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Isoquants, Isocosts and Cost Minimization Overhead

s

We define the production function as

y represents output

f represents the relationship between y and x

xj is the quantity used of the jth input

(x1, x2, x3, . . . xn) is the input bundle

n is the number of inputs used by the firm

Holding other inputs fixed, the production

function looks like this

y f (x1, x2, x3, . . . xn )

350

300

Output -y

250

200

150

100

50

0

0

2

4

6

8

10

12

Input -x

Marginal physical product

Marginal physical product is defined as the

increment in production that occurs when an

additional unit of a particular input is employed

Mathematically we define MPP as

Graphically marginal product looks like this

60

50

Output -y

40

30

20

10

0

1

2

3

4

5

6

7

8

9

10

11

-10

-20

-30

-40

Input -x

The Cost Minimization Problem

Pick y observe w1, w2, etc choose the least

cost xs

Isoquants

An isoquant curve in two dimensions

represents all combinations of two inputs that

produce the same quantity of output

The word isomeans same

while quant stand for quantity

Isoquants are contour lines of the production

function

If we plot in x1 - x2 space all combinations of

x1 and x2 that lead to the same (level) height

for the production function, we get contour

lines similar to those you see on a contour map

Isoquants are analogous to indifference curves

Indifference curves represent combinations of

goods that yield the same utility

Isoquants represent combinations of inputs that

yield the same level of production

Production function for the hay example

Another view

Yet another view (low xs)

With a horizontal plane at y 250,000

With a horizontal plane at 100,000

Contour plot

Another contour plot

There are many ways to produce 2,000 bales of hay

per hour

Workers Tractor-Wagons Total Cost AC

10 1 80 0.04 6.45 1.66 71.94 .03597 5.

48 2 72.8658 0.0364 3.667 3 82.0015 0.041

2.636 4 95.8167 0.0479 1.9786 5 111.872 .05

59

Plotting these points in x1 - x2 space we obtain

Or

Isoquant y 2,000

12

X

10

1

8

6

4

2

0

0

2

4

6

8

10

12

14

X

2

Cutting Plane for y 10,000

Isoquant for y 10,000

Only the negatively sloped portions of the

isoquant are efficient

Isoquant for y 10,000

x1 x2 output y -- 1 10,000 -- 2 10,000 -- 3

10,000 12.469 4 10,000 9.725 5

10,000 8.063 6 10,000 6.883 7 10,000 5.990 8 1

0,000 5.290 9 10,000

y 10,000

y 2, 000 y 10,000

Graphical representation

Isoquants y 2,000, y 10,000

14

12

x1

10

8

6

4

2

0

0

2

4

6

8

10

12

14

x

2

y 2,000, y 5,000, y 10,000

More levels

And even more

Comparison to full map

Slope of isoquants

An increase in one input (factor) requires a

decrease in the other input to keep total

production unchanged

Therefore, isoquants slope down (have a negative

slope)

Properties of Isoquants

Higher isoquants represent greater levels of

production

Isoquants are convex to the origin

This means that as we use more and more of an

input, its marginal value in terms of the

substituting for the other input becomes less and

less

Slope of isoquants

The slope of an isoquant is called the marginal

rate of (technical) substitution MR(T)S

between input 1 and input 2

The MRS tells us the decrease in the quantity of

input 1 (x1) that is needed to accompany a one

unit increase in the quantity of input 2 (x2),

in order to keep the production the same

The Marginal Rate of Substitution (MRTS)

14

12

x1

10

8

?

6

?

4

2

?

0

0

2

4

6

8

10

12

14

x

2

Algebraic formula for the MRS

The marginal rate of (technical) substitution

of input 1 for input 2 is

We use the symbol - y constant - to

remind us that the measurement is along a

constant production isoquant

Example calculations y 2,000

Workers Tractor-Wagons x1 x2 10 1 5.48 2

3.667 3 2.636 4 1.9786 5

Change x2 from 1 to 2

Example calculations y 2,000

Change x2 from 2 to 3

Workers Tractor-Wagons x1 x2 10 1 5.48 2

3.667 3 2.636 4 1.9786 5

More example calculations y 10,000

x1 x2 12.469 4 9.725 5 8.063 6 6.883 7 5.990 8

Change x2 from 5 to 6

A declining marginal rate of substitution

The marginal rate of substitution becomes larger

in absolute value as we have more of an input.

