Using Statistics to Optimize Cost and Labor

Statistics has been used in the optimization of cost and laborfor many years. It provides methodologies and principles that help business managers to minimize variances in operations. The process usually requires automation, and thanks to the various statistical programs we have today, data analysts can be able to explore and analyze different sets of company data to identify patterns and trends that may aid in making informed decisions regarding labor and cost optimization.

Combination of labor and capital

The optimal combination of Labor (L) and Capital (K) (represented by machines) for fixed output, is achieved through cost minimization. This optimal point is found where the slope of the Iso-cost is equal to the slope of the Isoquant curve. An iso-cost reflects the possible combination of production factors that hold the same total cost, while the isoquant is the possible combination of production factors that generate the same level of output (q).
The isoquant is represented by:
q=F(L,K)Where the output (q) is a function of the production factors Labour (L) and Capital (K). And the iso-cost is represented by
C=w*L+r*K
Where C is the Total fixed cost, L and K are the units of labor and machines, and w and r represent the price per unit of labor and capital respectively.
With all this information, and assuming that the given combinations of L and K are all the existing combinations to achieve the desired level of output, the optimal combination can be found where the C value is the lowest, knowing that:
w=2
r=3
so;
With a higher output of 20, both L and K are higher, and the iso-cost is also higher. The optimal input combination is 21 units of labor and 8 units of capital. 


Q=10 Q=20 Q=30 Q=40
Labor (L) Capital (K) Cost (C ) Labor (L) Capital (K) Cost (C ) Labor (L) Capital (K) Cost (C ) Labor (L) Capital (K) Cost (C )
1 14 44 9 14 60 21 14 84 39 14 120
3 14 48 11 14 64 25 14 92 45 14 132
8 14 58 15 14 72 30 14 102 53 14 148
12 14 66 21 14 84 33 14 108 60 14 162
15 14 72 28 14 98 39 14 120 66 14 174
21 14 84 39 14 120 45 14 132 72 14 186
39 14 120 57 14 156 56 14 154 83 14 208
            87 14 216      

Quantity (q) Fixed Cost (FC) Variable Cost (VC) Total Cost (TC) Average Variable Cost (AVC) Marginal Cost (MC)
10 42 2 44 0.2 -
20 42 18 60 0.9 1.6
30 42 42 84 1.4 2.4
40 42 78 120 1.95 3.6

Applied statistics

a) The marginal product of labor is the change in output resulting from each new unit of labor added:


Units of Labor (L) Output (q) Marginal Product (MPL) Marginal revenue Product (MRPL) Fixed Cost (FC) Variable Cost (VC) Total Cost (TC) Marginal Cost (MC) revenue  (r ) Marginal Revenue (MR) Profit (u)
1 5      $  50.00  $   10.00  $   60.00    $    10.00   $50.00
2 15 10 20  $  50.00  $   20.00  $   70.00  $    1.00  $    30.00  $    2.00 $40.00
3 30 15 30  $  50.00  $   30.00  $   80.00  $    0.67  $    60.00  $    2.00 $20.00
4 50 20 40  $  50.00  $   40.00  $   90.00  $    0.50  $  100.00  $    2.00 $10.00
5 65 15 30  $  50.00  $   50.00  $ 100.00  $    0.67  $  130.00  $    2.00 $30.00
6 77 12 24  $  50.00  $   60.00  $ 110.00  $    0.83  $  154.00  $    2.00 $44.00
7 86 9 18  $  50.00  $   70.00  $ 120.00  $    1.11  $  172.00  $    2.00 $52.00
8 94 8 16  $  50.00  $   80.00  $ 130.00  $    1.25  $  188.00  $    2.00 $58.00
9 98 4 8  $  50.00  $   90.00  $ 140.00  $    2.50  $  196.00  $    2.00 $56.00
10 96 -2 -4  $  50.00  $ 100.00  $ 150.00  $   (5.00)  $  192.00  $    2.00 $42.00

MPL increases with the first 4 employees, but it starts decreasing after the 5th. By the 10th employee, the MPL is negative, meaning that this new employee is causing the total output to decrease.
VC increases with each unit of labor added, and the marginal cost, like the marginal product of labor, gets worse with the 5th unit of labor added.
This firm, with its cost and production function as it is, sees positive profits when producing 50 units of output.
b) Maximization of profit through optimization of inputs comes directly from the relationship between iso-cost and isoquant. The maximization point is where the slopes of both curves are the same:
c) Either through labor or output, the maximization point is the same: 8 employees that produce 94 units of output. So, for this firm, it doesn´t matter if it chooses to maximize labor or output, because the firm is within a perfectly competitive market, with no control of either the output´s price or the cost of labor.
To get professional assistance with this area, connect with our providers of applied statistics assignment help.
Econometrics
If both firms collude, they will be behaving effectively as a monopoly, so the output that maximizes profit will be a function of: