Farmer Buck tasked William, his financial consultant, with creating a mathematical model that explains the details of ranching.

Assume the following:

1. Definitions: A heifer is an unbred cow. Herd size is adult pregnant female cows + adult nonpregnant female cows (does not include calves or bulls) and in other words, means the minimum population of cows Farmer Buck needs to maintain for his given goals throughout the year . A replacement cow is a one-year-old cow you receive when you trade your heifer to the auction house from a different ranch and a different father-bull (there is no cash inflow or cash outflow when trading a heifer for a replacement cow). A bull is a male cow. Gross income is total income/revenue before any expenses.

2. Model/formula: Assumes Farmer Buck has been in the ranching business for years already and only includes regular operational income and costs (i.e. no start-up costs, no purchases of cows, etc.). Every 25th bull cow in the herd requires an additional bull.

3. Expenses: In the middle of October of every year, all unvaccinated cattle over 7 months old are vaccinated. Calves under 7 months old are not vaccinated. If a cow is pregnant/lactating, that cow is not vaccinated. Vaccination fees are $50 per head (charged only by the cattle to be vaccinated). Yearly property tax is $4,000 and yearly property insurance is $2,000 independent of herd size. Farmer Buck spends $500/week on cow treats, independent of herd size, and $500/month on electricity, independent of herd size (for pumping water, exterior illumination, etc.).

4. Pastures: There are two pastures: the breeding pasture and the nursing pasture. The breeding pasture is for pregnant cows or lactating mothers with calves under 9 months. Once the calf is 9 months old, the calf and its mother will be moved from the nursery pasture to the breeding pasture. At twelve months old, the calf is traded for a one-year-old replacement cow. However, a calf will not become fertile until it is at least 15 months. Since replacement cows from the auction are assumed to be 12 months old, there is a three-month delay until the cow is fertile for breeding. When a replacement cow is brought to the ranch, she will be placed in the breeding pasture. There is a one-month delay for Farmer Buck to switch the cows from the nursing pasture to the breeding pasture.

5. Gestation: The gestation (pregnancy) period of a cow is 10 months, and a cow gives birth to only one calf at a time (no twins). The cow is bred (impregnated) in the middle of the month and delivers (gives birth) in the middle of the month. When a cow gives birth to a calf, she will not be bred again for the 9 months she is lactating (until the calf is weaned in 9 months). There is another one month delay for a bull to breed the cow. There is a 50% chance of a calf, when born, being a bull or a heifer. Only non-pregnant, non-lactating cows are bred (impregnated). Calves have a 20% mortality rate (poison, illness, etc.). A cow is fertile for the first eight years of its life (once it is 15 months old, of course), after which fertility rate drops to 20% for the next four years, and then becomes infertile. A bull can mate for the first ten years of its life (once it is 15 months old, of course) after which fertility rate drops to 20% for the next two years, and then becomes sterile.

6. Income: The market price Farmer Buck sells bulls to the auction house at is $2,000. This is also the price that he may buy a bull at. Selling bulls is the only source of income Farmer Buck has identified. Replacement cows are worth $1,500 for both sale and purchase, but Farmer Buck pays no cash for them, only trades-in his one-year-old heifers at a ratio of 1:1 for an equal amount of one-year-old replacement cows.

QUESTION 1: Provide a mathematical model/formula to directly associate time and growth of the herd size. If your herd size is 9 this year, calculate the herd size, gross income, and calves born per year for a span of the next eleven years, including this year. Assume the herd size this year has only one-year-old cows and the bull is one year old.

QUESTION 2:How will the ten-year-model change if all calves born this year are female (heifers)? If all calves born this year are males (bulls)?

QUESTION 3: Make a schedule of recommended expenditures/actions to make Farmer Buck’s ranch more profitable.