Optimal water allocation, , for each use is found using the first order conditions of the Lagrangian function noting that Legrange multipliers indicate the marginal economic value of having one additional unit of water into the system.

1)Mathematical Model: Using the following value functions, the constraints on water supply and the method of Legrange, derive the optimal allocation of water to each of the two uses of water as well as the value for the Legrange multiplier. Show all work using formulas in word. (8 points)

User 1 Value Function:

User 2 Value Function:

Constraint: (assume that all water available is used up between the two uses) & (non-negativity constraints)

a)Set up the optimization problem with choice variables and the Legrange multiplier (2 points)

b)Derive the first order conditions for (3 points)

c)Using the first order conditions in (b), solve for (3 points)