1. Isoquants, Isocosts and Cost Minimization
Overheads

2. 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

3. y = f (x1, x2, x3, . . . xn ) y Holding other inputs fixed,
the production function looks like this
y = f (x1, x2, x3, xn )
350
300
Output -y y
250
200
150
100 50 2 4 6 8 10 12
Input -x

4. 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

5. Mathematically we define MPP as

6. Graphically marginal product looks like this60 50
A MPP
Output -y 40 30 20 10
-10 1 2 3 4 5 6 7 8 9 10 11
-20
-30
-40
Input -x

7. choose the least cost x’s
The Cost Minimization Problem
Pick y; observe w1, w2, etc;
choose the least cost x’s

8. Isoquants The word “iso”means same while “quant” stand for quantity
An isoquant curve in two dimensions represents
all combinations of two inputs that produce the
same quantity of output
The word “iso”means same
while “quant” stand for quantity

9. 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

10. Production function for the hay example

11. Another view

12. Yet another view (low x’s)

13. With a horizontal plane at y = 250,000

14. With a horizontal plane at 100,000

15. Contour plot

16. Another contour plot

17. There are many ways to produce 2,000 bales of hay per hour
Workers Tractor-Wagons Total Cost AC

18. Plotting these points in x1 - x2 space we obtain

19. Or Isoquant y = 2,000 x2 = 4, x1 = 2.636 Isoquant y = 2000 X X 12 10 82 4 6 8 10 12 14 X 2

20. Cutting Plane for y = 10,000

21. Isoquant for y = 10,000

22. Only the negatively sloped portions
of the isoquant are efficient

23. Isoquant for y = 10,000 x1 x2 output y -- 1 10,000 -- 2 10,000
,000
,000
,000
,000
,000
,000
,000
,000
,000

24. y = 10,000

25. y = 2, y = 10,000

26. Graphical representation
Isoquants y = 2,000, y = 10,000 14 12
Isoquant y = 10000 x1 10
Isoquant y = 2000 8 6 4 2 2 4 6 8 10 12 14 x 2

27. y = 2,000, y = 5,000, y = 10,000

28. More levels

29. And even more

30. Comparison to full map

31. 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)

32. 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

33. 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

34.  The Marginal Rate of Substitution (MRTS)  x1 2 4 6 8 10 12 14 x 14
Isoquant y = 10000 x1 10
Isoquant y = 2000 8  6  4 2  2 4 6 8 10 12 14 x 2

35. 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

36. Example calculations y = 2,000
Workers Tractor-Wagons
x1 x2
10 1
Change x2 from 1 to 2

37. Example calculations y = 2,000
Change x2 from 2 to 3
Workers Tractor-Wagons
x1 x2
10 1

38. Change x2 from 5 to 6 More example calculations y = 10,000 x1 x2
Change x2 from 5 to 6

39. 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 units with a one unit increase in input 2
and keep production the same

40. 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

41. 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

42. Rearrange this expression by subtracting MPPx2  x2
from both sides,
Then divide both sides by MPPx1
Then divide both sides by  x2

43. 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

44. 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

45.  The Marginal Rate of Substitution (MRTS) x1 2 4 6 8 10 12 14 x 14 12
Isoquant y = 2000 8  6 4 2  2 4 6 8 10 12 14 x 2

46. Approx x1 x2 MRS MPP1 MPP2 12.4687 4.0000 --- 664.6851 2585.7400

47. Approx x1 x2 MRS MPP1 MPP2 12.4687 4.0000 --- 664.6851 2585.7400
x2 rises and MRS falls

48. Isocost lines Quantities of inputs - x1, x2, x3, . . .
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, . . .

49. Graphical representation
Cost = w1 = 6 w2 = 20 22 x 20 1 18 16 14 12 10 8 6 4 2 1 2 3 4 5 6 7 x 2

50. Slope of the isocost line
So the slope is -w2 / w1

51. Example
C = $120, w1 = 6.00, w2 = 20.00

52. Intercept of the isocost line
So the intercept is C / w1
With higher cost, the isocost line moves out

53. 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

54. Combine an isoquant with several isocost lines
Isocost lines for $20, $60, $120, $180, $240, $360

55. Consider C = 120 and C = 180
Isoquant y = 10000 24 X
Isocost 120 1 20
Isocost 180 16 12 8 4 3 4 5 6 7 8 9 10 11 12 13 X 2
At intersection there are opportunities for trade

56. Add C = 160 Isoquant y = 10000 X Isocost 120 Isocost 180 Isocost 160 X24
Isoquant y = 10000 X 20
Isocost 120
Isocost 180 1 16
Isocost 160 12 8 4 4 5 6 7 8 9 10 11 12 13 X 2

57. Add C = 154.6 Isoquant y = 10000 X Isocost 120 Isocost 180 Isocost 16024
Isoquant y = 10000 X 20
Isocost 120 1
Isocost 180 16
Isocost 160
Isocost 154.6 12 8 4 4 5 6 7 8 9 10 11 12 13 X 2

58. In review Isoquant y = 10000 X Isocost 120 Isocost 180 Isocost 16024
Isoquant y = 10000 X 20
Isocost 120 1
Isocost 180 16
Isocost 160
Isocost 154.6 12 8 4 4 5 6 7 8 9 10 11 12 13 X 2

59. 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

60. Slope of the isocost line
-w2 / w1
Slope of the isoquant

61. Optimality conditions
Slope of the isocost line = Slope of the isoquant
Substituting we obtain
The price ratio equals the ratio of marginal products

62. We can write this in a more interesting form
Multiply both sides by MPPx1
and then divide by w2

63. Graphical representation

64. 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

65. d. The slope of the isocost line and the slope
of the isoquant are equal at the optimum
e. The ratio of prices is equal to the ratio of
marginal products

66. 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

67. 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 MRS -w2 / w1

68. 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

69. w1 = 6
w2 = 20
C = 180 24 X 20 1
Isocost 180 16 12 10 8
6.66 4 3 4 5 6 7 8 9 10 11 12 13 X 2

70. 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
Isoquant y = 10000 x 20 1 16 12 8 4 4 5 6 7 8 9 10 11 12 13 x2

71. 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

72. 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
Isoquant y = 10000 x 20 1
Isocost 180 16
Isocost 160 12 8 4 4 5 6 7 8 9 10 11 12 13 x2

73. 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

74. When the slope of the isoquant is steeper
than the isocost line, the firm will move down the line 24
Isoquant y = 10000 x 20 1 16 12 8 4 4 5 6 7 8 9 10 11 12 13 x2

75. The End