A beekeeper and a farmer with an apple orchard are neighbors. This is convenient for the orchard owner
since the bees pollinate the apple trees: one beehive pollinates one acre of orchard. Unfortunately, there are not enough bees next door to pollinate the whole orchard and pollination costs are $10 per acre. The
beekeeper has total costs of TC = H
2 + 10 × H + 10 and marginal costs MC = 10 + 2 × H where H is the
number of hives. Each hive yields $20 worth of honey. (In the space below, draw a graph with H on the
horizontal axis and cost/benefit on the vertical axis.)
1. How many hives would the beekeeper maintain if operating independently of the farmer?
2. What is the socially efficient number of hives?
3. In the absence of transaction costs, what outcomes do you expect to arise from bargaining between
the beekeeper and the farmer?
1. If the beekeeper operates independently of the farmer, they will maintain a number of hives that maximize their profit. We can find the profit-maximizing level of production by setting the marginal cost (MC) equal to the marginal revenue (MR). The beekeeper's revenue per hive is $20 worth of honey, so MR = 20.
MC = 10 + 2 × H
20 = 10 + 2 × H
10 = 2 × H
Therefore, H = 5 hives.
2. The socially efficient number of hives would take into account the pollination benefits to the farmer. Given that pollination costs for the farmer are $10 per acre and each hive pollinates one acre of orchard, the farmer benefits by $10 for each additional hive. The marginal external benefit (MEB) is $10. The marginal social benefit (MSB) is the sum of MR and MEB, so MSB = 20 + 10 = 30.
We can find the socially efficient number of hives by setting MSB equal to MC:
30 = 10 + 2 × H
20 = 2 × H
Therefore, H = 10 hives.
3. In the absence of transaction costs, we would expect the beekeeper and the farmer to reach a mutually beneficial agreement through bargaining. The beekeeper would likely increase the number of hives to the socially efficient level of 10, as the farmer would be willing to pay for the additional pollination benefits provided by the extra hives. This would lead to an agreement in which the farmer compensates the beekeeper for the additional costs associated with maintaining more hives, while both parties benefit from the increased pollination of the orchard and production of honey.
To answer these questions, let's first understand the information given:
The beekeeper has total costs (TC) of H^2 + 10H + 10 and marginal costs (MC) of 10 + 2H, where H is the number of hives.
Each hive yields $20 worth of honey.
Now, let's proceed to answer the questions:
1. How many hives would the beekeeper maintain if operating independently of the farmer?
To determine the number of hives the beekeeper would maintain when operating independently, we need to find the level of hive production where the marginal cost equals the marginal benefit.
The marginal benefit for the beekeeper is the revenue generated from honey production, which is $20 per hive.
Setting the marginal cost equal to the marginal benefit:
10 + 2H = 20
Simplifying the equation:
2H = 10
H = 5
Therefore, if operating independently, the beekeeper would maintain 5 hives.
2. What is the socially efficient number of hives?
The socially efficient number of hives is the level at which the total social benefit is maximized. In this case, the total social benefit includes both the beekeeper's revenue from honey production and the farmer's benefit from pollination services.
The farmer's benefit from pollination is worth $10 per acre. Since one hive can pollinate one acre, the total benefit is equal to the number of hives multiplied by $10.
So, the total social benefit can be represented as 10H.
To find the socially efficient number of hives, we need to equate the marginal cost to the social benefit:
10 + 2H = 10H
Simplifying the equation:
2H = 10H - 10
8H = 10
H = 1.25
Since we can't have a fractional number of hives, the socially efficient number of hives is 1.
3. In the absence of transaction costs, what outcomes do you expect to arise from bargaining between the beekeeper and the farmer?
In the absence of transaction costs, the beekeeper and the farmer would likely negotiate to reach an agreement that maximizes their joint utility. They would consider the costs and benefits associated with each additional hive.
The beekeeper would want to increase the number of hives until the marginal cost of an additional hive equals the marginal benefit from honey production ($20). On the other hand, the farmer would consider the additional pollination benefit from each hive.
Negotiations might result in the beekeeper maintaining more hives than when operating independently (5 hives). However, the socially efficient outcome (1 hive) would likely not be achieved because negotiations may not take into account the farmer's pollination benefit unless explicitly addressed.
1. To find out how many hives the beekeeper would maintain if operating independently of the farmer, we need to calculate the beekeeper's profit-maximizing level of hives. The beekeeper's profit can be calculated as revenue minus costs.
The revenue from selling honey is given by $20 multiplied by the number of hives (H). So the revenue function is R = 20H.
The total cost function is TC = H^2 + 10H + 10.
To find the profit-maximizing level of hives, we need to find the level of hives (H) at which the difference between revenue and total cost is maximized. This occurs when marginal revenue (MR) equals marginal cost (MC).
MR is the derivative of the revenue function with respect to H, which in this case is constant at $20.
MC is the derivative of the total cost function with respect to H, which is given as MC = 10 + 2H.
Setting MR equal to MC, we have:
20 = 10 + 2H
Solving this equation, we get:
2H = 10
H = 5
Therefore, if the beekeeper is operating independently of the farmer, they would maintain 5 hives.
2. The socially efficient number of hives is the number that maximizes the total benefit to society. In this case, the total benefit consists of the value of pollination to the farmer and the value of honey production to the beekeeper, minus the cost of providing the additional hives for pollination.
The benefit to the farmer from pollination is $10 per acre of orchard, so the benefit function is B = 10H.
The total benefit function is TB = B + R, where R is the revenue function.
Substituting the revenue function R = 20H, we have:
TB = 10H + 20H
TB = 30H
To find the socially efficient number of hives, we need to maximize the total benefit function with respect to H. In this case, since TB is a linear function of H, the maximum occurs at the highest possible value of H.
Given the constraint that one beehive can pollinate one acre of orchard, the socially efficient number of hives is equal to the total number of acres in the orchard.
Unfortunately, the total number of acres in the orchard is not provided in the question, so we cannot determine the socially efficient number of hives without that information.
3. In the absence of transaction costs, bargaining between the beekeeper and the farmer would likely result in an outcome that is mutually beneficial. The farmer would be willing to pay the beekeeper for the pollination services, as it increases the yield of the orchard, while the beekeeper would benefit from the additional income.
The outcome of the bargaining would depend on the specific negotiation and the relative bargaining power of the two parties. Possible outcomes could include the farmer paying the beekeeper a fee per hive for pollination services, the beekeeper receiving a share of the increased profits from the increased yield, or some other mutually agreed-upon arrangement.