Derive the growth rates for the decentralized economy and

Question

The question is also in Barro,RJ, X,salai-martin "Economic Growth" 2nd edition book in page 235.

Spillovers from Average Capital per Worker

In the model presented in Section 4.3, assume that the firm's productivity parameter, Ai; depends on the economy's average capital per worker, K/L rather than on the aggregate capital stock, K. The production function is assumed to be Cobb-Douglas:

Y_i = A. (K_i)^α.(K/L)^1-α

Derive the growth rates for the decentralized economy and for the social planner. Comment on how the scale effect discussed in Section 4.3 does not appear with this new specifcacation.