Question

Two farmers produce corn on adjacent plots of land. Farmer 1 uses groundwater to irrigate her crop while farmer 2 uses surface water. When farmer 1 utilizes groundwater, she imposes an external cost on farmer 2, since groundwater use leads to depletion of the surface water that is available to farmer 2. In particular, the profit function for farmer 1 is ∏1 = 8z - 1/2z^2.  While the corn production function for farmer 2 is ∏2 = 10x - x^2 - 2z, where z is the quantity of groundwater used by farmer 1 and x is quantity of surface water used by farmer 2.

a. What profits will each farmer earn if they each use water inputs in an attempt to maximize their individual profit? Hint: to determine farmer 1’s optimal choice of , take the derivative of ∏1 with respect to z and set equal to zero. For farmer 2's choice of x, take the derivative of ∏2 with respect to x and set equal to zero (since they cant control z).

b. What profit would each farmer earn if water inputs are chosen so as to maximize the total profits from corn production across the two farms? Hint: total profits are equal to ∏1 + ∏2. You then need to take the partial derivative of this function with respect to x and z.

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Answer 1

The solution is given in the image attached. To summarize, we maximize the profit functions for farmer 1 by taking the derivative of with respect to z. Doing this we obtain z=8. Putting it back into the profit function() we get the maximum profit earned by farmer 1. We find it to be equal to 32.

We repeat the same process for farmer 2. Here we take the partial derivative of with respect to x. What is at variance here is the fact that we substitute the profit maximizing value of z. We do this because z appears in the profit equation of both farmer 1 and 2. Since both maximize their profits, we assume that farmer 1 maximizes first. This is why the profit maximizing z has been substituted in . This is the solution for part (a).

To solve part b, we consider .

We take the partial derivative of with respect to z and x and substitute the values in the equation .

Doing this we get =30 which is lower than in part a (=32) and =13 which is higher than that in part a (=9).