a. A tiny fishing village has 3 residents. Ann has a utility of 10, Bruce has...

a. A tiny fishing village has 3 residents. Ann has a utility of 10, Bruce has a utility of 6, and Charlie has a utility of 7. If the mayor uses a Rawlsian social welfare function, the social welfare of this tiny village would be

b. A mountain village owns a common pasture where villagers graze their goats. The cost to a goat owner of owning and caring for a goat is 4 EUR. The pasture gets overgrazed if too many goats share the pasture. The total revenue from all goats on the common pasture is f (g) = 48g - 2g², where g is the number of goats on the pasture. The town council notices that total profit from the pasture is not maximized if villagers are allowed to pasture goats for free. The council decides to allow a goat to use the common pasture only if its owner buys it a goat license. To maximize total profit (of villagers and council), how many EUR per goat should the council charge?

Solutions

Expert Solution

As per the Rawlsian social welfare function, social welfare of the society is equal to the utility of the worst-off member of society.

Therefore,

Social welfare of the village = min ( U_Ann , U_Bruce, U_Charlie )

Social welfare of the village = min ( 10, 6, 7 )

Social welfare of the village = 6

Consider the scenario without any license fees initially

Cost of owning and caring for each goat = 4 EUR

Total cost of owning and caring for "g" goats = 4g EUR

Total revenue derived from "g" goats = 48g - 2g²

Total profit ( P ) generated from "g" goats = 48g - 2g² - 4g

For profit ( P ) to be maximum, d ( P ) / dg = 0

=> d ( 48g - 2g² - 4g ) / dg = 0

=> 48 - 4g - 4 = 0

=> g = 11 goats

Therefore, for profits to be maximum, villagers will end up rearing 11 goats.

The profit will now be 48g - 2g² - 4g ( g = 11 )

=> Profit = 48*11 - 2*11² - 4*11

=> Profit = 242 EUR

Since the profits are positive, the villagers will keep on grazing their goats leading to overgrazing. To avoid this, let us assume that the council levies a charge "x" as a license fee per goat. The villagers will continue to graze their goats as long as their profits become zero.

=> 242 - 11x = 0

=> x = 22 EUR

Therefore, the council will charge 22 EUR per goat for the goat license and drive the profits earned to zero to avoid overgrazing of the pasture.

Rahul Sunny answered 1 year ago

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