Agglomeration can cause both positive and negative externalities. Suppose that the increase in population raises the...

Agglomeration can cause both positive and negative externalities. Suppose that the increase in population raises the productivity of workers since they can share their knowledge or know-how. In contrast, a higher population causes higher rent since lands become scarce. The total benefit from better productivity is given by TB = 6,000N^(1/3) whereas the total cost of higher rent is given by TC = 300N^(4/3), where N denotes the population of this city in terms of millions. Find the optimal population to maximize net social benefit.

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Expert Solution

TB = 6000N^1/3

TC = 300N^4/3

Net benefit: NB= TB - TC

=> NB = 6000N^1/3 - 300N^4/3

Differentiate NB with respect to N and and put it equal to zero.

=> (dNB / dN) = 6000*(1/3) N^1/3-1 - 300 *(4/3)N^{4/3 -1} = 0

=> 2000N^-2/3 - 400N^1/3 = 0

=> (2000 / N^2/3) = (400N^1/3)

=> (2000 / 400) = (N^2/3 * N^1/3)

=> 5 = N

=> N = 5

Optimal population size is 5 million which will maximize net social benefit.

Rahul Sunny answered 2 years ago

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