1. An Animated Instructional Module
for Teaching Production Economics
with the Aid of
Three-Dimensional Graphics
David L. Debertin
University of Kentucky Y
250 2 4
167 83 6 8 10 12 14 16 18 20 X1 20 18 16 14 12 10 X2 8 6 4

2. This instructional module is based on the
polynomial production function 2 3 2
y = x + x x + x x 1 2 1 1 2 3
x x x . 2 1 2
This production function was chosen
for a number of reasons.

3. 1. It possesses a region of increasing marginal returns,
a region of diminishing marginal returns
and a region of negative marginal returns
to the variable inputs
x and x . 1 2
dy = x x x . 2 1 1 2 dx 1 2
dy = x x x . 2 2 1 dx 2

4. 2. Since the function has a finite maximum,
there are "ring" isoquants, centered at the
maximum output level.
At the maximum output level,
dy and dy are both zero. dx dx 1 2
Maximum output is units
corresponding to
x = x = units 1 2

5. 3. There is a global point of profit maximization.
This point occurs at an output level
less than the global point of
profit maximization.
For example, if the price of the product (y)
is $1.00 per unit, the price of x is $3.00 1
per unit, and the price of x is $1.50 per unit, 2
then profit is maximum at x =15.21 and x = 15.71 1 2
units. Total Revenue is $234.85;
Total Cost for x and x is $69.20; 1 2
Profit is Total Revenue - Total Cost = $165.20

6. In the following sequence, the production surface
for the polynomial is sliced horizontally at
various levels.
The isoquant at each output level appears in red.
Note that isoquants at low output levels
are concave to the origin, but as the output
level increases, the isoquants become
convex to the origin.
You are looking at the 3-D surface from the
origin. x is at your right; x at your left. 1 2
Output (y) is measured on the
vertical axis.

7. Y
250
167 83 20 20 18 18 16 16 14 14 12 12 10 10 8 8 X2 6 X1 6 4 4 2 2

8. Y
250
167 83 20 20 18 18 16 16 14 14 12 12 10 10 8 8 X2 6 X1 6 4 4 2 2

9. Y
250
167 83 20 20 18 18 16 16 14 14 12 12 10 10 8 8 X2 6 X1 6 4 4 2 2

10. Y
250
167 83 20 20 18 18 16 16 14 14 12 12 10 10 8 8 X2 6 X1 6 4 4 2 2

11. Y
250
167 83 20 20 18 18 16 16 14 14 12 12 10 10 8 8 X2 6 X1 6 4 4 2 2

12. Y
250
167 83 20 20 18 18 16 16 14 14 12 12 10 10 8 8 X2 6 X1 6 4 4 2 2

13. Y
250
167 83 20 20 18 18 16 16 14 14 12 12 10 10 8 8 X2 6 X1 6 4 4 2 2

14. Y
250
167 83 20 20 18 18 16 16 14 14 12 12 10 10 8 8 X2 6 X1 6 4 4 2 2

15. Y Y
250
250
167
167 83 83 20 20 20 20 18 18 18 18 16 16 16 16 14 14 14 14 12 12 12 12 10 10 10 10 8 8 8 8 X2 X2 6 6 X1 X1 6 6 4 4 4 4 2 2 2 2

16. Y
250
167 83 20 20 18 18 16 16 14 14 12 12 10 10 8 8 X2 6 X1 6 4 4 2 2

17. Y Y
250
250
167
167 83 83 20 20 20 20 18 18 18 18 16 16 16 16 14 14 14 14 12 12 12 12 10 10 10 10 8 8 8 8 X2 X2 6 6 X1 X1 6 6 4 4 4 4 2 2 2 2

18. Y
250
167 83 20 20 18 18 16 16 14 14 12 12 10 10 8 8 X2 6 X1 6 4 4 2 2

19. Y Y
250
250
167
167 83 83 20 20 20 20 18 18 18 18 16 16 16 16 14 14 14 14 12 12 12 12 10 10 10 10 8 8 8 8 X2 X2 6 6 X1 X1 6 6 4 4 4 4 2 2 2 2

20. Y
250
167 83 20 20 18 18 16 16 14 14 12 12 10 10 8 8 X2 6 X1 6 4 4 2 2

21. Y Y
250
250
167
167 83 83 20 20 20 20 18 18 18 18 16 16 16 16 14 14 14 14 12 12 12 12 10 10 10 10 8 8 8 8 X2 X2 6 6 X1 X1 6 6 4 4 4 4 2 2 2 2

