The Island That Never Slows Down

Two neighbouring islands save the same share s of their output, lose the same share delta of their capital to wear each year, and see their workforces grow at the same rate n (fractions: 5 % is 0.05). Both start with the same capital per worker k0, and both accumulate capital per worker by the same rule

They differ in how capital turns into output per worker:

• Mardale has diminishing returns (the Solow model): y_t = A_{solow}\,k_t^{\alpha} with 0 < \alpha < 1.
• Kestry has none (an AK model, the simplest model of endogenous growth: think of "capital" as including knowledge and skills, so each unit adds as much as the last): y_t = A_{ak}\,k_t.

Write ak_overtakes(A_solow, alpha, A_ak, s, delta, n, k0, max_years) that returns a tuple (year, ak_growth):

• year is the first year t with 0 \le t \le max_years in which Kestry's output per worker is strictly greater than Mardale's, or None if that does not happen by year max_years;
• ak_growth is the growth rate of Kestry's output per worker from one year to the next (a fraction; it is the same every year, and may be zero or negative).

Examples

Input:  A_solow = 1, alpha = 0.5, A_ak = 0.5, s = 0.3, delta = 0.05, n = 0.01, k0 = 1, max_years = 100
Output: (20, 0.09)
Explanation: Kestry starts behind (0.5 against 1) but grows 9 % every year, while
Mardale's growth fades. In year 19 the outputs are 2.5708 and 2.7226; in year 20, 2.8022 and 2.7901.

Input:  A_solow = 1, alpha = 0.5, A_ak = 0.5, s = 0.3, delta = 0.05, n = 0.01, k0 = 1, max_years = 10
Output: (None, 0.09)

Input:  A_solow = 1, alpha = 0.5, A_ak = 0.15, s = 0.3, delta = 0.05, n = 0.01, k0 = 1, max_years = 100
Output: (None, -0.015)
Explanation: saving 0.3 * 0.15 = 0.045 of capital a year does not cover the 0.06 lost.

Constraints

• answers are compared with a tolerance of 1e-6

Goals

• Simulate a Solow economy and an AK economy side by side
• Find the first year one path overtakes the other, or report that it does not within a limit
• See that without diminishing returns growth never fades