Production with two variable inputs
Cobb Douglas production function for maize:
Y = 3.19X1^0.08 X2^0.12
Y maize yields in ton per ha
X1 quantity of phosphate application in kg per ha
X2 quantity of potash application in kg per ha
v1 R 1.75 price of a kg of phosphate
v2 R 3.35 price of a kg of potash
P R 2,150 price of a ton of maize
Question 1
Y = 3.19X1^0.08 X2^0.12
A 3.19
b1 0.08
b2 0.12
Combinations of (fertiliser) inputs and Y (maize yields)
X1\X2
1 2 3 4 5
1 3.19 3.47 3.64 3.77 3.87
2 3.37 3.66 3.85 3.98 4.09
3 3.48 3.79 3.97 4.11 4.23
4 3.56 3.87 4.07 4.21 4.32
5 3.63 3.94 4.14 4.29 4.40
6 3.68 4.00 4.20 4.35 4.47
7 3.73 4.05 4.25 4.40 4.52
8 3.77 4.09 4.30 4.45 4.57
9 3.80 4.13 4.34 4.49 4.61
10 3.84 4.17 4.38 4.53 4.65
11 3.86 4.20 4.41 4.56 4.69
12 3.89 4.23 4.44 4.60 4.72
13 3.92 4.26 4.47 4.63 4.75
14 3.94 4.28 4.50 4.65 4.78
15 3.96 4.31 4.52 4.68 4.81
16 3.98 4.33 4.54 4.70 4.83
17 4.00 4.35 4.57 4.73 4.85
18 4.02 4.37 4.59 4.75 4.88
19 4.04 4.39 4.61 4.77 4.90
20 4.05 4.41 4.63 4.79 4.92
21 4.07 4.42 4.64 4.81 4.94
22 4.08 4.44 4.66 4.82 4.96
23 4.10 4.46 4.68 4.84 4.97
24 4.11 4.47 4.69 4.86 4.99
25 4.13 4.48 4.71 4.87 5.01
26 4.14 4.50 4.72 4.89 5.02
27 4.15 4.51 4.74 4.90 5.04
28 4.16 4.53 4.75 4.92 5.05
29 4.18 4.54 4.76 4.93 5.07
30 4.19 4.55 4.78 4.95 5.08
31 4.20 4.56 4.79 4.96 5.09
32 4.21 4.57 4.80 4.97 5.11
33 4.22 4.59 4.81 4.98 5.12
34 4.23 4.60 4.83 5.00 5.13
35 4.24 4.61 4.84 5.01 5.14
36 4.25 4.62 4.85 5.02 5.15
37 4.26 4.63 4.86 5.03 5.17
38 4.27 4.64 4.87 5.04 5.18
39 4.28 4.65 4.88 5.05 5.19
,40 4.29 4.66 4.89 5.06 5.20
41 4.29 4.67 4.90 5.07 5.21
42 4.30 4.67 4.91 5.08 5.22
43 4.31 4.68 4.92 5.09 5.23
44 4.32 4.69 4.93 5.10 5.24
45 4.33 4.70 4.94 5.11 5.25
46 4.33 4.71 4.94 5.12 5.26
47 4.34 4.72 4.95 5.13 5.27
48 4.35 4.73 4.96 5.13 5.27
49 4.36 4.73 4.97 5.14 5.28
50 4.36 4.74 4.98 5.15 5.29
51 4.37 4.75 4.98 5.16 5.30
52 4.38 4.76 4.99 5.17 5.31
53 4.38 4.76 5.00 5.18 5.32
54 4.39 4.77 5.01 5.18 5.32
55 4.40 4.78 5.02 5.19 5.33
56 4.40 4.78 5.02 5.20 5.34
57 4.41 4.79 5.03 5.21 5.35
58 4.41 4.80 5.04 5.21 5.35
59 4.42 4.80 5.04 5.22 5.36
60 4.43 4.81 5.05 5.23 5.37
61 4.43 4.82 5.06 5.23 5.38
62 4.44 4.82 5.06 5.24 5.38
63 4.44 4.83 5.07 5.25 5.39
64 4.45 4.84 5.08 5.25 5.40
65 4.45 4.84 5.08 5.26 5.40
66 4.46 4.85 5.09 5.27 5.41
67 4.47 4.85 5.09 5.27 5.42
68 4.47 4.86 5.10 5.28 5.42
69 4.48 4.86 5.11 5.29 5.43
70 4.48 4.87 5.11 5.29 5.44
71 4.49 4.88 5.12 5.30 5.44
72 4.49 4.88 5.12 5.30 5.45
73 4.50 4.89 5.13 5.31 5.45
74 4.50 4.89 5.14 5.32 5.46
75 4.51 4.90 5.14 5.32 5.47
76 4.51 4.90 5.15 5.33 5.47
77 4.52 4.91 5.15 5.33 5.48
78 4.52 4.91 5.16 5.34 5.48
79 4.52 4.92 5.16 5.34 5.49
80 4.53 4.92 5.17 5.35 5.49