The amount of an input we can to give up and

keep production the same is greater, when we

already have a lot of it.

When the firm is using 10 units of x1, it can

give up 4.52 units with an increase of only 1

unit of input 2, and keep production the same

But when the firm is using only 5.48 units of x1,

it can only give up 1.813 units with a one unit

increase in input 2 and keep production the same

Slope of isoquants and marginal physical product

Marginal physical product is defined as the

increment in production that occurs when an

additional unit of a particular input is employed

Marginal physical product and isoquants

All points on an isoquant are associated with

the same amount of production

Hence the loss in production associated with

?x1 must equal the gain in production from ?x2 ,

as we increase the level of x2 and decrease the

level of x1

Rearrange this expression by subtracting MPPx2 ?

x2 from both sides,

Then divide both sides by MPPx1

Then divide both sides by ? x2

The left hand side of this expression is the

marginal rate of substitution of x1 for x2, so

we can write

So the slope of an isoquant is equal to the

negative of the ratio of the marginal

physical products of the two inputs at a given

point

The isoquant becomes flatter as we move to the

right, as we use more x2 (and its MPP declines)

and we use less x1 ( and its MPP increases)

So not only is the slope negative, but the

isoquant is convex to the origin

The Marginal Rate of Substitution (MRTS)

14

12

x1

10

8

?

6

4

2

?

0

0

2

4

6

8

10

12

14

x

2

Approx x1 x2 MRS MPP1 MPP2 12.4687 4.00

00 --- 664.6851 2585.7400 11.8528 4.1713 -3

.5946 739.5588 2465.2134 9.7255 5.0000 -2.5672 1

010.5290 2050.0940 9.3428 5.1972 -1.9411 1063.13

21 1975.4051 8.0629 6.0000 -1.5941 1254.9695 172

4.5840 6.9792 6.9063 -1.1959 1447.3307 1508.9951

6.8827 7.0000 -1.0291 1466.3867 1489.5380

Approx x1 x2 MRS MPP1 MPP2 12.4687 4.00

00 --- 664.6851 2585.7400 11.8528 4.1713 -3

.5946 739.5588 2465.2134 9.7255 5.0000 -2.5672 1

010.5290 2050.0940 9.3428 5.1972 -1.9411 1063.13

21 1975.4051 8.0629 6.0000 -1.5941 1254.9695 172

4.5840 6.9792 6.9063 -1.1959 1447.3307 1508.9951

6.8827 7.0000 -1.0291 1466.3867 1489.5380

x2 rises and MRS falls

Isocost lines

An isocost line identifies which combinations of

inputs the firm can afford to buy with a given

expenditure or cost (C), at given input prices.

Quantities of inputs - x1, x2, x3, . . .

Prices of inputs - w1, w2, w3, . . .