22. Y
250
167 83 20 20 18 18 16 16 14 14 12 12 10 10 8 8 X2 6 X1 6 4 4 2 2

23. Y Y
250
250
167
167 83 83 20 20 20 20 18 18 18 18 16 16 16 16 14 14 14 14 12 12 12 12 10 10 10 10 8 8 8 8 X2 X2 6 6 X1 X1 6 6 4 4 4 4 2 2 2 2

24. 2 4 6 8 10 12 14 16 18 20 X1 X2 Y 83
167
250

25. 2 4 6 8 10 12 14 16 18 20 X1 X2 Y 83
167
250 2 4 6 8 10 12 14 16 18 20 X1 X2 Y 83
167
250

26. In the following sequence, isoquants representing
various output levels appear in different colors
and the output level represented by each isoquant
corresponds with the key at the bottom of the
chart.

27. The budget constraint is represented by red
lines of constant slope P1/P2 where P1 is the
price of input x , and P2 is the price of input 1
x . 2
Increases in the amount of money available
for the purchase of inputs shift the budget
constraint outward.
Each budget constraint is tangent (just touches)
an isoquant.
These points, here marked by blue circles,
are where P1/P2 = the Marginal Rate of
Substitution of x for x . 1 2

28. Valid constrained output maximization points are
only those on isoquants that are convex to the
origin of the graph. Points on concave isoquants
are constrained output minimization points,
and are marked with a yellow X.
The expansion path connects valid points of
constrained output maximization, and is
shown in green.

29. Ridge line 1 connects all points of zero slope on the isoquants.
Ridge line 2 connects all points of infinite slope
on the isoquants.
Ridge lines, shown here in blue-green
intersect at the global point of
output maximization.
This occurs at x = x = and y = 1 2

30. Pseudo Scale Line 1 connects profit maximization
points for x holding x constant. 1 2
Each point on Pseudo Scale Line 1 is defined by
MPP x = P1/Py, where P1 is the price of x and 1 1
Py is the output price.
Pseudo Scale Line 2 connects profit maximization
points for x holding x constant. 2 1
Each point on Pseudo Scale Line 2 is defined by
MPP x = P2/Py, where P2 is the price of x and 2 2
Py is the output price.
Pseudo Scale Lines, shown here in orange,
Converge at the point of global profit maximization.

31. What happens to Pseudo Scale Lines and the
position of the Expansion Path when one of the
input prices changes is also shown.
First, P2 (the price of x ) is increased. 2
Pseudo Scale Line 2 moves in from Ridge Line 2,
and the Expansion Path now favors the use of the
now relatively cheaper input x . 1
Then P2 (the price of x ) is decreased. 2
Pseudo Scale Line 2 moves toward Ridge Line 2,
and the Expansion Path now favors the use of the
now relatively cheaper input x . 2
Pseudo scale line 1 has not moved in either case
as the price of x has not changed. 1

32. X2 20 18 16 14 12 10 8 6 4 2 2 4 6 8 X1 10 12 14 16 18 20 Y 24 47 71 95
119
142
166
190
213
237

33. X2 20 18 16 14 12 10 8 6 4 2 P1 P2 2 4 6 8 X1 10 12 14 16 18 20 Y 24 47 71 95
119
142
166
190
213
237

34. X2 20 18 16 14 12 10 8 6 X X 4 X 2 P1 P2 2 4 6 8 X1 10 12 14 16 18 20 Y 24 47 71 95
119
142
166
190
213
237

35. X2 20 18
Global
Output
Maximum 16
Expansion 14
Path 12 10 8 6 X X 4 X 2 P1 P2 2 4 6 8 X1 10 12 14 16 18 20 Y 24 47 71 95
119
142
166
190
213
237

36. X2 20 18
Global
Output
Maximum 16
Expansion 14
Path 12 10 8 6 X X 4 X 2 P1 P2 2 4 6 8 X1 10 12 14 16 18 20 Y 24 47 71 95
119
142
166
190
213
237

37. X2 20 18
Global
Output
Maximum 16
Ridge
Line 2
Expansion 14
Path 12 10 8 6 X X 4 X 2 P1
Ridge P2
Line 1 2 4 6 8 X1 10 12 14 16 18 20 Y 24 47 71 95
119
142
166
190
213
237

38. X2 20 18
Global
Output
Maximum 16
Ridge
Line 2
Expansion 14
Path 12
MPPx2=
Px2/Py 10 8 6 X X 4
MPPx1 = X
Px1/Py 2 P1
Ridge P2
Line 1 2 4 6 8 X1 10 12 14 16 18 20 Y 24 47 71 95
119
142
166
190
213
237

