1. (4,10) (8,14) Sometimes we can’t observe the production function in green. But we can observe how production changes between two discrete points. Slope of line between the two points is 1 In these cases, we measure marginal products by the rise/run between these two points. lbs nitrogen YieldYield 5.3. Marginal Product

2. Produce Wheat? Change in N (lbs / acre) Change in Wheat yield (bushels / acre)MP NO----- Yes0 →1023→25 0.20 Yes10→2025→30.5 0.55 Yes20→3030.5→33 0.25 Yes30→4033→35 0.20 Yes40→5035→36 0.10 Yes50→6036→36.20.02 MP = rise/run Going from 0 to 10 lbs N: MP = (25-23)/(10-0) = 2/10 = 0.2 MP = rise/run Going from 0 to 10 lbs N: MP = (25-23)/(10-0) = 2/10 = 0.2

3. 5.3. Marginal Product Produce Wheat? Change in N (lbs / acre) Change in Wheat yield (bushels / acre)MP NO----- Yes0 →1023→25 0.20 Yes10→2025→30.5 0.55 Yes20→3030.5→33 0.25 Yes30→4033→35 0.20 Yes40→5035→36 0.10 Yes50→6036→36.20.02 Stage 1 Stage 2 There is no Stage 3 1. Stage 1 begins at 0→10 lbs of nitrogen per acre and ends at 10→20 lbs. 2. Stage 2 begins at 10→20 lbs of nitrogen per acre and ends at 50→60 lbs. 3. Stage 3 begins at __________ lbs of nitrogen per acre and ends at ___________ lbs. There is no Stage 3 Note that the end of Stage 1 and beginning of Stage 2 can overlap.

4. 5.3. Marginal Product Produce Wheat? Change in N (lbs / acre) Change in Wheat yield (bushels / acre)MP NO----- Yes0 →1023→25 0.20 Yes10→2025→30.5 0.55 Yes20→3030.5→40 0.95 Yes30→4040→35 -0.50 Yes40→5035→33 -0.20 Yes50→6033→29-0.40 Stage 1 Stage 3 There is no Stage 2 4. Stage 1 begins at 0→10 lbs of nitrogen per acre and ends at 20→30 lbs. 5. Stage 2 begins at _____lbs of nitrogen per acre and ends at _____ lbs. There is no Stage 2. 6. Stage 3 begins 30→40 lbs of nitrogen per acre and ends at 50→60 lbs. Note that Stage 3 never overlaps with any other stage. Observe that this violates Stage 3, where MP is negative and should be falling (not rising) always.

5. 5.3. Marginal Product Produce Wheat? Change in N (lbs / acre) Change in Wheat yield (bushels / acre)MP NO----- Yes0 →1023→25 0.20 Yes10→2025→30.5 0.55 Yes20→3030.5→40 0.95 Yes30→4040→45 0.50 Yes40→5045→47 0.20 Yes50→6047→43-0.40 Stage 1 Stage 3 7. Stage 1 begins at 0→10 lbs of nitrogen per acre and ends at 20→30 lbs. 8. Stage 2 begins at 20→30 lbs of nitrogen per acre and ends at 40→50 lbs. There is no Stage 2. 9. Stage 3 begins 50→60 lbs of nitrogen per acre and ends at 50→60 lbs. Stage 2

6. 5.3. Marginal Product Consider feedlot stage, where cattle are fed a diet high in grain in a small pen until they are ready for slaughter. DOF = “days on feed”, the number of days the animal has been kept in a feedlot. Live-weight = the weight of the animal. Feedlots choose the DOF that maximizes profits.

7. TodayTomorrow DOF = days on feed LW = live-weight of steer Stage 2: the growth rate is decreasing as the calf ages—the input (DOF) is still productive, but becoming less productive. Stage 3: the cow is declining in weight—the input (DOF) hinders productivity. Stage 1: the growth rate is increasing as the calf ages— the input (DOF) is becoming more productive.

8. DOF = days on feed LW = live-weight of steer Without a formula for the production function, we can only measure marginal products by the rise/run between discrete points

9. The marginal product of Days On Feed (DOF) Cattle ID Live-weight (lbs) Days on feed (DOF) 124086011 124081240138 124081398180 124116081 124111210138 124111316180

10. The marginal product of Days On Feed (DOF) Cattle ID Live-weight (lbs) Days on feed (DOF) 124086011 124081240138 124081398180 124116081 124111210138 124111316180

11. The marginal product of Days On Feed (DOF) Cattle ID Live-weight (lbs) Days on feed (DOF) 124086011 124081240138 124081398180 124116081 124111210138 124111316180

12. MP = 4.36 – 0.0157(DOF) MP at 25 days: MP = 4.36-0.0157(25) = 3.97 MP at 200 days: MP = 4.36-0.0157(200) = 1.22

13. MP = 4.36 – 0.0157(DOF) Live- weight Days on Feed (DOF) At maximum weight the MP will equal zero. Solve for this. 4.36 – 0.0157(DOF) = 0 4.36 = 0.0157(DOF) 4.36/0.0157 = (DOF) (DOF) = 278 at maximum weight 278

14. MVP = marginal value product MP tells us the additional weight of cattle from feeding them one more day. MVP tells us the value of that additional weight. MVP = (MP)(price of cattle per lb) Suppose cattle price is $0.75 per lb MVP = {4.36 – 0.0157(DOF)}($0.75) MVP = 3.27 – 0.011775(DOF)

15. Input price We feed cattle one more day whenever the MVP is greater than the cost of one DOF Suppose it costs $1.41 to feed cattle each day Price of DOF is $1.41 So we feed another day whenever MVP > $1.41

16. Output price = $0.75/ lb; Price of DOF = 1.41 MVP = 3.27 – 0.011775(DOF) MVP at 150 DOF = 3.27 – 0.011775(150) = 1.5 MVP at 150 DOF = 3.27 – 0.011775(150) = 1.5 Price of one DOF is always $1.41 158

17. Applications There are computer programs that calculate the optimal days on feed for cattle Of different breeds Of different genotypes within a breed In different environments And a targeted meat quality