A village has n fishermen who exploit the sea nearby and sell their catch at market price (assumed constant). The extraction technology displays decreasing returns to scale and is a=e^-1/2 where e is the total effort.

1) What is the optimal behavior for the village?
2) What is the efficient behavior?

To find the optimal behavior for the village, we need to maximize its overall profit. The village's profit can be calculated by subtracting the total cost (effort) from the total revenue (catch sold at market price).

1) To get the optimal behavior, we need to maximize the profit function with respect to effort (e).

Profit = Total Revenue - Total Cost
Total Revenue = Market Price * Catch
Total Cost = effort

Given that the extraction technology displays decreasing returns to scale and is represented by a = e^(-1/2), we can substitute this into the profit function:

Profit = Market Price * Catch - effort

We want to find the value of effort (e) that maximizes the profit. To do this, we can take the derivative of the profit function with respect to effort (e) and set it equal to zero:

d(Profit)/d(e) = 0

Taking the derivative of the profit function with respect to e, we get:

d(Profit)/d(e) = Market Price * d(Catch)/d(e) - 1 = 0

Rearranging, we find:

d(Catch)/d(e) = 1/Market Price

This means that the optimal behavior for the village is to allocate effort (e) in such a way that the marginal catch (d(Catch)/d(e)) is equal to the inverse of the market price.

2) Efficient behavior refers to the allocation of resources that maximizes the overall welfare or well-being of the village. In this case, the most efficient behavior would entail maximizing the total catch of fish while taking into account the decreasing returns to scale imposed by the extraction technology.

To achieve efficient behavior, the village needs to allocate effort (e) in a way that maximizes the total catch. In other words, the village should allocate more effort in activities that yield higher marginal catch until the marginal catch is equalized across all fishermen.

This can be achieved by redistributing the effort among the fishermen until the marginal catch for each fisherman is equal to the overall average marginal catch. By doing so, the village can maximize its total catch while minimizing wasteful efforts.

Overall, the optimal behavior and efficient behavior for the village are to allocate effort (e) among the fishermen in a way that equalizes the marginal catch across all fishermen and maximizes the village's profit or total catch.