Question

Five farms are located along the Blue River, 2 wheat farms and 3 corn farms. The 2 wheat farms each have a property right to use 15 units of water from the river while the corn farms each have the right to use 10 ( in total there are water rights for 60 units). The profit functions below define the relationship between the units of water applied and the profits for each type of farm, where W_i represents units of water applied and $\Pi$_i profits by an individual farm on type i.

Wheat Farm: $\Pi ^{_{1}}=12W_{1}-.3W_{1}^{}2$

Corn Farm:

A. If the firms trade their property rights until they each have the quantity that is socially optimal (i.e the amount that ensures that the sum of profits to all corn and wheat producers are maximized) how many units of water will each farm type apply at the end of trading?

Answer 1

A) at eqm optimal allocation of permits, Marginal Benefit are Equalized across all firms

So differentiate the profit function to get MB function

So MB1 = 12 - .6W1

MB2 = 10 - .4W2

So at eqm , 12-.6W1 = 10-.4W2. & W1+W2 = 60

So 2 = .6W1 - .4W2

Now put W1 = 60-W2

So 2 = .6*(60-W2)-.4W2

2 = 36-.6W2 -.4W2

W2 = 34

& So W1* = 60-34 = 26

so wheat firm will apply 26 units & corn firm will apply 34 units