x

x

x

MPP

wp

wpMPP

wx

fp

x

MPP

wp

wpMPP

wx

fp

x

TFCxwxwxxpfxx

40

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

41

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.

42

Example 1 of Profit Maximization with Two Variable Inputs Suppose you have the following

production function: y = f(x1,x2) = 40x1

½ 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?

43

Example 1 of Profit Max with Two Variable Inputs Cont.

Summary of what is known: w1 = 1, w2 = 16 y = 40x1

½ x2½

p = 20

2

1

22

1

1

21,

40:subject to

100162021

xxy

xxyMaxxx

44

Example 1 of Profit Max with Two Variable Inputs Cont.

classin doneSolution

040

02

14016

02

1401

020

4010016120),2,1,(

2

1

22

1

1

2

1

22

1

12

2

1

22

1

11

2

1

22

1

121

xxy

xxx

xxx

y

yxxxxyxxy

45

Example 2 of Profit Max with Two Variable Inputs Cont.

Summary of what is known: w1 = 1, w2 = 16 y = 40x1

1/4 x21/4

p = 20

4

1

24

1

1

21,

40:subject to

100162021

xxy

xxyMaxxx

46

Example 2 of Profit Max with Two Variable Inputs Cont.

classin doneSolution

040

04

14016

04

1401

020

4010016120),2,1,(

4

1

24

1

1

4

3

24

1

12

4

1

24

3

11

4

1

24

1

121

xxy

xxx

xxx

y

yxxxxyxxy

47

Example 2: Finding the Profit Max Inputs Using the Production Function and MPPxi=wi/p

class.in doneSolution

20

16xx

4

140

20

16Set

20

1xx

4

140

20

1Set

xx4

140

xx4

140

100,16,1,20

xx40

4

1

24

3

1

2

4

1

24

3

1

1

4

3

24

1

1

4

1

24

3

1

21

4

1

24

1

1

2

1

2

1

p

wMPP

p

wMPP

MPP

MPP

TFCwwp

y

x

x

x

x

48

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)).

49

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.

50

Profit Maximization with Two Outputs and One Input Cont.

Xxx

xfy

xfy

TFCwxwxypypMaxyyxx

21

222

111

212211,,,

)(

)(

:subject to2121

51

First Order Conditions for the Constrained Profit Maximization Problem with Two Outputs

class.in discussedSolution

0

0)(

0)(

0

0

0

0

)())(())((

),,,,,,(

213

2222

1111

32

22

2

31

11

1

222

111

21322221111212211

3212121

xxX

yxf

yxf

x

fw

x

x

fw

x

py

py

xxXyxfyxfTFCwxwxypyp

xxyy

52

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

53

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

54

Example 1 of Profit Maximization with Two Outputs and One Input Suppose you have the following production

functions: y1 = f1(x1) = 300x1

1/3

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?

55

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 = 300x1

1/3

y2 = 300x21/3

21

3

1

22

3

1

11

2121,

9000

300

300

:subject to

1000421

xx

xy

xy

xxyyMaxxx

56

Example 1 of Profit Max with Two Outputs Cont.

class.in discussedSolution

09000

0300

0300

01001

01001

01

04

)9000()300()300(10004

),,,,,,(

213

23

1

22

13

1

11

33

2

222

33

2

111

22

11

21323

1

2213

1

112121

3212121

xx

yx

yx

xx

xx

y

y

xxyxyxxxyy

xxyy

57

Example 1: Finding the Profit Max Outputs Using MRPT = p1/p2

class.in doneSolution

yy300*9000

9000

1,4300

yx300

300

yx300

1

2

32

31

3

21

21

3

32

23

1

22

3

21

13

1

11

dy

dyMRPT

xx

pp

xy

xy

58

Example 1: Finding the Profit Max Inputs Using MVPy1 = MVPy2

class.in finishedSolution

8

x100x400

x100x1001*

x400x1004*

9000

1,4

x100x300

x100x300

21

3

2

23

2

1

21

3

2

23

2

2222

3

2

13

2

1111

21

21

3

2

22

223

1

22

3

2

11

113

1

11

xx

MVPMVP

MPPpMVP

MPPpMVP

xx

pp

dx

dyMPPy

dx

dyMPPy

yy

yy

yy

y

y