Isoquants, Isocosts and Cost Minimization

Overheads

We define the production function as

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

f(x) maxy

[y: (x, y) is an element of the production set]

y represents output

f(x1 , x2 , x3 , ) maxy ε P(x)

[y]

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

0

50

100

150

200

250

300

350

0 2 4 6 8 10 12

Input -x

Ou

tpu

t -y

y

Holding other inputs fixed,the production function looks like this

Marginal physical product

Marginal physical product is defined as the incrementin production that occurs when an additional unitof a particular input is employed

Mathematically we define MPP as

MPPxj MPPj

Δf(x1 , x2 , x3 , , xn )

Δxj

Graphically marginal product looks like this

-40

-30

-20

-10

0

10

20

30

40

50

60

1 2 3 4 5 6 7 8 9 10 11

Input -x

Ou

tpu

t -y

A MPP

The Cost Minimization Problem

C(y, w1,w2 , ) minx1 ,x2 , ,xn

Σn

i 1wixi such that y f(x1,x2 , xn)

Pick y; observe w1, w2, etc;choose the least cost x’s

Isoquants

An isoquant curve in two dimensions representsall combinations of two inputs that produce the same quantity of output

The word “iso”means 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 goodsthat yield the same utility

Isoquants represent combinations of inputs that yieldthe same level of production

Production function for the hay example

0 10 20 30 40 50 60 70

x1

015

3045

6075

x2

0100000200000300000

y

0 10 20 30 40 50 60 70

x1

015

3045

6075

x2

Another view

0

10

20

30

40

50

60

70

x1

0

15

30

45

60

75

x2

0200000

y

0

10

20

30

40

50

60

70

x1

0

15

30

45

60

75

x2

0200000

y

Yet another view (low x’s)

05

1015

2025

x1

0

5

10

15

20

25

x2

01500030000

75000

y

05

1015

2025

x1

0

5

10

15

20

25

x2

With a horizontal plane at y = 250,000

0 10 20 30 40 50 60 70

x1

015

3045

6075 x2

0100000200000300000

y

0 10 20 30 40 50 60 70

x1

015

3045

6075 x2

0 10 20 30 40 50 60 70

x1

015

3045

6075 x2

0100000200000300000

y

0 10 20 30 40 50 60 70

x1

015

3045

6075 x2

With a horizontal plane at 100,000

Contour plot

0 10 20 30 40 50 60 70x1

0

20

40

60

802x

0 10 20 30 40 50 60 70x1

0

20

40

60

80

2xAnother contour plot

There are many ways to produce2,000 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 0.03643.667 3 82.0015 0.0412.636 4 95.8167 0.04791.9786 5 111.872 .0559

Plotting these points in x1 - x2 space we obtain

0 2 4 6 8 10x1

0

2

4

6

8

10

2x

Or

Isoquant y = 2,000

0

2

4

6

8

10

12

0 2 4 6 8 10 12 14

X2

X1

Isoquant y = 2000

x2 = 4, x1 = 2.636

0 10 20 30 40 50 60 70

x1

015

3045

6075 x2

0100000200000300000

y

0 10 20 30 40 50 60 70

x1

015

3045

6075 x2

Cutting Plane for y = 10,000

0 10 20 30 40x1

0

10

20

30

40

2xIsoquant for y = 10,000

Only the negatively sloped portionsof the isoquant are efficient

0 10 20 30 40x1

0

10

20

30

40

2x

Isoquant for y = 10,000

x1 x2 output y-- 1 10,000-- 2 10,000-- 3 10,00012.469 4 10,0009.725 5 10,0008.063 6 10,0006.883 7 10,0005.990 8 10,0005.290 9 10,000

y = 10,000

0 2 4 6 8 10x1

0

2

4

6

8

102x

0 2 4 6 8 10x1

0

2

4

6

8

10

2x

y = 2, 000 y = 10,000

Graphical representation

Isoquants y = 2,000, y = 10,000

0

2

4

6

8

10

12

14

0 2 4 6 8 10 12 14

x 2

Isoquant y = 2000

Isoquant y = 10000

x1

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

0 2 4 6 8 10x1

0

2

4

6

8

10

2x

More levels

0 5 10 15 20 25 30x1

0

5

10

15

20

25

30

2x

And even more

0 10 20 30 40 50 60x1

0

10

20

30

40

50

60

2x

Comparison to full map

0 10 20 30 40 50 60 70x1

0

20

40

60

80

2x

Slope of isoquants

An increase in one input (factor) requires a decrease in theother input to keep total production unchanged

Therefore, isoquants slope down (have a negative slope)

Properties of Isoquants

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 forthe other input becomes less and less

Higher isoquants represent greater levels of production

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 increasein the quantity of input 2 (x2),

in order to keep the production the same

0

2

4

6

8

10

12

14

0 2 4 6 8 10 12 14

x2

Isoquant y = 2000

Isoquant y = 10000

x1

The Marginal Rate of Substitution (MRTS)