Graphical representation

Cost 120 w1 6 w2 20

22

20

x

1

18

16

14

12

10

8

6

4

2

0

0

1

2

3

4

5

6

7

x

2

Slope of the isocost line

So the slope is -w2 / w1

Example

C 120, w1 6.00, w2 20.00

Intercept of the isocost line

So the intercept is C / w1

With higher cost, the isocost line moves out

Isoquants and isocost lines

We can combine isoquants and isocost lines to

help us determine the least cost input combination

The idea is to be on the lowest isocost line that

allows production on a given isoquant

Combine an isoquant with several isocost lines

Isocost lines for 20, 60, 120, 180, 240, 360

Consider C 120 and C 180

24

X

20

1

16

12

8

4

0

3

4

5

6

7

8

9

10

11

12

13

X

2

At intersection there are opportunities for trade

Add C 160

24

X

20

1

16

12

8

4

0

4

5

6

7

8

9

10

11

12

13

X

2

Add C 154.6

24

X

20

1

16

12

8

4

0

4

5

6

7

8

9

10

11

12

13

X

2

In review

24

X

20

1

16

12

8

4

0

4

5

6

7

8

9

10

11

12

13

X

2

The least cost combination of inputs

The optimal input combination occurs where the

isoquant and the isocost line are tangent

Tangency implies that the slopes are equal

Slope of the isocost line

-w2 / w1

Slope of the isoquant

Optimality conditions

Slope of the isocost line Slope of the

isoquant

Substituting we obtain

The price ratio equals the ratio of marginal

products

We can write this in a more interesting form

Multiply both sides by MPPx1

and then divide by w2

Graphical representation

Statement of optimality conditions

a. The optimum point is on the isocost line

b. The optimum point is on the isoquant

c. The isoquant and the isocost line are

tangent at the optimum combination of x1 and x2

(No Transcript)

f. The marginal product of each input divided

by its price is equal to the marginal product

of every other input divided by its price

Example Table w1 6, w2 20

To get an x1, I can give up 3.33 x2 in terms of

cost

Approx x1 x2 MRS MPP1 MPP2 MPP1/w1 MPP2/w2 MR

S -w2 / w1 1.0000 -3.3333 2.0000

-3.3333 3.0000 -3.3333 12.4687 4.0000 66

4.6851 2585.7400 110.7809 129.2870 -3.8902 -3.3333

11.8528 4.1713 -3.5946 739.5588 2465.2134 123.259

8 123.2607 -3.3334 -3.3333 9.7255 5.0000 -2.5672 1

010.5290 2050.0940 168.4215 102.5047 -2.0287 -3.33

33 9.3428 5.1972 -1.9411 1063.1321 1975.4051 177.1

887 98.7703 -1.8581 -3.3333 8.0629 6.0000 -1.5941

1254.9695 1724.5840 209.1616 86.2292 -1.3742 -3.33

33 6.9792 6.9063 -1.1959 1447.3307 1508.9951 241.2

218 75.4498 -1.0426 -3.3333 6.8827 7.0000 -1.0291

1466.3867 1489.5380 244.3978 74.4769 -1.0158 -3.33

33 5.9898 8.0000 -0.8929 1663.9176 1305.9560 277.3

196 65.2978 -0.7849 -3.3333 5.2904 9.0000 -0.6994

1855.3017 1155.0760 309.2169 57.7538 -0.6226 -3.33

33 4.7309 10.0000 -0.5595 2044.1823 1026.1700 340.

6971 51.3085 -0.5020 -3.3333 4.2773 11.0000 -0.453

5 2232.4134 912.4680 372.0689 45.6234 -0.4087 -3.3

333 3.9071 12.0000 -0.3702 2420.9761 809.4240 403.

4960 40.4712 -0.3343 -3.3333 3.6042 13.0000 -0.302

9 2610.3937 713.8360 435.0656 35.6918 -0.2735 -3.3

333

Intuition for the conditions

The isocost line tells us the rate at which the

firm is able to trade one input for the

other, given their relative prices and total

expenditure

For example in this case the firm must give up 3

1/3 units of input 1 in order to buy a unit of

input 2

w1 6 w2 20 C 180

24

X

20

1

16

12

10

8

4

0

4

5

6

7

8

9

10

11

12

13

3

X

2

The isoquant tells us the rate at which the

firm can trade one input for the other and

remain at the same production level

24

x

20

1

16

12

8

4

0

4

5

6

7

8

9

10

11

12

13

x2

If there is any difference between the rate

at which the firm can trade one input for

another with no change in production and the

rate at which it is able to trade given relative

prices, the firm can always make itself better

off by moving up or down the isocost line

The isoquant tells us the rate at which the

firm can trade one input for the other and

remain at the same production level

24

x

20

1

16

12

8

4

0

4

5

6

7

8

9

10

11

12

13

x2

When the slope of the isoquant is steeper than

the isocost line, the firm will move down the line

When the slope of the isoquant is less steep than

the isocost line, the firm will move up the line

When the slope of the isoquant is steeper than

the isocost line, the firm will move down the line

24

x

20

1

16

12

8

4

0

4

5

6

7

8

9

10

11

12

13

x2

The End

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