A perfectly competitive firm operates in the short-run with labor as its only va
ID: 1225205 • Letter: A
Question
A perfectly competitive firm operates in the short-run with labor as its only variable factor. Its production function is: Q = -L3 + 10L2 + 88L where Q is output per week measured in tons and L is the number of workers employed. The weekly wage is $324 and the product sells for $3.24 per ton. (a) At what weekly output is marginal cost equal to average variable cost? (b) What is the minimum product price at which the firm will operate in the short-run? (c) How many workers should the firm employ to maximize profits?
Explanation / Answer
Use the table to analyse the behaviour of this firm
Q
L
WAGE
TC
AVC
MPL
VMPL
TR
PROFIT
0
0
324
0
0
0
97
1
324
324
3.34
97.00
314.28
314.28
-9.72
208
2
324
648
3.12
111.00
359.64
673.92
25.92
327
3
324
972
2.97
119.00
385.56
1059.48
87.48
448
4
324
1296
2.89
121.00
392.04
1451.52
155.52
565
5
324
1620
2.87
117.00
379.08
1830.60
210.60
672
6
324
1944
2.89
107.00
346.68
2177.28
233.28
763
7
324
2268
2.97
91.00
294.84
2472.12
204.12
832
8
324
2592
3.12
69.00
223.56
2695.68
103.68
873
9
324
2916
3.34
41.00
132.84
2828.52
-87.48
880
10
324
3240
3.68
7.00
22.68
2851.20
-388.80
847
11
324
3564
4.21
-33.00
-106.92
2744.28
-819.72
(a) The given firm is perfectly competitive so that its efficient scale, or the minimum of AVC must be occuring at a level where AVC = MC. Note the relationship between MC and MPL as well as AVC and APL (Average product of labor). The peak of MPL and APL corresponds to the trough of MC and AVC. This implies that the maximum value of APL will give the required number of workers to produce a level of output at which AVC is minimum
APL = Q/L = - L2 + 10L + 88
Minimum of APL implies the first derivative of APL being set equal to zero
d(- L2 + 10L + 88)/dL = 0
- 2L + 10 = 0
L = 5
Hence, the minimum of Average variable cost is found at a level of 5 units of labor inputs. When L = 5, output Q = -(5)3 + 10(5)2 + 88(5) = 565 tons.
At a level of 565 tons of weekly output is marginal cost equal to average variable cost.
(b) The minimum product price at which the firm will operate in the short-run is the minimum of AVC since the firm will not operate if the price falls below this level. This is the minimum threshold that the firm must receive to survive in the short run. So the minimum price is $2.87 per ton found as VC/Q = 324*5/565 = $2.87
(c) The given firm first allocate the number of labor to be hired and for that it must realize its marginal product of labor. At equilibrium level of output, the wage rate is equal to the value of the marginal product labor:
VMPL= w, where VMPL is nothing but the price of the marginal product of labor, P*MPL
Now MPL is the derivative of production function with respect to labor:
MPL = dQ/dL
= -3L2 + 20L + 88.
The optimal quantity of labor hired will be the one whose value is equal to the wage rate given to it:
P*MPL = w
MPL = w/p
-3L2 + 20L + 88 = 100
3L2 - 20L + 12 = 0
Solving this quadratic equation gives an optimal level of labor L* = 6 units that will maximize profits
Q
L
WAGE
TC
AVC
MPL
VMPL
TR
PROFIT
0
0
324
0
0
0
97
1
324
324
3.34
97.00
314.28
314.28
-9.72
208
2
324
648
3.12
111.00
359.64
673.92
25.92
327
3
324
972
2.97
119.00
385.56
1059.48
87.48
448
4
324
1296
2.89
121.00
392.04
1451.52
155.52
565
5
324
1620
2.87
117.00
379.08
1830.60
210.60
672
6
324
1944
2.89
107.00
346.68
2177.28
233.28
763
7
324
2268
2.97
91.00
294.84
2472.12
204.12
832
8
324
2592
3.12
69.00
223.56
2695.68
103.68
873
9
324
2916
3.34
41.00
132.84
2828.52
-87.48
880
10
324
3240
3.68
7.00
22.68
2851.20
-388.80
847
11
324
3564
4.21
-33.00
-106.92
2744.28
-819.72
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