Algebraic formula for the MRS

MRSx1,x2

Δx1

Δx2

y constant

The marginal rate of (technical) substitution ofinput 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

MRSx1,x2

Δx1

Δx2

y constant

10 5.481 2

4.52 1

4.52

Change x2 from 1 to 2Workers Tractor-Wagons

x1 x2

10 15.48 23.667 32.636 41.9786 5

Example calculations y = 2,000

Change x2 from 2 to 3

MRSx1,x2

Δx1

Δx2

y constant

5.48 3.6672 3

1.813 1

1.813

Workers Tractor-Wagons x1 x2

10 15.48 23.667 32.636 41.9786 5

More example calculations y = 10,000

MRSx1,x2

Δx1

Δx2

y constant

9.725 8.0635 6

1.662 1

1.662

Change x2 from 5 to 6

x1 x2

12.469 49.725 58.063 66.883 75.990 8

A declining marginal rate of substitution

The marginal rate of substitution becomes larger in absolute value as we have more of an input.

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 onlygive up 1.813 units with a one unit increase in input 2 and keep production the same

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.

Slope of isoquants and marginal physical product

Marginal physical product is defined as the incrementin production that occurs when an additional unitof a particular input is employed

MPPxj MPPj

Δf(x1 , x2 , x3 , , xn)

Δxj

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

MPPx1Δx1 MPPx2

Δx2 0

MPPx1 Δx1 MPPx2 Δx2

Rearrange this expression by subtracting MPPx2 x2

from both sides,

MPPx1 Δx1 MPPx2 Δx2 0

Then divide both sides by MPPx1

Δx1 MPPx2 Δx2

MPPx1Then divide both sides by x2

Δx1

Δx2

MPPx2MPPx1

The left hand side of this expression is the marginal rate of substitution of x1 for x2, so we can write

MRSx1x2

Δx1

Δx2

MPPx2

MPPx1

So the slope of an isoquant is equal tothe negative of the ratio of the marginal physicalproducts of the two inputs at a given point

The isoquant becomes flatter as we moveto the right, as we use more x2 (and itsMPP declines) and we use less x1 ( and itsMPP increases)

So not only is the slope negative, but the isoquantis convex to the origin

0

2

4

6

8

10

12

14

0 2 4 6 8 10 12 14

x2

Isoquant y = 2000

x1

The Marginal Rate of Substitution (MRTS)

Approx x1 x2 MRS MPP1 MPP2

12.4687 4.0000 --- 664.6851 2585.740011.8528 4.1713 -3.5946 739.5588 2465.2134 9.7255 5.0000 -2.5672 1010.5290 2050.0940 9.3428 5.1972 -1.9411 1063.1321 1975.4051 8.0629 6.0000 -1.5941 1254.9695 1724.5840 6.9792 6.9063 -1.1959 1447.3307 1508.9951 6.8827 7.0000 -1.0291 1466.3867 1489.5380

MRSx1,x2

Δx1

Δx2

y constant

9.3428 8.06295.1972 6

1.2799 0.8028

1.594

Approx x1 x2 MRS MPP1 MPP2

12.4687 4.0000 --- 664.6851 2585.740011.8528 4.1713 -3.5946 739.5588 2465.2134 9.7255 5.0000 -2.5672 1010.5290 2050.0940 9.3428 5.1972 -1.9411 1063.1321 1975.4051 8.0629 6.0000 -1.5941 1254.9695 1724.5840 6.9792 6.9063 -1.1959 1447.3307 1508.9951 6.8827 7.0000 -1.0291 1466.3867 1489.5380

MRSx1x2

Δx1

Δx2

MPPx2

MPPx1

x2 rises and MRS falls

Isocost lines

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

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

An isocost line identifies which combinations of inputsthe firm can afford to buy with a given expenditureor cost (C), at given input prices.

w1x1 w2x2 w3x3 wnxn C

Graphical representation

Cost = 120 w1 = 6 w2 = 20

02468

10121416182022

0 1 2 3 4 5 6 7

x2

x1

Slope of the isocost line

w1x1 w2x2 C

w1x1 C w2x2

x1 Cw1

w2

w1

x2

So the slope is -w2 / w1

Example

C = $120, w1 = 6.00, w2 = 20.00

6x1 20x2 120

6x1 120 20x2

x1 1206

206x2

20 3 13x2

Intercept of the isocost line

x1 Cw1

w2

w1

x2

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 linethat allows production on a given isoquant

