Question - Consider an infinite-horizon groundwater management problem where xt is the stock of groundwater at time t, yt is the quantity of groundwater extracted at time t, g(yt) gives the amount of total recharge to the aquifer, B(yt) is the benefit of extracting yt, and C(xt, yt) is the cost of extracting yt from a stock of size xt. Suppose that there are property rights to groundwater, but a large number of users. Groundwater used in agriculture is unregulated. Suppose further that:
- g(y)= A + θy
- B(y) = α√y
- C(x, y) = (γ - δx)y
(a) Determine the steady-state levels of x and y under myopic behavior (xc and yc). (This is not the social plannerâs problem.) Hint: you don't have to set up a Lagrangian if you already know what the right equilibrium conditions are.
(b) Now suppose that the government subsidizes electricity for farmers, which is the primary cost component of pumping groundwater. Thus, instead of producers paying costs C(x, y) to pump groundwater, they only pay a fraction Ï of those costs where 0 < Φ < 1. Calculate the resulting steady-state levels of x and y under myopic behavior. What is the effect of the electricity subsidy on extractions yc and stock levels xc? How do your answers compare to your answers in part (a) above? Be sure to show your work and provide an intuitive explanation for any differences you find.