Product and Cost Math Supplement This supplement will provide two tables of practice to show the link between input productivity and costs of output It will show the intuition that higher marginal productivity of labor MPL leads to lower marginal costs MC of output In addition lower MPL leads to higher MC Let s remember the following equations MC w MPL where w is the wage rate paid to workers MPL APL TP L TC TVC TFC ATC AVC AFC MC ATC AFC AVC TFC Q TVC Q TC Q Example 1 Assume that TFC 0 and that the wage rate is w 60 per worker Note that if TFC 0 then TC TVC and ATC AVC TP L TC Q L 0 1 2 3 4 5 6 7 8 Q 0 2 6 12 20 30 38 43 45 APL MPL TC 0 AVC MC a Fill in the APL column b Fill in the MPL column c Fill in the TC column d Fill in the AVC column e Fill in the MC column f Note the relationship between MPL and APL and the relationship between MC and AVC g Note the relationship between MPL and MC Product and Cost Math Supplement Walkthrough a b For parts a and b it is relatively simple to use the equations c But how do we and find the TC We can t really use our quantities or other costs since we don t have those yet For quantities we don t really see a linear pattern we can use However we do know the cost of labor is w 60 and we know the labor used in the production of the outputs shown Thus we can find TC by finding the labor costs associated with the labor inputs that were used to give us the outputs Each worker costs 60 so we can multiply 60 by the number of workers to get TC L 0 1 2 3 4 5 6 7 8 L 0 1 2 3 4 5 6 7 8 Q 0 2 6 12 20 30 38 43 45 Q 0 2 6 12 20 30 38 43 45 APL 2 3 4 5 6 6 333333 6 142857 5 625 MPL 2 4 6 8 10 8 5 2 APL 2 3 4 5 6 6 333333 6 142857 5 625 MPL 2 4 6 8 10 8 5 2 TC 0 TC 0 60 120 180 240 300 360 420 480 AVC AVC MC MC Remember When we graph the TC function we put Q instead of L on the horizontal This will give us an upward sloping but nonlinear TC function See the graphs for this example in a couple of pages Product and Cost Math Supplement AVC and the TVC Q d At this point problem d asks us to find AVC which can be filled in using fact that TFC 0 so TVC TC e Finally in part e MC can be found one of two ways We can either use MC MC or TC Q w MPL L 0 1 2 3 4 5 6 7 8 L 0 1 2 3 4 5 6 7 8 Q 0 2 6 12 20 30 38 43 45 Q 0 2 6 12 20 30 38 43 45 APL 2 3 4 5 6 6 333333 6 142857 5 625 MPL 2 4 6 8 10 8 5 2 TC 0 60 120 180 240 300 360 420 480 AVC 30 20 15 12 10 9 473684 9 767442 10 66667 MC APL 2 3 4 5 6 6 3333 6 1429 5 625 MPL 2 4 6 8 10 8 5 2 TC 0 60 120 180 240 300 360 420 480 AVC 30 20 15 12 10 9 4737 9 7674 10 667 MC 30 15 10 7 5 6 7 5 12 30 w 5 7 MC 60 8 or 180 12 240 20 TC Q For example look at L 4 and Q 20 MC f Note that MPL goes up and then down APL rises when MPL APL and APL falls when MPL APL The average follows the margin MC goes down and then up AVC falls when MC AVC and AVC rises when MC AVC The average follows the margin See the graphs on the next page g MPL and MC are inversely related As MPL rises workers 1 5 MC falls At the MPL max L 5 MC is at its minimum MC 6 at output Q 30 Then due to diminishing marginal product MPL starts to fall workers 6 8 and MC starts to rise See the graphs on the next page 60 8 MPL 5 7 Example 1 Graphs Product and Cost Math Supplement Example 2 Assume that TFC 0 and that the wage rate is w 500 per worker Note that if TFC 0 then TC TVC and ATC AVC Product and Cost Math Supplement L 0 1 2 3 4 5 6 7 8 Q 0 20 60 125 185 240 285 320 343 APL MPL TC 0 AVC MC a Fill in the APL column b Fill in the MPL column c Fill in the TC column d Fill in the AVC column e Fill in the MC column f Note the relationship between MPL and APL and the relationship between MC and AVC g Note the relationship between MPL and MC answers on next page Product and Cost Math Supplement Example 2 answers a b c d e See the table below L 0 1 2 3 4 5 6 7 8 Q 0 20 60 125 185 240 285 320 343 APL 20 30 41 667 46 25 48 47 5 45 714 42 875 MPL 20 40 65 60 55 45 35 23 TC 0 500 1000 1500 2000 2500 3000 3500 4000 12 16 667 AVC 25 MC 25 12 5 7 6923 10 811 8 3333 10 417 9 0909 10 526 11 111 10 938 14 286 11 662 21 739 f Note that MPL goes up and then down APL rises when MPL APL and APL falls when MPL APL The average follows the margin MC goes down and then up AVC falls when MC AVC and AVC rises when MC AVC The average follows the margin See the graphs below g MPL and MC are inversely related As MPL rises workers 1 3 MC falls At the MPL max L 3 MC is at its minimum MC 7 69 at output Q 125 Then due to diminishing marginal product MPL starts to fall workers 4 8 and MC starts to rise See the graphs below
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