Combine an isoquant with several isocost lines

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

0 2 4 6 8 10 12 14x1

0

5

10

15

202x

Consider C = 120 and C = 180

At intersection there are opportunities for trade

0

4

8

12

16

20

24

3 4 5 6 7 8 9 10 11 12 13

X2

X1

Isoquant y = 10000

Isocost 120

Isocost 180

Isocost 160

Isoquant y = 10000

0

4

8

12

16

20

24

4 5 6 7 8 9 10 11 12 13

X2

X1 Isocost 120

Isocost 180

Add C = 160

Isocost 160

Isocost 154.6

Isoquant y = 10000

0

4

8

12

16

20

24

4 5 6 7 8 9 10 11 12 13

X2

X1 Isocost 120

Isocost 180

Add C = 154.6

Isocost 160

Isocost 154.6

Isoquant y = 10000

0

4

8

12

16

20

24

4 5 6 7 8 9 10 11 12 13

X2

X1 Isocost 120

Isocost 180

In review

The least cost combination of inputs

The optimal input combination occurs wherethe 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

MRSx1x2

Δx1

Δx2

MPPx2

MPPx1

w2

w1

MPPx2

MPPx1

Optimality conditions

w2

w1

MRSx1x2

Δx1

Δx2

Substituting we obtain

w2

w1

MRSx1x2

Δx1

Δx2

MPPx2

MPPx1

The price ratio equals the ratio of marginal products

Slope of the isocost line = Slope of the isoquant

We can write this in a more interesting form

Multiply both sides by MPPx1

w2

w1

MPPx2MPPx1

and then divide by w2

MPPx1

w1

MPPx2x2

MPPx1 w2

w1

MPPx2

Graphical representation

0 2 4 6 8 10 12 14x1

0

5

10

15

20

2x

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

d. The slope of the isocost line and the slopeof the isoquant are equal at the optimum

w2

w1

MRSx1x2

MPPx2MPPx1

e. The ratio of prices is equal to the ratio ofmarginal products

w2

w1

MPPx2MPPx1

f. The marginal product of each input divided byits price is equal to the marginal product ofevery other input divided by its price

MPPx1

w1

MPPx2

w2

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

1.0000 -3.33332.0000 -3.33333.0000 -3.3333

12.4687 4.0000 664.6851 2585.7400 110.7809 129.2870 -3.8902 -3.333311.8528 4.1713 -3.5946 739.5588 2465.2134 123.2598 123.2607 -3.3334 -3.33339.7255 5.0000 -2.5672 1010.5290 2050.0940 168.4215 102.5047 -2.0287 -3.33339.3428 5.1972 -1.9411 1063.1321 1975.4051 177.1887 98.7703 -1.8581 -3.33338.0629 6.0000 -1.5941 1254.9695 1724.5840 209.1616 86.2292 -1.3742 -3.33336.9792 6.9063 -1.1959 1447.3307 1508.9951 241.2218 75.4498 -1.0426 -3.33336.8827 7.0000 -1.0291 1466.3867 1489.5380 244.3978 74.4769 -1.0158 -3.33335.9898 8.0000 -0.8929 1663.9176 1305.9560 277.3196 65.2978 -0.7849 -3.33335.2904 9.0000 -0.6994 1855.3017 1155.0760 309.2169 57.7538 -0.6226 -3.33334.7309 10.0000 -0.5595 2044.1823 1026.1700 340.6971 51.3085 -0.5020 -3.33334.2773 11.0000 -0.4535 2232.4134 912.4680 372.0689 45.6234 -0.4087 -3.33333.9071 12.0000 -0.3702 2420.9761 809.4240 403.4960 40.4712 -0.3343 -3.33333.6042 13.0000 -0.3029 2610.3937 713.8360 435.0656 35.6918 -0.2735 -3.3333

Example Table w1 = 6, w2 = 20

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

Intuition for the conditions

The isocost line tells us the rate at which the firmis 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

0

4

8

12

16

20

24

4 5 6 7 8 9 10 11 12 13

X2

X1

Isocost 180

w1 = 6w2 = 20C = 180

3

6.66

10

Isoquant y = 10000

0

4

8

12

16

20

24

4 5 6 7 8 9 10 11 12 13x2

x1

The isoquant tells us the rate at which the firmcan trade one input for the other and remainat the same production level

If there is any difference between the rate atwhich the firm can trade one input for anotherwith no change in production and the rateat which it is able to trade given relative prices,the firm can always make itself better off bymoving up or down the isocost line

Isocost 160

Isoquant y = 10000

0

4

8

12

16

20

24

4 5 6 7 8 9 10 11 12 13x2

x1

Isocost 180

The isoquant tells us the rate at which the firmcan trade one input for the other and remainat the same production level

When the slope of the isoquant is steeperthan the isocost line, the firm will move down the line

When the slope of the isoquant is less steepthan the isocost line, the firm will move up the line

0

4

8

12

16

20

24

4 5 6 7 8 9 10 11 12 13x2

x1

Isoquant y = 10000

When the slope of the isoquant is steeperthan the isocost line, the firm will move down the line

The End