81 4.53 4.93 5.17 5.35 5.50
82 4.54 4.93 5.18 5.36 5.51
83 4.54 4.94 5.18 5.36 5.51
84 4.55 4.94 5.19 5.37 5.52
85 4.55 4.95 5.19 5.38 5.52
86 4.56 4.95 5.20 5.38 5.53
87 4.56 4.96 5.20 5.39 5.53
88 4.56 4.96 5.21 5.39 5.54
89 4.57 4.96 5.21 5.39 5.54
90 4.57 4.97 5.22 5.40 5.55
91 4.58 4.97 5.22 5.40 5.55
92 4.58 4.98 5.23 5.41 5.56
93 4.58 4.98 5.23 5.41 5.56
94 4.59 4.99 5.23 5.42 5.57
95 4.59 4.99 5.24 5.42 5.57
96 4.60 4.99 5.24 5.43 5.58
97 4.60 5.00 5.25 5.43 5.58
98 4.60 5.00 5.25 5.44 5.58
, 99 4.61 5.01 5.26 5.44 5.59
100 4.61 5.01 5.26 5.45 5.59
8.40
7.90
7.40
6.90
Y (MAIZE YIELD T/HA)
6.40
5.90
5.40
4.90
4.40
3.90
3.40
96
91
86
81
76
71
66
61
56
51
46
41
36
31
26
21
X2 (QUANTITY OF POTASH APPLICATION IN KG PER HA)
3.40-3.90 3.90-4.40 4.40-4.90
Question 2
MRTS X1X2 = -(b1/b2)*(X2/X1)
b1 0.08
b2 0.12
X1 33
X2 51
MRTS = -1.03 at X1 = 33 and X2 = 51
Question 3
v1 R 1.60
v2 R 3.40
, Budget constraint X1 X2
C* = R165 0 49 R 165
103 0
C* = R185 0 54 R 185
116 0
C* = R215 0 63 R 215
134 0
Question 3
A 3.19
b1 0.08
b2 0.12
p R 2,150
v1 1.75
v2 3.35
Profit function
П = pY - v1X1 - v2X2 y = AX1^b1 X2^b2
П = pAX1^b1 X2^b2 - v1X1 - v2X2
Take first derivative of profit function with regards to each input
(b1 p AX1^(b1-1) X2^b2)/ v1 = ( b2 p AX1^b1 X2^(b2-1)) / v2 = 1
Profit maximising point will be on the expansion path --> X2 = (v1/v2)(b2/b1) X1
X2 = 0.78 X1
Cobb Douglas production function for maize:
Y = 3.19X1^0.08 X2^0.12
Y maize yields in ton per ha
X1 quantity of phosphate application in kg per ha
X2 quantity of potash application in kg per ha
v1 R 1.75 price of a kg of phosphate
v2 R 3.35 price of a kg of potash
P R 2,150 price of a ton of maize
Question 1
Y = 3.19X1^0.08 X2^0.12
A 3.19
b1 0.08
b2 0.12
Combinations of (fertiliser) inputs and Y (maize yields)
X1\X2
1 2 3 4 5
1 3.19 3.47 3.64 3.77 3.87
2 3.37 3.66 3.85 3.98 4.09
3 3.48 3.79 3.97 4.11 4.23
4 3.56 3.87 4.07 4.21 4.32
5 3.63 3.94 4.14 4.29 4.40
6 3.68 4.00 4.20 4.35 4.47
7 3.73 4.05 4.25 4.40 4.52
8 3.77 4.09 4.30 4.45 4.57
9 3.80 4.13 4.34 4.49 4.61
10 3.84 4.17 4.38 4.53 4.65
11 3.86 4.20 4.41 4.56 4.69
12 3.89 4.23 4.44 4.60 4.72
13 3.92 4.26 4.47 4.63 4.75
14 3.94 4.28 4.50 4.65 4.78
15 3.96 4.31 4.52 4.68 4.81
16 3.98 4.33 4.54 4.70 4.83
17 4.00 4.35 4.57 4.73 4.85
18 4.02 4.37 4.59 4.75 4.88
19 4.04 4.39 4.61 4.77 4.90
20 4.05 4.41 4.63 4.79 4.92
21 4.07 4.42 4.64 4.81 4.94
22 4.08 4.44 4.66 4.82 4.96
23 4.10 4.46 4.68 4.84 4.97
24 4.11 4.47 4.69 4.86 4.99
25 4.13 4.48 4.71 4.87 5.01
26 4.14 4.50 4.72 4.89 5.02
27 4.15 4.51 4.74 4.90 5.04
28 4.16 4.53 4.75 4.92 5.05
29 4.18 4.54 4.76 4.93 5.07
30 4.19 4.55 4.78 4.95 5.08
31 4.20 4.56 4.79 4.96 5.09
32 4.21 4.57 4.80 4.97 5.11
33 4.22 4.59 4.81 4.98 5.12