39. X2 20 18
Global
Output
Maximum 16
Ridge
Line
Global 2
Profit
Maximum
Expansion 14
Pseudo
Path
Scale
Line 2 12
MPPx2=
Px2/Py 10 8 6 X X
Pseudo 4
Scale Line 1
MPPx1 = X
Px1/Py 2 P1
Ridge P2
Line 1 2 4 6 8 X1 10 12 14 16 18 20 Y 24 47 71 95
119
142
166
190
213
237

40. X2 20 18
Global
Output
Maximum 16
Ridge
Line 2 14
Expansion
Global
Path
Profit 12
Maximum
Pseudo
Scale
Line 2 10
MPPx2=
Px2/Py 8 X 6 X
Pseudo 4
Scale Line 1
MPPx1 =
Px1/Py 2 P1
Ridge P2
Line 1 2 4 6 8 X1 10 12 14 16 18 20 Y 24 47 71 95
119
142
166
190
213
237

41. X2 20 18
Global
Global
Output
Profit
Maximum
Maximum 16
Ridge
Line 2
Pseudo
Scale
Line 2 14
MPPx2= 12 X
Px2/Py
Expansion
Path 10 8 6
Pseudo 4
Scale
Line 1
MPPx1 =
Px1/Py 2 P1
Ridge P2
Line X X 1 2 4 6 8 X1 10 12 14 16 18 20 Y 24 47 71 95
119
142
166
190
213
237

42. In the following sequence, vertical slices
of the production surface are made.
The slice is at the budget constraint angle
relative to the origin of the graph.
Each slice represents a different level
of the budget constraint.
The locus ABC represents the function being
maximized or minimized in the constrained
optimization problem.
If B is lower than A or C, then the function is
being minimized.
If B is higher than A or C, then the function is
being maximized.

43. Y
250
167 83 20 20 18 18 16 16 14 14 12 12 10 10 8 8 X2 6 X1 6 4 4 2 2

44. Y
250
167 83 20 20 18 18 16 16 14 14 A 12 12 B C 10 10 8 8 X2 6 X1 6 4 4 2 2

45. Y
250
167 83 20 20 A 18 18 C B 16 16 14 14 12 12 10 10 8 8 X2 6 X1 6 4 4 2 2

46. Y
250
167 83 A C B 20 20 18 18 16 16 14 14 12 12 10 10 8 8 X2 6 X1 6 4 4 2 2

47. Y
250
167 83 B A C 20 20 18 18 16 16 14 14 12 12 10 10 8 8 X2 6 X1 6 4 4 2 2

48. Y
250
167 B 83 A C 20 20 18 18 16 16 14 14 12 12 10 10 8 8 X2 6 X1 6 4 4 2 2

49. Y
250 B
167 83 A C 20 20 18 18 16 16 14 14 12 12 10 10 8 8 X2 6 X1 6 4 4 2 2

50. 2 4 6 8 10 12 14 16 18 20 X1 X2 Y 83
167
250 A B C

51. Y Y
250
250 B
167
167 83 83 A C 20 20 20 20 18 18 18 18 16 16 16 16 14 14 14 14 12 12 12 12 10 10 10 10 8 8 8 8 X2 X2 6 6 X1 X1 6 6 4 4 4 4 2 2 2 2

52. Global Output Max Global Profit Max Constrained Output Max Y 250 16783 20 20 18 18 16 16 14 14 12 12 10 10 8 8 X2 6 X1 6 4 4 2 2

53. 2 4 6 8 10 12 14 16 18 20 X1 X2 Y 83
167
250 A B C

54. 2 4 6 8 10 12 14 16 18 20 X1 X2 Y 83
167
250 A B C Y
250
167 83 20 20 18 18 16 16 14 14 12 12 10 10 8 8 X2 6 X1 6 4 4 2 2

55. The following sequences illustrate three
approaches for precisely locating Pseudo Scale
Lines and the point of Global Profit Maximization.
Profit maximization in the single input case
always occurs at the point where
Py MPPx = Px
or MPPx = Px/Py, where Px is the price of x.
Consider this relationship in the two-input
case.
Mppx = P1/Py 1
MPPx = P2/Py
, where 2
P1 = price of x ; P2 = price of x . 1 2

