Cost Function with Perfect Substitutes
The two horizontal axes represent the price of the two factors, the vertical axis is the total cost
1
Multiplant Cost Minimization-2 Plants
Firm's Problem:
M iny1,y2 C1(y1) + C2(y2)
Subject to : y1 + y2 = Y
Using the constraint to substitute out y2 :
First-order condition:
∂C1(y1) ∂C2(Y − y1)
=
∂y1
∂y1
i.e. the marginal cost of plant 2 is expressed as a function of the output
of plant 1, given the output target.
Example:
C1 = 16 + 6y12
and
C2 = 240 + 2y22
Then
M C1 = 12y1
M C2 = 4(Y − y1)
Suppose
Y = 20.
Then
12y1 = 4(20 − y1)
or
y1‘ = 5,
hence
y2 = 15.
2
0
marginal cost
100
200
300
Original marginal cost functions
0
5
10
15
20
25
output
marg. cost of plant 1
marg. cost of plant 2
marg. cost of plant 2
0
marginal cost
100
200
300
Output of Firm 1 Given Target of 20
0
5
10
15
20
25
output
marg. cost of plant 1
Other mc2 given output target
3
mc2 given output target
Multiplant Cost Minimization-3 Plants or More
With N plants, given an output target, we end up with
with the same number of unknown variables.
4
N − 1 equations
0
marginal cost
100
200
300
Three plants
0
5
10
15
20
25
output
marg. cost of plant 1
MC of plant 3
marg. cost of plant 2
0
Marginal Cost
100
200
300
Three Plants
0
5
10
15
20
Output
marg. cost of plant 1
marg. cost of plant 2
marg. cost of plant 2
MC of plant 3
With higher MC2 line plant 2 is never used: only 1 and 3
5
25
C1
C2
Output
22
40
70
112
166
232
310
400
502
616
742
880
1030
1192
1366
1552
1750
1960
2182
2416
2662
2920
3190
3472
3766
4072
242
248
258
272
290
312
338
368
402
440
482
528
578
632
690
752
818
888
962
1040
1122
1208
1298
1392
1490
1592
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
MC1
12
24
36
48
60
72
84
96
108
120
132
144
156
168
180
192
204
216
228
240
252
264
276
288
300
312
MC2
4
8
12
16
20
24
28
32
36
40
44
48
52
56
60
64
68
72
76
80
84
88
92
96
100
104
6
Units prod. Units prod.
by plant 1 by plant 2
4
1
8
2
12
3
16
5
20
6
7
9
10
11
13
14
15
17
18
19
MC2 Conditioned by Target
Y=20
Y=12
76
72
68
64
60
56
52
48
44
40
36
32
28
24
20
16
12
8
4
0
-4
-8
-12
-16
-20
-24
44
40
36
32
28
24
20
16
12
8
4
0
-4
-8
-12
-16
-20
-24
-28
-32
-36
-40
-44
-48
-52
-56