34 4.23 4.60 4.83 5.00 5.13
35 4.24 4.61 4.84 5.01 5.14
36 4.25 4.62 4.85 5.02 5.15
37 4.26 4.63 4.86 5.03 5.17
38 4.27 4.64 4.87 5.04 5.18
39 4.28 4.65 4.88 5.05 5.19
,40 4.29 4.66 4.89 5.06 5.20
41 4.29 4.67 4.90 5.07 5.21
42 4.30 4.67 4.91 5.08 5.22
43 4.31 4.68 4.92 5.09 5.23
44 4.32 4.69 4.93 5.10 5.24
45 4.33 4.70 4.94 5.11 5.25
46 4.33 4.71 4.94 5.12 5.26
47 4.34 4.72 4.95 5.13 5.27
48 4.35 4.73 4.96 5.13 5.27
49 4.36 4.73 4.97 5.14 5.28
50 4.36 4.74 4.98 5.15 5.29
51 4.37 4.75 4.98 5.16 5.30
52 4.38 4.76 4.99 5.17 5.31
53 4.38 4.76 5.00 5.18 5.32
54 4.39 4.77 5.01 5.18 5.32
55 4.40 4.78 5.02 5.19 5.33
56 4.40 4.78 5.02 5.20 5.34
57 4.41 4.79 5.03 5.21 5.35
58 4.41 4.80 5.04 5.21 5.35
59 4.42 4.80 5.04 5.22 5.36
60 4.43 4.81 5.05 5.23 5.37
61 4.43 4.82 5.06 5.23 5.38
62 4.44 4.82 5.06 5.24 5.38
63 4.44 4.83 5.07 5.25 5.39
64 4.45 4.84 5.08 5.25 5.40
65 4.45 4.84 5.08 5.26 5.40
66 4.46 4.85 5.09 5.27 5.41
67 4.47 4.85 5.09 5.27 5.42
68 4.47 4.86 5.10 5.28 5.42
69 4.48 4.86 5.11 5.29 5.43
70 4.48 4.87 5.11 5.29 5.44
71 4.49 4.88 5.12 5.30 5.44
72 4.49 4.88 5.12 5.30 5.45
73 4.50 4.89 5.13 5.31 5.45
74 4.50 4.89 5.14 5.32 5.46
75 4.51 4.90 5.14 5.32 5.47
76 4.51 4.90 5.15 5.33 5.47
77 4.52 4.91 5.15 5.33 5.48
78 4.52 4.91 5.16 5.34 5.48
79 4.52 4.92 5.16 5.34 5.49
80 4.53 4.92 5.17 5.35 5.49
81 4.53 4.93 5.17 5.35 5.50
82 4.54 4.93 5.18 5.36 5.51
83 4.54 4.94 5.18 5.36 5.51
84 4.55 4.94 5.19 5.37 5.52
85 4.55 4.95 5.19 5.38 5.52
86 4.56 4.95 5.20 5.38 5.53
87 4.56 4.96 5.20 5.39 5.53
88 4.56 4.96 5.21 5.39 5.54
89 4.57 4.96 5.21 5.39 5.54
90 4.57 4.97 5.22 5.40 5.55
91 4.58 4.97 5.22 5.40 5.55
92 4.58 4.98 5.23 5.41 5.56
93 4.58 4.98 5.23 5.41 5.56
94 4.59 4.99 5.23 5.42 5.57
95 4.59 4.99 5.24 5.42 5.57
96 4.60 4.99 5.24 5.43 5.58
97 4.60 5.00 5.25 5.43 5.58
98 4.60 5.00 5.25 5.44 5.58
, 99 4.61 5.01 5.26 5.44 5.59
100 4.61 5.01 5.26 5.45 5.59
8.40
7.90
7.40
6.90
Y (MAIZE YIELD T/HA)
6.40
5.90
5.40
4.90
4.40
3.90
3.40
96
91
86
81
76
71
66
61
56
51
46
41
36
31
26
21
X2 (QUANTITY OF POTASH APPLICATION IN KG PER HA)
3.40-3.90 3.90-4.40 4.40-4.90
Question 2
MRTS X1X2 = -(b1/b2)*(X2/X1)
b1 0.08
b2 0.12
X1 33
X2 51
MRTS = -1.03 at X1 = 33 and X2 = 51
Question 3
v1 R 1.60
v2 R 3.40
, Budget constraint X1 X2
C* = R165 0 49 R 165
103 0
C* = R185 0 54 R 185
116 0
C* = R215 0 63 R 215
134 0
Question 3
A 3.19
b1 0.08
b2 0.12
p R 2,150
v1 1.75
v2 3.35
Profit function
П = pY - v1X1 - v2X2 y = AX1^b1 X2^b2
П = pAX1^b1 X2^b2 - v1X1 - v2X2
Take first derivative of profit function with regards to each input
(b1 p AX1^(b1-1) X2^b2)/ v1 = ( b2 p AX1^b1 X2^(b2-1)) / v2 = 1
Profit maximising point will be on the expansion path --> X2 = (v1/v2)(b2/b1) X1
X2 = 0.78 X1