56. In this sequence, the budget constraint
C = $3.00 x x is represented by a 1 2
hyperplane of constant slope (since input prices
are constant.
The output quantity is multiplied by the output
price, assumed to be $1.00 per unit.
The hyperplane, shown in red, is raised
to locate the position of the Pseudo Scale Lines
and the global point of profit maximization.
MPPx = P1/Py MPPx = P2/Py. 1 2

57. Y
250
167 83 20 20 18 18 16 16 14 14 12 12 10 10 8 8 X2 6 6 4 X1 4 2 2

58. Y
250
167 83 20 20 18 18 16 16 14 14 12 12 10 10 8 8 X2 6 6 4 X1 4 2 2

59. Y
250
167 83 20 20 18 18 16 16 14 14 12 12 10 10 8 8 X2 6 6 4 X1 4 2 2

60. Global Output Maximum Y
Ridge Line 2
Ridge Line 1
250
Input Price
Hyperplane
167 83 20 20 18 18 16 16 14 14 12 12 10 10 8 8 X2 6 6 4 X1 4 2 2

61. Global Output Maximum Y
Global Profit Maximum
Ridge Line 2
250
Ridge Line 1
Input Price
Hyperplane
Pseudo Scale Line 2
167
Pseudo Scale Line 1 83 20 20 18 18 16 16 14 14 12 12 10 10 8 8 X2 6 6 4 X1 4 2 2

62. In this sequence, the gross revenue
surface is superimposed on the profit surface.
The gross revenue surface appears in white,
the profit surface in blue.
The global point of revenue maximization
and the global point of profit maximization
are both shown.
Py = $1.00; P1 = $3.00; P2 = $1.50

63. $
250
153 57
-40 20 20 18 18 16 16 14 14 12 12 10 10 8 8 X2 6 6 4 X1 4 2 2

64. Y $
250
250
153
153 57 57
-40
-40 20 20 18 18 16 16 14 14 12 12 10 10 8 8 X2 6 6 4 X1 4 2 2

65. $ Y
250
250
153
153 57 57
-40
-40 20 20 18 18 16 16 14 14 12 12 10 10 8 8 X2 6 6 4 X1 4 2 2
Total Revenue Surface
Profit Surface

66. Revenue,
Profit
Global Revenue Maximization $
Global Profit Maximization
250
250
153
153 57 57
-40
-40 20 20 18 18 16 16 14 14 12 12 10 10 8 8 X2 6 6 4 X1 4 2 2
Total Revenue Surface
Profit Surface

67. Revenue,
Profit
Global Revenue Maximization $
Global Profit Maximization
250
250
Ridge Line 2
Ridge Line 1
Pseudo Scale Line 2
153
153
Pseudo Scale Line 1 57 57
-40
-40 20 20 18 18 16 16 14 14 12 12 10 10 8 8 X2 6 6 4 X1 4 2 2
Total Revenue Surface
Profit Surface

68. In this sequence, isorevenue contour lines
represent points of constant total revenue.
They appear as dotted lines. Except for the units
change, they look just like isoquants.
Isoprofit contour lines are superimposed on the
same graph.
They are solid lines.
Ridge lines connect points of zero or infinite
slope on the isorevenue lines (isoquants).
These are shown in cyan.
Pseudo Scale Lines connect points of
zero or infinite slope on isoprofit lines.
They are shown in orange.

69. Isorevenue Lines Isoprofit Lines X2 20 18 16 14 12 10 8 6 4 2 2 4 6 82 4 6 8 10 12 14 16 18 20 X1
Isorevenue Lines
Isoprofit Lines

70. Isorevenue Lines Isoprofit Lines X2 2 4 6 8 10 12 14 16 18 20 20 18 162 4 6 8 10 12 14 16 18 20 20 18 16 14 12 10 8 6 4 2 2 4 6 8 10 12 14 16 18 20 X1
Isorevenue Lines
Isoprofit Lines

71. Isorevenue Lines Isoprofit Lines X2 2 4 6 8 10 12 14 16 18 20 20 182 4 6 8 10 12 14 16 18 20 20 18
Global Output Max 16 14 12 10 8 6 4 2 2 4 6 8 10 12 14 16 18 20 X1
Isorevenue Lines
Isoprofit Lines

72. Isorevenue Lines Isoprofit Lines X2 20 2 4 6 8 10 12 14 16 18 20 182 4 6 8 10 12 14 16 18 20 18
Global Output Max 16
Global Profit Max 14 12 10 8 6 4 2 2 4 6 8 10 12 14 16 18 20 X1
Isorevenue Lines
Isoprofit Lines