The republic of Lisk has sat at its Solow steady state for decades. Output per worker is y_t = A\,k_t^{\alpha} (0 < \alpha < 1), depreciation is delta, and with saving rate s and workforce growth n (fractions: 5 % is 0.05), capital per worker moves by
Then something changes: a thrift campaign moves the saving rate from s_old to s_new, and/or families get smaller and workforce growth moves from n_old to n_new.
The timing is exact:
- In year 0 capital per worker is the old steady state, k_0 = \left(s_{old} A / (\delta + n_{old})\right)^{1/(1-\alpha)}.
- The new parameters apply from year 0's saving onwards, so k_1 = k_0 + s_{new}\,y_0 - (\delta + n_{new})\,k_0, and so on.
Write after_shock(A, alpha, delta, s_old, n_old, s_new, n_new, T) that returns the list of growth rates of output per worker in years 1, \ldots, T: g_t = y_t / y_{t-1} - 1 (fractions; T numbers, an empty list for T = 0).
Press Run with plot(rates) (or plot the levels y_t): growth jumps up and then fades back to zero as Lisk approaches its new, higher steady state.
Examples
Input: A = 1, alpha = 0.5, delta = 0.05, s_old = 0.2, n_old = 0.05, s_new = 0.3, n_new = 0.05, T = 3
Output: [0.02469508, 0.02292962, 0.02132448]
Explanation: k0 = 4 and y0 = 2. Then k1 = 4 + 0.3 * 2 - 0.1 * 4 = 4.2,
so y1 = sqrt(4.2) = 2.04939: growth 2.47 %, then less each year.
Input: A = 2, alpha = 0.3, delta = 0.06, s_old = 0.25, n_old = 0.01, s_new = 0.25, n_new = 0, T = 3
Output: [0.00298956, 0.00284478, 0.00270790]
Explanation: slower workforce growth also raises capital per worker, more gently.
Constraints
- answers are compared with a tolerance of
1e-6; do not round
Goals
- Start a simulation at the old steady state and run it under new parameters
- Compute year-by-year growth rates of output per worker
- See that a higher saving rate raises growth only for a while