Question

Producer 1 is a beekeeper with a profit function of:

u₁(y) = 1000y - 5y²
y: number of beehives
Producer 2 is the neighbor of Producer 1 and an apple grower with a profit function of:
u₂(y) = 50000 + 200y
Producer 2's profit is positively affected by the number of beehives because the bees pollinate the apple flowers.
a. Calculate producer 1's profit maximizing level of output y*.

Answer 1

The profit maximizing level of output, y*, for Producer 1 is 100 beehives.

Step-by-step explanation:

Producer 1's profit maximizing level of output, y*, can be found by determining the quantity of output at which total revenue exceeds total cost by the largest amount.

To find this level of output, we need to find the quantity at which the difference between total revenue and total cost is largest. This quantity will correspond to the maximum profit level.

Given the profit function u₁(y) = 1000y - 5y², we can calculate the quantity of output that corresponds to maximum profit by finding the derivative of the profit function and setting it equal to zero.

Taking the derivative of u₁(y) with respect to y, we get:

u₁'(y) = 1000 - 10y

Setting u₁'(y) equal to zero, we can solve for y:

1000 - 10y = 0

10y = 1000

y = 100

So, the profit maximizing level of output, y*, for Producer 1 is 100 beehives.